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In this paper, we presented a simple and robust algorithm for QPE using compressed sensing. For the single eigenvalue estimation (i.e., QPE), we rigorously established its Heisenberg-limit scaling in Theorem 2 and numerically demonstrated its performance compared to other state-of-the-art QPE algorithms in Sec. 4. Our algorithm has a smaller average error when the initial overlap is large, provided that the runtime costs (Ttotal,Tmax)subscript𝑇totalsubscript𝑇max(T_{\mathrm{total}},T_{\mathrm{max}})( italic_T start_POSTSUBSCRIPT roman_total end_POSTSUBSCRIPT , italic_T start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT ) are approximately the same. Similarly to QMEGS, our algorithm is non-adaptive, which means we can perform all the measurements first, and then focus on the classical post-processing part. As a comparison, RPE-inspired algorithms are usually adaptive [12, 13]. Our algorithm requires a rather small size of 𝒯𝒯\mathcal{T}caligraphic_T on a discrete set. In Fig. 3, for a signal of length N∈(100,600)𝑁100600N\in(100,600)italic_N ∈ ( 100 , 600 ), our algorithm only requires ≈101absentsuperscript101\approx 10^{1}≈ 10 start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT different time samples while MM-QCELS requires ≈102absentsuperscript102\approx 10^{2}≈ 10 start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT number of time samples from a continuous region. Our numerical tests also show that QMEGS only requires ≈101absentsuperscript101\approx 10^{1}≈ 10 start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT different samples to obtain an accurate estimation. As mentioned in the end of Sec. 2.3, there is a close connection between QMEGS and algorithm OMP, and we conjecture that it is possible to analyze QMEGS using the framework of compressed sensing.
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Compressed sensing, in essence, is similar to non-adaptive MM-QCELS. Both methods aim to fit the sampled data using a signal ansatz. However, they differ in sampling strategies and optimization objectives. In MM-QCELS, times are sampled from a continuous probability distribution, and the cost function minimizes the total empirical error in the time domain. In contrast, compressed sensing samples times uniformly at random from a discrete set, and the cost function minimizes the 1-norm of the signal in the frequency domain.
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For a Heisenberg-limited QPE algorithm with maximal runtime Tmaxsubscript𝑇T_{\max}italic_T start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT, if the size of the time samples needed is 𝒪(polylogTmax)𝒪polysubscript𝑇\mathcal{O}(\mathrm{poly}\log T_{\max})caligraphic_O ( roman_poly roman_log italic_T start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT ), the sampling is considered sparse in our paper. In this work, we design a non-adaptive algorithm that only requires discrete and sparse sampling of times without sacrificing performance. Informally, our main result can be stated as follows.
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In discrete sampling protocols, would it be possible to shorten the maximal runtime by biased sampling of times? What is the limitation in the discrete scenario? Can we achieve a similar improvement to the Gaussian filter method in [18]?
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In our numerical experiments, the test of another sampling (Algorithm 3) is actually unnecessary. Is it possible to show this analytically as well?
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The same strategy is used to compute density_matrixT(v)subscriptdensity_matrix𝑇𝑣\texttt{density\_matrix}_{T}\left(v\right)density_matrix start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT ( italic_v ).
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We compare the asymptotic computational costs of the density matrix algorithm and the baseline algorithm that utilizes canonicalization, as discussed in Section 3.2.1. In Theorem 6.1, we establish that the density matrix algorithm can be more efficient in approximating a general tensor network as an embedding tree. The cost of the density matrix algorithm is upper-bounded by that of the canonicalization-based algorithm. This efficiency arises from the fact that the density matrix algorithm does not need to explicitly contract the partition embedded in each tree vertex into a tensor.
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We will show in Section 6.4 that the cost of the density matrix algorithm is upper-bounded by the canonicalization-based algorithm.
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In Algorithm 4, the computation of each density matrix occurs only once. Considering that the embedding tree T𝑇Titalic_T is limited to being a rooted binary tree, there are at most three density matrices to be calculated for each vertex v𝑣vitalic_v in the embedding tree. Below we bound the asymptotic computational cost of the density matrix algorithm using memoization, and we show that for a given embedding ϕitalic-ϕ\phiitalic_ϕ, the cost will be upper-bounded by the algorithm that uses canonicalization, justifying the efficiency of the algorithm.
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Specifically, we show in Section 6.4 that the asymptotic computational cost of the algorithm is upper-bounded by the cost of the canonicalization-based algorithm, and we show in Section 8 that for many input tensor networks, the proposed algorithm substantially reduces the overall execution time.
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Fig. 7 examines the effects of the parameter cOsubscript𝑐Oc_{\rm O}italic_c start_POSTSUBSCRIPT roman_O end_POSTSUBSCRIPT on the time-domain profiles. Here, for small values of time t𝑡titalic_t, both scalar and electromagnetic perturbations exhibit very similar profiles, with minimal differences in their oscillation patterns. However, as time progresses, variations in oscillation frequencies become more noticeable, particularly for larger values of t𝑡titalic_t. Despite these differences, the damping rates for both types of perturbations appear relatively unaffected by changes in cOsubscript𝑐Oc_{\rm O}italic_c start_POSTSUBSCRIPT roman_O end_POSTSUBSCRIPT, suggesting that this parameter influences the frequency of oscillations more than the rate at which the perturbations decay. For the range of cOsubscript𝑐Oc_{\rm O}italic_c start_POSTSUBSCRIPT roman_O end_POSTSUBSCRIPT values considered, the impact on damping seems marginal, indicating that cOsubscript𝑐Oc_{\rm O}italic_c start_POSTSUBSCRIPT roman_O end_POSTSUBSCRIPT primarily affects the oscillatory dynamics rather than the overall stability.
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The analysis of the perturbation potential provides valuable insights into the nature of the QNMs for both scalar and electromagnetic perturbations. The variations in the potential with respect to parameters such as the multipole moment l𝑙litalic_l, the model parameter α𝛼\alphaitalic_α, the coupling parameter cOsubscript𝑐Oc_{\rm O}italic_c start_POSTSUBSCRIPT roman_O end_POSTSUBSCRIPT, and the black hole charge Q𝑄Qitalic_Q reveal how sensitive the QNMs are to these factors. This understanding is crucial for predicting the stability and dynamical response of black holes to perturbations, particularly in the context of different gravity models or in the presence of additional fields. One may note that with Q=0𝑄0Q=0italic_Q = 0, one can easily obtain the corresponding potentials discussed in Ref. [35]. However, we have noticed that the charge of the black hole i.e. Q𝑄Qitalic_Q has a noticeable impact on the QNMs spectrum as predicted by the variation of the corresponding potential. In the presence of Q𝑄Qitalic_Q, the peak value of the potential increases and the variation is not similar to that of the Reissner-Nordström black hole. Hence, the QNMs spectrum seems to be different from that of a standard Reissner–Nordström black hole.
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The charge parameter Q𝑄Qitalic_Q has a significant and nonlinear impact on the QNM spectrum of black holes. As shown in Figs. 13 and 14, both the oscillation frequency (real part of QNMs) and the damping rate (imaginary part of QNMs) increase as Q𝑄Qitalic_Q increases. This behavior holds true for both scalar and electromagnetic perturbations, although the precise values of the frequencies and decay rates differ depending on the type of perturbation. In the case of electromagnetic perturbations, the oscillation frequencies and damping rates are generally lower than those observed for scalar perturbations. The nonlinear nature of the relationship between Q𝑄Qitalic_Q and the QNMs suggests that even small changes in the charge parameter can have a pronounced effect on the behavior of the black hole.
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Finally, in Fig. 8, the time-domain profiles are plotted with varying values of the charge parameter Q𝑄Qitalic_Q. The results indicate that the black hole charge has a noticeable effect on the oscillation frequency for both scalar and electromagnetic perturbations. As Q𝑄Qitalic_Q increases, the oscillation frequencies increase, suggesting that a more highly charged black hole causes perturbations to oscillate more rapidly. The damping rate, however, increases only slightly for both types of perturbations, implying that while the charge of the black hole influences the oscillatory behavior, it has a relatively minor impact on the overall rate at which the perturbations decay. This finding highlights that the black hole charge primarily affects the dynamic response of the system without drastically altering its stability over time.
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Physically, the increase in oscillation frequency with higher Q𝑄Qitalic_Q can be understood in the context of the black hole’s enhanced electric field. As the black hole’s charge increases, the strength of its electromagnetic field grows, which leads to more tightly bound perturbations. This results in higher frequency oscillations for perturbing fields around the black hole. Furthermore, the increase in the damping rate with Q𝑄Qitalic_Q indicates that the perturbations decay more rapidly, meaning that the charged black hole tends to settle down faster after being perturbed. The faster decay is likely due to the increased energy stored in the electromagnetic field, which enhances the dissipation of perturbing waves.
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In addition, a number of potential improvement may be anticipated. The usage of permutations on the residual indices in step (vi) in Subsection II.3 is a posterior collection. It may be feasible to be performed as a prior with the step (iiid) of using linear combinations of excitation operators. That would lead to a simplification of the workflow. Adopting orthogonal relations to accelerate convergence for different spin states knowles2000erratum may be beneficial for general-order PSA-CC methods.
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In the present work, we reported the further formulations for the linear combinations of the projection manifolds, the hash-table canonicalization algorithm, and the numerical results for the previous general-order open-shell CC method based on the PSA scheme wang2024general . The energy differences between the present approaches and the spin-orbital CC methods are below 0.1 kcal mol-1. After further optimizations the PSA-CC methods such as factorizations and merging equations with active indices, the current approach is expected to be an improvement with lower prefactors of computational costs for the spin-orbital based general-order CC methods for open-shell systems.
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We further notice the differences between PSA and spin-orbital CC are mainly at the CCSD levels. For example, in the OH molecule, the difference between the PSA-T[1|2]delimited-[]conditional12[1|2][ 1 | 2 ]R[1|2]delimited-[]conditional12[1|2][ 1 | 2 ]-CCSD and the spin-orbital CCSD is about 6 ×\times× 10-5 a.u., that is similar to the difference between the PSA-T[1|2|3]delimited-[]123[1|2|3][ 1 | 2 | 3 ]R[1|2|3]delimited-[]123[1|2|3][ 1 | 2 | 3 ]-CCSDT and the spin-orbital CCSDT, 8 ×\times× 10-5 a.u.. Likewise, the difference between PSA-T[1|2|3]delimited-[]123[1|2|3][ 1 | 2 | 3 ]R[1|2|3]delimited-[]123[1|2|3][ 1 | 2 | 3 ]-CCSDTQ and spin-orbital CCSDTQ is about 7 ×\times× 10-5 a.u. These data suggest further improving the PSA accuracy may be more effective to increase the spin adaptations at the CCSD levels, e.g., the T[2|2]delimited-[]conditional22[2|2][ 2 | 2 ] level, than the higher-order CC expansions such as CCSDT and CCSDTQ. This is similar to the compound methods martin1999towards ; tajti2004heat ; yang2014ab ; karton2017w4 ; karton2022quantum . Namely, smaller basis sets with higher-order CC levels are effective to obtain results beyond the chemical accuracy. Since the size consistency issue hirata2015mutual is practically unimportant at the PSA-T[1|2]delimited-[]conditional12[1|2][ 1 | 2 ]R[1|2]delimited-[]conditional12[1|2][ 1 | 2 ]-CCSD level Heckert:06 , this aspect may not be effective in the higher-order methods.
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Adapting spatial orbitals in open-shell CC methods bartlett1995coupled ; christiansen2006coupled ; Bartlett:07 ; crawford2007introduction ; Bartlett:09 ; bartlett2012coupled ; Lyakh:12 ; bartlett2024perspective ; Helgaker:00book ; Harris:99 ; Olsen:00 ; Hirata:00 ; hirata2003tensor ; Kallay:00 ; kallay2001higher ; auer2006automatic ; parkhill2010sparse ; engels2011fully ; evangelista2022automatic ; Stanton:14 ; Stanton:15 can reduce the prefactors of computational costs comparing to the spin-orbital approaches Bartlett:09 at least in principle, since the dimension the vector space of interest from the spatial indices is smaller than the spin-orbital indices for a given basis set Helgaker:00book . Different from the closed-shell approaches scuseria1987closed ; scuseria1988efficient ; lee1988efficient ; koch1990coupled ; hampel1992comparison ; Helgaker:00book ; Bartlett:09 ; Stanton:14 ; matthews2014non ; Stanton:15 ; wang2018simple ; matthews2019diagrams , the contractions between active orbitals in open-shell systems complicate the formulations and caused open-shell spatial-orbital CC methods remain a challenge in quantum chemistry lindgren1978coupled ; nakatsuji1977cluster ; nakatsuji1978cluster_a ; nakatsuji1978cluster ; mukherjee1979hierarchy ; nakatsuji1979cluster ; nakatsuji1983cluster ; haque1984application ; mukherjee1989use ; Janssen:91 ; Knowles:93 ; jayatilaka1993open ; Li:94 ; neogrady1994spin ; Jeziorski:95a ; Neogrady:95 ; nooijen1996many ; nooijen1996general ; urban1997spin ; Szalay:97 ; Szalay:00 ; Li:98 ; Jeziorski:99 ; nooijen2001towards ; grotendorst2000modern ; knowles2000erratum ; Helgaker:00book ; Heckert:06 ; Datta:08 ; datta2011state ; Datta:13 ; datta2014analytic ; gunasekera2024multireference ; datta2015communication ; datta2019accurate ; herrmann2020generation ; herrmann2022analysis ; herrmann2022correctly ; folkestad2023entanglement . Accuracy beyond the chemical accuracy (1 kcal mol-1) can be important in predicting reaction selectivities krieger2021impact ; laplaza2024overcoming . Higher-order CC methods (beyond triple excitations in the cluster operator) martin1999towards ; tajti2004heat ; yang2014ab ; karton2017w4 ; karton2022quantum is one of the few methods Bartlett:09 ; zahrt2019prediction ; thorpe2024factorized at present mcardle2020quantum can reach this level of accuracy for general many-electron systems.
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The number of resulting equations can be large at higher order PSA-CC and spin adaptations. This could slow down the computations. Though for a given basis set, the finite dimensional vector space for the computations from the spatial orbitals is smaller than from the spin orbitals. One may then expect a further optimized PSA-CC will be more efficient than the spin-orbital approaches. Nevertheless, it may be necessary to advance further improvement including factorizations kucharski1991recursive ; kallay2001higher ; hirata2003tensor ; parkhill2010sparse ; lai2012effective ; pfeifer2014faster ; manzer2017general ; engels2011fully . For example, in the present PSA-CC, there are many contractions involving active indices
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Although the above AFM band-folding analysis based on the tight-binding model helps us to characterize the momentum-dependent pseudogap, it requires the abrupt collapse of the ΔksubscriptΔ𝑘\Delta_{k}roman_Δ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT parameter around (π𝜋\piitalic_π, 0), indicating the breakdown of the mean-field-type band-folding picture. Furthermore, the key aspects of the experimental results are the absence of band folding and of the spectral-weight suppression across the folded BZ boundary in the antinodal lower band (Figs. 1f and g). Even more important is the gap opening away from the folded BZ boundary in the antinodal region (Figs. 3b–d). While the gap positions nearly trace the AFM BZ boundary around (π/2𝜋2\pi/2italic_π / 2, π/2𝜋2\pi/2italic_π / 2), they deviate from the AFM BZ boundary as the cut approached (π𝜋\piitalic_π, 0). The gap positions in k space plotted in Fig. 3e follows the original Fermi surface before the AFM band folding, on which the gossamer Fermi surface Xu et al. (2023) and the electron pocket are formed and are smoothly connected, rather than the AFM BZ boundary. These spectral features are incompatible with the slow, long-range AFM correlation scenarios and suggest the importance of more short-ranged, dynamical AFM correlation in creating the pseudogap.
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suggests that the Mott physics is at the origin of the pseudogap of electron-doped cuprates through the enhanced electron correlations without long-range AFM correlations, as in the case of the hole-doped cuprates Maier et al. (2002); Civelli et al. (2005); Kyung et al. (2006); Ferrero et al. (2009); Sakai et al. (2010, 2013).
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As for the global momentum dependence of the pseudogap, the ARPES results reported by Matsui et al. Matsui et al. (2005) and Ikeda et al. Ikeda et al. (2009) have indicated the reduction of pseudogap magnitude upon approaching (π𝜋\piitalic_π, 0). This momentum dependence cannot be understood within the simple AFM band-folding picture, which predicts a constant gap magnitude on the AFM BZ boundary. Most of theoretical studies, either with weak-coupling Kyung et al. (2004) or strong-coupling treatment Sénéchal and Tremblay (2004), have not explicitly shown the momentum dependence of the AFM gap, whereas a variational Monte-Carlo study based on the two-dimensional t𝑡titalic_t-t′superscript𝑡′t^{\prime}italic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT-t"𝑡"t"italic_t "-J𝐽Jitalic_J model by Chou and Lee Chou and Lee (2008) and a variational cluster-perturbation theory study of the Hubbard model by Sénéchal et al. Sénéchal et al. (2005) have shown that the gap decreases as one goes from the hot spot to (π𝜋\piitalic_π, 0). The failure of the simple AFM band-folding picture and the previous theoretical models to explain the salient features of the ARPES data should contain crucial information about the nature of the electron-electron and spin-spin correlations not only in the electron-doped cuprates but also in the hole-doped cuprates, and calls for further investigations.
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So far, the pseudogap of the electron-doped cuprates has been observed through the measurements of optical conductivity Onose et al. (2001); Wang et al. (2006), scanning tunneling spectroscopy Zimmers et al. (2007), and ARPES Armitage et al. (2001); Matsui et al. (2005); Park et al. (2007); Matsui et al. (2007); Richard et al. (2007); Ikeda et al. (2009); Song et al. (2012); Horio et al. (2016).
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One scenario to explain the overall band structure without assuming the long-range AFM order is the nodeless pseudogap proposed by Sakai et al. Sakai et al. (2010, 2013), based on a CDMFT calculation Kotliar et al. (2001) on the Hubbard model. They
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However, as highlighted in discussions of entanglement entropy, the AdS/CFT correspondence inherently features nonlocality[9].
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Then, the training can be interpreted as a correction due to quantum effects corresponding to nonlocality.
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This nonlocality, which arises from quantum mechanical effects, is also expected to permute or spread information beyond the classical distortions of time and space mentioned earlier.
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Note that our discussion of nonlocality concerns how quantum effects can spread or mix information in ways that exceed classical correlations.
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However, note that nonlocality stemming from quantum effects manifests as the mixing and spreading of information, and it can be partially emulated even within classical CA models.
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The main purpose of this paper is to prove (1.2.2) for M=N−1𝑀𝑁1M=N-1italic_M = italic_N - 1 (Theorem 4.5.2).
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with monodromy q𝑞qitalic_q. Recall that q𝑞qitalic_q is assumed to be a transcendental complex number.
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use the Weil genericity of q𝑞qitalic_q assumption. Anyway, ℰℰ{\mathcal{E}}caligraphic_E is rigid and contains all the
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In this paper we deal with the case M=N−1𝑀𝑁1M=N-1italic_M = italic_N - 1 for generic q𝑞qitalic_q. The advantage of the M=N−1𝑀𝑁1M=N-1italic_M = italic_N - 1 assumption is that in this case the group UM,Nsubscript𝑈𝑀𝑁U_{M,N}italic_U start_POSTSUBSCRIPT italic_M , italic_N end_POSTSUBSCRIPT is
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In other words, assuming that q𝑞qitalic_q is transcendental111Probably the assumption that q𝑞qitalic_q is not
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The time the GAN post-processing takes is negligible in comparison, taking 0.35 seconds per model year on a V100 GPU and 37.17 seconds on a single CPU. The quantile mapping is similarly efficient taking 0.59 seconds per model year on a CPU.
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This can be explained by the fact that the precipitation of the future scenario lies well outside the training distribution.
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Down-scaling applications that increase the resolution of the ESM could be a direction for future research.
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A potential direction for future research could be to develop a stochastic model that directly learns the uncertainties.
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taking ERA5 reanalysis data as ground truth. Note that any other, and especially purely observational, precipitation dataset with sufficient temporal resolution could readily be used instead.
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|\langle\pi|\varphi\rangle|^{2},italic_λ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( | italic_φ ⟩ ) = start_UNDERACCENT | italic_π ⟩ end_UNDERACCENT start_ARG max end_ARG | ⟨ italic_π | italic_φ ⟩ | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ,
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}{\mathcal{N}^{2}}<\lambda^{2}| ⟨ italic_π | over~ start_ARG italic_φ end_ARG ⟩ | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = divide start_ARG 1 end_ARG start_ARG caligraphic_N start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG | ⟨ italic_π | italic_φ ⟩ + italic_θ ⟨ italic_π | italic_η ⟩ | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = divide start_ARG italic_λ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG caligraphic_N start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG < italic_λ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT
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|\langle\pi|\varphi\rangle|^{2},italic_λ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( | italic_φ ⟩ ) = start_UNDERACCENT | italic_π ⟩ end_UNDERACCENT start_ARG max end_ARG | ⟨ italic_π | italic_φ ⟩ | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ,
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While computing λ2superscript𝜆2\lambda^{2}italic_λ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT for a generic pure state is, in principle, difficult
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G(|φ⟩)=1−λ2(|φ⟩)𝐺ket𝜑1superscript𝜆2ket𝜑G(|\varphi\rangle)=1-\lambda^{2}(|\varphi\rangle)italic_G ( | italic_φ ⟩ ) = 1 - italic_λ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( | italic_φ ⟩ )
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A different argument to prove chiral symmetry breaking using ’t Hooft anomaly matching was pursued independently by Schwimmer in Ref. Schwimmer (1982). He considered an SU(Nf|Nf)𝑆𝑈conditionalsubscript𝑁𝑓subscript𝑁𝑓SU(N_{f}|N_{f})italic_S italic_U ( italic_N start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT | italic_N start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT ) superalgebra whose grading acts so that odd supergenerators mix massless states with opposite helicity. Although strictly speaking SU(Nf|Nf)𝑆𝑈conditionalsubscript𝑁𝑓subscript𝑁𝑓SU(N_{f}|N_{f})italic_S italic_U ( italic_N start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT | italic_N start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT ) is not a symmetry of a QCD-like theory, it can be used as a classification group. Schwimmer argued that, for a purely baryonic massless spectrum, irreducible representations of SU(Nf|Nf)𝑆𝑈conditionalsubscript𝑁𝑓subscript𝑁𝑓SU(N_{f}|N_{f})italic_S italic_U ( italic_N start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT | italic_N start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT ) satisfy PMC trivially, but they cannot furnish an integral solution of AMC.
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It would be desirable to extend the work of Schwimmer to the case of a massless spectrum with exotics, and prove that a generic PMC solution can be always written in terms of irreps of SU(Nf|Nf)𝑆𝑈conditionalsubscript𝑁𝑓subscript𝑁𝑓SU(N_{f}|N_{f})italic_S italic_U ( italic_N start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT | italic_N start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT ). In this way one would obtain an algebraic proof of chiral symmetry breaking valid for any Nf>2subscript𝑁𝑓2N_{f}>2italic_N start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT > 2. We leave such study to future work.
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In order to elevate this simple and elegant argument to the level of a proof of chiral symmetry breaking, one should demonstrate that it works for a generic spectrum with exotics and that it gives the most general solution of PMC, i.e. that every PMC solution is a collection of superalgebra irreps. This evidence is still missing.
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The alternative approach proposed by Schwimmer in Ref. Schwimmer (1982) makes use of the SU(Nf|Nf)𝑆𝑈conditionalsubscript𝑁𝑓subscript𝑁𝑓SU(N_{f}|N_{f})italic_S italic_U ( italic_N start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT | italic_N start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT ) superalgebra and does not rely on Nfsubscript𝑁𝑓N_{f}italic_N start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT-independence. As such, it is a promising candidate for a purely algebraic proof of chiral symmetry breaking. We have explicitly verified that, in the case of QCD-like theories with Nc=3subscript𝑁𝑐3N_{c}=3italic_N start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT = 3 and 5, the superalgebra solution coincides with the generic solution of PMC equations if one assumes a purely baryonic spectrum of massless states. This result holds for any number of flavors Nf>2subscript𝑁𝑓2N_{f}>2italic_N start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT > 2, i.e. even in presence of equivalent tensors.
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We have shown that the form of PMC[Nf,1]subscript𝑁𝑓1[N_{f},1][ italic_N start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT , 1 ] is Nfsubscript𝑁𝑓N_{f}italic_N start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT-dependent in general, which implies that PMC[Nf+1,1]PMCsubscript𝑁𝑓11\text{PMC}[N_{f}+1,1]PMC [ italic_N start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT + 1 , 1 ] are linearly independent with respect to PMC[Nf,1]PMCsubscript𝑁𝑓1\text{PMC}[N_{f},1]PMC [ italic_N start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT , 1 ]. This is in turn an obstruction to the Nfsubscript𝑁𝑓N_{f}italic_N start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT-independence of the solution of PMC. However, Nfsubscript𝑁𝑓N_{f}italic_N start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT-independence can be still obtained if one imposes suitable restrictions on the spectrum of possible massless states.
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(c) Fusion and Fission of half-skyrmions, accompanying annihilation and creation of -1/2 disclinations. This process occurs in a diffusionless manner and becomes the dominant contribution to the structural relaxation as shown in Fig. 5.
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A notable feature arising from half-skyrmion dynamics is the fusion and fission of half-skyrmions, as marked in Fig. 3(a) and (b), where two half-skyrmions merge into a large single half-skyrmion (fusion) and vice versa (fission). This process is schematically described in Fig. 3(c), accompanied by the annihilation and creation of two disclinations with vorticity q=−1/2𝑞12q=-1/2italic_q = - 1 / 2 at the vertex of three half-skyrmions. The vorticity q𝑞qitalic_q is a topological invariant defined as the molecular rotation angle along a closed loop: q=12π∮𝑑ℓ⋅∇ϕ𝑞12𝜋contour-integral⋅differential-dbold-ℓ∇italic-ϕq=\frac{1}{2\pi}\oint~{}d\mbox{\boldmath$\ell$}\cdot\nabla\phiitalic_q = divide start_ARG 1 end_ARG start_ARG 2 italic_π end_ARG ∮ italic_d bold_ℓ ⋅ ∇ italic_ϕ, where ϕ=tan−1(ny/nx)italic-ϕsuperscript1subscript𝑛𝑦subscript𝑛𝑥\phi=\tan^{-1}(n_{y}/n_{x})italic_ϕ = roman_tan start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_n start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT / italic_n start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ) denotes the molecular angle in xy𝑥𝑦xyitalic_x italic_y-plane. q=−1/2𝑞12q=-1/2italic_q = - 1 / 2 for the disclination in Fig. 3(d), whereas q=1𝑞1q=1italic_q = 1 in half-skyrmions in Fig. 1(d). When fusion occurs, the number of half-skyrmions (q=1𝑞1q=1italic_q = 1) decreases by one, whereas two disclinations (q=−1/2𝑞12q=-1/2italic_q = - 1 / 2) annihilate. Hence, the total vorticity is conserved.
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(c) Fusion and Fission of half-skyrmions, accompanying annihilation and creation of -1/2 disclinations. This process occurs in a diffusionless manner and becomes the dominant contribution to the structural relaxation as shown in Fig. 5.
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(d) Schematic representation of half-skyrmions (left) and skyrmions (right). Molecular orientation becomes perpendicular (parallel) to the center in half-skyrmions (skyrmions). The vorticity q𝑞qitalic_q and skyrmion number Nsksubscript𝑁skN_{\rm sk}italic_N start_POSTSUBSCRIPT roman_sk end_POSTSUBSCRIPT read (q,Nsk)=(1,1/2)𝑞subscript𝑁sk112(q,N_{\rm sk})=(1,1/2)( italic_q , italic_N start_POSTSUBSCRIPT roman_sk end_POSTSUBSCRIPT ) = ( 1 , 1 / 2 ) for half-skyrmions, while (1,1)11(1,1)( 1 , 1 ) for skyrmions.
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(d) Schematic illustration of annihilation process of disclinations with vorticity q=−1/2𝑞12q=-1/2italic_q = - 1 / 2.
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D
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There are two viable possibilities: the heavy neutrinos Nisubscript𝑁𝑖N_{i}italic_N start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT can be either lighter or heavier than the new scalars.
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The expected number of heavy neutrinos produced at the interaction point and decaying with a displacement of at least x1subscript𝑥1x_{1}italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and at most x2subscript𝑥2x_{2}italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, can be obtained by combining the production and decay of heavy neutrinos 222The vertex displacement x𝑥xitalic_x is defined as the distance between the primary vertex where the heavy neutrino (N𝑁Nitalic_N or N¯¯𝑁\bar{N}over¯ start_ARG italic_N end_ARG) with finite lifetime was produced, and its secondary decay vertex. .
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In this case oscillations can occur between the members of the quasi-Dirac pair, preventing the cancellations that would otherwise suppress the LNV signal. Assuming this oscillation is effective, which holds for an ideal detector as long as ΔM≫ΓNmuch-greater-thanΔ𝑀subscriptΓ𝑁\Delta M\gg\Gamma_{N}roman_Δ italic_M ≫ roman_Γ start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT (see Sec. 5), we now discuss the cross sections for all possible LNV final states coming from the production channels e+e−→NN¯,e−γ→NH−formulae-sequence→superscript𝑒superscript𝑒𝑁¯𝑁→superscript𝑒𝛾𝑁superscript𝐻e^{+}e^{-}\to N\overline{N},\,e^{-}\gamma\to NH^{-}italic_e start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT → italic_N over¯ start_ARG italic_N end_ARG , italic_e start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT italic_γ → italic_N italic_H start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT and e+e−→H+H−→superscript𝑒superscript𝑒superscript𝐻superscript𝐻e^{+}e^{-}\to H^{+}H^{-}italic_e start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT → italic_H start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT italic_H start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT. In Sec. 5, we discuss in detail the impact of heavy neutrino-antineutrino oscillations on the LNV rates, taking into account the finite detector size.
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We first discuss the novel production mechanisms for the heavy neutrinos and scalars, and then discuss their decay modes and resulting collider signatures.
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Therefore, the ratio between the LNC and LNV decay modes when the heavy quasi-Dirac neutrinos are produced as real particles is given by [16, 20]
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C
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However, we show that the conventional approach is not enough to define the interactions between two open, coupled non-equilibrium thermodynamic systems. Therefore, we introduce new interaction schemes and theorems to define the correct interaction equations. Below, for simplicity, we label dark matter and dark energy as systems 1111 and 2222, respectively.
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The continuity equation for a non-equilibrium system 1111 which interacts with the system 2222 should be presented in the form as
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Change of the source X𝑋Xitalic_X can be defined as X=Γ1x2𝑋subscriptΓ1subscript𝑥2X=\Gamma_{1}x_{2}italic_X = roman_Γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT where x2subscript𝑥2x_{2}italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is the energy flux from system 1 to the 2, Γ1subscriptΓ1\Gamma_{1}roman_Γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT denotes the energy transfer rates, i.e., the system 1111 which determines the energy amount which is transferred the from system 2222 to the system 1111. In the present study, based on this Lemma, we present a new interaction schema and new theorems.
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As seen from Fig.3, the energy x1subscript𝑥1x_{1}italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT flows from system 1 to system 2 and acts on system 2. Similarly, x2subscript𝑥2x_{2}italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT flows from the system 2 to the system 1 and acts on the system 1. While x1subscript𝑥1x_{1}italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT causes a cyclic dynamics within the system 2, and x2subscript𝑥2x_{2}italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT also causes a cyclic dynamics on the system 1.
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The interactions between two coupled non-equilibrium systems can be obtained from the energy conservation law of statistical thermodynamics and is given in a simple form as
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A
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A similar accuracy metrics was suggested in the study of IBD and LTID [47] for the harmonic potential case. We use it for the free particle case as well as the same harmonic one (in the following section). Criteria like (59) have been applied only for the range such that |Cp∗(τ∗)|>0.01superscriptsubscript𝐶𝑝superscript𝜏0.01\lvert C_{p}^{*}\left(\tau^{*}\right)\rvert>0.01| italic_C start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) | > 0.01, ignoring smaller values with spurious digits which are more sensitive to implementation details than to the integrator structure itself.
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The harmonic potential case is a basic model for microscopical interactions and correspondingly, to validate the quality of auto-correlations [47]. In addition, several real systems can be approximated at first order by such interactions in the case of small departures from equilibrium positions. The corresponding Langevin equation is given by:
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As the simulated systems exhibit the hallmark of scale separation, computational strategies rely on the reduction of fast degrees of freedom in favor of retaining the slow, important ones. A similar consideration apply to representing systems in contact with a heat reservoir that does not need to be represented in full detail. The implicit solvent model [17] and the usage of effective thermostats to sample states from the canonical ensemble [1] consider the motion of particles within a stochastic environment formed by other particles represented in effective terms. As a result, the molecular trajectory is approximated by Langevin dynamics (LD) [1, 18] featuring frictional and random forces, the latter encoded by a delta-correlated stationary Gaussian process [19], while the Brownian dynamics refers to dynamics in presence of explicit effective hydrodynamic forces [18]. The equations of motion of Langevin or Brownian dynamics are not phenomenological but derived through rigorous multiscale theoretical analysis.
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A similar accuracy metrics was suggested in the study of IBD and LTID [47] for the harmonic potential case. We use it for the free particle case as well as the same harmonic one (in the following section). Criteria like (59) have been applied only for the range such that |Cp∗(τ∗)|>0.01superscriptsubscript𝐶𝑝superscript𝜏0.01\lvert C_{p}^{*}\left(\tau^{*}\right)\rvert>0.01| italic_C start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) | > 0.01, ignoring smaller values with spurious digits which are more sensitive to implementation details than to the integrator structure itself.
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Here and hereinafter, the subscript sim differentiates numerical values unless expressed explicitly. The expression is a version of the diffusion equation [45] in the case of a particle at rest at time τ∗=0superscript𝜏0\tau^{*}=0italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = 0 that is assigned a given initial velocity and in absence of thermal noise (in the so-called ”kick” experiment). We intend to reproduce the evolution for the time interval τ∗=[0,10]superscript𝜏010\tau^{*}=[0,10]italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = [ 0 , 10 ]. Testing this time range was performed for a small time step Δ∗superscriptΔ\Delta^{*}roman_Δ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT=0.01. The precision parameter is taken to be:
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A
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The advantages of studying the hadronic final state are as follows. The large hadronic branching ratio and significant mass difference between VLQ and the scalar significantly boost the top quark. As a result, reconstructing the top quark as a boosted fatjet additionally provides the inherent properties of the jet. Furthermore, we can have additional handel using jet substructure variables.
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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. The kinematic cuts (C1-C4) are described in the text. After applying the C4 cut, the remaining events are passed for the multivariate analysis.
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(C4) pruned mass of the two leading jets MJ0,MJ1>120subscript𝑀subscript𝐽0subscript𝑀subscript𝐽1120M_{J_{0}},M_{J_{1}}>120italic_M start_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , italic_M start_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT > 120 GeV. After applying the preselection cut (C1), we find V𝑉Vitalic_V+jets (V=Z,W𝑉𝑍𝑊V=Z,Witalic_V = italic_Z , italic_W) are the principal background while tt¯𝑡¯𝑡t\bar{t}italic_t over¯ start_ARG italic_t end_ARG+jets is the subdominant background. However, after a b-tag within J0subscript𝐽0J_{0}italic_J start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT or J1subscript𝐽1J_{1}italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and demanding large fatjet masses, we found tt¯𝑡¯𝑡t\bar{t}italic_t over¯ start_ARG italic_t end_ARG+jets becomes the primary background, while V𝑉Vitalic_V+jets are the subdominant. Applying all those cuts, we still retain a substantial number of signal events while the background reduces significantly.
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All the signal and background processes are passed through all these event selection criteria up to C4 before passing events to MVA. We create two separate signal and background classes. The combined background is the weighted combination of all the different background processes. Each signal and background class is randomly divided into 50%percent5050\%50 % for training and the rest 50%percent5050\%50 % for testing. We use boosted decision tree (BDT) algorithm and choose a set of kinematic variables from a wider collection of variables for MVA. The variables with high relative importance distinguishing the signal class from the background class are preferable. Table 5 lists the relative importance of the various kinematic variables involved in the MVA. The left (signal) and right (background) tables of Figure 9 show the linear correlation coefficients among the variables employed in MVA for BP1. The optimized BDT hyperparameters used in our analysis are outlined in Table 6.
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The normalized distribution of the BDT response for test and train samples of both signal (BP1) and background classes is plotted on the left side of Figure 10. We find signal and background are well separated. With the cut applied to the BDT output, the signal and background efficiency, as well as the statistical significance (NSNS+NBsubscript𝑁𝑆subscript𝑁𝑆subscript𝑁𝐵\frac{N_{S}}{\sqrt{N_{S}+N_{B}}}divide start_ARG italic_N start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT end_ARG start_ARG square-root start_ARG italic_N start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT + italic_N start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT end_ARG end_ARG) for 139 fb−1superscriptfb1\text{fb}^{-1}fb start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT data, are presented in the right plot of Figure 10. Before applying any cuts to the BDT output, Table 7 shows the number of signals (NSbcsuperscriptsubscript𝑁𝑆𝑏𝑐N_{S}^{bc}italic_N start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT) and background (NSMsubscript𝑁𝑆𝑀N_{SM}italic_N start_POSTSUBSCRIPT italic_S italic_M end_POSTSUBSCRIPT) events for various BPs. It also shows the expected number of signal events (NSsubscript𝑁𝑆N_{S}italic_N start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT) and background events (NBsubscript𝑁𝐵N_{B}italic_N start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT) that remain after applying an optimal cut (BDToptsubscript𝑇𝑜𝑝𝑡T_{opt}italic_T start_POSTSUBSCRIPT italic_o italic_p italic_t end_POSTSUBSCRIPT) to the BDT output. We optimize each of the three BPs separately. The sixth column displays the statistical significance of the signal at 139 (300) fb−1superscriptfb1\text{fb}^{-1}fb start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT luminosity, incorporating Delphes simulation without systematics. The subsequent column provides an estimated statistical significance, accounting for a 5%percent55\%5 % systematic consideration. The last column illustrates the signal-to-background ratio.
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A
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Temperature-dependent magnetic susceptibility was measured using a 7 Tesla Cryogenic Ltd SQUID (superconducting quantum interference device) magnetometer with a Helium-3 probe from 0.3 K to 2 K and with a Helium-4 probe from 2 K to 300 K. For the Helium-3 measurements, a small crystalline YbZn2GaO5 sample of 1.04 mg was mounted on a silver sample holder and the Helium-4 measurement, 11.90 mg of YbZn2GaO5 single crystal was used. The crystals were oriented using the Laue diffraction method and the regular shape of the single crystals was obtained using a wire saw. The magnetic measurements were performed under an applied magnetic field parallel (H𝐻Hitalic_H∥parallel-to\parallel∥c) and perpendicular (H𝐻Hitalic_H⟂perpendicular-to\perp⟂c) to the crystallographic c-directions of YbZn2GaO5. The isothermal magnetization measurements along both directions of YbZn2GaO5 single crystal sample were performed using a VSM (vibration sample magnetometer) in PPMS up to 14 Tesla of the applied magnetic field (see Fig. S1).
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The inelastic neutron scattering (INS) experiments were performed on the Fine-Resolution Fermi Chopper Spectrometer (SEQUOIA) [49] and the Cold Neutron Chopper Spectrometer (CNCS) [50] at the Spallation Neutron Source (SNS), Oak Ridge National Laboratory. For the SEQUOIA experiment, 6.5 g of pure powder samples of YbZn2GaO5 and LuZn2GaO5 were used with incident neutron energies of Ei=subscript𝐸𝑖absentE_{i}=italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 80 and 120 meV at temperatures of T=𝑇absentT=italic_T = 5 and 100 K. The phonon contributions in the YbZn2GaO5 spectrum were subtracted using isostructural non-magnetic LuZn2GaO5. For the CNCS experiment, 10 pieces of high-quality single crystal samples of YbZn2GaO5 with a total mass of ∼similar-to\sim∼ 1.8 g were co-aligned within 1.5∘ using a Laue X-ray back-scattering diffractometer and mounted along (hk0)ℎ𝑘0(hk0)( italic_h italic_k 0 ) scattering plane on an oxygen-free copper sample holder (see Fig. S1). The measurements were carried out in a dilution refrigerator with a base temperature of 0.1 K. A neutron-absorbing Cd foil was placed at the bottom of the holder to reduce the background from the sample holder. The measurements were conducted at the base temperature and 45 K under zero-field with an incident neutron energy of Ei=subscript𝐸𝑖absentE_{i}=italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT =3.32 meV. The sample was rotated with an increment of 1∘, with a range of -180∘ to 180∘. The data were analyzed using the HORACE software and were folded 3 times along the high symmetry axis [0H𝐻Hitalic_H0], [H¯¯𝐻\bar{H}over¯ start_ARG italic_H end_ARGH𝐻Hitalic_H0], and [H𝐻Hitalic_H00] into a 60 ∘ sector in the reciprocal space to improve statistics. For the constant energy slice, the folded data were cut, duplicated, and recombined to restore 360∘ coverage for the purpose of presentation. To better extract the magnetic signal of interest, a comparative analysis was conducted between the 0.1 K and 45 K spectra. The 45 K spectrum was normalized to 0.1 K data and used as background which then was subtracted from the 0.1 K spectrum.
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Neutron Powder Diffraction (NPD) measurement of YbZn2GaO5 was collected using the General Materials Diffractometer (GEM) at the ISIS Neutron and Muon Source (Rutherford Appleton Laboratory, United Kingdom) [48]. The sample was loaded into an 8 mm diameter vanadium sample holder, and the NPD pattern was collected at room temperature, with a 15/40 mm (horizontal/vertical) beam size.
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Temperature-dependent magnetic susceptibility was measured using a 7 Tesla Cryogenic Ltd SQUID (superconducting quantum interference device) magnetometer with a Helium-3 probe from 0.3 K to 2 K and with a Helium-4 probe from 2 K to 300 K. For the Helium-3 measurements, a small crystalline YbZn2GaO5 sample of 1.04 mg was mounted on a silver sample holder and the Helium-4 measurement, 11.90 mg of YbZn2GaO5 single crystal was used. The crystals were oriented using the Laue diffraction method and the regular shape of the single crystals was obtained using a wire saw. The magnetic measurements were performed under an applied magnetic field parallel (H𝐻Hitalic_H∥parallel-to\parallel∥c) and perpendicular (H𝐻Hitalic_H⟂perpendicular-to\perp⟂c) to the crystallographic c-directions of YbZn2GaO5. The isothermal magnetization measurements along both directions of YbZn2GaO5 single crystal sample were performed using a VSM (vibration sample magnetometer) in PPMS up to 14 Tesla of the applied magnetic field (see Fig. S1).
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The poly-crystalline sample of YbZn2GaO5 was synthesized using a solid-state reaction route. The high-purity precursors of Yb2O3 (99.9%percent\%%), Ga2O3 (99.9%percent\%%), and ZnO (99.9%percent\%%) with 5%percent\%% excess ZnO were used and mixed thoroughly and then pressed into a pellet. The pellets were sintered at 1275 \tccentigrade\tccentigrade\tccentigrade for 36 hours with intermediate grinding to obtain a pure phase of YbZn2GaO5. The phase purity is confirmed using Powder X-ray diffraction (PXRD) (see Fig. S1). The pure powder sample and around 10%percent\%% excess of ZnO were mixed and pressed into a cylindrical rod using a hydrostatic pressure of 700 bar. Single crystals of YbZn2GaO5 were grown using the optical floating-zone technique in the presence of a 10-bar oxygen atmosphere. A transparent single-grain crystal was successfully obtained, with a cleaved facet plane along [001] which was confirmed from Laue X-ray diffraction (see Fig. S1).
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B
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The expressions of Df0subscript𝐷subscript𝑓0D_{f_{0}}italic_D start_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT, Da0subscript𝐷subscript𝑎0D_{a_{0}}italic_D start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT, and Dσ0subscript𝐷subscript𝜎0D_{\sigma_{0}}italic_D start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT
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Accordingly, BESIII has recently measured the semileptonic Ds+superscriptsubscript𝐷𝑠D_{s}^{+}italic_D start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT decays BESIII:2023wgr ; BESIII:2021drk .
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have been extracted in Refs. Flatte:1976xu ; Flatte:1976xv ; Baru:2004xg ; Abele:1998qd ; Bugg:2008ig ; LHCb:2014ooi ; LHCb:2014vbo
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Abele:1998qd ; BESIII:2020hfw , and LHCb:2019sus ; Oller:2004xm ; Cheng:2022vbw ; BES:2004mws ; Zou:1993az , respectively:
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The S0→M1M2→subscript𝑆0subscript𝑀1subscript𝑀2S_{0}\to M_{1}M_{2}italic_S start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT → italic_M start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT strong decay is defined as Cheng:2022vbw
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C
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ΠlSsubscriptsuperscriptΠ𝑆𝑙\displaystyle\Pi^{S}_{l}roman_Π start_POSTSUPERSCRIPT italic_S end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT
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−r212S¯ijrS¯jkrS¯kir,superscript𝑟212superscriptsubscript¯𝑆𝑖𝑗𝑟superscriptsubscript¯𝑆𝑗𝑘𝑟superscriptsubscript¯𝑆𝑘𝑖𝑟\displaystyle-\frac{r^{2}}{12}\overline{S}_{ij}^{r}\overline{S}_{jk}^{r}%
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12∂τijr∂(r2)=12∇2τijr+A¯ikrA¯jkr,τijr=0=0,formulae-sequence12superscriptsubscript𝜏𝑖𝑗𝑟superscript𝑟212superscript∇2superscriptsubscript𝜏𝑖𝑗𝑟superscriptsubscript¯𝐴𝑖𝑘𝑟superscriptsubscript¯𝐴𝑗𝑘𝑟superscriptsubscript𝜏𝑖𝑗𝑟0012\frac{\partial\tau_{ij}^{r}}{\partial(r^{2})}=\frac{1}{2}\nabla^{2}\tau_{ij}%
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r248ω¯irω¯jrS¯ijr,superscript𝑟248superscriptsubscript¯𝜔𝑖𝑟superscriptsubscript¯𝜔𝑗𝑟superscriptsubscript¯𝑆𝑖𝑗𝑟\displaystyle\frac{r^{2}}{48}\overline{\omega}_{i}^{r}\overline{\omega}_{j}^{r%
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−112∫0r2(S¯ikθS¯jkθ¯ϕ−S¯ikθ¯ϕS¯jkθ¯ϕ)S¯ijrdθ2,112superscriptsubscript0superscript𝑟2superscript¯superscriptsubscript¯𝑆𝑖𝑘𝜃superscriptsubscript¯𝑆𝑗𝑘𝜃italic-ϕsuperscript¯superscriptsubscript¯𝑆𝑖𝑘𝜃italic-ϕsuperscript¯superscriptsubscript¯𝑆𝑗𝑘𝜃italic-ϕsuperscriptsubscript¯𝑆𝑖𝑗𝑟differential-dsuperscript𝜃2\displaystyle-\frac{1}{12}\int_{0}^{r^{2}}\left(\overline{\overline{S}_{ik}^{%
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A
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The concept of extra dimensions and brane worlds has been a topic of interest for many years. Initially, the extra dimensions are supposed to be compactified on the scale of the Planck length Antoniadis (1990); Arkani-Hamed, Dimopoulos, and Dvali (1998); Antoniadis et al. (1998). While in the Randall-Sundrum brane model Randall and Sundrum (1999a, b), the extra dimensions were considered infinite, enhancing the possibility of their experimental detection. Our study focuses on the physical phenomena that arise from these extra dimensions. For example, a massless field in the bulk can acquire mass through these extra dimensions, manifesting as a series of Kaluza-Klein (KK) modes in the brane Karch and Randall (2001); Youm (2001); Ichinose (2002); Wang (2002); Langlois and Sasaki (2003); Mukhopadhyaya et al. (2004); Melfo, Pantoja, and Tempo (2006); Davies and George (2007); Guo and Ma (2008); Liu et al. (2008); Koley, Mitra, and SenGupta (2009); Alencar et al. (2010); Cruz, Tahim, and Almeida (2010); Yang et al. (2012); Das and SenGupta (2013); Jones et al. (2013); Chakraborty and SenGupta (2014); Zhao, Xie, and Zhong (2014); Sousa et al. (2014); Vaquera-Araujo and Corradini (2015); Zhong et al. (2019); Chen, Guo, and Liu (2021); Li et al. (2023); Tan, Guo, and Liu (2022); Tan et al. (2023a, b); Wan and Liu (2023); Zhong, Yang, and Liu (2022). This has profound implications for the U(1)𝑈1U(1)italic_U ( 1 ) gauge field.
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The key to gauge invariance of the effective action is the interrelation between masses of the KK modes with the vector-scalar and scalar-scalar coupling constants. Building on our prior research Fu et al. (2016a, b, 2020); Fu, Zhong, and Liu (2019); Fu, Zhao, and Sun (2022); Fu (2022), we found that the effective action is gauge invariant in branes described by a conformal metric. However, the existing literature presents solutions for 6D branes with non-conformal metrics. Our research aims to extend the investigation of gauge invariance to these 6D brane models. In this scenario, the metric (the geometry) of the brane is different, all the coupling constants will change. Therefore, we will first reconsider whether the gauge invariance is maintained under such variations.
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It is well known that the Stueckelberg mechanism Ruegg and Ruiz-Altaba (2004) or Higgs mechanism Higgs (1964) is necessary to preserve the gauge invariance of massive gauge field. These two mechanisms respectively introduce a Stueckelberg field or a Goldstone boson to maintain the symmetry of the gauge field. We observe that for a higher-dimensional massless U(1)𝑈1U(1)italic_U ( 1 ) gauge field, there are two types of KK modes present in the brane: vector and scalar. Due to the scalar KK modes, these massive vector modes are formulated gauge invariant Fu et al. (2016a, b). We have investigated how the number of extra dimensions, the dimensionality of the brane, and the coupling between the bulk field and other scalar fields, such as the dilaton field, affect this gauge invariance Fu, Zhong, and Liu (2019); Fu et al. (2020); Fu (2022).
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To address this, we begin by reviewing the method for studying the gauge invariance of the effective action of a bulk free U(1)gauge field. From this, we show that the relationships between the coupling coefficients will determine the gauge invariance. Next, we consider a brane model with the following general line-element:
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This study begun with an examination of the methodology for deriving the effective action of a massless bulk U(1)𝑈1U(1)italic_U ( 1 ) gauge field through a general KK decomposition within branes of codimension 2. The effective action implies the existence of two distinct types of scalar KK modes that couple with the massive vector modes. While we have established that the effective action maintains gauge invariance in brane models with a conformal metric. However, the solvable 6D branes are usually constructed within the non-conformal metrics. By comparing the EOM for the KK modes (deriving from two ways), we revealed that to preserve the gauge invariance of the effective action in these 6D branes, certain constraints on the brane’s geometry must be introduced.
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Our model has rich phenomenology. Direct detection experiments (χN→χN→𝜒𝑁𝜒𝑁\chi N\to\chi Nitalic_χ italic_N → italic_χ italic_N mediated by A′superscript𝐴′A^{\prime}italic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT) like PandaX-II Yang et al. (2021) constrain the kinetic mixing between A′superscript𝐴′A^{\prime}italic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT and photons to ϵ≲10−11less-than-or-similar-toitalic-ϵsuperscript1011\epsilon\lesssim 10^{-11}italic_ϵ ≲ 10 start_POSTSUPERSCRIPT - 11 end_POSTSUPERSCRIPT for mχ∼𝒪(10)similar-tosubscript𝑚𝜒𝒪10m_{\chi}\sim\mathcal{O}(10)italic_m start_POSTSUBSCRIPT italic_χ end_POSTSUBSCRIPT ∼ caligraphic_O ( 10 ) GeV and mA′∼𝒪(10)similar-tosubscript𝑚superscript𝐴′𝒪10m_{A^{\prime}}\sim\mathcal{O}(10)italic_m start_POSTSUBSCRIPT italic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ∼ caligraphic_O ( 10 ) MeV. We note that the introduction of the additional Higgs would loop-induced kinetic mixing of the visible photon and dark photon. However, it has been found that the loop induced kinetic mixing of the visible photon and dark photon is highly suppressed Koren and McGehee (2020), which appears at five loops in our model. The combination of loop factors and coupling suppression provides sufficient suppression to maintain consistency with current limits.
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where σVsubscript𝜎𝑉\sigma_{V}italic_σ start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT is the viscosity cross section Tulin et al. (2013b); Cline et al. (2014); Yang and Yu (2022) and v0subscript𝑣0v_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is the velocity dispersion. Analytic formulae of DM scattering cross section at different parameter regions corresponding to different coupling strength and kinetic energy have been given in the literature Tulin et al. (2013b); Feng et al. (2010); Buckley and Fox (2010); Kahlhoefer et al. (2017); Khrapak et al. (2003); Cyr-Racine et al. (2016); Colquhoun et al. (2021); Girmohanta and Shrock (2022, 2023), and implemented in public package CLASSICS Colquhoun et al. (2021), which is used in our calculation. The fit result will be given in Section IV.
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Recently, the NANOGrav, CPTA, EPTA and PPTA collaborations have presented new observations of stochastic gravitational waves (GWs) using pulsar timing arrays (PTAs) Agazie et al. (2023); Xu et al. (2023); Antoniadis et al. (2023); Reardon et al. (2023). In particular, the NANOGrav Agazie et al. (2023) and CPTA Xu et al. (2023) collaborations report a signal significance ∼4σsimilar-toabsent4𝜎\sim 4\sigma∼ 4 italic_σ for the Hellings-Downs correlation curve. These observations provide compelling evidence for the presence of stochastic GWs with a peak frequency around 10−8superscript10810^{-8}10 start_POSTSUPERSCRIPT - 8 end_POSTSUPERSCRIPT Hz. While the standard interpretation has been inspiraling supermassive black hole binaries (SMBHBs), alternative explanations such as a first-order phase transition (FOPT) remain viable. It is known that a GW signal at 10−8superscript10810^{-8}10 start_POSTSUPERSCRIPT - 8 end_POSTSUPERSCRIPT Hz implies a FOPT at the MeV scale, and a Bayesian analysis of the NANOGrav data even favors the FOPT model over the baseline SMBHB model Afzal et al. (2023). Therefore, this observation potentially shows the first evidence of signals from the early Universe prior to the Big Bang nucleosynthesis (BBN) and Cosmic Microwave Background (CMB). Studies on the theoretical models to explain the previous NANOGrav data can be found in Refs. Ellis and Lewicki (2021); Blasi et al. (2021); Addazi et al. (2021); Buchmuller et al. (2020); Addazi et al. (2021); Vaskonen and Veermäe (2021); De Luca et al. (2021); Domènech and Pi (2022); Addazi et al. (2021); Kohri and Terada (2021); Bian et al. (2021); Nakai et al. (2021); Ratzinger and Schwaller (2021); Borah et al. (2021); Freese and Winkler (2022); Bringmann et al. (2023); Madge et al. (2023); Kobakhidze et al. (2017); Arunasalam et al. (2018); Chiang and Lu (2021); Ashoorioon et al. (2022); Nakai et al. (2021); Bigazzi et al. (2021); Freese and Winkler (2023).
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Overall, for dwarf galaxies or galaxies with a DM average velocity ⟨v⟩delimited-⟨⟩𝑣\langle v\rangle⟨ italic_v ⟩ around 10−2001020010-20010 - 200 km/s, the cross section σ/mDM𝜎subscript𝑚DM\sigma/m_{\text{DM}}italic_σ / italic_m start_POSTSUBSCRIPT DM end_POSTSUBSCRIPT needs to be within the range 𝒪(1)−𝒪(10)𝒪1𝒪10\mathcal{O}(1)-\mathcal{O}(10)caligraphic_O ( 1 ) - caligraphic_O ( 10 ) cm2/g Zavala et al. (2013); Elbert et al. (2015). However, for galaxy clusters where ⟨v⟩∼2000similar-todelimited-⟨⟩𝑣2000\langle v\rangle\sim 2000⟨ italic_v ⟩ ∼ 2000 km/s, current fit result favors σ/mDM∼0.2similar-to𝜎subscript𝑚DM0.2\sigma/m_{\text{DM}}\sim 0.2italic_σ / italic_m start_POSTSUBSCRIPT DM end_POSTSUBSCRIPT ∼ 0.2 cm2/g Sagunski et al. (2021). It has been shown that a velocity-dependent σ/mDM𝜎subscript𝑚DM\sigma/m_{\text{DM}}italic_σ / italic_m start_POSTSUBSCRIPT DM end_POSTSUBSCRIPT can solve small-scale problems at different systems Kochanek and White (2000); Andrade et al. (2021); Elbert et al. (2018); Fry et al. (2015); Yoshida et al. (2000a); Moore et al. (2000); Zavala et al. (2013); Elbert et al. (2015); Rocha et al. (2013); Peter et al. (2013); Kaplinghat et al. (2016); Tulin and Yu (2018); Vogelsberger et al. (2014); Dooley et al. (2016); Dave et al. (2001); Robertson et al. (2019); Spergel and Steinhardt (2000); Colin et al. (2002); Vogelsberger et al. (2012); Harvey et al. (2015); Burkert (2000); Sagunski et al. (2021); Yoshida et al. (2000b); Yang et al. (2023a).
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Furthermore, the very distinct gravothermal evolution of SIDM halo, which start with core formation-expansion and followed by core collapse Balberg et al. (2002); Koda and Shapiro (2011); Essig et al. (2019); Huo et al. (2020); Nishikawa et al. (2020); Sameie et al. (2020); Kahlhoefer et al. (2019); Turner et al. (2021); Zeng et al. (2022); Outmezguine et al. (2023); Yang and Yu (2022); Yang et al. (2023b, a); Nadler et al. (2023), leave imprint on star formation history Robles et al. (2019); Sameie et al. (2021); Vargya et al. (2022); Burger et al. (2022) and supermassive black holes seeding Balberg and Shapiro (2002); Pollack et al. (2015); Feng et al. (2021).
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Table 1: Saturated THz amplitude and central frequency of the THz pulse emitted from heterostructures CoFeB (2nm)/NM.
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To gain deeper insights into the fabricated THz emitters, we analyzed the correlation between the amplitude of the emitted THz field at 1 THz and the varying thicknesses of the CoFeB layer in stacks with specified NM layers, see Fig. 3 (a) and (b). The THz emitters composed of pure Pt layers demonstrate the highest THz amplitude among the emitters. Specifically, the stack with a 2 nm Pt layer exhibits a THz signal amplitude that is twice as high as the stack with a 2 nm PtBi layer. However, the emitter with PtBi, as shown in Figs. 3 (c) and (d), exhibits a wider bandwidth of approximately 0.35 THz compared to other stacks, along with a higher central frequency of the THz signal. Interestingly, it demonstrates a significant central frequency shift of approximately 0.3 THz when compared to the emitter with 2 nm Pt. These findings highlight the crucial role of the NM layer material in influencing THz characteristics and suggest the potential advantages of the PtBi stack for applications requiring broader bandwidth and higher frequencies, despite a lower THz amplitude. When evaluating the trade-off between THz signal strength and bandwidth, it is crucial to consider the specific requirements of the intended application, as different applications may prioritize either a higher THz signal strength or a broader bandwidth based on their unique needs and constraints. This observation suggests potential advantages in applications that benefit from higher frequency THz signals, including high-resolution imaging, spectroscopy, communications, medical diagnostics, and etc. The values of the saturated THz amplitudes and THz peak position measured for different emitters are presented in Table 1.
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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. (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. (d) The THz bandwidth as a function of CoFeB-layer thicknesses.
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Dependence of THz Signal Amplitude, Central Frequency, and Bandwidth on Heterostructure Thickness and NM Materials: Fig. 2 illustrates the emitted THz signal as a function of delay time and laser spot position for distinct samples. Accordingly, irrespective of the type of metal layers, no THz emission is detected at the thin end of the wedge, where the CoFeB thickness is 0 nm. As the thickness of the CoFeB layer increases, the amplitude of the emitted THz signal exhibits growth, peaking at around 2 nm of CoFeB thickness. It is noteworthy that beyond this critical point, further increases in CoFeB thickness do not have a significant impact on the amplitude of the THz signal. This behavior can be attributed to the progressive diffusion of an increasing number of electrons into the NM layer as the CoFeB layer thickness increases up to 2 nm. The influx of electrons enhances the induced spin current, thereby generating stronger THz radiation. However, beyond a 2 nm thickness of the CoFeB layer, the presence of heightened structural defects, electron scattering, and resistance within the magnetic layer hinders further amplification of the THz signal amplitude. As a result, the THz signal reaches a plateau beyond this critical thickness. Furthermore, the results indicate that the maximum THz amplitude is attained when utilizing a 2 nm Pt layer as the heavy metal component of the emitter. The decrease in THz amplitude with increasing Pt thickness can be attributed to the absorption and attenuation of THz waves by the Pt layer, which will be discussed in the subsequent section.
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To optimize spintronic THz emitters performance, we conducted a comparative study on SiO22{}_{2}start_FLOATSUBSCRIPT 2 end_FLOATSUBSCRIPT/CoFeB/NM heterostructures, where the CoFeB layer thickness varied from 0 to 5nm, and different heavy metals (Pt, W, Au) and alloys (Pt%92{}_{\%92}start_FLOATSUBSCRIPT % 92 end_FLOATSUBSCRIPTBi%8{}_{\%8}start_FLOATSUBSCRIPT % 8 end_FLOATSUBSCRIPT and Ag%90{}_{\%90}start_FLOATSUBSCRIPT % 90 end_FLOATSUBSCRIPTBi%10{}_{\%10}start_FLOATSUBSCRIPT % 10 end_FLOATSUBSCRIPT) served as the NM layer. Our investigation revealed a critical threshold at 2nm thickness for the CoFeB layer, beyond which the emitted THz signal amplitude saturated. Furthermore, the heterostructure with 2nm CoFeB and 2nm Pt exhibited the highest THz signal amplitude. Surprisingly, despite the Pt%92{}_{\%92}start_FLOATSUBSCRIPT % 92 end_FLOATSUBSCRIPTBi%8{}_{\%8}start_FLOATSUBSCRIPT % 8 end_FLOATSUBSCRIPT (2nm) emitter exhibiting only half the THz amplitude compared to the Pt (2nm) emitter, it demonstrated a higher central THz frequency and the largest THz bandwidth among all the emitters investigated in our study. This study provide a solid foundation for future studies on different compositions of Pt1−x1𝑥{}_{1-x}start_FLOATSUBSCRIPT 1 - italic_x end_FLOATSUBSCRIPTBix𝑥{}_{x}start_FLOATSUBSCRIPT italic_x end_FLOATSUBSCRIPT alloy aimed at achieving even higher wideband and frequency capabilities in THz emitters. To gain a deeper understanding of the behavior exhibited by the different emitters, we conducted time-domain THz spectroscopy to analyze their conductivity and impedance characteristics. Our findings demonstrate that thicker NM layers exhibit higher electron concentration and lower mobility, resulting in lower impedance and subsequently lower THz amplitude. The anomalous behavior of PtBi, characterized by a negative imaginary part of conductivity according to our applied model, highlights the need for in-depth investigations and potential modifications of the model to better comprehend and explain the unique properties exhibited by this material.
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In the analysis that follows, we only consider the parameters that fall within the red areas, where we have two local maxima that describe from left to right.
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In this situation, we may have 1 or 2 local maxima of the effective potential for different parameters.
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In the analysis that follows, we only consider the parameters that fall within the red areas, where we have two local maxima that describe from left to right.
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When we select the appropriate value of λ𝜆\lambdaitalic_λ to construct the U(r)𝑈𝑟U(r)italic_U ( italic_r ) as shown in Fig. 2(a), there are two local maxima of the effective potential.
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Choosing different values of the parameters, we may have from 00 to 3333 local maxima of the effective potential, as shown in the left image of Fig. 5.
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We note that the KDE’s bandwidth acts like a control parameter with potential room for user-controlled fine-tuning in the computation of its value, somewhat analogous to the σ𝜎\sigmaitalic_σ of the narrow Gaussian method. However, unlike the narrow Gaussian method where σ𝜎\sigmaitalic_σ can in principle be chosen to be anything, the bandwidth of the KDE is computed directly from the properties of the samples (such as the number of samples and dimensionality of the parameter space) under reasonable assumptions regarding the true density. Thus, the user’s choice is restricted to a number of discrete such assumptions (for example, Scott’s rule [38], Silverman’s rule [37], etc.). Furthermore, the effects of choosing a bandwidth on the estimated density (and hence the remainder of the inference) is limited in the sense that the covariance matrix of the Gaussians is determined from the samples themselves with the bandwidth only acting as a scaling parameter that is usually of order unity. This additionally restricts the effects of user-controlled fine-tuning on the inference as compared to the narrow Gaussian method wherein the width of the Gaussian that approximates the delta function is completely determined by the user’s choice. A more detailed discussion of this comparison between the two methods in the context of the MDC can be found in Sec. III.6 . For our chosen bandwidth approximation scheme (Scott’s rule) the KDE method can be seen to perform extremely well in the MDC. For these reasons, we choose the KDE method for our final results on the SME constraints.
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To carry out parameter estimation for each event that passes our selection criteria, we use public data [19, 20] from GWTC-1 through GWTC-3. We use lalinference_mcmc [18, 21, 22, 23], which implements Markov chain Monte Carlo (MCMC) with the Metropolis-Hastings algorithm and lalinference_nest, which implements nested sampling to run the Bayesian parameter estimation [18, 24, 25]. For our purposes of extracting vgsubscript𝑣𝑔v_{g}italic_v start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT distributions, these two algorithms generate comparable results. We use the publicly available power spectral densities and calibration envelopes from the LIGO Scientific, Virgo and KAGRA (LVK) Collaboration in our analysis. In this paper, we use a uniform prior in vgsubscript𝑣𝑔v_{g}italic_v start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT between 0.1c0.1𝑐0.1c0.1 italic_c and 10c10𝑐10c10 italic_c. When the vgsubscript𝑣𝑔v_{g}italic_v start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT posterior rails against the prior, we increase the upper limit of the prior by another 10c10𝑐10c10 italic_c. The broadest prior we use is from 0.1c0.1𝑐0.1c0.1 italic_c to 30c30𝑐30c30 italic_c, which we only use for one event, GW190929_012149. For parameters such as binary masses and spins, we use the same uniform and isotropic priors as those used by the LVK [5, 3, 4]. We choose a distance prior that is proportional to luminosity distance squared, similar to Ref. [5]. We do not use the more complicated cosmological priors used by Refs. [3, 4]. For O1 and O2 events, we use the posterior samples from Ref. [7], which used the IMRPhenomPv2 [26, 27, 28] waveform for all events except for the binary neutron star (BNS) event GW170817, which was analyzed with the TaylorF2 waveform [29, 30, 31, 32, 33, 34]. For most O3 events, we use the IMRPhenomD waveform [26, 27], which is an aligned spin waveform model for black-hole binaries. We do not use the more sophisticated IMRPhenomPv2 model for these events since in the context of our study, we do not expect any significant change in vgsubscript𝑣𝑔v_{g}italic_v start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT measurements to result from the additional intricacies of the more sophisticated model. We have verified this lack of change for a subset of these events and hence chosen to stick to the IMRPhenomD model consistently for all O3 events except for GW190521. For additional discussion of this point, see Appendix A. For the extremely high-mass binary black hole (BBH) event GW190521, we use the NRSur7dq4 waveform [35], which is one of the waveform models used by Ref. [36] for inferring this event’s source properties. We note that IMRPhenomPv2, IMRPhenomD, and NRSur7dq4 are all waveform models with inspiral, merger, and ringdown.
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Because of these considerations, we conclude that the weighted SVD-assisted random draw method produces constraints that are unreliable and are likely to be underestimates of the true uncertainties in the measurement of SME coefficients. We verify this claim by testing this method against its Bayesian counterparts in a MDC that we describe later in this work. The results of the MDC show that the samples of SME parameters produced by this method are concentrated in a narrow region around the true values of the parameters, which also coincide with the peaks of the posterior distributions inferred by the Bayesian methods. Therein lies the merit of this method in the present context and its potential to serve as a rapid consistency check for the Bayesian methods. Furthermore, this method is extremely fast and computationally cheap and hence can be used to quickly find the narrow region in the parameter space inside which the peak of the posteriors lies. The stochastic MCMC sampling employed by our Bayesian methods is expected to converge much faster if the MCMC chains are initialized near the maxima of the posterior being sampled. Thus the SVD-assisted random draw method can be used to optimize the MCMC sampling used in our Bayesian methods with significant speed-up gains for narrowly peaked SME posteriors. Given the large number of events expected to be observed in O4 and the width of the Bayesian intervals we compute using our current set of events, the posterior distributions of the SME coefficients can be expected to be very narrow post O4, and hence lead to a drastic increase in the computational cost and latency of the Bayesian methods being applied to such a dataset. This will likely make the optimization of the Bayesian methods as offered by the SVD-assisted random draw method a necessary tool in the near future.
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We note that the KDE’s bandwidth acts like a control parameter with potential room for user-controlled fine-tuning in the computation of its value, somewhat analogous to the σ𝜎\sigmaitalic_σ of the narrow Gaussian method. However, unlike the narrow Gaussian method where σ𝜎\sigmaitalic_σ can in principle be chosen to be anything, the bandwidth of the KDE is computed directly from the properties of the samples (such as the number of samples and dimensionality of the parameter space) under reasonable assumptions regarding the true density. Thus, the user’s choice is restricted to a number of discrete such assumptions (for example, Scott’s rule [38], Silverman’s rule [37], etc.). Furthermore, the effects of choosing a bandwidth on the estimated density (and hence the remainder of the inference) is limited in the sense that the covariance matrix of the Gaussians is determined from the samples themselves with the bandwidth only acting as a scaling parameter that is usually of order unity. This additionally restricts the effects of user-controlled fine-tuning on the inference as compared to the narrow Gaussian method wherein the width of the Gaussian that approximates the delta function is completely determined by the user’s choice. A more detailed discussion of this comparison between the two methods in the context of the MDC can be found in Sec. III.6 . For our chosen bandwidth approximation scheme (Scott’s rule) the KDE method can be seen to perform extremely well in the MDC. For these reasons, we choose the KDE method for our final results on the SME constraints.
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We present the results of the KDE method upon its use in the analysis of the events marked * listed in tables I through III (except for GW170817 for which we do not use the fixed posteriors due to the inability of the KDE to estimate very narrow densities) in Fig. 4 as our final result.
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(b) False-color scanning electron microscopy of a 2×2222\times 22 × 2 germanium quantum dot device, nominally identical to the one used here.
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Next, we manipulate the spins by two simultaneous microwave pulses on plunger gates P2𝑃2P2italic_P 2 and P4𝑃4P4italic_P 4, with a duration tpsubscript𝑡pt_{\mathrm{p}}italic_t start_POSTSUBSCRIPT roman_p end_POSTSUBSCRIPT and microwave frequencies fP2subscript𝑓𝑃2f_{P2}italic_f start_POSTSUBSCRIPT italic_P 2 end_POSTSUBSCRIPT and fP4subscript𝑓𝑃4f_{P4}italic_f start_POSTSUBSCRIPT italic_P 4 end_POSTSUBSCRIPT.
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We focus on the spin qubits Q1𝑄1Q1italic_Q 1 and Q2𝑄2Q2italic_Q 2, while Q3𝑄3Q3italic_Q 3 and Q4𝑄4Q4italic_Q 4 remain in their ground state.
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We analyze three resonance lines [dashed lines in Fig. 2(b)] resulting from bichromatic rotation of Q1𝑄1Q1italic_Q 1 and Q2𝑄2Q2italic_Q 2.
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We operate Q1𝑄1Q1italic_Q 1 and Q2𝑄2Q2italic_Q 2 with microwave bursts applied to P2𝑃2P2italic_P 2 and P4𝑃4P4italic_P 4.
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Finally, we can compare our numerics with the numerics on spreading of wave packets in the following way. Since we see a hint of a deviation from the τ∝λ−4proportional-to𝜏superscript𝜆4\tau\propto\lambda^{-4}italic_τ ∝ italic_λ start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT law, one could ask how hard it would be to probe the same nonlinearities as we do in a spreading experiments. It is harder because spreading numerics needs a larger volume than our decorrelation numerics. More concretely, to see our smallest nonlinearity in the setup [4, 5], we would have to simulate the system at least 102superscript10210^{2}10 start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT times longer than done in [4, 5], (assuming that the spreading does not slow down compared to the t1/6superscript𝑡16t^{1/6}italic_t start_POSTSUPERSCRIPT 1 / 6 end_POSTSUPERSCRIPT law), see Appendix H.
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We define the decorrelation time τ𝜏\tauitalic_τ as the time at which η𝜂\etaitalic_η reaches a fixed value η0subscript𝜂0\eta_{0}italic_η start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT (for clarity, we also use the notation τη0subscript𝜏subscript𝜂0\tau_{\eta_{0}}italic_τ start_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT when needed).
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We provide for the first time a direct comparison between numerical observations and mathematical bounds for the KG chain. As an alternative to the speed of spreading or the diffusion constant, we focus on the decorrelation time in thermal equilibrium as it is more accessible to a proper mathematical treatment and to numerics.
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We have run the dynamics up to a time t=1010𝑡superscript1010t=10^{10}italic_t = 10 start_POSTSUPERSCRIPT 10 end_POSTSUPERSCRIPT for the smallest values of λ𝜆\lambdaitalic_λ.
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Let us fix an arbitrary threshold value 0<q<10𝑞10<q<10 < italic_q < 1 for the decorrelation parameter η𝜂\etaitalic_η, as in Figure 3.
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Instead, we do investigate how much the dynamics at positive temperature and at weak interaction is slowed down by its proximity to a localized system. We will explain later how this connects to an interesting ongoing debate, but first, we introduce the model and set the stage.
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We focus here on the dynamics at λ=0𝜆0\lambda=0italic_λ = 0: The chain is harmonic and the system is decomposed into a set of |ΛL|subscriptΛ𝐿|\Lambda_{L}|| roman_Λ start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT | independent modes.
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The disordered Klein-Gordon chain with a quartic interaction is a prototypical example of an interacting classical many-body system,
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The fate of Anderson localization in the presence of genuine many-body interactions, or anharmonicity, is a matter of considerable interest and debate,
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It is also important to consider sufficiently generic interactions, see e.g. [12] for an example of a many-body system that, despite probably being chaotic, exhibits subdiffusive transport.
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We consider strained Si/SiGe devices. Because of the mismatch of lattice constant between the materials, a force develops at their interfaces, resulting in a displacement field 𝒖(𝒓)𝒖𝒓\boldsymbol{u}(\boldsymbol{r})bold_italic_u ( bold_italic_r ) for the atoms. Consequently, in equilibrium the lattice constants of the two materials match at the interface; this is referred to as the pseudomorphic condition.
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In linear elasticity theory \citeSKosevich1986sm the change of lengths in a deformed body is given by the strain tensor
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To calculate the effect of strain due to lattice mismatch, we simulate the strain tensor elements εijsubscript𝜀𝑖𝑗\varepsilon_{ij}italic_ε start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT in our devices by solving the differential equation in Eq. (S5) numerically. In particular, we use the FEM implemented in COMSOL Multiphysics ® \citeSCOMSOLsm. For the lifting of the valley degeneracy, we pay particular attention to the shear strain component which is the main source of the large valley gap ΔΔ\Deltaroman_Δ in our fins. By rotating the coordinate system such that the z𝑧zitalic_z axis is aligned with the [110]delimited-[]110[110][ 110 ] growth direction the strain tensor becomes
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Our system comprising two materials with different lattice constants ai(𝒓)subscript𝑎𝑖𝒓a_{i}(\boldsymbol{r})italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( bold_italic_r ) and two different elastic stiffness tensors Cijkl(𝒓)subscript𝐶𝑖𝑗𝑘𝑙𝒓C_{ijkl}(\boldsymbol{r})italic_C start_POSTSUBSCRIPT italic_i italic_j italic_k italic_l end_POSTSUBSCRIPT ( bold_italic_r ) can be simulated by linear elasticity theory by introducing the equivalent body force \citeSMengistu2016sm
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In the Supplemental Material we provide more details on the simulation of the strain tensor in our devices via the finite element method and continuum elasticity theory. We show results on the strain tensor components that cause the localization of the electron wave function in certain areas of the device cross section. We also provide a more detailed analysis of the electric field dependence of the valley splitting which demonstrates that a precisely aligned electric field is not required. Furthermore, we analyze the effect of atomistic disorder at the interface between the Si and the SiGe alloy and demonstrate that in contrast to planar heterostructures, disorder has no effect on our fins. Also a modification of the cross section of the triangular fin do not spoil the large valley splitting predicted in the main text. Additionally, we present simulations of fins with other cross sections, different from the one showed in the main text. In these systems, we find a valley splitting similar to the one for the setups discussed in the main text, thus corroborating our claim that fine-tuning of the cross section shape is not required to reach large values of the valley splitting.
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A
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While this paper has demonstrated implementations for parallel collimator SPECT and LM-TOF PET, there exist many other modalities not presently included in the software, including (but not limited to) diverging/converging/pinhole SPECT, various forms of computed tomography, (CT), magnetic resonance (MR) imaging, and compton camera (CC) imaging. It is hoped that the modular structure and flexibility of PyTomography will encourage contributions from various research groups interested in particular imaging systems or reconstruction algorithms. Since the library is open source, newly developed functionality can be easily tested and verified by many independent research groups with their own private data. The immediate goals for future development in PyTomography at the time of publication include:
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This work describes the python library PyTomography and highlights specific use cases in SPECT and PET imaging. The software architecture facilitates the development of different imaging modalities and likelihoods that can all interface with the same reconstruction algorithms. The goal of this research was to create a simple-to-use and computationally-efficient library for medical image reconstruction, where system modeling and reconstruction techniques are implemented, shared, and explored by experts in the community.
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Overall, the aim of this study is to introduce the community to the PyTomography software project, demonstrate its current capabilities and flexibility, and to encourage community involvement in the continuation of its development. Example applications are explored in SPECT and PET reconstruction, though the library also has capabilities for clinical CT reconstruction as well. In what follows, we elaborate on our methods and results for PyTomography.
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To this end, we developed the python library PyTomography with the priorities to: (i) implement standard and traditional imaging modalities and reconstruction algorithms, (ii) disseminate recent research developments, such as the deep image prior [8], and (iii) encourage community involvement via extensive documentation and user-friendly tutorials. Based on the motivations previously described, the main functionality of the library has been developed using PyTorch. Other libraries are used for more particular tasks; the python library paralleproj [19] is used for line integral computation in PET projectors. While the present focus of PyTomography is the development and validation of reconstruction algorithms for SPECT and PET, implementation of other imaging modalities and reconstruction algorithms can be added using the building blocks provided.
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In this work, the software architecture of PyTomography was demonstrated and use cases were explored on both SPECT and PET data. The efficient reconstruction times, shown in Table 3, can enable researchers to perform extensive studies on digital/physical phantom and patient data that may include multiple patients, noise realizations, and reconstruction algorithms. A detailed discussion on each use case is included to highlight example analyses that can be performed with the software.
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A
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In the main text, we show that magnetoelastic couplings in the triangular lattice with D3 symmetry lead to non-zero integer band Chern numbers.
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To understand the effects of the lattice symmetry, we study the topological classification of the equivalent phononic system
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In the main text, we show that magnetoelastic couplings in the triangular lattice with D3 symmetry lead to non-zero integer band Chern numbers.
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For the triangular lattice with D3 symmetry, the symmetry-adapted form [46] of J𝐽Jitalic_J-tensor is
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Below in Fig. S1 we plot the dispersion with the color showing the chirality for K2=0subscript𝐾20K_{2}=0italic_K start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0, and we find the bands are modified by the magnetoelastic effects but are not fully gapped and the chirality is zero for any 𝐤𝐤\mathbf{k}bold_k.
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A
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The predicted effects of PSOI are highly sensitive to the Rashba constant, α𝛼\alphaitalic_α, which naturally drives the search for materials that i) exhibit giant Rashba effect, and ii) allow for ample tunability of the Rashba splitting by the external electric field. Promising candidates include thin layers of Bi2Se3subscriptBi2subscriptSe3\mathrm{Bi_{2}Se_{3}}roman_Bi start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT roman_Se start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT and KTaO3subscriptKTaO3\mathrm{KTaO_{3}}roman_KTaO start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT 999In typical experiments, the layered materials are proximated by metallic gates. One then should take into account the emergent image-potential-induced PSOI, which leads to effects similar to those considered in the present paper [32, 39, 40]., graphene on TMDs, van der Waals materials with heavy adatoms, and engineered heterostructures at the LaAlO3/SrTiO3subscriptLaAlO3subscriptSrTiO3\mathrm{LaAlO_{3}/SrTiO_{3}}roman_LaAlO start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT / roman_SrTiO start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT interface [1, 87].
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The predicted effects of PSOI are highly sensitive to the Rashba constant, α𝛼\alphaitalic_α, which naturally drives the search for materials that i) exhibit giant Rashba effect, and ii) allow for ample tunability of the Rashba splitting by the external electric field. Promising candidates include thin layers of Bi2Se3subscriptBi2subscriptSe3\mathrm{Bi_{2}Se_{3}}roman_Bi start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT roman_Se start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT and KTaO3subscriptKTaO3\mathrm{KTaO_{3}}roman_KTaO start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT 999In typical experiments, the layered materials are proximated by metallic gates. One then should take into account the emergent image-potential-induced PSOI, which leads to effects similar to those considered in the present paper [32, 39, 40]., graphene on TMDs, van der Waals materials with heavy adatoms, and engineered heterostructures at the LaAlO3/SrTiO3subscriptLaAlO3subscriptSrTiO3\mathrm{LaAlO_{3}/SrTiO_{3}}roman_LaAlO start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT / roman_SrTiO start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT interface [1, 87].
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In this section we work out consequences of the phenomenological Rashba spin-orbit coupling, Eq. (1), for electron-electron interaction Hamiltonian. To this end consider a 3D Rashba system without structure inversion asymmetry. A 2D case is a simple particular case of this consideration, which is briefly mentioned in the end. We do not consider the effects of disorder in our analysis. Furthermore, since a bulk electric field is not allowed in a 3D electronic system, it might be tempting to conclude that the Rashba spin-orbit interaction term, as given in Eq. (1), is inconsequential. However, such conclusion is premature because of the fluctuating electric fields created by electron-electron interactions. For weakly screened Coulomb interactions, considered below, these fluctuating Coulomb fields can reach magnitudes on the order of 107superscript10710^{7}10 start_POSTSUPERSCRIPT 7 end_POSTSUPERSCRIPT–108superscript10810^{8}10 start_POSTSUPERSCRIPT 8 end_POSTSUPERSCRIPT V/cm, which are comparable to the built-in electric fields responsible for giant Rashba splitting in van der Waals materials [3].
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The notion of pair spin-orbit interactions (PSOI) [31] represents a conceptual shift from these traditional SOI mechanisms. Unlike the single-particle RSOI, which arises from external electric fields, PSOI originates directly from the Coulomb fields of conduction band electrons. Therefore, PSOI does not rely on structural inversion asymmetry. This key insight — that the electric field, 𝓔𝓔\bm{\mathcal{E}}bold_caligraphic_E, in Eq. (1) can arise from the Coulomb forces — opens up an entirely new research avenue. This coupling directly affects the two-body interactions between electrons, rendering them both spin- and momentum-dependent. As a result, the modified two-body interactions lead to phenomena, distinctly different from both the conventional single-body RSOI and usual two-body interactions.
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Yet, a significant gap persists in reliably estimating the Rashba coupling constant, particularly concerning the response of layered materials to in-plane electric fields. The Rashba coupling in promising 3D systems remains largely unknown. Many existing estimates rely on ARPES measurements, which mainly probe the Rashba effect from built-in i.e. normal electric fields. Crucially, direct ab initio calculations of Rashba SOI from in-plane fields in 2D systems are lacking. This presents a dual challenge: while it complicates precise predictions of the effect’s magnitude, it also opens an exciting new research avenue of searching for materials with a giant, tunable Rashba effect.
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D
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1Ωh−2M−M2(1−1−ε2)>0.1subscriptΩℎ2𝑀superscript𝑀211superscript𝜀20\dfrac{1}{\Omega_{h}}-2M-M^{2}(1-\sqrt{1-\varepsilon^{2}})>0.divide start_ARG 1 end_ARG start_ARG roman_Ω start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT end_ARG - 2 italic_M - italic_M start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( 1 - square-root start_ARG 1 - italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) > 0 .
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According to the relation (52), when ε→0→𝜀0\varepsilon\rightarrow 0italic_ε → 0, the black hole is the near extreme case, and when ε=0𝜀0\varepsilon=0italic_ε = 0, the black hole is the extreme case. Using the relation (52), the condition (51) for destroying the event horizon of the black hole is simplified as
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For the case of near-extreme black holes, that is, when the dimensionless parameter ε→0→𝜀0\varepsilon\to 0italic_ε → 0, for the cold dark matter halo-black hole, its result can be obtained by calculating the conditional expression (53) as
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while for the ultralight dark matter halo-black hole, its result can be obtained by calculating the conditional expression (53) as
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Combining formulas (85), (86) and (87), since the time interval dt𝑑𝑡dtitalic_d italic_t, the parameter ε𝜀\varepsilonitalic_ε, and the dark matter parameters k1subscript𝑘1k_{1}italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT or k2subscript𝑘2k_{2}italic_k start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT are all first-order small quantities, it can be obtained that: in the absence of dark matter, the spacetime degenerates into a near-extreme Kerr black hole, and the result at this time is M2′−J′>0M^{{}^{\prime}2}-J^{{}^{\prime}}>0italic_M start_POSTSUPERSCRIPT start_FLOATSUPERSCRIPT ′ end_FLOATSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_J start_POSTSUPERSCRIPT start_FLOATSUPERSCRIPT ′ end_FLOATSUPERSCRIPT end_POSTSUPERSCRIPT > 0. This indicates that the event horizon of the near-extreme Kerr black hole cannot be destroyed by the scalar field, and this result is consistent with the conclusion that the event horizon of the near-extreme Kerr black hole cannot be destroyed by the scalar field. In the presence of dark matter, regardless of the influence of the cold dark matter halo or the light dark matter halo, the result is still M2′−J′>0M^{{}^{\prime}2}-J^{{}^{\prime}}>0italic_M start_POSTSUPERSCRIPT start_FLOATSUPERSCRIPT ′ end_FLOATSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_J start_POSTSUPERSCRIPT start_FLOATSUPERSCRIPT ′ end_FLOATSUPERSCRIPT end_POSTSUPERSCRIPT > 0. This further demonstrates that when considering the coupling effect between the near-extreme Kerr black hole and dark matter, the event horizon of the near-extreme Kerr black hole remains stable and will not be destroyed by the scalar field. Therefore, this analysis result further supports the weak cosmic censorship conjecture.
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B
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In Fig. 8, the score for the random acting agents is approximately 00, since the cooperative sequential coin game is a zero-sum game. The evolutionary-trained VQC approache, however, perform significantly better leading to an average score around 7777. The total coins collected depicted in Fig. 9(a) suggests that in contrast to the random agents, the VQC agents successfully learn to collect coins. The own coins collected correlates with number of collected coins (Fig. 9(b)). In Fig. 9(c) we can see that in neither case the cooperation increases over time. In summary, trained agents performed significantly better than random on all metrics, indicating that the training was successful.
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As depicted in Fig. 8, a better result is achieved by the VQC approach compared to random. On this basis, we compare the performance of this VQC approach to that of a neural network with a comparable number of parameters.
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In our experiments, we showed that our VQC approach performs significantly better compared to a neural network with a similar amount of trainable parameters. Compared to the larger neural network, we can see that the VQC approach achieves similar results, showing the effectiveness of using VQCs in a MAQRL environment. We can reduce the number of parameters by 97.88%percent97.8897.88\%97.88 % using the VQC approach compared to the similarly good neural network. In comparison to previous works [Chen et al., 2022], we used recombination in addition to mutation in our evolutionary algorithm, which performed worse than mutation alone in the tested setting. Additionally, we used more layers for the VQCs than previous works [Chen et al., 2022], as they have yielded better results in the experiments.
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In the previous section, we observed that a neural network with the same number of parameters as the VQC of our approach cannot match the VQC’s performance. We will now compare the results with a neural network that has significantly more parameters. Again, mutation only is used for the evolution of subsequent generations in both cases and the mutation power is σ=0.01𝜎0.01\sigma=0.01italic_σ = 0.01. We chose a fully connected NN with two hidden layers of size 64646464, resulting in a parameter count of 6788678867886788.
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Here we exploit the higher expressive power of VQCs compared to conventional neural networks [Chen et al., 2022]. Similar to [Chen et al., 2022], we define the expressive power as the capacity to represent particular functions with a constrained number of parameters. Note that the VQC has 148148148148 parameters (3 * 6 * 8 + 4). The neural network uses two hidden layers with dimension 3 and 4 respectively, resulting in a parameter count of 147147147147. Both the neural network and the VQC, are trained with mutation only with mutation power σ=0.01𝜎0.01\sigma=0.01italic_σ = 0.01. In Fig. 8, we can see that the neural network reward fluctuates in the range of 2.52.52.52.5 to 3333. As previously discussed in the last section, the VQC approach exhibits a slow learning curve leading to a significant higher score therefore consequently outperforming this neural network. The inferior performance can be explained by the small number of hidden units and parameters present in neural networks. Typically, the number of hidden units is chosen much higher. Further evidence of the neural network’s deficiency is provided by the average number of coins collected. As shown in Fig. 9(a), the NN’s number of collected coins is below the average score of the random agents until generation 115115115115 and after that slightly over it. In comparison, the VQC with the same number of parameters collects two times as many coins on average. The neural network is able to outperform random agents on the basis of its collected own coins, what can be seen in Fig. 9(b), leading to the better performance regarding the own coin rate (Fig. 9(c)). In terms of collected own coins and the own coin rate, the neural network performs significantly worse than the VQC with nearly the same number of parameters. A neural network with this few hidden units and, consequently, parameters is not able learn in the coin game environment successfully. This demonstrates the power of VQCs for RL archiving significantly higher with the same amount of parameters.
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A
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There are several steps to solving the optimization problem as shown in Fig. 3. In this section, we will discuss the process of solving the optimization problem on IBM’s superconducting (gate based) quantum computers, and bench-marking them with quantum simulators and classical tools like CPLEX and Simulated Annealing.
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In this paper, we are proposing a turn based model in a 3D lattice for the protein folding problem. The peptide chain is coarse grained into a C-alpha backbone model where each bead represents the C-alpha atom of individual amino acids in the peptide sequence. Further, the interaction model could either be in the classical HP form, where only H beads have interaction, or it could be in the MJ form where interaction strengths of all beads are considered. With N𝑁Nitalic_N beads we will have N−1𝑁1N-1italic_N - 1 turns, and every turn requires 6666 qubits for a 3D lattice. The number of qubits required in total is 6(N−1)6𝑁16(N-1)6 ( italic_N - 1 ). We could follow the assumption given in [6], and consider the first turn as given. The number of qubits required would then be 6(N−2)6𝑁26(N-2)6 ( italic_N - 2 ). Although the method proposed here uses more resources in terms of qubits, it provides for increased degrees of freedom in terms of the turns the amino acid chain can take. The method allows for diagonal movements both in a given plane (diagonal in a 2D square) as well as along a steric diagonal (the longest diagonal in a 3D cube). The chain while going from one bead to the next is capable of taking 6666 turns along the usual three axes (±x,±yplus-or-minus𝑥plus-or-minus𝑦\pm x,\pm y± italic_x , ± italic_y and ±zplus-or-minus𝑧\pm z± italic_z), 12121212 turns along the diagonals in the three axial planes (4444 each in the xy𝑥𝑦xyitalic_x italic_y, yz𝑦𝑧yzitalic_y italic_z and zx𝑧𝑥zxitalic_z italic_x planes), and 8888 turns along the steric diagonals. Overall, the degrees of freedom available is 26262626. For a C-alpha backbone model on this 3D lattice, the internal angles can take 45°, 54.74°, 90°, 125.26°, 135°or 180°and these are possible between any two consecutive backbone bonds. The pseudo-angles between three consecutive C-alpha atoms in an actual protein varies between 80°to 155°[13, 14], which is effectively covered by this 3D lattice. It is also important to note that the HP model of protein in cubic 3D lattice has a shortcoming termed as the parity problem - it is impossible to have contacts between two H beads if they are both in even or odd positions in the sequence. By extending the 3D lattice to include the diagonal this problem is effectively solved. The formulations result in the presence of up to 6-local terms (terms involving 6 variables), but we show that with suitable addition of constraints, we can limit the terms to 2-locals (QUBO formulation), without compromising on the degrees of freedom. The Quantum algorithm we have used is VQE. We also tried out the usage of cVARs as was done in [9]. We used Qiskit runtime estimators as well as samplers (sampling VQE) to get the desired bit-strings [15]. The quantum algorithms were run on both simulators as well as real IBM Quantum hardwares [16]. The maximum number of beads we have used so far is 20202020, requiring a total of 114114114114 qubits. We used 127 qubit IBM quantum hardwares to run our algorithms. We compare results given by quantum algorithms with the ones given by classical simulated annealing algorithms and IBM CPLEX.
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This exercise has to be carried out for all pairs of (i,j)𝑖𝑗(i,j)( italic_i , italic_j ). This will penalize all non-adjacent beads from having equal sets of coordinates. This kind of a constraint (non-overlap in any one dimension) ensures minimum contradiction with the main objective function, which tries to bring the beads closer. There is an element of randomness here in selecting the dimension over which the non-overlap happens and this will also get reflected in the final result. To account for this inherent randomness in a classical setting, the best outcome could be selected from a population of sample runs. The equivalent Hamiltonian operator (of the best outcome) could then be forwarded to be run on a quantum hardware.
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To run it in a Quantum hardware or simulator using quantum algorithms, we need to generate the Hamiltonian operator from the formulation. The detailed steps are discussed in the following sections. As discussed above, there is randomness in the formulation when the overlap and diagonal crossing constraints are taken into account. To get the best results, we can solve the optimization problem multiple times and select the best outcome from a population of solution vectors. Doing the same exercise in a Quantum hardware is not feasible, given the limited hardware time available. To get the best results in a Quantum hardware, we could use the Hamiltonian operator corresponding to the best classical outcome. We would call this classically enhanced, as shown in Fig. 3. One can also do a direct single run to generate the Hamiltonian operator from a one time formulation.
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In this paper we have discussed a novel turn based protein folding formulation that can be run on Quantum computing hardwares using variational algorithms. The highlight of the formulation is in the increased degrees of freedom that a turn has while going from one protein molecule to the next. We ran our formulation using both classical optimization algorithms as well as quantum algorithms in machines from IBM Quantum, and the results were compared. Hydrophobic collapse, the most predominant factor in the phenomena of protein folding, could be captured in both Quantum and classical simulations. Quantum algorithms were found to give better results when the sequences had lesser number of contacts or when there were many adjacent H beads. For sequences having higher number of contacts, the classical algorithms gave slightly better results. We used both Sampler as well as Estimator primitives in Qiskit runtime to estimate the energy values. In conjunction with VQE, we also used the CVaR technique to zero down on the most ideal bitstring. We went as far as to use 114114114114 qubits in a 127127127127 qubit machine (ibm_cusco) for a 20202020 bead system. With a reasonably shallow circuit, it is possible to get a large number of meaningful outcomes in combinatorial optimization problems.
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C
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For diagnostics, we employ Frequency-Resolved Optical Gating (FROG) and an Optical Spectrum Analyzer
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(OSA) to measure the temporal and spectral profiles, respectively. Additionally, a Time-Stretch Dispersive
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Stretched temporal profiles recorded by the oscilloscope for modes of {1}1\mathrm{\{1\}}{ 1 }, {2}2\mathrm{\{2\}}{ 2 }, and {3}3\mathrm{\{3\}}{ 3 }, respectively.
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respectively. The stretched temporal profiles demonstrate robustness over 250250250250 round trips and a dip is clearly observed at the center of the
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FGVD and nonlinearity. (a) and (d): The normalized temporal intensity and gradient (−∂I/∂t𝐼𝑡-\partial I/\partial t- ∂ italic_I / ∂ italic_t). (b) and (e): The corresponding temporal phase and
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A
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The previous paper Yahiaoui2018 proposed the EIT from bright-bright modes coupling, which is similar to Fig. 2 (b). However, our CIT is demonstrated in Fig. 2 (c). Note that Fig. 2 (b) and (c) have different polarization. Therefore, in our paper, two dark modes can not be excited by external field without bright mode (CW). Thus, the physics behind CIT and EIT from bright-bright modes coupling are entirely different.
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The CIT dip comes from the interference of dark modes due to asymmetric dark modes. The interference of dark modes can provide the Fano shape in the transmission spectrum, as shown in Fig. 2 (b). Fig. 2 (b) demonstrates the transmission spectrum of dark modes with x-polarized THz wave without bright mode and this configuration is equivalent to the excitation of dark modes by bright mode’s coupling with y-polarized THz wave. This Fano shape of the transmission spectrum is the BIC shape due to the asymmetric dark modes. The physics behind BICs comes from the interference of different structures’ radiated waves, providing the sub-radiated mode (destructive interference) and super-radiated mode (constructive interference) which causes the Fano shape. The sub-radiated mode suppresses the excitation of dark modes and it leads that the bright mode can not coupled to dark modes which causes the CIT dip in the transmission spectrum. Therefore, our CIT comes from the coupling between the bright mode and interference between the dark modes.
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Figure 6: The electric field (left) and current density field (right) at CIT frequency, with corresponding to our example as shown in the red line in Fig. 2 (c) at around 0.72 THz.
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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). 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.
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For better understanding, We plot the electric field and current density field at CIT frequency, as shown in fig. 6 with corresponding to around 0.72 THz for the red line in Fig. 2 (c). As we can obtain from the results, the dark modes present two opposite modes which provide destructive interference between the dark modes at CIT frequency. Therefore, the bright mode can not couple to dark modes at this frequency due to the suppression of dark modes, which remains some current density and energies. Thus, the bright mode can be re-excited by external THz wave and it causes the CIT dip in the transmission spectrum.
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D
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Although our examples were restricted to the examination of two bodies, there is nothing in the analysis presented here that does not immediately extend to a greater number of bodies. We are eager to see these techniques used to describe many-body elastic interactions, though it may be that the simpler far-field interactions will prove more useful as a starting point for suspension configurations.
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This paper is organized as follows. We begin in §2 with a review of the mathematical model, including a discussion of boundary conditions and surface tractions, and we recall from Ref. [12] the effective boundary technique that allows for the solution of a weak (finite) anchoring problem based on the solution of a strong (infinite) anchoring problem with a slightly different boundary. Analytical solutions for two immersed bodies are then provided in §3. Two worked examples of multiple-body interactions are then presented, which demonstrate the above methodology for determining the two-dimensional director field at equilibrium, including the selection of the topological charges and defect positions on the body surfaces. The first of these two examples is given in §4, where we investigate two immersed cylinders with tangential anchoring, which includes the case of a single cylinder near an infinite wall as a limiting case. We consider a more involved example in §5, the interactions between two triangular prisms, where we again provide formulae for the body forces and torques, and observe how defect positioning and particle interactions are orientation-dependent, reproducing experimental findings. When the distance between the bodies is large, asymptotically valid approximations may be derived, as described in §6. Finally, in §7, we provide a closing summary and directions of future applications.
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Helpful conversations with Nicholas Abbott, Thomas Powers, and Raghav Venkatraman are gratefully acknowledged. SES acknowledges the UW–Madison Office of the Vice Chancellor for Research and Graduate Education and funding from the Wisconsin Alumni Research Foundation.
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We begin with a description of the general problem, and recall the relevant structure developed for the case of a single immersed body [12]. Consider a two-dimensional nematic liquid crystal outside N𝑁Nitalic_N simply-connected bodies, as illustrated in Fig. 1 for N=2𝑁2N=2italic_N = 2. The liquid crystal domain and the boundary of the k𝑘kitalic_kth body are denoted by D𝐷Ditalic_D and ∂Dksubscript𝐷𝑘\partial D_{k}∂ italic_D start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT, respectively. Assuming the one-constant approximation, the director angle, θ(x,y)𝜃𝑥𝑦\theta(x,y)italic_θ ( italic_x , italic_y ), is described by the Dirichlet free energy ℱsurface≔K|∇θ|2/2≔subscriptℱsurface𝐾superscript∇𝜃22\mathcal{F}_{\mathrm{surface}}\coloneqq K|\nabla\theta|^{2}/2caligraphic_F start_POSTSUBSCRIPT roman_surface end_POSTSUBSCRIPT ≔ italic_K | ∇ italic_θ | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2, where K𝐾Kitalic_K is the single Frank elastic constant. In general there are distinct elastic moduli penalizing LC bend and splay deformations, but they tend to be comparable [8, 104], and the single constant model is often used to simplify mathematical analysis [18].
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Among the most alluring (and analytically challenging) features of liquid crystals are the prevalence of topological singularities, which satisfy global conservation laws [46, 3]. The locations of the defects on the surface or in the fluid depend on the relationship between the bulk elastic energy and the surface anchoring conditions on any domain boundaries. In addition to focusing elastic stress on immersed surfaces, topological defects are important sites in biological settings for the onset of cell death and extrusion [73], layer formation [14], cell accumulation [37], cell sorting [5], and morphogenesis [52, 92, 97]. They have also been considered for directed self-assembly [55, 96] and control [62, 28, 47, 49, 25]. Analytical insight into defect positioning and its consequences for locally stored elastic energy is, thus, of broad interest.
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B
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The momentum-averaged positions 𝔼(𝐳(t))𝔼𝐳𝑡\mathbb{E}({\bf z}(t))blackboard_E ( bold_z ( italic_t ) ) and velocities ddt𝔼(𝐳(t))𝑑𝑑𝑡𝔼𝐳𝑡\frac{d}{dt}\mathbb{E}({\bf z}(t))divide start_ARG italic_d end_ARG start_ARG italic_d italic_t end_ARG blackboard_E ( bold_z ( italic_t ) ) are discrete analogs of momentum-averaged position and momentum velocity in electromagnetism, where they emerge from the virial density in an application of the virial theorem to electromagnetic pulses [11]. Whereas cov(𝐫(t),𝐳(t))cov𝐫𝑡𝐳𝑡\mathrm{cov}({\bf r}(t),{\bf z}(t))roman_cov ( bold_r ( italic_t ) , bold_z ( italic_t ) ) is frequently referred to as the selection term in the Price equation [5], the connection to the virial theorem suggests that it is better described as a selection rate (Table 1). Similarly, the transmission term 𝔼(d𝐳(t)dt)𝔼𝑑𝐳𝑡𝑑𝑡\mathbb{E}\left(\frac{d{\bf z}(t)}{dt}\right)blackboard_E ( divide start_ARG italic_d bold_z ( italic_t ) end_ARG start_ARG italic_d italic_t end_ARG ) is more accurately a transmission rate. Most significantly, while the dynamical interpretation typically relies on associating force to natural selection, drift, migration, or mutation [23], the equivalence between the virial theorem and the Price equation, suggests that force is more naturally associated to fitness. The correspondence of force to a rate of change is not surprising, since force is also a rate of change, specifically the rate of change of momentum. This stands more in line with the statistical interpretation of evolutionary theory [54, 53, 29], which, among several critiques of the dynamical interpretation, finds fault with the analogies of biological processes such as mutation with forces in physics, arguing that the physical forces are causal in a way that processes such as selection or mutation are not [54]. However, the analogy of population growth with force can be viewed as consistent with the dynamical interpretation; for example, population growth can directly affect DNA polymorphism patterns [55]. Moreover, the virial theorem in the setting of the ecological simple harmonic oscillator affirms [23] in noting that “natural selection turns out to be more similar to forces such as friction and elastic forces rather than the more canonical gravitation.”
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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. The extension of (7) to (8) is interesting in its own right, and should be interesting to develop in future work. Moreover, Theorem 5 shows that subpopulations subject to distinct maternal effects can generate arbitrarily complex population dynamics, thereby affirming the main thesis of [18].
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The momentum-averaged positions 𝔼(𝐳(t))𝔼𝐳𝑡\mathbb{E}({\bf z}(t))blackboard_E ( bold_z ( italic_t ) ) and velocities ddt𝔼(𝐳(t))𝑑𝑑𝑡𝔼𝐳𝑡\frac{d}{dt}\mathbb{E}({\bf z}(t))divide start_ARG italic_d end_ARG start_ARG italic_d italic_t end_ARG blackboard_E ( bold_z ( italic_t ) ) are discrete analogs of momentum-averaged position and momentum velocity in electromagnetism, where they emerge from the virial density in an application of the virial theorem to electromagnetic pulses [11]. Whereas cov(𝐫(t),𝐳(t))cov𝐫𝑡𝐳𝑡\mathrm{cov}({\bf r}(t),{\bf z}(t))roman_cov ( bold_r ( italic_t ) , bold_z ( italic_t ) ) is frequently referred to as the selection term in the Price equation [5], the connection to the virial theorem suggests that it is better described as a selection rate (Table 1). Similarly, the transmission term 𝔼(d𝐳(t)dt)𝔼𝑑𝐳𝑡𝑑𝑡\mathbb{E}\left(\frac{d{\bf z}(t)}{dt}\right)blackboard_E ( divide start_ARG italic_d bold_z ( italic_t ) end_ARG start_ARG italic_d italic_t end_ARG ) is more accurately a transmission rate. Most significantly, while the dynamical interpretation typically relies on associating force to natural selection, drift, migration, or mutation [23], the equivalence between the virial theorem and the Price equation, suggests that force is more naturally associated to fitness. The correspondence of force to a rate of change is not surprising, since force is also a rate of change, specifically the rate of change of momentum. This stands more in line with the statistical interpretation of evolutionary theory [54, 53, 29], which, among several critiques of the dynamical interpretation, finds fault with the analogies of biological processes such as mutation with forces in physics, arguing that the physical forces are causal in a way that processes such as selection or mutation are not [54]. However, the analogy of population growth with force can be viewed as consistent with the dynamical interpretation; for example, population growth can directly affect DNA polymorphism patterns [55]. Moreover, the virial theorem in the setting of the ecological simple harmonic oscillator affirms [23] in noting that “natural selection turns out to be more similar to forces such as friction and elastic forces rather than the more canonical gravitation.”
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Ultimately, analogies between the Price equation and the virial theorem point towards potentially productive directions for exploration in both biology and physics. The statistical framing of the virial theorem in (5) highlights phenomena that may have been overlooked in the physics realm. For example, the first term on the right-hand side of (5), namely cov(𝐫(t),𝐳(t))cov𝐫𝑡𝐳𝑡\mathrm{cov}({\bf r}(t),{\bf z}(t))roman_cov ( bold_r ( italic_t ) , bold_z ( italic_t ) ), can be understood to quantify the extent of the Yule-Simpson effect [33, 56, 45], which describes a situation where within-group trends can be reversed upon averaging. In biology, the Price equation has the potential to be used more widely as a tool. Although it has been hailed as a unifying framework for researchers [25], one that “can serve as a heuristic principle to formulate and systematize different theories and models in evolutionary biology” [26], the emphasis on its use has been more oriented toward understanding how it generalizes specific equations, rather than applying it for biological discovery. For example, the Price equation can be used to derive the Breeder’s equation [4, 57], Fisher’s fundamental theorem [37, 35], the house of cards approximation for genetic variance at mutation-selection balance [57, 50], and many other formulas and identities in genetics [57, 40]. However, it has been referred to as a tautology and a vacuous statement without application. In [51] the Price equation is described as a theorem that establishes that “If the left-hand side is computed as suggested in [36], and the right-hand side too, then they are equal.” This critique of the Price equation, namely that it does not and cannot serve as a tool, stands in contradiction to evidence from physics, where the mathematically equivalent virial theorem has been understood as a powerful tool since its use to discover dark matter in 1933 [58]. The manifold applications of the virial theorem [28] suggest that there is still much to gain from the application of the Price equation as a tool for biology. In fact, the equivalence we have demonstrated between the Price equation and the virial theorem shows that the description of missing heritability as dark matter [27] may be understood to be more than just an informal analogy between mysteries in genetics and astronomy.
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In genetics, the natural time increment to consider is discrete (generation), whereas in physics continuous-time is more natural. Thus, the discrete Price equation pertains to change in a trait after a single generation, whereas the virial theorem is formulated with continuous-time, and is additionally time averaged. However, the less intuitive forms of these equations that arise from the correspondences derived above may yield important insights. For example, the perspective of the virial theorem as a special case of the equipartition theorem [34] may be fruitful in evolutionary biology [31]. Translation between biology and physics via the virial theorem and the Price equation may also accelerate discovery of generalizations. While the stochastic Price equation in evolution [41] and the stochastic virial theorem in astronomy [7] were discovered independently, their similarity suggests other generalizations could similarly parallel each other. Moreover, the virial theorem has been applied in a variety of fields (for example economics [2]), meaning that understanding its relationship to the Price equation could be relevant beyond physics and biology.
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A
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Nevertheless, providing reference implementations avoids duplication efforts in writing file writers and parsers, aiming at obtaining a robust library that can be easily reused, maintained, and where bugs can be quickly resolved. To efficiently address this goal, it is useful to select a popular language such as Python; this can also facilitate external contributions.
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The concept of MLWF can be actually extended to multiparticle Bloch states and has been recently applied to excitons, which are two-particle correlated e-h excitations and where maximal localization can be defined with respect to an average e-h coordinate in real space [171].
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The efficient interpolation in reciprocal space of k𝑘kitalic_k-dependent quantities is arguably the most common application of WFs, enabling the calculation of simple (e.g., the band structure) or complex (e.g., electron-phonon coupling) electronic-structure properties. A large part of this review is devoted to the fundamentals of WF interpolation (Sec. III.3) and their applications, including ballistic transport (Sec. III.4), Berry-phase related properties (Sec. III.5) and electron-phonon interactions (Sec. III.7). As discussed in more detail in Sec. III.3, the reason for such widespread set of applications (not all of them covered in this review) is that WFs can be easily applied to any generic operator that is local in reciprocal space, i.e., any lattice-periodic operator. More generally, we note that even some non-local operators in reciprocal space (e.g. containing the position operator, which is not lattice periodic and transforms into k𝑘kitalic_k derivatives) can also be interpolated, see e.g. Sec. III.5 on Berryology. Equally important is that WFs allow reproducing the correct band connectivity: in particular, avoided crossings are not mistaken for actual crossings. This distinguishes Wannier interpolation from other methods based on direct Fourier interpolation of the energy eigenvalues. In other words, WFs allow to exploit the fundamental locality (“nearsightedness” according to Kohn [94, 95, 209]) of the electronic structure and the related exponential localization of WFs to construct a potentially exact representation of an operator in real space, such that any interpolation back to reciprocal space is exact as well. The procedure is also systematic as WFs are guaranteed to exist and the convergence is exponential with the linear sampling density [307, 54, 308]; prefactors and coefficients might depend on electronic-structure properties such as the band gap, and on the specific operator under consideration.
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Nevertheless, we stress that the concept of a common parsing library is not limited to the Python language, but can also be applied to other emerging
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Nevertheless, providing reference implementations avoids duplication efforts in writing file writers and parsers, aiming at obtaining a robust library that can be easily reused, maintained, and where bugs can be quickly resolved. To efficiently address this goal, it is useful to select a popular language such as Python; this can also facilitate external contributions.
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C
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={u∈H˙1(ℝd):u is radially symmetric}.absentconditional-set𝑢superscript˙𝐻1superscriptℝ𝑑𝑢 is radially symmetric\displaystyle=\{u\in\dot{H}^{1}(\mathbb{R}^{d}):u\text{ is radially symmetric}\}.= { italic_u ∈ over˙ start_ARG italic_H end_ARG start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ) : italic_u is radially symmetric } .
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Note that the restriction q<2∗𝑞superscript2q<2^{*}italic_q < 2 start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT in (2) is necessary, namely the quadratic form h[u]ℎdelimited-[]𝑢h[u]italic_h [ italic_u ] is really weaker than ‖∇u‖L22superscriptsubscriptnorm∇𝑢superscript𝐿22\|\nabla u\|_{L^{2}}^{2}∥ ∇ italic_u ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. Here we are interested in a replacement of (2) which covers the critical power q=2∗𝑞superscript2q=2^{*}italic_q = 2 start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT, with the expense that the L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-norm is replaced by the energy associated with the inverse square potential. We have
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In the radial case, we can work directly with the quadratic form h[u]ℎdelimited-[]𝑢h[u]italic_h [ italic_u ] in (1). We have
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Let us show that (4) fails if d≥4𝑑4d\geq 4italic_d ≥ 4 or if θ≠1/3𝜃13\theta\neq 1/3italic_θ ≠ 1 / 3. First, the necessity of θ≤1/d𝜃1𝑑\theta\leq 1/ditalic_θ ≤ 1 / italic_d can be seen from the radial case as explained in Theorem 3 (i). To be precise, we consider the example in (18) with ε>0𝜀0\varepsilon>0italic_ε > 0 small. In this case, u𝑢uitalic_u is radially symmetric decreasing, and hence (28) holds.
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In this short article we are interested in interpolation inequalities involving the quadratic form h[u]ℎdelimited-[]𝑢h[u]italic_h [ italic_u ] and Lpsuperscript𝐿𝑝L^{p}italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT-norms of u𝑢uitalic_u. A classical result in this direction is the Gagliardo–Nirenberg type inequality
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B
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{s}(r^{\prime},\vartheta^{\prime})\,e^{i\ell(\vartheta-\vartheta^{\prime})}\,.start_ROW start_CELL ( italic_R start_POSTSUBSCRIPT roman_P , ± end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( roman_F ) end_POSTSUPERSCRIPT ( italic_λ ) bold_italic_ψ ) start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_r , italic_ϑ ) = ∫ start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT roman_d italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∫ start_POSTSUPERSCRIPT 2 italic_π end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT roman_d italic_ϑ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT roman_ℓ ∈ blackboard_Z end_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT | roman_ℓ + italic_α | end_POSTSUBSCRIPT ( ∓ italic_i square-root start_ARG italic_λ end_ARG ( italic_r ∧ italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ) italic_K start_POSTSUBSCRIPT | roman_ℓ + italic_α | end_POSTSUBSCRIPT ( ∓ italic_i square-root start_ARG italic_λ end_ARG ( italic_r ∨ italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ) italic_ψ start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_ϑ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) divide start_ARG italic_e start_POSTSUPERSCRIPT italic_i roman_ℓ ( italic_ϑ - italic_ϑ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUPERSCRIPT end_ARG start_ARG 2 italic_π end_ARG end_CELL end_ROW start_ROW start_CELL = divide start_ARG italic_i end_ARG start_ARG 4 end_ARG ∫ start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT roman_d italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∫ start_POSTSUPERSCRIPT 2 italic_π end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT roman_d italic_ϑ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT roman_ℓ ∈ blackboard_Z end_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT | roman_ℓ + italic_α | end_POSTSUBSCRIPT ( ± square-root start_ARG italic_λ end_ARG ( italic_r ∧ italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ) italic_H start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT | roman_ℓ + italic_α | end_POSTSUBSCRIPT ( ± square-root start_ARG italic_λ end_ARG ( italic_r ∨ italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ) italic_ψ start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_ϑ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) italic_e start_POSTSUPERSCRIPT italic_i roman_ℓ ( italic_ϑ - italic_ϑ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUPERSCRIPT . end_CELL end_ROW
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To begin with, from (2.21), (2.22) and (2.23) (see also [OLBC10, §10.27 and §10.29(ii)]), we deduce that
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To say more, using (2.8), (2.9) and (3.8), together with the Bessel connection formula [OLBC10, Eq. 10.27.8], for any 𝐪∈ℂ4𝐪superscriptℂ4\mathbf{q}\in\mathbb{C}^{4}bold_q ∈ blackboard_C start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT, we infer
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In the second line we have used the connection formulas reported in [OLBC10, Eqs. 10.27.6 and 10.27.8].
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Notice that the kernel in (2.22) is slightly different, compared to [AT98], as we write it using the modified Bessel functions of second kind Iν,Kνsubscript𝐼𝜈subscript𝐾𝜈I_{\nu},K_{\nu}italic_I start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT , italic_K start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT in place of the Bessel and Hankel functions Jν,Hν(1)subscript𝐽𝜈superscriptsubscript𝐻𝜈1J_{\nu},H_{\nu}^{(1)}italic_J start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT , italic_H start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT (this is obtained using the connection formulas [OLBC10, Eqs. 10.27.6 and 10.27.8]).
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C
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This could potentially signal the PDW origin of superconductivity in these strongly correlated quantum materials as well.
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The finite interaction range R0subscript𝑅0R_{0}italic_R start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT in this case has the same electrostatic origin as the Thomas-Fermi screening.
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In the repulsive case, V(r)>0𝑉𝑟0V(r)>0italic_V ( italic_r ) > 0, Eq. (1) models a screened Coulomb interaction, where R0subscript𝑅0R_{0}italic_R start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT plays role of the Thomas-Fermi screening length scaling as λFγ−1/(D−1)≫λFmuch-greater-thansubscript𝜆𝐹superscript𝛾1𝐷1subscript𝜆𝐹\lambda_{F}\gamma^{-1/(D-1)}\gg\lambda_{F}italic_λ start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT italic_γ start_POSTSUPERSCRIPT - 1 / ( italic_D - 1 ) end_POSTSUPERSCRIPT ≫ italic_λ start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT at weak interaction coupling γ≪1much-less-than𝛾1\gamma\ll 1italic_γ ≪ 1.
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Importantly, the electron-electron interaction, U(τ,𝒓)𝑈𝜏𝒓U(\tau,\bm{r})italic_U ( italic_τ , bold_italic_r ), within our model is instantaneous and assumed to have a finite range R0subscript𝑅0R_{0}italic_R start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT that significantly exceeds the average inter-electron distance given by the Fermi wavelength, λFsubscript𝜆𝐹\lambda_{F}italic_λ start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT,
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It might be possible to change the interaction range R0subscript𝑅0R_{0}italic_R start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT in these materials in a controlled way by gates, which affect the screening, and thereby test our predictions.
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D
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Here, v(z,ω)𝑣𝑧𝜔v(z,\omega)italic_v ( italic_z , italic_ω ) is the speed of light in the medium, and 𝒢(z,K,ω)𝒢𝑧𝐾𝜔\mathcal{G}(z,K,\omega)caligraphic_G ( italic_z , italic_K , italic_ω ) the direction-dependent optical density of states specified in Appendix A.
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In the calculations of optical cavities in Sec. III.1, the photon numbers are calculated by making direct use of the FED description deployed in Ref. 12. This description is well suited for calculating the net emission of a single layer assumed to have a given temperature and quasi-Fermi level separation. In Sec. III.2, Eq. (1) is instead solved iteratively with the drift-diffusion equations to arrive at self-consistent electro-optical solutions. To do this, it is beneficial to have an optical model which directly allows a device-wide and position-dependent excitation level that can even change inside single material layers. For this, we define a generalized radiative transfer (RT) equation for the photon numbers written as
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To recap, the objective of this Subsection is to study whether the total emission of extremely thin layers can be boosted with cavity effects and by how much. To do that, Eq. (1) is solved for structures illustrated in Fig. 1, with the structure in Fig. 1(a) representing a thin-film intracavity device including both an emitter and an absorber layer, and Fig. 1(b) representing a reference case, where a thin GaAs layer simply emits radiation into free space consisting of AlGaAs. In this Section, we always have a constant quasi-Fermi level separation of 1.3 eV and temperature of 300 K in the emitter layer. Similar to Ref. 12, the absorber layer in Fig. 1(a) is assumed to have significantly weaker excitation (lower applied bias and/or temperature), such that its own emission can be considered negligible. The permittivities deployed in the calculations are given in Refs. 20; 21; 22, and the permittivity of Ag is multiplied by 100 similarly as in Ref. 12 to decrease mirror losses and enable focusing on the optical processes that take place inside the cavity. Photon energies between 1.41 and 1.65 eV are considered, and the K𝐾Kitalic_K values span from 0 to 1.05 NGaAsk0subscript𝑁𝐺𝑎𝐴𝑠subscript𝑘0N_{GaAs}k_{0}italic_N start_POSTSUBSCRIPT italic_G italic_a italic_A italic_s end_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, where NGaAssubscript𝑁𝐺𝑎𝐴𝑠N_{GaAs}italic_N start_POSTSUBSCRIPT italic_G italic_a italic_A italic_s end_POSTSUBSCRIPT is the refractive index of GaAs and k0subscript𝑘0k_{0}italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is the wave number in vacuum, both dependent on ℏωPlanck-constant-over-2-pi𝜔\hbar\omegaroman_ℏ italic_ω.
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The dependence of the optical power on the cavity length is first illustrated in Fig. 2 for the intracavity structure of Fig. 1(a) with a single example. Here, the thicknesses of the GaAs emitter and absorber layers are 20 and 60 nm, respectively. As in Ref. 12, the emitted and absorbed optical powers depend on the cavity length. Here, the maximum emission and absorption are reached at a total cavity length of approximately 250 nm. The optical power emitted by a similarly thick GaAs layer in the reference structure of Fig. 1(b) is also plotted as a constant dotted line in Fig. 2. It can be seen that at the maximum, the emission in the cavity is roughly 35 % stronger than in the reference structure.
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The net RG rates [Rradsubscript𝑅𝑟𝑎𝑑R_{rad}italic_R start_POSTSUBSCRIPT italic_r italic_a italic_d end_POSTSUBSCRIPT in Eq. (1)] discussed in the previous paragraph for applied biases 0.8 V and 1.0 V are illustrated in more detail in Fig. 7. In Fig. 7, Rrad(z)subscript𝑅𝑟𝑎𝑑𝑧R_{rad}(z)italic_R start_POSTSUBSCRIPT italic_r italic_a italic_d end_POSTSUBSCRIPT ( italic_z ) is shown as a function of position under illumination at applied biases (a) 0.8 V and (b) 1.0 V. Note that the curves shown in Fig. 6(b) were obtained by integrating the curves of Fig. 7 over position. In Fig. 7(a), Rradsubscript𝑅𝑟𝑎𝑑R_{rad}italic_R start_POSTSUBSCRIPT italic_r italic_a italic_d end_POSTSUBSCRIPT is positive close to the top surface of the solar cell and negative further away from it (towards the left in the figure). This is essentially the same net photon recycling process that we reported in Ref. 14. Interestingly, there is also non-negligible emission from the AlGaAs layer at the left side of the figure. This is due to sunlight absorbed by the AlGaAs layer, which results in a quasi-Fermi level separation of 1.23 eV there (see band diagrams in Appendix D). This is associated with emission from AlGaAs even with the drift-diffusion currents removing excess electrons and holes towards lower-bandgap layers and contacts.
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A
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Moreover, the eigenspectra of open cavities show both continuous and discrete branches [18, 19] and a non-singular transformation is required so that their topology is preserved [20, 21].
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Finally, from a numerical standpoint, the computation of resonances in open systems is complicated by the reflection of the leaky waves at the necessarily finite-grid boundaries. To tackle this issue, we use Perfectly Matched Layers (PML), whose analysis and performance for spectral problems have been extensively documented in the literature, see, e.g., [22, 23, 24, 25, 26].
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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. 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. 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.
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Perhaps the most striking and well-studied effect enabled by TA is cloaking, allowing perfect concealment of an arbitrary object using a singular transformation. However, another strategy proposed in the work by Li et al. [14], the so-called carpet cloak, requires non-singular transformations, making this route more suited for the practical implementation of the equivalent properties. Twinning closed cavities through TA has been recently developed by Lenz et al. in [15], where the discrete spectrum of a closed domain with Dirichlet boundary conditions is successfully matched. Here we consider unbounded open domains; this is not straightforward as spectral problems for open cavities allow for the leakage of energy into the unbounded medium, have complex eigenspectra and further complications for both theoretical and numerical aspects; we are unaware of attempts to achieve isospectral domains in open systems in wave physics and this opens the way to, for instance, twinning optical waveguides.
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The stretch of the geometry induces an anisotropic density whose principal components are aligned with the radial and the tangential directions of the cavity, similarly to the invisibility cloak for axisymmetric obstacles see, e.g., [38]. This situation lends itself to the use of a layered arrangement of two homogeneous and isotropic fluids for achieving the effective properties we need. The following analytical relations hold [38] for the effective properties:
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A
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0italic_q = 2 italic_k start_POSTSUBSCRIPT italic_p italic_r italic_o italic_b italic_e end_POSTSUBSCRIPT ≈ 0). At the very low frequency (<<<100 GHz) a Brillouin mode is expected to be detected in the SL (similarly to the one observed in STO) but is not since its characteristic period of oscillation (typically the one observed for STO) is longer than the time of flight of the acoustic phonons in the SL. More importantly, there is a second region (ωZCsubscript𝜔𝑍𝐶\omega_{ZC}italic_ω start_POSTSUBSCRIPT italic_Z italic_C end_POSTSUBSCRIPT) where modes can be detected and values of around 1.1-1.2 THz are expected. These frequency are actually consistent with the detected mode f2subscript𝑓2f_{2}italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT with ωZC/2π∼f2similar-tosubscript𝜔𝑍𝐶2𝜋subscript𝑓2\omega_{ZC}/2\pi\sim f_{2}italic_ω start_POSTSUBSCRIPT italic_Z italic_C end_POSTSUBSCRIPT / 2 italic_π ∼ italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. Such ωZCsubscript𝜔𝑍𝐶\omega_{ZC}italic_ω start_POSTSUBSCRIPT italic_Z italic_C end_POSTSUBSCRIPT zone centered mode coming from the mode folding governed by the periodic chemical composition BFO-LFO (ΛΛ\Lambdaroman_Λ) is the typical one that is conventionally observed in the coherent phonon spectra of many semiconductor [2, 4, 5, 7, 8, 9, 10] and ferroelectric [15, 16, 17] superlattices. Very importantly, we show that the mode f1subscript𝑓1f_{1}italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT=0.75 THz cannot be explained by the conventional mode folding driven by the chemical super-order. In the theoretical calculation, a mode in the region 0.5-0.7 THz is predicted, but it is a zone edge mode (ωZEsubscript𝜔𝑍𝐸\omega_{ZE}italic_ω start_POSTSUBSCRIPT italic_Z italic_E end_POSTSUBSCRIPT), thus not detected according to the optical detection selection rule. We discuss in the following the origin of this mode.
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It is first worth mentioning that magnon mode at 0.7 THz has already been reported in BiFeO33{}_{3}start_FLOATSUBSCRIPT 3 end_FLOATSUBSCRIPT nanostructures [26] arising from a magnon mode folding caused by the magnetic cycloid. Such a mode can be excluded in our case since we have a confined and strained multiferroic nanostructure where the cycloid is no more present as discussed in the literature [26]. We have confirmed the absence of the cycloid with polarized neutron diffraction in a BFO(12)/LFO(12) superlattice (see Supplementary Note 3).
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In this work, by analysing electron and X-ray diffraction patterns, we reveal the existence of rich multiple structural orders in a BiFeO33{}_{3}start_FLOATSUBSCRIPT 3 end_FLOATSUBSCRIPT/LaFeO33{}_{3}start_FLOATSUBSCRIPT 3 end_FLOATSUBSCRIPT (BFO/LFO) superlattice. We show that besides the known in-plane orthorhombic distortion and antiferroelectric-like order [18, 19], a long-range out-of-plane super-order is revealed. We evidence a supercell parameter which approaches the double (∼2Λsimilar-toabsent2Λ\sim 2\Lambda∼ 2 roman_Λ) of the chemical superlattice parameter. The latter one (ΛΛ\Lambdaroman_Λ) being solely defined by the thickness of the double layer BFO/LFO. On the basis of a first principle calculation, we assign this new order to a specific sequence of FeO66{}_{6}start_FLOATSUBSCRIPT 6 end_FLOATSUBSCRIPT octahebra tilt that makes two adjacent BFO layers non-equivalent. We arrive then to a super-period BFO(1)/LFO/BFO(2)/LFO corresponding to a structural period of ∼2Λsimilar-toabsent2Λ\sim 2\Lambda∼ 2 roman_Λ. We then discuss how such super-order can affect the THz acoustic phonon spectrum of the superlattice. For this purpose, we performed time-resolved optical pump-probe experiments. We show that with these BiFeO33{}_{3}start_FLOATSUBSCRIPT 3 end_FLOATSUBSCRIPT/LaFeO33{}_{3}start_FLOATSUBSCRIPT 3 end_FLOATSUBSCRIPT superlattices it is indeed possible to optically generate and detect at room temperature a 1.2 THz longitudinal coherent acoustic phonon mode which is consistent with mode-folded phonons arising from the chemical order. Such a folded mode was already revealed in different oxide superlattices like in Pb(Zr,T)O33{}_{3}start_FLOATSUBSCRIPT 3 end_FLOATSUBSCRIPT-SrRuO33{}_{3}start_FLOATSUBSCRIPT 3 end_FLOATSUBSCRIPT [15, 16] and BaTiO33{}_{3}start_FLOATSUBSCRIPT 3 end_FLOATSUBSCRIPT-SrTiO33{}_{3}start_FLOATSUBSCRIPT 3 end_FLOATSUBSCRIPT [17] superlattices. More interestingly, we detect a new mode centered at 0.7 THz which cannot be explained by taking into account solely the superlattice chemical order. We attribute this new mode to a secondary mode-folding process in agreement with the existence of a nearly double super-cell revealed by the structural analysis. These finding opens new avenue to tailor the spectrum of particles (electron, phonon, electromagnon) in multiferroic superlattices with a controllable mode-folding.
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The origin of this mode at 0.7 THz might have another origin. Different hypothesis can be discussed. First of all, one could envision also a shear acoustic mode folding. Considering the shear velocity of around 3000 m.s−11{}^{-1}start_FLOATSUPERSCRIPT - 1 end_FLOATSUPERSCRIPT [30, 33, 38, 39], one can expect a zone center mode at around 0.7 THz (calculation not shown). But, generating a shear acoustic phonon requires that the system has the proper symmetry, i.e. that an in-plane symmetry breaking exists (i.e. absence of a symmetry plane perpendicular to the surface). This condition is fulfilled in the bulk single-domain BiFeO33{}_{3}start_FLOATSUBSCRIPT 3 end_FLOATSUBSCRIPT, with rhombohedral symmetry (3m3𝑚3m3 italic_m) [30, 33, 38] or in single domain monoclinic BiFeO33{}_{3}start_FLOATSUBSCRIPT 3 end_FLOATSUBSCRIPT thin film [39]. This in-plane symmetry breaking is usually well correlated with optical birefringence in both aformentioned cases as a witness of crystal anisotropy [30, 33, 39]. Such optical birefringence is absent in the BFO/LFO superlattice (see Supplementary Note 4). This is an indication, that at the level of an optical beam focus, the BFO/LFO superlattice is isotropic. This is consistent with the microscopic organisation revealed by electron microscopy where we see the coexistence of so-called conventional and non-conventional regions (see discussion in Structural analysis Section). We do not have a single domain ferroelectric/multiferroic structure in our superlattice. Even if at the level of the unit cell, light can generate shear acoustic waves, the multidomain state might lead to an average effect at the macroscopic level. So we think then that we can exclude that the mode at 0.7 THz comes from the generation and detection of coherent transverse mode.
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Neutron diffraction data were collected on the long wavelength time-of-flight diffractometer WISH [41] located at the second target station of the ISIS Pulsed Neutron and Muon Source, United Kingdom. By nature, neutron diffraction allows the average structure of the whole superlattice (and the substrate) to be measured. The experimental setup used was similar to our previous work on BiFeO33{}_{3}start_FLOATSUBSCRIPT 3 end_FLOATSUBSCRIPT thin films [42] (Fig. 6a) but with higher spatial instrumental resolution (divergence 0.17∘{}^{\circ}start_FLOATSUPERSCRIPT ∘ end_FLOATSUPERSCRIPT instead of 0.25∘{}^{\circ}start_FLOATSUPERSCRIPT ∘ end_FLOATSUPERSCRIPT) to allow better detection of possible peak splitting. The superlattice under investigation has a BFO and LFO layers twice thicker than the superlattice discussed in the main text. This permits to enhance the neutron signal. Despite the fact that the structure appears antiferromagnetic, implying BiFeO3 should therefore not host a cycloid, we collected neutron diffraction data for two different rotations. Both datasets show no peak splitting either spatially or in d-spacing (along the time of flight direction) as shown in Fig. 6b but instead reveal a single sharp peak for each orientation at the exact same d-spacing. The results are consistent with a simple G-type antiferromagnetic structure with a propagation vector [121212121212\frac{1}{2}\frac{1}{2}\frac{1}{2}divide start_ARG 1 end_ARG start_ARG 2 end_ARG divide start_ARG 1 end_ARG start_ARG 2 end_ARG divide start_ARG 1 end_ARG start_ARG 2 end_ARG] in cubic setting. This is in contrast to our previous measurements on thin films of BiFeO33{}_{3}start_FLOATSUBSCRIPT 3 end_FLOATSUBSCRIPT grown on orthorhombic DyScO33{}_{3}start_FLOATSUBSCRIPT 3 end_FLOATSUBSCRIPT substrate where clear peak splitting was observed spatially and in d-spacing [42], corresponding to the tick green marks on Fig. 6b, from which the period of the modulation could be easily extracted. The fact that only one peak is observed in both datasets also reinforces the point that there is no polar direction in the system. These results show and confirm that the strong confinement prevent the formation of the cycloid and therefore magnon can no more be folded. In particular, the magnon mode at 0.7 THz cannot exist. This constraint will be all the more strong in a superlattice having even thinner BFO and LFO layers.
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A
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In this section, we formulated the transport coefficients, specifically shear and bulk viscosities, for medium QGP following the theoretical framework developed in Ref. [46]. It is important to emphasize that this formulation is conducted in the limit of zero chemical potential. Furthermore, the analysis assumes the system to be in a state of near-local equilibrium, characterized by a spatially varying temperature distribution, T(x)𝑇𝑥T(x)italic_T ( italic_x ), and a local flow velocity, u(x)𝑢𝑥u(x)italic_u ( italic_x ).
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With the obtained values of the fit parameters in the Table 1, the thermodynamic variables are plotted as a function of the scaled temperature (solid line) in the Fig. (3) for the gluonic plasma as well as for the medium QGP along with the corresponding lattice findings. A decent match with the lattice data can be observed across the relevant temperature range.
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For the medium slightly out of equilibrium, the distribution function f(x,p)𝑓𝑥𝑝f(x,p)italic_f ( italic_x , italic_p ) could be written as
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Now, while formulating the bulk viscosity for gluons, the coefficients 𝒜1subscript𝒜1\mathcal{A}_{1}caligraphic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT(corresponding to ϵ1subscriptitalic-ϵ1\epsilon_{1}italic_ϵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT) and 𝒜2subscript𝒜2\mathcal{A}_{2}caligraphic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT (corresponding to ϵ2subscriptitalic-ϵ2\epsilon_{2}italic_ϵ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT) are needed to be determined. The uniqueness of these coefficients is given by 𝒜1parsuperscriptsubscript𝒜1𝑝𝑎𝑟\mathcal{A}_{1}^{par}caligraphic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_p italic_a italic_r end_POSTSUPERSCRIPT and 𝒜2parsuperscriptsubscript𝒜2𝑝𝑎𝑟\mathcal{A}_{2}^{par}caligraphic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_p italic_a italic_r end_POSTSUPERSCRIPT, where ``par"``𝑝𝑎𝑟"``par"` ` italic_p italic_a italic_r " signifies the particular solution of the coefficient, which are realized for as long as the Landau frame condition is satisfied (see Ref. [46]), the detail of which could also be found in the Appendix[A]. Therefore, solving the Boltzmann equation (Eq. (47)) with the only change in the form of the distribution function, we get
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It could also be realized that the deviation from the local equilibrium is related to the function ϕ(x,p)italic-ϕ𝑥𝑝\phi(x,p)italic_ϕ ( italic_x , italic_p ) as
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B
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}-2\,t\,a-c_{x}=0.italic_b + italic_v start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT = 0 , 2 italic_a + italic_b start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT = 0 , 2 italic_b italic_v start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT - italic_t italic_b - italic_c = 0 , 4 italic_a italic_v start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT - 2 italic_t italic_a - italic_c start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT = 0 .
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In terms of the matrix Y1subscript𝑌1Y_{1}italic_Y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT appearing in (3.16), we have
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In the first line we use (4.14), in the second one we use (3.13), and in the last one we use Proposition 3.9.
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We use the first three to determine a,b,c𝑎𝑏𝑐a,b,citalic_a , italic_b , italic_c in terms of v𝑣vitalic_v and its derivatives, as claimed in (4.5).
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In terms of (4.14) and cαsubscript𝑐𝛼c_{\alpha}italic_c start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT defined in (3.33), we have
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C
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For our quasicrystal, there are five reflection planes (ℛℛ{\cal{R}}caligraphic_R) dividing the system into two mirror-symmetric halves. They bisect the edges of the central decagon at right angles; see Fig. 1(a). The flux associated with the plaquettes intersecting ℛℛ{\cal{R}}caligraphic_R are fixed by property (i). For the remaining plaquettes, we divide the entire system into ten reflection-symmetric sectors. Within each sector involving Npsubscript𝑁𝑝N_{p}italic_N start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT plaquettes not intersected by the ℛℛ{\cal{R}}caligraphic_R, we can identify a total of 2Npsuperscript2subscript𝑁𝑝2^{N_{p}}2 start_POSTSUPERSCRIPT italic_N start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT end_POSTSUPERSCRIPT distinct flux sectors. We find the configuration with the minimum ground state energy, EGSsubscript𝐸GSE_{\textnormal{GS}}italic_E start_POSTSUBSCRIPT GS end_POSTSUBSCRIPT, after sampling over all distinct flux configurations, keeping reflection-symmetric {uij}subscript𝑢𝑖𝑗\{u_{ij}\}{ italic_u start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT } and constraining to the physical Hilbert space with ∑mnm=0subscript𝑚subscript𝑛𝑚0\sum_{m}n_{m}=0∑ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT = 0. We observe that EGSsubscript𝐸GSE_{\textnormal{GS}}italic_E start_POSTSUBSCRIPT GS end_POSTSUBSCRIPT is minimized when every plaquette has canonical flux (see definition above and Fig. 1 (a)); we dub this the canonical flux sector. By varying the anisotropy associated with the Kitaev exchange (i.e. Jz/J,Jx=Jy=Jsuperscript𝐽𝑧𝐽superscript𝐽𝑥superscript𝐽𝑦𝐽J^{z}/J,J^{x}=J^{y}=Jitalic_J start_POSTSUPERSCRIPT italic_z end_POSTSUPERSCRIPT / italic_J , italic_J start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT = italic_J start_POSTSUPERSCRIPT italic_y end_POSTSUPERSCRIPT = italic_J) we have confirmed that the canonical flux sector continues to remain the ground state; see Appendix C. We have performed the numerical flux samplings up to generation #4#4\#4# 4 (where the total number of sites=340absent340=340= 340) and confirmed that the above result continues to remain valid. For higher generations, the sampling procedure becomes increasingly computationally intense, and so we assume that the ground state will continue to conform to the canonical flux sector. We next turn to the nature of the excitations above the canonical ground state flux sector.
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The boundary states, on the other hand, are strongly localized along the edges leading to a nearly size-independent IPR. To quantitatively study the relation between the boundary states and the local imbalance, we introduce ΔN=∑regions|NA1−NA2|Δ𝑁subscriptregionssubscript𝑁subscript𝐴1subscript𝑁subscript𝐴2\Delta N=\sum_{\textnormal{regions}}|N_{A_{1}}-N_{A_{2}}|roman_Δ italic_N = ∑ start_POSTSUBSCRIPT regions end_POSTSUBSCRIPT | italic_N start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT - italic_N start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT |, where NA1,A2subscript𝑁subscript𝐴1subscript𝐴2N_{A_{1},A_{2}}italic_N start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT represents the number of sites belonging to each sublattice in each of the 10 regions partitioned by the reflection planes (Fig. 1(c)) Day-Roberts et al. (2020). In Fig. 3(d), we plot the ratio between the number of boundary states and ΔNΔ𝑁\Delta Nroman_Δ italic_N with increasing system size. For generations #4#4\#4# 4 (Nsites≤340)subscript𝑁sites340(N_{\textnormal{sites}}\leq 340)( italic_N start_POSTSUBSCRIPT sites end_POSTSUBSCRIPT ≤ 340 ) and higher, the ratio converges to a finite value ∼0.3similar-toabsent0.3\sim 0.3∼ 0.3 (red dashed line), demonstrating the close connection between the boundary states and the local imbalance. Note that both the localized states and the local imbalance are restricted to the region near the boundary. The zero-energy states are fragile against perturbations that do not preserve the local imbalance. For the analysis of E=0𝐸0E=0italic_E = 0 states, the 2−limit-from22-2 -coordinated sites are paired up by infinitesimal couplings to preserve the original boundary condition. Connecting these sites via a non-zero coupling eventually gets rid of these E=0𝐸0E=0italic_E = 0 states.
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Due to the imbalance (e.g. without any loss of generality, more A1subscript𝐴1A_{1}italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT sites compared to A2subscript𝐴2A_{2}italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT), the number of rows and columns in G𝐺Gitalic_G are different, leading to some eigenstates with zero eigenvalue. The corresponding E=0𝐸0E=0italic_E = 0 states are then localized on the majority sublattice (A1)subscript𝐴1(A_{1})( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ). A prominent example of this phenomenon arises on the Penrose quasicrystal Kohmoto and Sutherland (1986); Arai et al. (1988), which hosts E=0𝐸0E=0italic_E = 0 states throughout the bulk since the system can be partitioned into decoupled sectors by “membranes” that preclude any zero-energy states Flicker et al. (2020); Day-Roberts et al. (2020). In contrast, for our tri-coordinated quasicrystal, only the boundary exhibits a local imbalance leading to localization along the edge. We have also obtained boundary states with a small non-zero energy, manifest in the intermediate peaks over a finite range of energy 0<E<Δf0𝐸subscriptΔ𝑓0<E<\Delta_{f}0 < italic_E < roman_Δ start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT (Fig. 1(d)). These states remain distributed near the boundary, but are not strictly localized.
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Figure 3: Distribution of the fermionic wave function for (a) the lowest-E𝐸Eitalic_E bulk state, and (b) two representative boundary states associated with the marked arrows in Fig. 1(d), respectively. The blob size denotes the probability of finding the state at that site, and the color (red/blue) represents the sublattice. (c) Finite size scaling of the IPR. The bulk IPR is computed for the lowest-energy bulk state, while the boundary IPR is obtained by averaging over all boundary states. (d) The ratio between the number of boundary states and local imbalance, ΔNΔ𝑁\Delta Nroman_Δ italic_N, as a function of system size.
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The Hamiltonian defined on any finite system reveals an interesting excitation spectrum; see, for instance, Fig. 1(d) for the density of states (DOS) at the isotropic point. While a naive interpretation might lead us to conclude the existence of a gapless phase (with E=0𝐸0E=0italic_E = 0) over a broad range of parameters, a careful analysis reveals that the bulk and boundary states need to be disentangled first in order to obtain the actual phase diagram.
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D
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The number N𝑁Nitalic_N of single diodes for Diode A is about 300, and that for Diode B is 7, respectively.
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On the other hand, under the negative-illumination, the linear curve moves up due to the photocurrent generation.
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The current sources represent photocurrent generation, and the parallel resistances are shunt resistances of the p-n junctions.
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Similarly, the zero-bias resistance RDsubscript𝑅𝐷R_{D}italic_R start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT is the sum of the resistances in the series p-n junctions, such as RD=N×rDsubscript𝑅𝐷𝑁subscript𝑟𝐷R_{D}=N\times r_{D}italic_R start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT = italic_N × italic_r start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT.
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The photocurrent is given by short-circuit current ISCsubscript𝐼𝑆𝐶I_{SC}italic_I start_POSTSUBSCRIPT italic_S italic_C end_POSTSUBSCRIPT.
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B
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Most recently, the 15-year pulsar timing data collected by the North American Nanohertz Observatory for Gravitational Waves (NANOGrav) presented convincing evidence for a low-frequency Gravitational Wave Background (GWB) [11], potentially originating from PBHs [12]. Future space-based GW interferometers like LISA, BBO, and DECIGO [13, 14] are anticipated to detect similar background signals.
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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. Inflation explains critical aspects of the universe’s structure, such as its immense size, uniformity, isotropy, and overall geometry. 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.
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PBHs have been proposed as potential constituents of dark matter, accounting for anywhere from a fraction to the entirety of its abundance. One of the primary mechanisms for PBH formation in the early universe involves the amplification of the primordial curvature perturbation spectrum.
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When the perturbation associated with the PBH enters the horizon, the PBH’s mass is dictated by the horizon’s mass. The relationship between the scale of the perturbation and the mass of the PBH during formation is expressed as follows,
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The fractional abundance of the primordial black holes as a function of their mass is shown in Fig. 6. We have used the benchmark point parameters of table 3. For only the first two BPs, the PBHs can explain dark matter in totality. The shaded regions correspond to the various observational constraints [95, 96, 17].
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B
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