liangzid/robench2025a_test_all_category_setphysicsSCP-s-50

shuffled_text stringlengths 280 1.57k	A stringclasses 6 values	B stringclasses 6 values	C stringclasses 6 values	D stringclasses 6 values	label stringclasses 4 values
A: The rest of the paper is organized as followsB: We then introduce our QPE algorithm based on compressed sensing in Sec. 3 and prove several analytical results, including its Heisenberg-limit scaling. We also numerically test the performance of our algorithm and compare it to previous works in Sec. 4. In the final section, we summarize several open problems and potential future research directions in Sec. 5. C: We start with preliminaries about QEEP, sparse Fourier transformation and compressed sensing in Sec. 2	ABC	ABC	BAC	ACB	Selection 4
A: Additionally, for both CATN [61] and SweepContractor [14], it is essential to carefully select the binary tree/MPS structures and permutations (i.e., a mapping from tensor modes onto binary tree vertices) [41]B: These choices should yield accurate low-rank approximations while enabling efficient subsequent contractionsC: However, previous works such as [35, 61, 14] have not systematically explored these parts of the design space.	ABC	CBA	BCA	CBA	Selection 1
A: This shift suggests that the presence of charge enhances the black hole’s ability to trap perturbations near the horizon, leading to higher QNM frequencies. The charge Q𝑄Qitalic_Q thus plays a dual role in both increasing the potential barrier and reducing the effective radius where the perturbation is concentrated, which may result in a more complex QNM spectrum. B: The second panel of Fig. 2 shows the variations of the potential with respect to the charge Q𝑄Qitalic_Q of the black holeC: The black hole charge has a pronounced effect on the potential behavior. As Q𝑄Qitalic_Q increases, the peak value of the potential rises significantly, and the radius r𝑟ritalic_r corresponding to the maximum of the potential shifts closer to the event horizon	ACB	CAB	ACB	BCA	Selection 2
A: Those equations from DECC need to be canonicalized termwise. This generation scheme implies one could build a hash-table thatB: Nevertheless, the time-demanding feature of the half-naive algorithm is partially due to the large number of non-canonicalized equations, that are generated from the permutations in the direct evaluation of coupling coefficient (DECC) wang2018simple algorithmC: We notice that though the double-coset algorithm wang2018simple can substantially reduce the computational cost over the half-naive canonicalization algorithm (generates symmetry equivalent equations and minimize the dummy indices)	CBA	BAC	ABC	CAB	Selection 1
A: (1991) cannot account for the present ARPES resultsB: Note that in the present CDMFT calculations, AFM order is absent while all the short-range charge and spin correlations within the 2×2222\times 22 × 2 cluster are fully taken into account.C: Even spin-wave theory beyond the mean-field AFM ordered state Marsiglio et al	CBA	CAB	BCA	CAB	Selection 3
A: Otherwise, it becomes difficult to define a time evolution law by slicing the spacetime into equal-time surfacesB: Moreover, Poincaré symmetry itself is originally defined in Minkowski space. In the context of AdS/CFT, two Minkowski hypersurfaces can be found in the near-extremal D3-brane solution in decoupling limit.C: To encode information about physical processes in a cellular automaton (CA) and maintain Poincaré symmetry, the spacetime under consideration must be Minkowski	ABC	BAC	ABC	BCA	Selection 4
A: Let us note that the q=1𝑞1q=1italic_q = 1 case discussed above is a special case of a very general set of conjectures due to D. Ben-Zvi, Y. Sakellaridis and A. Venkatesh; those conjectures were in fact motivated by known results about automorphic L𝐿Litalic_L-functionsB: Thus in some sense at the moment the only motivation for the equivalences (1.2.1) and (1.2.2) comes from mathematical physics. C: However, to the best of our knowledge, it is not known how to extend those general conjectures to the “quantum” (i.e. general q𝑞qitalic_q) case	BAC	BCA	BCA	ACB	Selection 4
A: (\APACyear2017)]B: The discriminator architecture is based on the PatchGAN [Isola \BOthersC: Denoting a 4×4444\times 44 × 4 convolutional layer with k𝑘kitalic_k filters, instance normalization (except for the first layer), leaky ReLU with slope 0.20.20.20.2 and a stride of 2222 with Ck. The full architecture of the discriminator is	ACB	BCA	CBA	BAC	Selection 4
A: It turns out that the proof does not rely on the particular product state structure of \|π⟩ket𝜋\ket{\pi}\| start_ARG italic_π end_ARG ⟩, so any figure of merit based on maximizing the overlap with pure states from some subset can be optimized with our methodB: The proof is given in Appendix A and comes with an interesting featureC: This turns the algorithm into a powerful and tool with a universal applicability. Indeed, we adapt it to the dimensionality of entanglement (see Appendix E), the stabilizer rank (Appendix D), matrix product states (Appendix E) as well as to the preparability in quantum networks (Appendix F). Remarkably, here novel states are found which are more distant to any network state as those known so far.	ABC	CBA	BCA	BAC	Selection 4
A: Ref. Bolognesi et al. (2021) for a recent review and references therein). Such line of research is extremely interesting and might lead to additional insight on the occurrence of the ordinary chiral symmetry breaking phase in QCD-like theories where more traditional arguments like those used in this paper fail. B: The notion of global symmetries has been recently generalized Gaiotto et al. (2015), providing some new perspective on strongly-coupled gauge dynamics. Among the many papers on this subject that appeared in the literature, those which use ’t Hooft anomaly matching of generalized symmetries include studies analyzing the vacuum of gauge theories at non-vanishing theta angle Gaiotto et alC: (2017, 2018), constraining the dynamics in the space of coupling constants Córdova et al. (2020a, b), constraining the phase diagram of QCD at finite temperature and/or finite density Tanizaki et al. (2018), excluding an exotic chiral symmetry breaking phase of QCD Tanizaki (2018), constraining the infrared dynamics of QCD-like theories with quarks in higher-dimensional representations with non-trivial N-ality Anber and Poppitz (2019) and of chiral gauge theories (see e.g	ACB	CAB	ABC	ACB	Selection 2
A: The molecules constituting the bond-broken half-skyrmions are also displayedB: (d-f) Spatial distribution of bond-breaking (in black lines) in liquid (d), hexatic (e), and crystalline (f) phasesC: Open circles and the background color are drawn in the same manner as (a-c). (g-i) Structure factors of half-skyrmions undergoing fusion-fission (g), half-skyrmion displacements (h), and bond-breaking (i). Broken curves in (h,i) denote fitting by the Ornstein-Zernike function to yield the correlation length ξ𝜉\xiitalic_ξ.	CAB	CBA	ABC	BAC	Selection 4
A: It was shown that the solutions obtained from the coupled density equations already show that the dynamics of the system is chaotic. In his approach, different components or kinds are consuming each otherB: which is the same expression as in Refs. [55, 56]C: However, this approach may violate the conservation laws. Therefore, we say that this interaction method is incomplete from the point of view of statistical thermodynamics. We will present a new method that covers the full thermodynamic behavior of the interacting system.	ABC	BAC	ACB	ACB	Selection 2
A: A well-known integrator was introduced by Brooks, Brünger, and Karplus, known as BBK [38]B: It is similar to the Verlet integrator [39], also known as the Störmer method or velocity Verlet integrator depending on a specific representationC: An alternative name for BBK is the generalized Verlet [1] which means that the original Verlet method is supplemented by a discretized Dirac delta (stochastic force) and friction terms (a momentum-dependent force):	CBA	CAB	ACB	ABC	Selection 4
A: The kinematic cuts (C1-C4) are described in the textB: After applying the C4 cut, the remaining events are passed for the multivariate analysis. C: Table 4: After applying various kinematic event selection cuts, signal and background events (in fb) indicate the efficiency for each set of cuts to reduce the backgrounds	BAC	CAB	BCA	ABC	Selection 3
A: In conclusion, we have successfully synthesized and characterized high-quality single crystals of YbZn2GaO5, a Yb-based triangular lattice system that displays no observable intrinsic chemical disorder. Our ac-susceptibility measurements rule out the possibility of a spin-glass ground stateB: Furthermore, our specific heat measurements reveal a quadratic temperature dependence of the magnetic component at ultra-low temperatures, as well as linear-T𝑇Titalic_T dependence as function of magnetic field, providing strong experimental evidence for the presence of a U(1) Dirac QSL state on a triangular lattice in YbZn2GaO5. Additionally, our INS investigation of YbZn2GaO5 shows gapless, continuum-like spectra at the high-symmetry M𝑀Mitalic_M and K𝐾Kitalic_K points, but not at the ΓΓ\Gammaroman_Γ point, confirming the potential existence of a U(1) Dirac QSL ground state in this material, and these experimental findings align well with theoretical interpretationsC: Consequently, we firmly believe that YbZn2GaO5 represents a highly promising candidate for the elusive U(1) Dirac QSL, and we advocate for further exploration using a diverse range of theoretical and experimental techniques. Furthermore, we have recently uncovered a substantial correlation between the predicted Dirac spin liquid spectrum for the superconducting cuprate pseudogap [47] and the physical properties observed in our compound. This discovery shines a spotlight on the potential realization of a Dirac QSL in YbZn2GaO5, underscoring its significance within the field.	CAB	CBA	BAC	ABC	Selection 4
A: ReλsubscriptRe𝜆\mathrm{Re}_{\lambda}roman_Re start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT is the Reynolds number based on the Taylor microscaleB: Nisubscript𝑁𝑖N_{i}italic_N start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and Lisubscript𝐿𝑖L_{i}italic_L start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT are the number ofC: the simulations	CAB	BCA	ACB	CBA	Selection 2
A: This work was also supported by the National Natural Science Foundation of China (Grants No. 11305119), the Natural Science Basic Research Plan in Shaanxi Province of China (NoB: 23JSQ001, No. 2020JM-198) and the Fundamental ResearchC: This work was supported by the Natural Science Foundation of Shaanxi Province (No. 2022JQ-037)	CBA	CBA	BCA	CAB	Selection 3
A: Validating the SIDM paradigm requires crucial searches for evidence of the existence of the dark sector.B: These difficulties arise in understanding the behavior and distribution of dark matter (DM) within galaxies and galaxy clustersC: To address these challenges, the self-interacting dark matter (SIDM) paradigm has emerged. It suggests that dark matter particles can interact through short-range forces mediated by a new particle called “dark photon” or “dark mediator,” typically with a mass at the MeV range	BAC	ABC	CAB	ACB	Selection 3
A: (d) The THz bandwidth as a function of CoFeB-layer thicknesses. B: Figure 3: (a) The amplitude of the THz signal at a frequency of 1 THz as a function of CoFeB-layer thicknesses for stacks containing different NM layers and varying thicknesses. The amplitude is measured in attovolt-secondsC: (b) A magnified scale of the THz signal for stacks comprising NM layers Ru, Au, and AgBi. (c) The amplitude of the emitted THz signal as a function of frequency for emitters with a constant CoFeB thickness of 2 nm	CAB	BCA	ABC	ABC	Selection 1
A: In the blue area, there are also two local maxima, but the local maximum on the right is greater than the one on the leftB: In the red area, there are two local maxima that decrease from left to rightC: In the remaining white area, there is only one local maximum.	ABC	CAB	CBA	BAC	Selection 4
A: This approach is well suited to experiments that take data as the laboratory is boosted and rotated. B: One approach involves directly fitting a single data stream to a model involving all of the coefficients in the family.222For a recent example of this approach in the gravity sector, see Ref. [45]C: A number of approaches to simultaneously estimate multiple coefficients exist in the literature [45, 46, 7, 13]	CBA	BAC	CAB	BCA	Selection 1
A: This effect, known as the Autler-Townes effect or ac Stark shift, has been observed in quantum optics and in strongly driven superconducting qubits [25, 26]B: It is at the basis of control strategies for highly coherent solid-state qubits [27]C: In particular, the continuous driving can decouple the spin from background magnetic field noise and thus extend their coherence [28, 29].	CAB	BAC	ACB	ABC	Selection 4
A: Before proceeding, we note that the above restriction to positive temperature is importantB: Indeed, if we consider a system close to the ground state, i.eC: close to the configuration of minimal energy, then the phenomenology is different and KAM tori can persist in the presence of small perturbations, see [19, 26].	ACB	CBA	CAB	ABC	Selection 4
A: This fin comprises a Ge semicircle on top of a Si substrate with a Si outer shell, where the electron is localizedB: This fin resembles the triangular fin discussed in the main text if the triangle has a round tip. In practice, a device without sharp tips is more realistic to be grown since the large strain values in the triangular device might cause instabilities of the structure or dislocations. C: We support these claims here by simulating a semicircular fin, see Fig. S6	CBA	BCA	ABC	BAC	Selection 2
A: The data was then reconstructed using LM-TOF OSEM for up to 27 iterations with 3 subsets with attenuation, random, and scatter modelingB: A 9 minute acquisition (high count) was simulated in GATE that included all relevant physical phenomenon in a typical PET scanC: PSF modeling was implemented via 3 dimensional blurring in object space using a Gaussian kernel with a full width half max of 3 mm.	BAC	CBA	ABC	ABC	Selection 1
A: Below in Fig. S2, we show a general bond-dependent spin Hamiltonian with S=3/2𝑆32{S=3/2}italic_S = 3 / 2, J=−1.3𝐽1.3{J=-1.3}italic_J = - 1.3 meV, Δ=0.5Δ0.5{\Delta=0.5}roman_Δ = 0.5, J±=−0.4subscript𝐽plus-or-minus0.4{J_{\pm}=-0.4}italic_J start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT = - 0.4 meV, Jz±=0.2subscript𝐽limit-from𝑧plus-or-minus0.2{J_{z\pm}=0.2}italic_J start_POSTSUBSCRIPT italic_z ± end_POSTSUBSCRIPT = 0.2 meV, Beff=1subscript𝐵eff1{B_{\text{eff}}=1}italic_B start_POSTSUBSCRIPT eff end_POSTSUBSCRIPT = 1 meV, ℏ⁢ω0=7.7Planck-constant-over-2-pisubscript𝜔07.7{\hbar\omega_{0}=7.7}roman_ℏ italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 7.7 meV,B: However, since the key element to the chiral and topological excitation in this magnon-phonon coupled system is the magnetoelastic couplings, introducing these non-zero bond-dependent interactions will only modify the magnon dispersion but not change the main conclusions in the main textC: As shown in Eq. (S8), the bond-dependent interaction Jz±subscript𝐽limit-from𝑧plus-or-minusJ_{z\pm}italic_J start_POSTSUBSCRIPT italic_z ± end_POSTSUBSCRIPT (and J±subscript𝐽plus-or-minusJ_{\pm}italic_J start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT) is generally allowed under D3⁢h⁢(D3)subscript𝐷3ℎsubscript𝐷3D_{3h}(D_{3})italic_D start_POSTSUBSCRIPT 3 italic_h end_POSTSUBSCRIPT ( italic_D start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) symmetry	CAB	CBA	CAB	ABC	Selection 2
A: Appearance of both Rashba and Darwin terms warrants some elaboration. The very notion of the SOI is an attribute of a simplified single-band portrayal of the electron dynamics, accomplished via the projection of a microscopic multi-band Hamiltonian onto the conduction bandB: Through this projection, an effective Hamiltonian emerges with the SOI and Darwin terms, reflecting the contribution of the valence band(s) to the conduction band electrons dynamics. In particular, the local scalar potential disturbs both valence and conduction bandsC: Due to the momentum-dependent nature of the rotation from local orbitals to the band representation, it leads to the gradient correction to the conduction-band density response. The RSOI term in Eq. (3) reflects the spin-dependent part of this gradient correction, while the Darwin term (4) — the spin-independent one.	BAC	ABC	BAC	CBA	Selection 2
A: However, considering that the influence of dark matter on the space - time structure is usually relatively weak, approximate solutions can be gradually obtained through iterative methodsB: For this purpose, we start from the analytical solution of the event horizon of the classical Kerr black hole. In the standard Kerr metric, the position of the event horizon is given by the following analytical expression: C: It can be seen from equation (11) that there is no exact analytical solution for this equation	BCA	ACB	CAB	CBA	Selection 1
A: Action 0 indicates a step to the north, action 1 a step to the south, action 2 a step to the west, and action 3 a step to the east. To prevent the VQC from choosing illegal actions, the expected values are normalized to the interval [0,1]01[0,1][ 0 , 1 ] and masked with environmental regulations.B: Each cell of the 3×3333\times 33 × 3 gridworld can contain either agent 1, agent 2, a red or blue coin. Empty grids need not be included in the observation, as movement can occur on them without consequence at any timeC: An agent may select from four possible actions, each of which is only possible if the ensuing movement does not lead outside the 3×3333\times 33 × 3 gridworld. The numerical actions range from 0 to 3	BCA	CBA	CAB	ABC	Selection 3
A: It is particularly effective in solving integer programming (IP) and mixed-integer programming (MIP) problems [20]. Qiskit has libraries (QuadraticProgram) in its optimization module that can convert model objects to a form that can be read by CPLEX. B: We have also used IBM CPLEX to obtain classical resultsC: IBM CPLEX is a powerful optimization tool widely used to solve a variety of complex problems	BAC	ABC	CAB	BCA	Selection 3
A: Natural Sciences and Engineering Research Council of Canada (NSERC) via the Canada Research Chair (CRC) of D.V.S.; the Fonds de Recherche du Québec Nature et Technologies (FRQNT) via Institut Transdisciplinaire d’ Information Quantique Transdisciplinary Institute for Quantum Information (INTRIQ); PBEEE/Bourses de court séjour de recherche ou perfectionnement from FRONT; Canada Research Chair Program (CRC) of E. K., NRC-uOttawa Joint Center for Extreme Quantum Photonics (JCEP); Mitacs Accelerate Program; Israel Science Foundation via grant NoB: M. European Union’s Horizon Europe research and innovation programme under grant agreement No. 101070700 (project MIRAQLS).C: 1695/22 by B. A	ACB	BCA	BCA	BAC	Selection 1
A: As we can see from the results, all the maximum frequencies of Fano shape in BIC in Fig. 2 (b) correspond to CIT frequencies in Fig. 2 (c) respectively and these results verify our analysis based on the coupling between the bright mode and the interference of dark modes.B: Thus, the overall dark modes can not be excited by bright mode at the maximum frequency of Fano shape and it leads that the coupling between bright mode and dark modes becomes much smaller. Therefore, the energy of bright mode can not flow to dark modes due to the suppression of dark modes. Consequently, bright mode can excite by external THz wave again, which causes the CIT effect, as shown in the blue, green and red lines of Fig. 2 (c)C: It is remarkable that when two metamaterial structures are slightly different, the metamaterial BIC starts to appear with corresponding to the blue, green and red lines of Fig. 2 (b). The typical transmission spectrum of quasi-BIC is the Fano shape and the overall dark modes become suppressed due to the interference of dark modes at the maximum point of Fano shape	ABC	BAC	CAB	CBA	Selection 4
A: Corners are natural locations for topological defects to reside in order to reduce the total elastic energy in the LC, just as the Kutta condition selects the circulation (by the placement of a surface stagnation point at a sharp edge) in potential flow theory [12]. In general, the final effective-boundary defect is not found at a corner due to the constraint of horizontal alignment in the far-fieldB: An example of a director field found in the manner above is shown as blue curves in Fig. 4, whilst the effective-boundary defects are shown as red dots. Note that three defects are located at the corners of the effective boundaries, whilst the fourth sits along one straight edgeC: Thus, one triangle is allowed to have both defects at corners (with 3 possible pairs) and the other triangle is only allowed one defect at a corner (again, with 3 possible choices). Each of the 3×3×2=18332183\times 3\times 2=183 × 3 × 2 = 18 possible combinations of corners for the effective-boundary defects to be located correspond to local energy minimizing configurations. Determining precisely which three corners yield the global energy minimum requires direct comparison. The LC configuration around a single regular polygon has also been studied using a reduced Landau–de Gennes framework [35, 34].	BCA	BAC	CAB	CAB	Selection 2
A: Moreover, Theorem 5 shows that subpopulations subject to distinct maternal effects can generate arbitrarily complex population dynamics, thereby affirming the main thesis of [18]. B: The extension of (7) to (8) is interesting in its own right, and should be interesting to develop in future workC: In particular, considerations of the analogies between biology and physics via the virial theorem led us to generalize the work of [18] and to derive the ecological simple harmonic oscillator, which to our knowledge is the first example of such an oscillator that emerges solely from maternal effects and does not require a predator-prey or other more sophisticated model	CAB	BAC	ACB	CBA	Selection 4
A: WFs have several profound connections with quantum-geometrical and topological aspects of the electronic structure [436]; some of them are discussed in Sec. III.5 and III.6B: A prime example of geometrical—and in some circumstances also topological—quantity is the electric polarization of periodic solids, which can be calculated in reciprocal space as a Berry phase [436] (see Sec. III.5).C: In the following, we refer to topological properties as a subset of geometrical properties that are quantized and hence represented by integer topological invariants robust to certain classes of perturbations	ACB	CAB	BAC	BAC	Selection 1
A: Frank for helpful suggestions, leading to our consideration of Lorentz norms in Theorem 2. We also thank Hajer Bahouri, Morris Brooks, Albert Cohen, Patrick Gérard, Kihyun Kim, Mathieu Lewin, Hoai-Minh Nguyen, Julien Sabin, Nikita Simonov, Thomas Sørensen, Jakob Stern, Hanne van den Bosch and Jean van Schaftingen for interesting and inspiring discussions.B: We are grateful to Rupert LC: Part of the work has been done when CD was a visiting reseacher at the Institut des Hautes Études and she would like to thank Laure Saint-Raymond for her support and hospitality	ABC	BCA	CBA	ABC	Selection 3
A: The latter are also going to be used to investigate spectral and scattering properties in § 3.3B: This section is devoted to the proofs of the results previously stated. The first result we prove is the classification of self-adjoint extensions: we start by addressing the quadratic forms (2.11) in § 3.1 and then apply Krein’s theory § 3.2 to get an alternative parametrization of the family, through the expression of the resolventsC: Finally, in § 3.4 and § 3.5 we study the symmetry properties of the extensions and the relation with the self-adjoint realization of the Dirac operator with an AB flux, respectively.	BCA	CBA	BAC	BCA	Selection 3
A: Valerii Kozin for valuable comments.B: Leggett Institute for Condensed Matter Theory, Urbana (USA), during the inauguration workshop New Horizons in Condensed Matter Physics, November, 2023, which stimulated this work. We also thank DrC: We wish to acknowledge hospitality at The Anthony J	ACB	BAC	CAB	CBA	Selection 4
A: First of all, it can be seen in Fig. 8(a) that the IV curves from the two different simulations are again essentially equal to each other (on linear scale and with this zoom level). Moreover, here the short-circuit current is practically equal to the solar generation rate expressed in mA/cm2. Removing the AlGaAs layer therefore removes the related electron leakage in the simulations, even if a few percent of the incoming solar light is still absorbed in the GaInP layer next to the rear contactB: The net emission rate is also shown for this case in Fig. 8(b), calculated similarly as in Fig. 6(b). Interestingly, also here there is a notable difference between the RTDD and IRTDD case only at the small voltages but not at the maximum power point of 1.0 V.C: To check whether modifying the layer structure somewhat would cause differences between the RTDD and IRTDD simulations, Fig. 8 repeats the information of Fig. 6 for the device shown in Fig. 5, but with the bottommost AlGaAs layer removed	BCA	BAC	CAB	ABC	Selection 1
A: The scheme validated through this work suggests a viable way to design the shape of twin open cavities, i.e. those cavities intended to resonate with the same spectrum of a reference cavity, but having a different geometryB: The upshot is that one could take designs from the literature that use constant parameter acoustic fluids as the reference and then transform then to other more convenient geometries. The resultant cavity/surface would have some region with anisotropic fluid within it but, as demonstrated, this could be replaced by layered or discrete homogeneous media as effective replacements. C: We have chosen examples to display the versatility of the approach, i.e. a resonant Helmholtz cavity, which is a good test of the approach having both trapped and leaky modes to reproduce and then a case from periodic systems, which is non-resonant but now has sharp corners	ACB	BAC	ABC	ABC	Selection 1
A: The light polarization was controlled with a half-plate (light polarization angle). The single domain BiFeO33{}_{3}start_FLOATSUBSCRIPT 3 end_FLOATSUBSCRIPT thin film with a thickness of 180 nm clearly shows an optical birefringence (Fig. 7a) as previously discussed [39]. On the opposite the optical reflectivity measured on the BFO/LFO superlattice does not show any detectable birefringence (Fig. 7b). B: above the band gap of BFO and LFOC: The optical reflectivity measurement has been performed with a photon energy of 3 eV, i.e	ACB	ABC	ACB	CBA	Selection 4
A: These coefficients can be obtained by solving the Boltzmann equationB: It is evident from Eq. (44) and Eq. (45) that to determine the temperature dependence of shear and bulk viscosities, it is essential to calculate the scalar coefficients 𝒜𝒜\mathcal{A}caligraphic_A and 𝒞𝒞\mathcal{C}caligraphic_CC: In this work, we solved the Boltzmann equation by the relaxation time approximation (RTA) approach in order to determine the shear and bulk viscosities.	CAB	BAC	CAB	ACB	Selection 2
A: To find the ground state configuration, we perform an extensive sampling of all the distinct flux configurations numericallyB: We leverage a “generalized” version of Lieb’s theorem Macris and Nachtergaele (1996); Jaffe and Pedrocchi (2014); Chesi et al. (2013) that reduces dramatically the number of samplings required to identify the ground state. C: Given that the quasicrystal consists of distinct polygons, it is not obvious a priori whether the ground state can be flux-free, and if not, which polygons host non-zero fluxes	ACB	BCA	CBA	CAB	Selection 2
A: It gives rise to primordial density fluctuations, acting as the foundational “seeds” for the creation of the cosmic structures we observe today, including galaxies and galaxy clusters. These primordial density fluctuations, in the standard inflationary scenario, originate from quantum fluctuations of a scalar field during the inflationary epoch. Additionally, inflation triggers the amplification of tensor perturbations in the spacetime metric, resulting in the generation of primordial gravitational waves. B: Inflation explains critical aspects of the universe’s structure, such as its immense size, uniformity, isotropy, and overall geometryC: In the vast landscape of cosmology, inflation emerges as a crucial concept that sheds light on fundamental characteristics of the observable universe at large scales	ABC	CAB	BCA	CBA	Selection 4

**A**: The rest of the paper is organized as follows**B**: We then introduce our QPE algorithm based on compressed sensing in Sec. 3 and prove several analytical results, including its Heisenberg-limit scaling. We also numerically test the performance of our algorithm and compare it to previous works in Sec. 4. In the final section, we summarize several open problems and potential future research directions in Sec. 5. **C**: We start with preliminaries about QEEP, sparse Fourier transformation and compressed sensing in Sec. 2

ABC

BAC

ACB

Selection 4

**A**: Additionally, for both CATN [61] and SweepContractor [14], it is essential to carefully select the binary tree/MPS structures and permutations (i.e., a mapping from tensor modes onto binary tree vertices) [41]**B**: These choices should yield accurate low-rank approximations while enabling efficient subsequent contractions**C**: However, previous works such as [35, 61, 14] have not systematically explored these parts of the design space.

ABC

CBA

BCA

CBA

Selection 1

**A**: This shift suggests that the presence of charge enhances the black hole’s ability to trap perturbations near the horizon, leading to higher QNM frequencies. The charge Q𝑄Qitalic_Q thus plays a dual role in both increasing the potential barrier and reducing the effective radius where the perturbation is concentrated, which may result in a more complex QNM spectrum. **B**: The second panel of Fig. 2 shows the variations of the potential with respect to the charge Q𝑄Qitalic_Q of the black hole**C**: The black hole charge has a pronounced effect on the potential behavior. As Q𝑄Qitalic_Q increases, the peak value of the potential rises significantly, and the radius r𝑟ritalic_r corresponding to the maximum of the potential shifts closer to the event horizon

ACB

CAB

ACB

BCA

Selection 2

**A**: Those equations from DECC need to be canonicalized termwise. This generation scheme implies one could build a hash-table that**B**: Nevertheless, the time-demanding feature of the half-naive algorithm is partially due to the large number of non-canonicalized equations, that are generated from the permutations in the direct evaluation of coupling coefficient (DECC) wang2018simple algorithm**C**: We notice that though the double-coset algorithm wang2018simple can substantially reduce the computational cost over the half-naive canonicalization algorithm (generates symmetry equivalent equations and minimize the dummy indices)

CBA

BAC

ABC

CAB

Selection 1

**A**: (1991) cannot account for the present ARPES results**B**: Note that in the present CDMFT calculations, AFM order is absent while all the short-range charge and spin correlations within the 2×2222\times 22 × 2 cluster are fully taken into account.**C**: Even spin-wave theory beyond the mean-field AFM ordered state Marsiglio et al

CBA

CAB

BCA

CAB

Selection 3

**A**: Otherwise, it becomes difficult to define a time evolution law by slicing the spacetime into equal-time surfaces**B**: Moreover, Poincaré symmetry itself is originally defined in Minkowski space. In the context of AdS/CFT, two Minkowski hypersurfaces can be found in the near-extremal D3-brane solution in decoupling limit.**C**: To encode information about physical processes in a cellular automaton (CA) and maintain Poincaré symmetry, the spacetime under consideration must be Minkowski

ABC

BAC

ABC

BCA

Selection 4

**A**: Let us note that the q=1𝑞1q=1italic_q = 1 case discussed above is a special case of a very general set of conjectures due to D. Ben-Zvi, Y. Sakellaridis and A. Venkatesh; those conjectures were in fact motivated by known results about automorphic L𝐿Litalic_L-functions**B**: Thus in some sense at the moment the only motivation for the equivalences (1.2.1) and (1.2.2) comes from mathematical physics. **C**: However, to the best of our knowledge, it is not known how to extend those general conjectures to the “quantum” (i.e. general q𝑞qitalic_q) case

BAC

BCA

ACB

Selection 4

**A**: (\APACyear2017)]**B**: The discriminator architecture is based on the PatchGAN [Isola \BOthers**C**: Denoting a 4×4444\times 44 × 4 convolutional layer with k𝑘kitalic_k filters, instance normalization (except for the first layer), leaky ReLU with slope 0.20.20.20.2 and a stride of 2222 with Ck. The full architecture of the discriminator is

ACB

BCA

CBA

BAC

Selection 4

**A**: It turns out that the proof does not rely on the particular product state structure of |π⟩ket𝜋\ket{\pi}| start_ARG italic_π end_ARG ⟩, so any figure of merit based on maximizing the overlap with pure states from some subset can be optimized with our method**B**: The proof is given in Appendix A and comes with an interesting feature**C**: This turns the algorithm into a powerful and tool with a universal applicability. Indeed, we adapt it to the dimensionality of entanglement (see Appendix E), the stabilizer rank (Appendix D), matrix product states (Appendix E) as well as to the preparability in quantum networks (Appendix F). Remarkably, here novel states are found which are more distant to any network state as those known so far.

ABC

CBA

BCA

BAC

Selection 4

**A**: Ref. Bolognesi et al. (2021) for a recent review and references therein). Such line of research is extremely interesting and might lead to additional insight on the occurrence of the ordinary chiral symmetry breaking phase in QCD-like theories where more traditional arguments like those used in this paper fail. **B**: The notion of global symmetries has been recently generalized Gaiotto et al. (2015), providing some new perspective on strongly-coupled gauge dynamics. Among the many papers on this subject that appeared in the literature, those which use ’t Hooft anomaly matching of generalized symmetries include studies analyzing the vacuum of gauge theories at non-vanishing theta angle Gaiotto et al**C**: (2017, 2018), constraining the dynamics in the space of coupling constants Córdova et al. (2020a, b), constraining the phase diagram of QCD at finite temperature and/or finite density Tanizaki et al. (2018), excluding an exotic chiral symmetry breaking phase of QCD Tanizaki (2018), constraining the infrared dynamics of QCD-like theories with quarks in higher-dimensional representations with non-trivial N-ality Anber and Poppitz (2019) and of chiral gauge theories (see e.g

ACB

CAB

ABC

ACB

Selection 2

**A**: The molecules constituting the bond-broken half-skyrmions are also displayed**B**: (d-f) Spatial distribution of bond-breaking (in black lines) in liquid (d), hexatic (e), and crystalline (f) phases**C**: Open circles and the background color are drawn in the same manner as (a-c). (g-i) Structure factors of half-skyrmions undergoing fusion-fission (g), half-skyrmion displacements (h), and bond-breaking (i). Broken curves in (h,i) denote fitting by the Ornstein-Zernike function to yield the correlation length ξ𝜉\xiitalic_ξ.

CAB

CBA

ABC

BAC

Selection 4

**A**: It was shown that the solutions obtained from the coupled density equations already show that the dynamics of the system is chaotic. In his approach, different components or kinds are consuming each other**B**: which is the same expression as in Refs. [55, 56]**C**: However, this approach may violate the conservation laws. Therefore, we say that this interaction method is incomplete from the point of view of statistical thermodynamics. We will present a new method that covers the full thermodynamic behavior of the interacting system.

ABC

BAC

ACB

Selection 2

**A**: A well-known integrator was introduced by Brooks, Brünger, and Karplus, known as BBK [38]**B**: It is similar to the Verlet integrator [39], also known as the Störmer method or velocity Verlet integrator depending on a specific representation**C**: An alternative name for BBK is the generalized Verlet [1] which means that the original Verlet method is supplemented by a discretized Dirac delta (stochastic force) and friction terms (a momentum-dependent force):

CBA

CAB

ACB

ABC

Selection 4

**A**: The kinematic cuts (C1-C4) are described in the text**B**: After applying the C4 cut, the remaining events are passed for the multivariate analysis. **C**: Table 4: After applying various kinematic event selection cuts, signal and background events (in fb) indicate the efficiency for each set of cuts to reduce the backgrounds

BAC

CAB

BCA

ABC

Selection 3

**A**: In conclusion, we have successfully synthesized and characterized high-quality single crystals of YbZn2GaO5, a Yb-based triangular lattice system that displays no observable intrinsic chemical disorder. Our ac-susceptibility measurements rule out the possibility of a spin-glass ground state**B**: Furthermore, our specific heat measurements reveal a quadratic temperature dependence of the magnetic component at ultra-low temperatures, as well as linear-T𝑇Titalic_T dependence as function of magnetic field, providing strong experimental evidence for the presence of a U(1) Dirac QSL state on a triangular lattice in YbZn2GaO5. Additionally, our INS investigation of YbZn2GaO5 shows gapless, continuum-like spectra at the high-symmetry M𝑀Mitalic_M and K𝐾Kitalic_K points, but not at the ΓΓ\Gammaroman_Γ point, confirming the potential existence of a U(1) Dirac QSL ground state in this material, and these experimental findings align well with theoretical interpretations**C**: Consequently, we firmly believe that YbZn2GaO5 represents a highly promising candidate for the elusive U(1) Dirac QSL, and we advocate for further exploration using a diverse range of theoretical and experimental techniques. Furthermore, we have recently uncovered a substantial correlation between the predicted Dirac spin liquid spectrum for the superconducting cuprate pseudogap [47] and the physical properties observed in our compound. This discovery shines a spotlight on the potential realization of a Dirac QSL in YbZn2GaO5, underscoring its significance within the field.

CAB

CBA

BAC

ABC

Selection 4

**A**: ReλsubscriptRe𝜆\mathrm{Re}_{\lambda}roman_Re start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT is the Reynolds number based on the Taylor microscale**B**: Nisubscript𝑁𝑖N_{i}italic_N start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and Lisubscript𝐿𝑖L_{i}italic_L start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT are the number of**C**: the simulations

CAB

BCA

ACB

CBA

Selection 2

**A**: This work was also supported by the National Natural Science Foundation of China (Grants No. 11305119), the Natural Science Basic Research Plan in Shaanxi Province of China (No**B**: 23JSQ001, No. 2020JM-198) and the Fundamental Research**C**: This work was supported by the Natural Science Foundation of Shaanxi Province (No. 2022JQ-037)

CBA

BCA

CAB

Selection 3

**A**: Validating the SIDM paradigm requires crucial searches for evidence of the existence of the dark sector.**B**: These difficulties arise in understanding the behavior and distribution of dark matter (DM) within galaxies and galaxy clusters**C**: To address these challenges, the self-interacting dark matter (SIDM) paradigm has emerged. It suggests that dark matter particles can interact through short-range forces mediated by a new particle called “dark photon” or “dark mediator,” typically with a mass at the MeV range

BAC

ABC

CAB

ACB

Selection 3

**A**: (d) The THz bandwidth as a function of CoFeB-layer thicknesses. **B**: Figure 3: (a) The amplitude of the THz signal at a frequency of 1 THz as a function of CoFeB-layer thicknesses for stacks containing different NM layers and varying thicknesses. The amplitude is measured in attovolt-seconds**C**: (b) A magnified scale of the THz signal for stacks comprising NM layers Ru, Au, and AgBi. (c) The amplitude of the emitted THz signal as a function of frequency for emitters with a constant CoFeB thickness of 2 nm

CAB

BCA

ABC

Selection 1

**A**: In the blue area, there are also two local maxima, but the local maximum on the right is greater than the one on the left**B**: In the red area, there are two local maxima that decrease from left to right**C**: In the remaining white area, there is only one local maximum.

ABC

CAB

CBA

BAC

Selection 4

**A**: This approach is well suited to experiments that take data as the laboratory is boosted and rotated. **B**: One approach involves directly fitting a single data stream to a model involving all of the coefficients in the family.222For a recent example of this approach in the gravity sector, see Ref. [45]**C**: A number of approaches to simultaneously estimate multiple coefficients exist in the literature [45, 46, 7, 13]

CBA

BAC

CAB

BCA

Selection 1

**A**: This effect, known as the Autler-Townes effect or ac Stark shift, has been observed in quantum optics and in strongly driven superconducting qubits [25, 26]**B**: It is at the basis of control strategies for highly coherent solid-state qubits [27]**C**: In particular, the continuous driving can decouple the spin from background magnetic field noise and thus extend their coherence [28, 29].

CAB

BAC

ACB

ABC

Selection 4

**A**: Before proceeding, we note that the above restriction to positive temperature is important**B**: Indeed, if we consider a system close to the ground state, i.e**C**: close to the configuration of minimal energy, then the phenomenology is different and KAM tori can persist in the presence of small perturbations, see [19, 26].

ACB

CBA

CAB

ABC

Selection 4

**A**: This fin comprises a Ge semicircle on top of a Si substrate with a Si outer shell, where the electron is localized**B**: This fin resembles the triangular fin discussed in the main text if the triangle has a round tip. In practice, a device without sharp tips is more realistic to be grown since the large strain values in the triangular device might cause instabilities of the structure or dislocations. **C**: We support these claims here by simulating a semicircular fin, see Fig. S6

CBA

BCA

ABC

BAC

Selection 2

**A**: The data was then reconstructed using LM-TOF OSEM for up to 27 iterations with 3 subsets with attenuation, random, and scatter modeling**B**: A 9 minute acquisition (high count) was simulated in GATE that included all relevant physical phenomenon in a typical PET scan**C**: PSF modeling was implemented via 3 dimensional blurring in object space using a Gaussian kernel with a full width half max of 3 mm.

BAC

CBA

ABC

Selection 1

**A**: Below in Fig. S2, we show a general bond-dependent spin Hamiltonian with S=3/2𝑆32{S=3/2}italic_S = 3 / 2, J=−1.3𝐽1.3{J=-1.3}italic_J = - 1.3 meV, Δ=0.5Δ0.5{\Delta=0.5}roman_Δ = 0.5, J±=−0.4subscript𝐽plus-or-minus0.4{J_{\pm}=-0.4}italic_J start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT = - 0.4 meV, Jz±=0.2subscript𝐽limit-from𝑧plus-or-minus0.2{J_{z\pm}=0.2}italic_J start_POSTSUBSCRIPT italic_z ± end_POSTSUBSCRIPT = 0.2 meV, Beff=1subscript𝐵eff1{B_{\text{eff}}=1}italic_B start_POSTSUBSCRIPT eff end_POSTSUBSCRIPT = 1 meV, ℏ⁢ω0=7.7Planck-constant-over-2-pisubscript𝜔07.7{\hbar\omega_{0}=7.7}roman_ℏ italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 7.7 meV,**B**: However, since the key element to the chiral and topological excitation in this magnon-phonon coupled system is the magnetoelastic couplings, introducing these non-zero bond-dependent interactions will only modify the magnon dispersion but not change the main conclusions in the main text**C**: As shown in Eq. (S8), the bond-dependent interaction Jz±subscript𝐽limit-from𝑧plus-or-minusJ_{z\pm}italic_J start_POSTSUBSCRIPT italic_z ± end_POSTSUBSCRIPT (and J±subscript𝐽plus-or-minusJ_{\pm}italic_J start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT) is generally allowed under D3⁢h⁢(D3)subscript𝐷3ℎsubscript𝐷3D_{3h}(D_{3})italic_D start_POSTSUBSCRIPT 3 italic_h end_POSTSUBSCRIPT ( italic_D start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) symmetry

CAB

CBA

CAB

ABC

Selection 2

**A**: Appearance of both Rashba and Darwin terms warrants some elaboration. The very notion of the SOI is an attribute of a simplified single-band portrayal of the electron dynamics, accomplished via the projection of a microscopic multi-band Hamiltonian onto the conduction band**B**: Through this projection, an effective Hamiltonian emerges with the SOI and Darwin terms, reflecting the contribution of the valence band(s) to the conduction band electrons dynamics. In particular, the local scalar potential disturbs both valence and conduction bands**C**: Due to the momentum-dependent nature of the rotation from local orbitals to the band representation, it leads to the gradient correction to the conduction-band density response. The RSOI term in Eq. (3) reflects the spin-dependent part of this gradient correction, while the Darwin term (4) — the spin-independent one.

BAC

ABC

BAC

CBA

Selection 2

**A**: However, considering that the influence of dark matter on the space - time structure is usually relatively weak, approximate solutions can be gradually obtained through iterative methods**B**: For this purpose, we start from the analytical solution of the event horizon of the classical Kerr black hole. In the standard Kerr metric, the position of the event horizon is given by the following analytical expression: **C**: It can be seen from equation (11) that there is no exact analytical solution for this equation

BCA

ACB

CAB

CBA

Selection 1

**A**: Action 0 indicates a step to the north, action 1 a step to the south, action 2 a step to the west, and action 3 a step to the east. To prevent the VQC from choosing illegal actions, the expected values are normalized to the interval [0,1]01[0,1][ 0 , 1 ] and masked with environmental regulations.**B**: Each cell of the 3×3333\times 33 × 3 gridworld can contain either agent 1, agent 2, a red or blue coin. Empty grids need not be included in the observation, as movement can occur on them without consequence at any time**C**: An agent may select from four possible actions, each of which is only possible if the ensuing movement does not lead outside the 3×3333\times 33 × 3 gridworld. The numerical actions range from 0 to 3

BCA

CBA

CAB

ABC

Selection 3

**A**: It is particularly effective in solving integer programming (IP) and mixed-integer programming (MIP) problems [20]. Qiskit has libraries (QuadraticProgram) in its optimization module that can convert model objects to a form that can be read by CPLEX. **B**: We have also used IBM CPLEX to obtain classical results**C**: IBM CPLEX is a powerful optimization tool widely used to solve a variety of complex problems

BAC

ABC

CAB

BCA

Selection 3

**A**: Natural Sciences and Engineering Research Council of Canada (NSERC) via the Canada Research Chair (CRC) of D.V.S.; the Fonds de Recherche du Québec Nature et Technologies (FRQNT) via Institut Transdisciplinaire d’ Information Quantique Transdisciplinary Institute for Quantum Information (INTRIQ); PBEEE/Bourses de court séjour de recherche ou perfectionnement from FRONT; Canada Research Chair Program (CRC) of E. K., NRC-uOttawa Joint Center for Extreme Quantum Photonics (JCEP); Mitacs Accelerate Program; Israel Science Foundation via grant No**B**: M. European Union’s Horizon Europe research and innovation programme under grant agreement No. 101070700 (project MIRAQLS).**C**: 1695/22 by B. A

ACB

BCA

BAC

Selection 1

**A**: As we can see from the results, all the maximum frequencies of Fano shape in BIC in Fig. 2 (b) correspond to CIT frequencies in Fig. 2 (c) respectively and these results verify our analysis based on the coupling between the bright mode and the interference of dark modes.**B**: Thus, the overall dark modes can not be excited by bright mode at the maximum frequency of Fano shape and it leads that the coupling between bright mode and dark modes becomes much smaller. Therefore, the energy of bright mode can not flow to dark modes due to the suppression of dark modes. Consequently, bright mode can excite by external THz wave again, which causes the CIT effect, as shown in the blue, green and red lines of Fig. 2 (c)**C**: It is remarkable that when two metamaterial structures are slightly different, the metamaterial BIC starts to appear with corresponding to the blue, green and red lines of Fig. 2 (b). The typical transmission spectrum of quasi-BIC is the Fano shape and the overall dark modes become suppressed due to the interference of dark modes at the maximum point of Fano shape

ABC

BAC

CAB

CBA

Selection 4

**A**: Corners are natural locations for topological defects to reside in order to reduce the total elastic energy in the LC, just as the Kutta condition selects the circulation (by the placement of a surface stagnation point at a sharp edge) in potential flow theory [12]. In general, the final effective-boundary defect is not found at a corner due to the constraint of horizontal alignment in the far-field**B**: An example of a director field found in the manner above is shown as blue curves in Fig. 4, whilst the effective-boundary defects are shown as red dots. Note that three defects are located at the corners of the effective boundaries, whilst the fourth sits along one straight edge**C**: Thus, one triangle is allowed to have both defects at corners (with 3 possible pairs) and the other triangle is only allowed one defect at a corner (again, with 3 possible choices). Each of the 3×3×2=18332183\times 3\times 2=183 × 3 × 2 = 18 possible combinations of corners for the effective-boundary defects to be located correspond to local energy minimizing configurations. Determining precisely which three corners yield the global energy minimum requires direct comparison. The LC configuration around a single regular polygon has also been studied using a reduced Landau–de Gennes framework [35, 34].

BCA

BAC

CAB

Selection 2

**A**: Moreover, Theorem 5 shows that subpopulations subject to distinct maternal effects can generate arbitrarily complex population dynamics, thereby affirming the main thesis of [18]. **B**: The extension of (7) to (8) is interesting in its own right, and should be interesting to develop in future work**C**: In particular, considerations of the analogies between biology and physics via the virial theorem led us to generalize the work of [18] and to derive the ecological simple harmonic oscillator, which to our knowledge is the first example of such an oscillator that emerges solely from maternal effects and does not require a predator-prey or other more sophisticated model

CAB

BAC

ACB

CBA

Selection 4

**A**: WFs have several profound connections with quantum-geometrical and topological aspects of the electronic structure [436]; some of them are discussed in Sec. III.5 and III.6**B**: A prime example of geometrical—and in some circumstances also topological—quantity is the electric polarization of periodic solids, which can be calculated in reciprocal space as a Berry phase [436] (see Sec. III.5).**C**: In the following, we refer to topological properties as a subset of geometrical properties that are quantized and hence represented by integer topological invariants robust to certain classes of perturbations

ACB

CAB

BAC

Selection 1

**A**: Frank for helpful suggestions, leading to our consideration of Lorentz norms in Theorem 2. We also thank Hajer Bahouri, Morris Brooks, Albert Cohen, Patrick Gérard, Kihyun Kim, Mathieu Lewin, Hoai-Minh Nguyen, Julien Sabin, Nikita Simonov, Thomas Sørensen, Jakob Stern, Hanne van den Bosch and Jean van Schaftingen for interesting and inspiring discussions.**B**: We are grateful to Rupert L**C**: Part of the work has been done when CD was a visiting reseacher at the Institut des Hautes Études and she would like to thank Laure Saint-Raymond for her support and hospitality

ABC

BCA

CBA

ABC

Selection 3

**A**: The latter are also going to be used to investigate spectral and scattering properties in § 3.3**B**: This section is devoted to the proofs of the results previously stated. The first result we prove is the classification of self-adjoint extensions: we start by addressing the quadratic forms (2.11) in § 3.1 and then apply Krein’s theory § 3.2 to get an alternative parametrization of the family, through the expression of the resolvents**C**: Finally, in § 3.4 and § 3.5 we study the symmetry properties of the extensions and the relation with the self-adjoint realization of the Dirac operator with an AB flux, respectively.

BCA

CBA

BAC

BCA

Selection 3

**A**: Valerii Kozin for valuable comments.**B**: Leggett Institute for Condensed Matter Theory, Urbana (USA), during the inauguration workshop New Horizons in Condensed Matter Physics, November, 2023, which stimulated this work. We also thank Dr**C**: We wish to acknowledge hospitality at The Anthony J

ACB

BAC

CAB

CBA

Selection 4

**A**: First of all, it can be seen in Fig. 8(a) that the IV curves from the two different simulations are again essentially equal to each other (on linear scale and with this zoom level). Moreover, here the short-circuit current is practically equal to the solar generation rate expressed in mA/cm2. Removing the AlGaAs layer therefore removes the related electron leakage in the simulations, even if a few percent of the incoming solar light is still absorbed in the GaInP layer next to the rear contact**B**: The net emission rate is also shown for this case in Fig. 8(b), calculated similarly as in Fig. 6(b). Interestingly, also here there is a notable difference between the RTDD and IRTDD case only at the small voltages but not at the maximum power point of 1.0 V.**C**: To check whether modifying the layer structure somewhat would cause differences between the RTDD and IRTDD simulations, Fig. 8 repeats the information of Fig. 6 for the device shown in Fig. 5, but with the bottommost AlGaAs layer removed

BCA

BAC

CAB

ABC

Selection 1

**A**: The scheme validated through this work suggests a viable way to design the shape of twin open cavities, i.e. those cavities intended to resonate with the same spectrum of a reference cavity, but having a different geometry**B**: The upshot is that one could take designs from the literature that use constant parameter acoustic fluids as the reference and then transform then to other more convenient geometries. The resultant cavity/surface would have some region with anisotropic fluid within it but, as demonstrated, this could be replaced by layered or discrete homogeneous media as effective replacements. **C**: We have chosen examples to display the versatility of the approach, i.e. a resonant Helmholtz cavity, which is a good test of the approach having both trapped and leaky modes to reproduce and then a case from periodic systems, which is non-resonant but now has sharp corners

ACB

BAC

ABC

Selection 1

**A**: The light polarization was controlled with a half-plate (light polarization angle). The single domain BiFeO33{}_{3}start_FLOATSUBSCRIPT 3 end_FLOATSUBSCRIPT thin film with a thickness of 180 nm clearly shows an optical birefringence (Fig. 7a) as previously discussed [39]. On the opposite the optical reflectivity measured on the BFO/LFO superlattice does not show any detectable birefringence (Fig. 7b). **B**: above the band gap of BFO and LFO**C**: The optical reflectivity measurement has been performed with a photon energy of 3 eV, i.e

ACB

ABC

ACB

CBA

Selection 4

**A**: These coefficients can be obtained by solving the Boltzmann equation**B**: It is evident from Eq. (44) and Eq. (45) that to determine the temperature dependence of shear and bulk viscosities, it is essential to calculate the scalar coefficients 𝒜𝒜\mathcal{A}caligraphic_A and 𝒞𝒞\mathcal{C}caligraphic_C**C**: In this work, we solved the Boltzmann equation by the relaxation time approximation (RTA) approach in order to determine the shear and bulk viscosities.

CAB

BAC

CAB

ACB

Selection 2

**A**: To find the ground state configuration, we perform an extensive sampling of all the distinct flux configurations numerically**B**: We leverage a “generalized” version of Lieb’s theorem Macris and Nachtergaele (1996); Jaffe and Pedrocchi (2014); Chesi et al. (2013) that reduces dramatically the number of samplings required to identify the ground state. **C**: Given that the quasicrystal consists of distinct polygons, it is not obvious a priori whether the ground state can be flux-free, and if not, which polygons host non-zero fluxes

ACB

BCA

CBA

CAB

Selection 2

**A**: It gives rise to primordial density fluctuations, acting as the foundational “seeds” for the creation of the cosmic structures we observe today, including galaxies and galaxy clusters. These primordial density fluctuations, in the standard inflationary scenario, originate from quantum fluctuations of a scalar field during the inflationary epoch. Additionally, inflation triggers the amplification of tensor perturbations in the spacetime metric, resulting in the generation of primordial gravitational waves. **B**: Inflation explains critical aspects of the universe’s structure, such as its immense size, uniformity, isotropy, and overall geometry**C**: In the vast landscape of cosmology, inflation emerges as a crucial concept that sheds light on fundamental characteristics of the observable universe at large scales

ABC

CAB

BCA

CBA

Selection 4