diff --git "a/0tAzT4oBgHgl3EQftv2j/content/tmp_files/load_file.txt" "b/0tAzT4oBgHgl3EQftv2j/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/0tAzT4oBgHgl3EQftv2j/content/tmp_files/load_file.txt" @@ -0,0 +1,1181 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf,len=1180 +page_content='Tower of quantum scars in a partially many-body localized system Michael Iversen and Anne E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Nielsen Department of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C, Denmark Isolated quantum many-body systems are often well-described by the eigenstate thermalization hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' There are, however, mechanisms that cause different behavior: many-body localization and quantum many-body scars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Here, we show how one can find disordered Hamiltonians hosting a tower of scars by adapting a known method for finding parent Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Using this method, we construct a spin-1/2 model which is both partially localized and contains scars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We demonstrate that the model is partially localized by studying numerically the level spacing statistics and bipar- tite entanglement entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' As disorder is introduced, the adjacent gap ratio transitions from the Gaussian orthogonal ensemble to the Poisson distribution and the entropy shifts from volume-law to area-law scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We investigate the properties of scars in a partially localized background and compare with a thermal background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' At strong disorder, states initialized inside or outside the scar subspace display different dynamical behavior but have similar entanglement entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We demon- strate that localization stabilizes scar revivals of initial states with support both inside and outside the scar subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Finally, we show how strong disorder introduces additional towers of approximate scar states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' INTRODUCTION The eigenstate thermalization hypothesis (ETH) de- scribes how isolated quantum systems reach thermal equilibrium [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The hypothesis is a statement about generic quantum many-body systems and has been veri- fied for a wide variety of physical models [3–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Despite the effectiveness of ETH, several phenomena are known to cause non-thermal behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' One such mechanism is many-body localization (MBL) [14–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' MBL appears in many-body interacting sys- tems with local disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' When the disorder strength is sufficiently strong, it causes a change in the structure of the energy eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' An extensive set of quasi- local integrals of motion (LIOM) emerges and the en- ergy eigenstates localize [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Consequently, all en- ergy eigenstates behave non-thermally and MBL repre- sents a strong violation of ETH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' While this phenomenon is well-established for finite systems, the stability of MBL in the thermodynamic limit is still an open question [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Another mechanism leading to non-thermal behav- ior was discovered in experiments with kinetically con- strained Rydberg atoms [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The atoms were arranged with strong nearest neighbor interactions so the simul- taneous excitation of neighboring atoms was prohibited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' When initializing the system in the N´eel state, observ- ables displayed abnormal persistent oscillations – con- trary to the predictions by ETH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Subsequent theoreti- cal works uncovered that the revivals were caused by a small number of non-thermal eigenstates dubbed quan- tum many-body scars (QMBS) [22–25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' These scar states have approximately equal energy spacing so any initial state in the scar subspace displays revivals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The scar states are uncommon and represent a vanishingly small part of an otherwise thermalizing spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Therefore, QMBS represent a weak violation of ETH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In this work, we realize both ETH-breaking mecha- nisms simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We study a one-dimensional dis- ordered spin-1/2 chain hosting a tower of QMBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' As the disorder strength is increased, the model transitions from the thermal phase to being partially localized while pre- serving the scar states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In earlier works, a single scar state was embedded in an otherwise MBL spectrum [26– 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Our work adds to these studies by considering a full tower of QMBS in an MBL spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The presence of multiple scar states, enables us to study the effect of lo- calization on the dynamical revivals characteristic of scar states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Using this model, we demonstrate how scar states can be distinguished from a localized background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We also find two phenomena originating from the interplay between QMBS and localization: disorder stabilization of scar revivals and disorder induced approximate scars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' II A, we summarize the model by Iadecola and Schecter which is the starting point of our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' II B, we explain how we find Hamiltonians having a set of scar states with equal energy spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' II C, we use this method to determine all local 1- and 2-body Hamiltonians for the tower of scar states in the Iadecola and Schecter model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' III A, we show that a subset of these Hamiltoni- ans partially localize as disorder is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We quan- tify the partial localization as a special structure in the energy eigenstates and compare with results from exact diagonalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We verify the localization by studying the level spacing statistics in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' III B and the entan- glement entropy in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' III C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' IV, we show that the fidelity between initial states and the corresponding time evolved states can be utilized to distinguish the scar states from the partially localized background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We fur- ther show that the bipartite entanglement entropy is an ineffective tool for distinguishing scar states from a par- tially localized background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' V, we demonstrate how scar revivals are stabilized by strong disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' VI, we uncover additional towers of approximate scar states which emerge as disorder is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Finally, we summarize our results in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='01681v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='dis-nn] 4 Jan 2023 2 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' MODEL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Model by Iadecola and Schecter We take the model by Iadecola and Schecter as our starting point [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Consider a one-dimensional spin- 1 2 chain of even length L with periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The local Hilbert space on each site is described by the eigenkets |↑⟩ and |↓⟩ of the Pauli z-matrix, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' ˆσz |↑⟩ = |↑⟩ and ˆσz |↓⟩ = − |↓⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The model by Iadecola and Schecter is given by ˆH0 = L � i=1 � λ(ˆσx i − ˆσz i−1ˆσx i ˆσz i+1) + ∆ˆσz i + J ˆσz i ˆσz i+1 � , (1) with λ, ∆, J ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' All indices are understood as modulo L, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' the index i+L is identified as i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The operators ˆσx i , ˆσy i and ˆσz i are the Pauli matrices acting on site i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The first term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (1) flips the spin si at site i if its nearest neighbors are in different states, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' si−1 ̸= si+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The second term is a magnetic field along the z-direction with strength ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The third term represents nearest neighbor interactions with strength J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Two adjacent spins in different states represent a do- main wall, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' ↑↓ or ↓↑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The Hamiltonian conserves the number of domain walls Ndw because only spins with dif- ferent neighbors are allowed to change their state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Fur- thermore, the Hamiltonian is invariant under spatial in- version and translation, but these symmetries are broken when disorder is introduced in section III and we will not consider them any further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' For nonzero values of λ, ∆ and J, the energy eigen- states are thermal except for a small number of ETH- violating scar states grouped into two towers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Through- out this work, we only focus on one of these towers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' This tower contains L/2+1 eigenstates and the n-th state |Sn⟩ is constructed by acting n times with the operator ˆQ† on the “all-spin-down” state |Sn⟩ ∝ � ˆQ†�n |↓↓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' ↓⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (2) The operator ˆQ† is given by ˆQ† = L � i=1 (−1)i ˆP ↓ i−1ˆσ+ i ˆP ↓ i+1, (3) where ˆσ+ i = (ˆσx i +iˆσy i )/2 is the raising operator and ˆP ↓ i = (ˆ1 − ˆσz i )/2 is the local projection onto spin down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The n-th scar state has energy En = 2(∆ − 2J)n + (J − ∆)L, number of domain walls Ndw = 2n and generally appears central in the spectrum after resolving all symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Since the scar states are equally spaced in energy, any initial state in the scar subspace displays the dynami- cal revivals characteristic of QMBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Furthermore, it was shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' [29] that the bipartite entanglement en- tropy of the scar states displays logarithmic scaling with system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Determining Hamiltonians All eigenstates of ˆH0 located near the middle of the spectrum are thermal except the scar states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We wish to extend the model so the scar states are embedded in a MBL background instead of a thermal background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' MBL is possible in systems with quench disorder, and it has been realized in numerous models by e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' introduc- ing a disordered magnetic field [17], bond-disorder [30] or disordered nearest-neighbor interactions [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Unfor- tunately, disorder cannot be introduced naively to the Hamiltonian ˆH0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' When promoting any parameter to be- ing site-dependent λ → λi, ∆ → ∆i or J → Ji, the scar states are no longer eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Therefore, disorder must be introduced through new terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In this section, we uncover all local few-body Hamiltonians which share the scar states as eigenstates and maintain equal energy spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In the next section, we show that a subset of these Hamiltonians are partially localized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We search for local Hamiltonians following Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The set of 2L×2L Hermitian operators form a vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Most of these operators are long-ranged, contain many-body interactions and are difficult to realize in ex- periments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Therefore, we restrict ourselves to Hamiltoni- ans containing local 1- and 2-body Hermitian operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' This subspace is spanned by the operator basis B2 = � ˆσa i ���a ∈ {x, y, z}, i ∈ ZL � ∪ � ˆσa i ˆσb i+1 ���a, b ∈ {x, y, z}, i ∈ ZL � , (4) where ZL = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' , L} are the first L integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' This subspace is considerably smaller than the full operator vector space and has dimension |B2| = 12L where | · | denotes the number of elements in a set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Any local 1- or 2-body interacting Hamiltonian can be expressed as a linear combination of the basis elements ˆH = |B2| � i=1 αiˆhi, ˆhi ∈ B2, (5) where αi ∈ R are free coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' To simplify no- tation, we collect the coefficients in a vector α = (α1, α2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' , α|B2|)T where T is the transpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We search for the vector of parameters α so the re- sulting Hamiltonian has |Sn⟩ as eigenstates for n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' , L/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The scar state |Sn⟩ is an eigenstate of ˆH if and only if the energy variance of |Sn⟩ is exactly zero ⟨Sn| ˆH2|Sn⟩ − ⟨Sn| ˆH|Sn⟩ 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (6) Inserting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (5), the expression becomes αT Cnα = 0, (7) where Cn is the quantum covariance matrix [Cn]ij = ⟨Sn|ˆhiˆhj|Sn⟩ − ⟨Sn|ˆhi|Sn⟩ ⟨Sn|ˆhj|Sn⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (8) 3 Equation (7) is satisfied when the vector of coefficients lies in the null space of the quantum covariance matrix α ∈ Null(Cn), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Cnα = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We ensure all scar states |Sn⟩ are simultaneously eigenstates of ˆH by demanding the vector of coefficients α lies in the null space of ev- ery covariance matrix α ∈ Null(C0) ∩ Null(C1) ∩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' ∩ Null(CL/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' While this condition ensures all scar states are eigenstates of ˆH, they are not necessarily equally spaced in energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Equal energy spacing is established by imposing another set of requirements ⟨Sn+2| ˆH|Sn+2⟩ − ⟨Sn+1| ˆH|Sn+1⟩ = ⟨Sn+1| ˆH|Sn+1⟩ − ⟨Sn| ˆH|Sn⟩ , (9) for all n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' , L/2 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Inserting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (5), we find Gα = 0, (10) where we introduce the rectangular matrix of energy gap differences [G]ij = ⟨Si+2|ˆhj|Si+2⟩ − 2 ⟨Si+1|ˆhj|Si+1⟩ + ⟨Si|ˆhj|Si⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (11) We observe that the scar states are equally spaced in en- ergy when the coefficient vector resides in the null space of the gap matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In summary, the scar states appear as eigenstates of the Hamiltonian with equal energy spacing when the vector of coefficients lies in the intersection α ∈ L/2 � n=0 Null(Cn) ∩ Null(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (12) It is straightforward to determine this subspace numeri- cally since the scar states are known analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Note however, that while the matrices Cn and G are com- plex, we only search for real vectors α ∈ R|B2| (for com- plex vectors α ∈ C|B2|, the linear combination in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (5) is not necessarily Hermitian).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We find real coeffi- cient vectors by stacking the real and imaginary parts of the matrices (Re(C0), Im(C0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' , Re(CL/2), Im(CL/2), Re(G), Im(G))T and determining the null space of the resulting rectangular matrix by e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' singular value de- composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The vectors αi produced by this numerical method are typically dense, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' have few nonzero entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' As a con- sequence, the corresponding operator � i αiˆhi is difficult to interpret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We overcome this difficulty by noting that if {αi|i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='} lies in the null space Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (12), then any linear combination of these vectors also lies in the null space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We apply a heuristic algorithm to determine sparse vectors in the subspace [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Generalized models We apply the numerical method for system sizes L = 8, 10, 12, 14 and for all sizes find L + 4 linearly indepen- dent vectors αi satisfying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The corresponding (i) ˆHz = �L i=1 ˆσz i (ii) ˆDi = ˆσz i + ˆσz i+1 + ˆσz i ˆσz i+1, for i ∈ ZL (iii) ˆHodd zz = �L/2 i=1 ˆσz 2i−1ˆσz 2i (iv) ˆHalt xz = �L i=1(−1)i(ˆσx i ˆσz i+1 + ˆσz i ˆσx i+1) (v) ˆHalt yz = �L i=1(−1)i(ˆσy i ˆσz i+1 + ˆσz i ˆσy i+1) TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Local 1- and 2-body operators which have |Sn⟩ for n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' , L/2 as energy eigenstates with equal energy spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The operators are determined by applying the nu- merical method presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' II B and Appendix A proves the statement rigorously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' operators are summarized in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The first operator ˆHz was already present in the initial model Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (1) and adds nothing new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The L operators ˆDi act locally on sites i and i+1 and represent good candidates for adding quench disorder into the model in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Indeed, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' III, we demonstrate the system partially localizes when introducing sufficiently strong disorder via these opera- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The third operator ˆHodd zz represents an interaction between every odd site and its right neighbor with equal interaction strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The fourth and fifth operators ˆHalt xz and ˆHalt yz flip spins with the sign of the term determined by the nearest neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Using the numerical method, we rediscover the 1- and 2-body terms of the model in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (1) by starting from the scar states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' As noted above, the operator ˆHz was already present in the original model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Furthermore, the third term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (1) is a linear combination of the oper- ators in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' I: �L i=1 ˆσz i ˆσz i+1 = �L i=1 ˆDi − 2 ˆHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Hence, the operators in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' I only represent L + 2 non-trivial extensions to the initial model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The numerical method presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' II B finds all operators in the operator subspace span(B2) hosting the tower of scars for finite L (up to length L = 14 in our case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' However, in principle, the scar states may not be eigenstates of these operators at larger L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Therefore, in Appendix A we prove analytically for all even L that the scar states remain eigenstates with equal energy spacing for all operators in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The method from Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' II B can be extended by in- cluding all 3-body terms to the basis B3 = B2 ∪ {ˆσa i ˆσb i+1ˆσc i+2 ���a, b, c ∈ {x, y, z}, i ∈ ZL}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' This results in a myriad of new operators – including the first term from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Hence, with a large enough operator basis, the numerical method fully recovers the original model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Since long-ranged many-body interactions are less rele- vant experimentally, we will not explore this possibility any further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Finally, we remark that the effectiveness of this ap- proach is highly non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' For an eigenstate of a generic local Hamiltonian, it is unlikely for another local Hamil- 4 tonian to exist that shares the same eigenstate [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Con- trary to this, we find a large subspace of local Hamiltoni- ans sharing a full tower of scar states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We attribute the effectiveness of our study to the analytical structure of the scar states, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (2) and (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Our methods are not expected to be valuable starting from generic eigenstates but may be equally effective in other scarred models with similar amount of structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' MANY-BODY LOCALIZATION In the last section, we determined a subspace of Hamil- tonians with the scar states |Sn⟩ as eigenstates equally spaced in energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Now, we study a concrete Hamiltonian from this subspace ˆH = ˆH0 + L � i=1 di ˆDi, (13) with di chosen randomly from the uniform probability distribution di ∈ [−W, W] where W > 0 is the disorder strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The action of ˆDi is given by ˆDi |s1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' sisi+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' sL⟩ = � 3 |s1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' sisi+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' sL⟩ , if si = si+1 = ↑ − |s1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' sisi+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' sL⟩ , otherwise (14) The operator ˆDi is related to the projection operators through ˆDi = 4 ˆP ↑ i ˆP ↑ i+1 − ˆ1 with ˆP ↑ i = (ˆ1 + ˆσz i )/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We remark that Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' [29] also observes that the operator ˆP ↑ i ˆP ↑ i+1 preserves the scar states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The model conserves the number of domain walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The dimension of the symmetry sector containing Ndw do- main walls is given by the binomial coefficient 2( L Ndw ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We generally consider the largest symmetry sector with Ndw = 2⌊L/4⌋ domain walls where ⌊·⌋ is the function rounding down to the nearest integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Partial many-body localization A physical system may transition to the MBL phase when disorder is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' MBL is usually real- ized with the disorder term in the Hamiltonian acting uniquely on each basis state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Consequently, a complete set of LIOMs emerge and all energy eigenstates are fully described by their eigenvalues of the LIOMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The situation is slightly different in our model because the disorder term � i di ˆDi treats some basis states the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The operator ˆDi is only sensitive to whether spins i and i+1 are both up (it acts identically on states where spins i and i+1 are ↓↓, ↓↑ or ↑↓).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Therefore, the operator � i di ˆDi has the same action on product states with all consecutive spin-ups placed identically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We do not expect these to localize in the usual sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Instead, we anticipate the spectrum to separate into fully MBL eigenstates and partially localized eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' This structure is most easily described when the prod- uct states |s1s2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' sL⟩ are relabeled to reflect the ac- tion of � i di ˆDi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In this spirit, we define |Ndw, D, n⟩ as a simultaneous eigenstate of the ˆDi’s with eigenvalues D = (D1, D2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' DL) where Di ∈ {−1, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We will refer to D as the disorder indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' As discussed above, the state |s1s2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' sL⟩ is not fully described by D since mul- tiple states can have the same eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Therefore, we further label the states by their number of domain walls Ndw and introduce a dummy index n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' , N (Ndw) D to distinguish states with identical Ndw and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' For in- stance, if two states |s1s2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' sL⟩ and |s′ 1s′ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' s′ L⟩ have the same number of domain walls Ndw and disorder indices D, then they are relabeled as |Ndw, D, n⟩ for n = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Note that some labelings are invalid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Consider the vector of eigenvalues D = (3, −1, 3, 3) for a small system L = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The “3”s imply all spins are up, while the “−1” entail at least one spin is down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In the following, we study a single symmetry sector and hence omit the Ndw index for clar- ity but reintroduce it in Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' V and VI when studying multiple symmetry sectors at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Upon introducing strong disorder, we expect LIOMs to emerge which are localized on the operators ˆDi and en- ergy eigenstates are characterized by their eigenvalues of the LIOMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Therefore, we expect the energy eigenstates to be close to linear combinations of product states with the same disorder indices |ED,m⟩ ≈ ND � n=1 αmn |D, n⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (15) with αmn ∈ R and m = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' , ND.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' This expression is an approximation rather than an equality due to an exponentially small overlap with states |D′, n⟩ with dif- ferent disorder indices D′ ̸= D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The special case ND = 1 corresponds to the disorder term acting uniquely on the basis state |D, 1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We expect the corresponding energy eigenstate |ED,1⟩ ≈ |D, 1⟩ to be MBL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' For ND > 1, the states {|ED,m⟩ |m = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' , ND} are only partially MBL since the LIOMs do not fully describe each state and all additional structure is captured by the extra in- dex m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The above considerations are verified in numerical sim- ulations by considering a system of size L = 8 at strong disorder W = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Figure 1 illustrates the norm squared overlap of all energy eigenstates |ED,m⟩ with the prod- uct states |D, n⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The (i, j)-th pixel displays the norm squared overlap between the i-th product state and j- th energy eigenstate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The product states on the second axis are sorted according to ND.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The energy eigenstates are reordered to allow the diagonal shape in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In the upper left corner of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 1, each eigenstate has high overlap with a single product state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Numerical analysis reveals that these product states exactly coincide with those being fully described by their disorder indices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' ND = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' These results support the claim that such eigen- 5 |ED,m⟩ |D, n⟩ 1 2 3 4 20 ND (a) (b) (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 |⟨D, n|ED,m⟩|2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The norm squared overlap of the energy eigenstates with the product states | ⟨D, n|ED,m⟩ |2 for system size L = 8, disorder strength W = 10 and parameters λ = ∆ = J = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The color of pixel (i, j) displays the overlap between the i’th product state and the j’th eigenstate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The product states are sorted into ascending order according to ND.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The second axis on the right hand side groups the product states according to ND.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The insets show eigenstates with significant weight on (a) two, (b) three and (c) four product states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The figure verifies that all energy eigenstates are approximately linear combinations of product states with the same disorder indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' states fully localize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The next eigenstates shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 1(a) each has significant overlap with exactly two product states of the same disorder indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The pattern contin- ues: we find eigenstates that are linear combinations of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 1(b) three, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 1(c) four, and (bottom right corner) twenty product states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In each case, the product states have the same disorder indices and hence correspond to {|D, n⟩ |n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' , ND} for ND = 3, 4, 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' These ob- servations are not restricted to L = 8, but seem universal at all system sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' For larger system sizes, the number and sizes of the blocks increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Finally, we note that the scar state within the considered symmetry sector is located in the block ND = 20 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The scar state is generally an equal weight linear combination of prod- uct states with the maximum ND.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' This fact will play an important role when we explore the system dynamics in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Next, we discuss how the eigenstates are distributed in energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The magnetization MD = � i σz i of a prod- uct state |D, n⟩ is fixed by the symmetry sector Ndw E Thermal |Sn⟩ Partial MBL |ED1,m1⟩ |ED2,1⟩ |ED2,2⟩ |ED2,3⟩ |ED3,m3⟩ |ED4,m4⟩ |ED5,m5⟩ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Sketch of the spectrum in the thermal phase (left) and in the partially localized phase (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In the thermal phase, the energy levels follow the Wigner-Dyson surmise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' As disorder is introduced, the spectrum experiences partial localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Eigenstates with similar indices D are near de- generate and the spectrum forms clusters of such eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The scar state lies in the largest of these clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' and disorder indices D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Likewise, the number N (↑↑,↓↓) D of adjacent spins pointing in the same direction (↑↑ or ↓↓) and the number N (↑↓,↓↑) D of adjacent spins point- ing in opposite directions (↑↓ or ↓↑) are also fully de- termined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Therefore, the terms ∆ � i ˆσz i , J � i ˆσz i ˆσz i+1 and � i di ˆDi have the same action on all product states with the same number of domain walls and disorder in- dices: {|D, n⟩ |n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' , ND}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' At strong disorder, the energy of an eigenstate is approximately ED,m ≈ ∆MD + J(N (↑↑,↓↓) D − N (↑↓,↓↑) D ) + � i diDi with a small correction that depends on the value of the m index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The slight additional contribution originates from the term � i λ(ˆσx i − ˆσz i−1ˆσx i ˆσz i+1) and scales with λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Consequently, at large disorder, the set of eigenstates {|ED,m⟩ |m = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' , ND} are near degenerate and form clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' A scar state resides in the largest of these clusters in all symmetry sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Figure 2 illustrates the spectral struc- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Note that Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 2 is highly idealized to highlight the structure described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In practice, it is highly likely for two or more clusters to overlap making the structure less apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Spectral statistics The distribution of energy gaps distinguishes the ther- mal and MBL phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Let Ei be the energies of the Hamiltonian in ascending order and δi = Ei+1 − Ei ≥ 0 the i-th energy gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In the thermal phase, the number of energy levels in an interval [E, E+∆E] is known to follow the Wigner-surmise [36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In particular, it follows the Gaussian orthogonal ensemble (GOE) since the model in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (13) is time-reversal invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' On the other hand, the number of energy levels in an interval follows the Pois- son distribution in the MBL phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Since our model only partially localizes, we review how the Poisson distribu- tion accurately describes the MBL phase and investigate T6 the validity of these arguments in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Consider two adjacent eigenstates with energies Ei and Ei+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' At large disorder, the energy of these states are dominated by the disorder term � i di ˆDi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' If the states have differ- ent disorder indices |ED,m⟩ and |ED′,m′⟩, then their en- ergies originate from different linear combinations of the random numbers di: � i diDi ≈ � i diD′ i with Di ̸= D′ i for some i’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Consequently, the eigenstates “arrive” at this energy independently of each other and hence fol- low the Poisson distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' These arguments are no longer valid when two adjacent eigenstates have the same disorder indices and different m indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In this case, we expect the level spacing distribution to follow GOE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Thus, the distribution of energy levels still identifies the transition to partial localization if we only consider level spacings between eigenstates of different disorder indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Instead of working directly with the level spacing dis- tribution, it is convenient to analyze the adjacent gap ratio since it removes the need for unfolding the spec- trum [37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The adjacent gap ratio is defined by [16] ri = min(δi, δi+1) max(δi, δi+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (16) This quantity is bounded by the interval ri ∈ [0, 1] and follows the distributions below in the thermal and MBL phases respectively [39] PGOE(r) = 27 4 r(1 + r) (1 + r + r2)5/2 , (17a) PPoisson(r) = 2 (1 + r)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (17b) The mean values of the distributions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (17) are given by ⟨r⟩GOE = 2(2 − √ 3) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='536 and ⟨r⟩Poisson = 2 ln 2 − 1 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='386.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Figure 3(a) illustrates the mean adjacent gap ratio as a function of disorder strength for different system sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We average the adjacent gap ratio over 2 × 103 disor- der realizations for L = 8, 103 disorder realizations for L = 10, 12, 14 and 500 disorder realizations for L = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' For each disorder realization, we average over all energies in the interval Ei ∈ [E(q=1/3), E(q=2/3)] where E(q) is the q-th quantile of the energy distribution for the current disorder realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' For system size L = 16, we average over the 103 energies closest to (Emin + Emax)/2 where Emin and Emax are the smallest and largest energies in the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The errorbars indicate two standard devi- ations of the average when assuming a Gaussian distri- bution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' As discussed above, the distribution of adjacent gap ratios only converges to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (17b) if the analysis is restricted to adjacent energy levels with different disor- der indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In practice, however, it is unlikely for two neighboring eigenstates to have the same disorder indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Furthermore, the likelihood of neighboring eigenstates having the same disorder indices decreases rapidly with system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' With this in mind, we study the mean adja- cent gap ratio using all eigenstates in the central third of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We verify the considerations above by also computing the mean adjacent gap ratio using only adja- cent eigenstates with different disorder indices at large disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' For each energy gap δi = Ei+1 −Ei, we inspect the eigenstates |ED,m⟩ and |ED′,m′⟩ corresponding to the energies Ei and Ei+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' At large disorder, the disorder in- dices D are accurately determined by computing which D yields �ND m=1 | ⟨D, m|ED,m⟩ |2 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The mean of the adjacent gap ratio is then restricted to energy gaps with D ̸= D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' For small system sizes, there is a large differ- ence between the two methods, but the difference is seen to be small for large systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The mean adjacent gap ratio agrees well with the GOE value at weak disorder 0 <∼ W <∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' As the disorder strength is increased, the mean adjacent gap ratio de- creases and ultimately approaches the Poisson value at 5 <∼ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The agreement of data with the GOE and Pois- son values improves with increasing system size and the transition between the thermal and localized phase be- comes steeper for larger systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Figures 3(b)-(d) illustrate the adjacent gap ratio dis- tribution at (b) weak disorder W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='46, (c) intermedi- ate disorder strength W = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='27 and (d) strong disorder W = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The figures display the distributions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (17) for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' As expected, the data agrees with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (17a) at weak disorder and (17b) at strong disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Fig- ure 3 indicates the system transitions from the thermal phase to being partially localized as disorder is intro- duced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Bipartite entanglement entropy In this section, we further verify the transition from the thermal phase to partial localization by studying the bipartite entanglement entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We separate the system into a left part L containing the first L/2 sites and a right part R containing the remaining sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The reduced density matrix of the left part is obtained by tracing out the right part ρL = TrR(ρ) (18) where ρ is the density matrix of the full system and TrR(·) is the partial trace over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The entanglement entropy between the left and right halves is given by, S = − TrL � ρL ln(ρL) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (19) In the thermal phase, we expect eigenstates near the center of the spectrum to display volume-law scaling with system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Specifically, the entropy is approximately described by the Page value SPage = [L ln(2) − 1]/2 [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' On the other hand, the entanglement entropy displays area-law scaling for MBL eigenstates [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' While some eigenstates in our model are fully MBL, others are only partially localized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Hence, the precise scaling behavior of the entanglement entropy is not clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Nonetheless, we expect the entropy of partially localized eigenstates to grow slower with system size than thermal eigenstates 7 P(r) (b) Poisson GOE P(r) (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 r P(r) (d) 0 1 2 3 4 5 6 W 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='50 ⟨r⟩ (a) GOE Poisson L = 8 10 12 14 16 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (a) Mean adjacent gap ratio ⟨r⟩ (solid line) as a function of disorder strength W for different system sizes L with parameters λ = ∆ = J = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The shaded areas display two standard deviations on the estimate of ⟨r⟩ when assuming a Gaussian distribution of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' For L = 8, the adjacent gap ratio is averaged over 2 × 103 disorder realizations, for L = 10, 12, 14 we use 103 disorder realizations and for L = 16 we use 500 disorder realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' For system sizes L = 8, 10, 12, 14, we average over all energies Ei ∈ [E(q=1/3), E(q=2/3)] where E(q) is the q-th quantile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' For system size L = 16, we average over the 103 energies closest to (Emin + Emax)/2 where Emin and Emax are the smallest and largest energies in the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' At low disorder 0 <∼ W <∼ 1, the system is thermal and ⟨r⟩ coincides with the Gaussian orthogonal ensemble ⟨r⟩GOE ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='536 (upper dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' At strong disorder 5 <∼ W, the mean adjacent gap ratio agrees with the Poisson distribution ⟨r⟩Poisson ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='386 (lower dotted line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The agreement between data and the GOE and Poisson values improves with system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Additionally, the transition from the thermal phase to partial localization happens more rapidly as a function of disorder strength for larger system sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The figure also illustrates the mean adjacent gap ratio when only averaging over neighboring energy eigenstates with different disorder indices (dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The errorbars show two standard deviations on the estimate of the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' This average coincides with the naive calculation at large system sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The figure also shows the adjacent gap ratio distribution for L = 16 at (b) weak disorder W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='46, (c) intermediate disorder strength W = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='27 and (d) strong disorder W = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' These plots include the distributions Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (17a) (dashed curve) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (17b) (dotted curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The data agrees with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (17a) at weak disorder and transitions to the distribution (17b) at strong disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' and we use the entropy to identify the onset of partial localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Figure 4(a) shows the entropy of the eigenstate with energy closest to (Emin + Emax)/2 as a function of disor- der strength W for different system sizes L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Each data point represents the average entropy over 103 disorder realizations with errorbars displaying two standard devi- ations of the mean when assuming a Gaussian distribu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' For low disorder, the entanglement entropy scales linearly with the system size and hence agrees with the expected volume-law scaling in the thermal phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Ad- ditionally, the entropy approaches the Page value with increasing system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' At large disorder, the entropy seems to be roughly independent of system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Thus, the scaling of entropy is consistent with area-law for par- tially localized eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The sudden shift in scaling behavior of the entropy verifies the transition from the thermal phase to partial localization at strong disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The transition point is identified by analyzing the variance of entanglement en- tropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Figure 4(b) illustrates the sample variance of the entropy over 103 disorder realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The variance dis- plays a peak when the system transitions from volume- law to area-law scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' DISTINGUISHABLE FEATURES OF SCAR STATES IN A PARTIALLY LOCALIZED BACKGROUND Scar states are commonly distinguished from a thermal background by their low entanglement and oscillatory dy- namics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In this section, we show that oscillatory dynam- ics can also be utilized to distinguish scar states from a partially localized background, while entanglement en- tropy turns out not to be an effective tool to identify the scar states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Entanglement entropy The entanglement entropy of the scar states scales log- arithmically with system size [29], while thermal states display volume-law scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Therefore, the entanglement 8 0 1 2 3 4 5 ⟨S⟩ (a) L = 8 10 12 14 16 0 2 4 6 W 0 1 Var(S) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (a) Average bipartite entanglement entropy of the eigenstate closest to the center of the spectrum ⟨S⟩ as a func- tion of disorder strength W for different system sizes L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The entropy is averaged over 103 disorder realizations with system parameters λ = ∆ = J = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Errorbars display two stan- dard deviations on the estimate of average entropy assuming a Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' At low disorder, the entropy displays volume-law scaling with system size and approaches the Page value (dashed lines) as expected in the thermal phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' At large disorder, the entropy follows area-law scaling with sys- tem size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (b) Variance of bipartite entanglement entropy of the eigenstate closest to the center of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The variance is computed from 103 disorder realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' As the disorder strength is increased, the variance displays a sudden peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' This indicates a transition from the thermal phase to partial localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The peak becomes higher at larger sys- tem sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' entropy provides a way to identify the scar states in a thermal background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Figure 5(a) illustrates the entropy as a function of energy of a thermal system with size L = 14 and disorder strength W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The thermal states form a narrow arc with maximum in the middle of the spectrum while the scar state appears as an out- lier at much lower entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The situation is different in a partially localized background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Figure 5(b) illustrates the entropy as a function of energy at strong disorder W = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' As discussed above, partially localized eigen- states are weakly entangled making it difficult to identify the scar state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We conclude that entanglement entropy is an ineffective tool for distinguishing scar states from a partially localized background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 ϵ 0 2 4 S (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 ϵ (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The entanglement entropy S as a function of nor- malized energy ϵ = (E − Emin)/(Emax − Emin) where Emin and Emax are the smallest and largest energies in the spec- trum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Lighter (darker) colors indicate lower (higher) den- sity of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (a) We consider a thermal system of size L = 14, disorder strength W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5 and system parameters λ = ∆ = J = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In the thermal phase, the energy eigenstates form a narrow band with maximum at the center of the spec- trum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The scar state (inside the green ring) is easily identified since it appears isolated below the curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (b) We consider a partially localized system at strong disorder W = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The en- ergy eigenstates are spread out at low entropy with the scar state embedded among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The entanglement entropy is hence not an effective tool to distinguish the scar state from a partially localized background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Fidelity States initialized in the scar subspace distinguish them- selves from a thermal background by displaying persis- tent dynamic revivals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We now show that this behavior also enables the identification of scar states from a par- tially localized background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We quantify the dynamics of quantum systems by the fidelity F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Let |ψ(0)⟩ be the initial state and |ψ(t)⟩ = e−i ˆ Ht |ψ(0)⟩ the time evolved state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The fidelity is given by F(t) = | ⟨ψ(0)|ψ(t)⟩ |2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (20) The time evolution of fidelity is most clearly understood by considering the overlap of the initial state with all energy eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Let |φi⟩ be the i-th energy eigenstate with corresponding energy Ei and let ci be the inner product between the i-th energy eigenstate and the initial state ci = ⟨φi|ψ(0)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The relation between ��delity and the expansion coefficients ci is highlighted by rewriting the fidelity according to F(t) = � i |ci|4 + � i̸=j |ci|2|cj|2ei(Ei−Ej)t (21) It is clear from this expression that the dynamics of fi- delity is sensitive to the distribution of |ci|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We generally display this distribution along with the fidelity for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We demonstrate the different dynamical behavior of the thermal and partial MBL phases by initializing a sys- tem of size L = 14 in a product state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' First, we consider 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 F (a) ⟨F T⟩ ⟨F MBL⟩ Fscar 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 F (b) 0 1 2 3 4 5 t/Tscar 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 F (c) −25 0 25 Ei 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='1 |ci|2 (d) −50 0 50 Ei 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5 (e) −20 0 Ei 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='1 (f) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (a) The average fidelity of a random product state in a thermal system at disorder strength W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (b) The aver- age fidelity in a partially localized system at disorder strength W = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The system is initialized in a product state which fully localizes (solid line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' For comparison, the system is ini- tialized in a random product state which only partially local- izes |ψ(0))⟩ = |D, n⟩ with ND = 5 (dashed line), 10 (dashed dotted line) and 35 (dotted line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (c) The system is initialized in the scar subspace at any disorder strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The average fidelity is in all cases calculated over 103 disorder realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The bottom panel displays the distribution of expansion coef- ficients |ci|2 across energy in a single disorder realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (d) For the thermal phase W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (e) For partial MBL W = 10 with initial state |ψ(0)⟩ = |D, n⟩ for ND = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (f) For the initial state being an equal weight linear combination of the scar states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' a thermal system at disorder strength W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The initial state is chosen as a random product state with all product states having the same probability of being drawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We ensure the initial state resides outside the scar subspace by drawing a new product state if the first has non-zero overlap with a scar state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We consider 103 disorder realizations and draw a random product state in each realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In the i-th realization, the fidelity is computed as a function of time Fi(t) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 6(a) shows the average fidelity ⟨F(t)⟩ = 10−3 �103 i=1 Fi(t) over all re- alizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Figure 6(d) shows the expansion coefficients |ci|2 of a single disorder realization following the Gaus- sian distribution as expected [42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Since the initial state has large overlap with many different eigenstates, the second sum in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (21) rapidly vanishes due to can- cellation between terms with different phase factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' As a consequence, the fidelity quickly decreases and satu- rates at Fi(t) ≈ � i |ci|4 ≈ 0 at long times Tscar ≪ t for all disorder realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' These considerations agree with the observed time evolution of the average fidelity in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 6(a) which rapidly decreases to a value near zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Next, we consider the same setup when the system is partially localized at large disorder W = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' As discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' III A, the spectrum separates into fully MBL eigenstates and partially localized eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Conse- quently, the dynamics depend greatly on the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The solid blue line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 6(b) is the average fidelity over 103 disorder realizations when initialing the sys- tem in a random product state which fully localizes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' |ψ(0)⟩ = |D, n⟩ with ND = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Fully MBL eigenstates have significant overlap with only one product state, and the average fidelity remains far from zero at all times as observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 6(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We note that a stronger disor- der strength is needed to achieve MBL in larger systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Therefore, the average fidelity saturates significantly be- low unity in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 6(b) even though all product states with ND = 1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 1 are near identical to an energy eigen- state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The average fidelity saturates closer to unity at larger disorder strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' When the initial state is chosen as a product state that only partially localizes, it has significant overlap with multiple eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Consequently, the average fidelity drops closer to zero as illustrated by the dashed and dot- ted curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 6(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' For these curves, we choose the initial state randomly as |ψ(0)⟩ = |D, n⟩ with ND = 5, 10 and 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' These initial states have significant support on up to ND eigenstates causing the average fidelity to de- crease with increasing ND.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Figure 6(e) illustrates the distribution of |ci|2 for a single disorder realization for a random initial state |ψ(0)⟩ = |D, n⟩ with ND = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The distribution is more sparse than the thermal case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Finally, we consider the initial state being a linear com- bination of scar states |ψscar⟩ = 1 � L 2 + 1 L/2 � n=0 |Sn⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (22) When the initial state is chosen within the scar subspace, the equal energy spacing causes the fidelity to display persistent periodic revivals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In particular, for the equal weight linear combination in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (22), the fidelity is given by Fscar(t) = 1 L 2 + 1 � 1 + 2 L/2 � n=1 � 1 − n L 2 + 1 � cos(n∆Et) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (23) Revivals occur at times tℓ = Tscarℓ = 2πℓ ∆Escar where ℓ ∈ N and ∆Escar is the energy spacing between consecutive scar states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Figure 6(c) illustrates the fidelity of this initial state and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 6(f) shows the distribution of the expansion coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 10 In the thermal phase, states initialized respectively in- side and outside the scar subspace behave differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The fidelity of states outside the scar subspace quickly drops to zero, while any linear combination of scar states display persistent revivals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In our analysis, we specifi- cally initialized the system as a product state, but the same conclusions hold for generic linear combinations of product states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In a partially localized background, the average fidelity distinguishes between states with sup- port inside and outside the scar subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The average fidelity of partially localized states saturates while scar states display revivals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Again, our analysis concerns the special case of initializing the system as a random prod- uct state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' If instead the initial state is a generic linear combination of a large number of product states, the sec- ond term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (21) will generally vanish due to phase cancellation, and the average fidelity saturates near zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' While this is true for generic linear combinations, there exists particular states where the phase cancellation hap- pens exceptionally slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We discuss these special initial states in section VI and how to distinguish them from the scar states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Summing up, the average fidelity represents an effective tool for identifying scar states in both a ther- mal and localized background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Finally, we remark that the fidelity of individual dis- order realizations are enough to distinguish initial states with support inside and outside the scar subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' This statement is simple in the thermal phase where initial states outside the scar subspace rapidly converges to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' At large disorder, the fidelity of individual disorder re- alizations may oscillate rapidly contrary to the average fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' However, these oscillations are generally com- posed of frequencies different from the scar revivals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The amplitude of the oscillations are also typically different from the scar revivals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Thus, the scar states can be dis- tinguished from a partially localized background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' DISORDER STABILIZATION OF SCAR REVIVALS We study the dynamics of initial states with support both inside and outside the scar subspace across all sym- metry sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In this case, we generally expect the scar revivals to diminish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The scar revivals are stabi- lized when the initial state only has support on product states with the same disorder indices as the scar states D0 = (−1, −1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' , −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We demonstrate this behavior by initializing the system in a generic state only having support on product states with disorder indices D0 |ψstable⟩ = 1 Nstable � |ψscar⟩ + � Ndw,n β(Ndw) n |Ndw, D0, n⟩ � , (24) where Nstable is a normalization constant and β(Ndw) n are drawn randomly from the interval β(Ndw) n ∈ [0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5/ � N (Ndw) D0 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We reintroduce the index Ndw to de- scribe product states with the same disorder indices in different symmetry sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The time evolution of fidelity is investigated at weak and strong disorder in 103 real- izations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The coefficients β(Ndw) n are redrawn in each dis- order realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Figure 7(a) displays the disorder aver- aged fidelity for a thermal system and a partially local- ized system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In both cases, the average fidelity displays persistent revivals with the revival amplitude decaying and eventually saturating at a value around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The fidelity amplitude quickly decays for a thermal system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The explanation can be found by studying the expansion coefficients |ci|2 as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 7(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Be- cause the system is thermal, the initial state has support on many energy eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Consequently, terms with different phases quickly cancel causing the fidelity ampli- tude to saturate almost immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' At large disorder, the fidelity amplitude decays at a much slower rate and only saturates alongside the ther- mal graph after many revivals t ∼ 7Tscar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We under- stand this behavior by recalling the spectral structure at large disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' First, recall that the energy eigenstates {|ED0,m⟩ |m = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' , ND0} are near degenerate and only have significant overlap with product states of the same disorder indices as described in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' There- fore, the second term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (24) can be rewritten as a sum of near degenerate eigenstates, ND0 � n=1 β(Ndw) n |Ndw, D0, n⟩ ≈ ND0 � m=1 γ(Ndw) m |ENdw,D0,m⟩ , (25) with γ(Ndw) m = � n β(Ndw) n ⟨ENdw,D0,m|Ndw, D0, n⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Fur- thermore, the scar states themselves are described by the disorder indices D0, so the eigenstates |ENdw,D0,m⟩ are close in energy to a scar state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Consequently, the eigenstates outside the scar subspace having large over- lap with |ψstab⟩ are always close in energy to a scar state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We sketch this structure in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 8 where the eigenstates |ENdw,D0,m⟩ have similar energy to the scar states for all Ndw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' These considerations agree with the observed distribution of |ci|2 for a single disorder realization illus- trated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 7(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The expansion coefficients are sharply peaked around the scar states and consequently the can- cellation of terms with different phases takes place at much larger times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In this way, the partially localized background stabi- lizes the scar revivals by rearranging the support outside the scar subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The stabilization takes place whenever the initial state is predominantly a linear combination of product states with the same disorder indices as the scar states D0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' If product states with other disorder indices D′ ̸= D0 are included, the stabilization will be less pro- nounced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 11 10−4 10−2 100 |ci|2 (b) −50 0 50 E 10−4 10−2 100 |ci|2 (c) 0 2 4 6 8 t/Tscar 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='00 F (a) W = 10 W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' A system of size L = 14 with parameters ∆ = 1, J = 5, λ = 1 is initialized according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (24) in the thermal phase at disorder strength W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5 and the partial MBL phase at disorder strength W = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (a) The average fidelity over 103 disorder realizations when the system is thermal and partially MBL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The disorder protects the scar revivals and the fidelity amplitude decays much slower compared to the thermal case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The right panels illustrate the distribution of expansion coefficients |ci|2 over energy Ei for a single disorder realization at disorder strength (b) W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5 and (c) W = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The distribution of the expansion coefficients is wide in the thermal phase and consists of narrow peaks near the scar states in the localized phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' E Symmetry sectors ∆Escar ∆Escar ∆Escar Ndw0 Ndw1 Ndw2 Ndw3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' At large disorder, the initial state Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (24) has significant overlap with a small number of energy eigenstates (black lines) as sketched in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' These eigenstates ap- pear in clusters around the energy of the scar states (green lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' A single cluster exists in every symmetry sector and the energy gap between two adjacent clusters equals the en- ergy gap between scar states ∆Escar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' DISORDER INDUCED APPROXIMATE SCARS Additional approximate scar states emerge as disorder is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' These approximate scars appear because some symmetry sectors contain energy eigenstates with the same disorder indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' For instance, the eigenstates |E2,D,1⟩ ≈ |↑↑↓↓↓↓⟩ and |E4,D,m⟩ ≈ αm1 |↑↑↓↑↓↓⟩ + αm2 |↑↑↓↓↑↓⟩ for m = 1, 2 have the same disorder in- dices D = (3, −1, −1, −1, −1, −1) but different number of domain walls Ndw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Recall from Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' III A that the en- ergy of an eigenstate at large disorder is approximately given by,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' ENdw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='m ≈ ∆MNdw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='D + J � N (↑↑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='↓↓) Ndw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='D − N (↑↓,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='↓↑) Ndw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='D � + � i diDi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (26) If an eigenstate |ENdw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='m⟩ is described by the values MNdw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' N (↑↑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='↓↓) Ndw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='D and N (↑↓,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='↓↑) Ndw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='D ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' then another eigenstate |ENdw+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='m⟩ with Ndw + 2 domain walls and identical disorder indices D is described by MNdw+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='D = MNdw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='D + 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (27a) N (↑↑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='↓↓) Ndw+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='D = N (↑↑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='↓↓) Ndw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='D − 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (27b) N (↑↓,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='↓↑) Ndw+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='D = N (↑↓,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='↓↑) Ndw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='D + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (27c) Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (26) and (27), one can show the energy dif- ference between two eigenstates with the same disorder indices D and number of domain walls ND and ND + 2 is approximately ENdw+2,D,m − ENdw,D,m ≈ ∆Escar, (28) where ∆Escar = 2(∆−2J) is the energy gap between the scar states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' This calculation demonstrates that towers of approximate scar states appear in the spectrum as disorder is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We demonstrate how the appearance of approximate scars generates non-trivial dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The system is ini- tialized in a generic linear combination of product states with disorder indices D1 = (3, −1, −1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' , −1) |ψinduced D1 ⟩ = 1 Ninduced � Ndw,n ζ(Ndw) n |Ndw, D1, n⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (29) The coefficients are chosen randomly from the interval ζ(Ndw) n ∈ [0, 1] and Ninduced is a normalization constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 12 0 1 2 3 4 5 t/Tscar 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 F (a) 0 1 2 3 4 5 t/Tscar 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 (b) 0 1 2 3 4 5 t/Tscar 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 (c) −50 0 50 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='025 |ci|2 (d) −50 0 50 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='1 (e) −50 0 50 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='2 (f) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The average fidelity of the initial state Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (29) over 103 disorder realizations for system size L = 14 with parameters λ = ∆ = 1, J = 5 at disorder strength (a) W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5, (b) W = 5 and (c) W = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The shaded areas show the interquartile range (middle 50%) of the disorder realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The corresponding distribution of expansion coefficients |ci|2 of a single disorder realization at disorder strength (d) W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5, (e) W = 5 and (f) W = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' At weak disorder, the initial state has significant overlap with many energy eigenstates and the average fidelity quickly decays to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' As the disorder strength is increased, the initial state has significant overlap with a small number of energy eigenstates with equal energy spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Consequently, the average fidelity shows persistent revivals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We study this initial state because, at large disorder, it is a linear combination of an approximate scar tower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We consider 103 disorder realizations at different disor- der strengths and the fidelity is computed for each re- alization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Figure 9(a) displays the average fidelity of a thermal system at weak disorder W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In this case, there is nothing special about the initial state in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (29) and it quickly decays to zero similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 6(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The dynamical behavior changes remarkably as the disorder strength is increased as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 9(b)-(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' At stronger disorder, the initial state Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (29) has large over- lap with eigenstates that are approximately equidistant in energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Consequently, the average fidelity oscillates with a period given by the energy gap Tscar = 2π ∆Escar .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The revival amplitude increases with disorder strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The shaded area in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 9(a)-(c) displays the interquar- tile range of disorder realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Figures 9(d)-(f) shows the expansion of the initial state in energy eigenstates at (d) weak disorder W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5, (e) strong disorder W = 5 and (f) very strong disorder W = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' As expected, the initial state is distributed over a wide range of eigenstates in the thermal phase similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 6(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' As the disor- der strength increases, the initial state has higher and higher overlap with eigenstates in an approximate tower of equidistant states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Figure 9 demonstrates that it is possible to observe revivals from generic linear combinations of the states {|Ndw, D, n⟩ |Ndw = 0, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='} at large dis- order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' However, the effects may be enhanced by choosing the initial state more carefully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The initial state in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (29) is, in some sense, the worst case scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' When all product states with disorder indices D are included in the sum, the initial state generally has significant overlap with all relevant energy eigenstates {|ENdw,D,m⟩ |Ndw = 0, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' m = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' This causes a large spread in the distribution of |ci|2 resulting in a faster decay of the av- erage fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' If instead, we consider an initial state with exactly one product state from each symmetry sector, the spread of |ci|2 is smaller | ˜ψinduced D1 ⟩ = 1 � L 2 − 1 � |↑↑↓↓↓↓↓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' ↓⟩ + |↑↑↓↑↓↓↓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' ↓⟩ + |↑↑↓↑↓↑↓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' ↓⟩ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' + |↑↑↓↑↓↑ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' ↓↑↓↓⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (30) Figure 10(a) shows the average fidelity of this initial state over 103 disorder realizations at strong disorder W = 10 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 10(b) displays the distribution of |ci|2 for a single realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' As expected, the distribution of |ci|2 is narrower and the revival amplitude larger compared to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The initial states Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (29) and (30) display revivals similar to the scar states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' However, one may distinguish these initial states from the scar subspace by noting that the average fidelity in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 9 and 10 decays to zero, while the amplitude in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 6(c) and 7 remain strictly larger than zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The different dynamical behavior is caused by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (29) and (30) being composed of approximate scar towers while the original scars |Sn⟩ are exactly equally spaced in energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 13 0 2 4 t/Tscar 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 F (a) −50 0 50 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content='1 |ci|2 (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (a) Average fidelity of the initial state Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (30) over 103 disorder realizations with system size L = 14 and parameters λ = ∆ = 1, J = 5 and W = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The shaded area displays the interquartile range of the disorder realiza- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The average fidelity displays persistent revivals with larger amplitude compared to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (b) Expansion of the initial state across energy eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The coefficients |ci|2 are sharply peaked around certain energies which are approx- imately equally spaced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' CONCLUSION Building on a known method to find parent Hamil- tonians, we proposed a way to determine Hamiltonians hosting a tower of QMBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Starting from the model by Iadecola and Schecter, we used this method to identify all local 1- and 2-body Hamiltonians of the scar tower |Sn⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Among these Hamiltonians, we found operators fa- cilitating the implementation of local disorder while pre- serving the scar states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' When introducing disorder, the mean level spacing statistics shifts from the GOE to the Poisson distribution and the entanglement entropy goes from volume-law to area-law scaling with system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We conclude the system transitions from the thermal phase to being partially localized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' A theory describing the par- tially localized eigenstates was developed and verified nu- merically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In total, we determined a system hosting a tower of scar states with the remaining spectrum being either thermal or partially localized depending on the disorder strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We studied the properties of scar states embedded in a localized spectrum and compared with the corresponding features in a thermal spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In contrast to thermal systems, the bipartite entanglement entropy does not en- able the identification of scar states in a localized back- ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The average fidelity, on the other hand, effec- tively identifies the scar subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We investigated the effect of localization on initial states with support both inside and outside the scar sub- space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' For a thermal system, the fidelity displays persis- tent revivals with rapidly decreasing amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In con- trast, the revival amplitude decays slower for a partially localized system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Hence, partial localization stabilizes the persistent revivals of states initialized partly outside the scar subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Finally, we demonstrated how additional approximate scar states emerge as disorder is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' When ini- tializing the system as a superposition of these states, the average fidelity displays revivals with the same period as the true scar states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' While this effect does not rely on fine-tuning the initial state, the revivals are amplified by choosing the initial state appropriately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work has been supported by the Carlsberg Foun- dation under grant number CF20-0658.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Appendix A: Proof that |Sn⟩ are eigenstates of all operators in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' I with equal energy spacing In section II C, we found L + 4 operators having the scar states as eigenstates equidistantly spaced in energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Since this analysis was carried out for finite system sizes L = 8, 10, 12, 14, the validity of this statement is not guaranteed for larger system sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In this appendix, we rigorously prove the scar states |Sn⟩ are equally spaced eigenstates of all operators in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Since the scar states are constructed iteratively by applying the operator Q†, we generally prove this statement using proof by induc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' First, we consider the operator ˆHz = � i ˆσz i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The lowest scar state |S0⟩ = |↓↓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' ↓⟩ is trivially an eigen- state of ˆHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' A straightforward calculation shows that [ ˆHz, ˆQ†] = 2 ˆQ† and by induction all other scar states are eigenstates because ˆHz |Sn+1⟩ ∝ ˆHz ˆQ† |Sn⟩ = � Ez,n ˆQ† + 2 ˆQ†� |Sn⟩ = � Ez,n + 2 � |Sn+1⟩ , (A1) where ˆHz |Sn⟩ = Ez,n |Sn⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The scar states are also equally spaced in energy En+1,z − En,z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' A simi- lar argument holds for ˆHodd zz since [ ˆHodd zz , ˆQ†] = −4 ˆQ† where the energy gap between scar states is −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Next, we consider the operators ˆDi = ˆσz i + ˆσz i+1 + ˆσz i ˆσz i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Recall that ˆDi is related to the projection oper- ators through ˆDi = 4 ˆP ↑ i ˆP ↑ i+1 − ˆ1 where ˆP ↑ i = (ˆ1 + ˆσz i )/2 projects site i onto spin-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' First note that ˆDi |S0⟩ = (4 ˆP ↑ i ˆP ↑ i+1 − ˆ1) |↓↓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' ↓⟩ = − |↓↓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' ↓⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' A simple calcu- lation shows that ˆDi commutes with ˆQ† by noting that 14 ˆP ↑ i ˆP ↓ i = 0 [ ˆDi, ˆQ†] = 4 L � j=1 (−1)j� ˆP ↓ j−1[ ˆP ↑ i , ˆσ+ j ] ˆP ↓ j+1 ˆP ↑ i+1 + ˆP ↑ i ˆP ↓ j−1[ ˆP ↑ i+1, ˆσ+ j ] ˆP ↓ j+1 � = 4(−1)i� ˆP ↓ i−1ˆσ+ i ˆP ↓ i+1 ˆP ↑ i+1 − ˆP ↑ i ˆP ↓ i ˆσ+ i+1 ˆP ↓ i+2 � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (A2) Thus, for all scar states we have ˆDi |Sn⟩ = − |Sn⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Alter- natively, one may note that |Sn⟩ by construction does not contain adjacent sites being spin-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Therefore, ˆP ↑ i ˆP ↑ i+1 naturally annihilates the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Next, we consider the operator ˆHalt xz .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Before studying the action of ˆHalt xz on the scar states, we prove by in- duction that the commutator [ ˆHalt xz , ˆQ†] annihilates |Sn⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The commutator is given by [ ˆHalt xz , ˆQ†] = L � i=1 � 2 � ˆP ↓ i ˆσ+ i+1ˆσ− i+2 − ˆσ+ i ˆσ+ i+1 ˆP ↓ i+2 � + i � ˆP ↓ i ˆσ+ i+1ˆσy i+2 + ˆσy i ˆσ+ i+1 ˆP ↓ i+2 + ˆσz i ˆσy i+1ˆσ+ i+2 ˆP ↓ i+3 − ˆP ↓ i ˆσ+ i+1ˆσy i+2ˆσz i+3 �� , (A3) where ˆP ↓ i = (ˆ1 − ˆσz i )/2 is the local projection onto spin- down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' By direct calculation, one can show the lowest scar state is annihilated by this expression [ ˆHalt xz , ˆQ†] |S0⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' A lengthy, yet straightforward, calculation also shows the nested commutator vanishes � [ ˆHalt xz , ˆQ†], ˆQ†� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' We now prove by induction that the commutator annihilates all scar states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Assume [ ˆHalt xz , ˆQ†] |Sn⟩ = 0 and consider, [ ˆHalt xz , ˆQ†] |Sn+1⟩ ∝ [ ˆHalt xz , ˆQ†] ˆQ† |Sn⟩ = � ˆQ†[ ˆHalt xz , ˆQ†] + � [ ˆHalt xz , ˆQ†], ˆQ†�� |Sn⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (A4) Having shown this intermediate result, we prove by in- duction that the operator ˆHalt xz annihilates the scar states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' First we show the operator ˆHalt xz annihilates |S0⟩ ˆHalt xz |S0⟩ = L � i=1 (−1)i(ˆσx i ˆσz i+1 + ˆσz i ˆσx i+1) |↓↓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' ↓⟩ = L � i=1 (−1)i+1(ˆσx i + ˆσx i+1) |↓↓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' ↓⟩ = 0, (A5) where the second term cancels the first after changing summation index i + 1 → i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Next, we show by induction that the n-th scar state is annihilated by ˆHalt xy .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Assume ˆHalt xz annihilates |Sn⟩ and consider ˆHalt xz |Sn+1⟩ ∝ ˆHalt xz ˆQ† |Sn⟩ = ( ˆQ† ˆHalt xz + [ ˆHalt xz , ˆQ†]) |Sn⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (A6) The first term vanishes by assumption and the second term is exactly what we considered in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (A4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' In total, we conclude ˆHalt xy has |Sn⟩ as eigenstates equidistantly separated in energy (with zero energy spacing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' Finally we consider the operator ˆHalt yz .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' One can prove this operator annihilates the scar states using similar ar- guments to above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' The commutator is given by [ ˆHalt yz , ˆQ†] =i L � i=1 � 2 � ˆP ↓ i ˆσ+ i+1ˆσ− i+2 + ˆσ+ i ˆσ+ i+1 ˆP ↓ i+2 � − ˆσx i ˆσ+ i+1 ˆP ↓ i+2 − ˆP ↓ i ˆσ+ i+1ˆσx i+2 + ˆP ↓ i ˆσ+ i+1ˆσx i+2ˆσz i+3 − ˆσz i ˆσx i+1ˆσ+ i+2 ˆP ↓ i+3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQftv2j/content/2301.01681v1.pdf'} +page_content=' (A7) Using 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