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Driven by the significant advantages offered by quantum computing, research in quantum machine learning has increased in recent years. While quantum speed-up has been demonstrated in some applications of quantum machine learning, a comprehensive understanding of its underlying mechanisms for improved performance remains elusive. Our study fills this gap by examining the expressibility of quantum circuits integrated within a convolutional neural network (CNN). Through numerical training on the MNIST dataset, our hybrid quantum-classical CNN model exhibited superior feature selection capabilities and significantly reduced the required training steps compared to the classical CNN. To understand the root of this enhanced performance, we conducted an analytical investigation of the functional expressibility of quantum circuits and derived a quantum activation function. We demonstrated that this quantum activation is more efficient in selecting important features and discarding unimportant information of input images. These findings not only deepen our comprehension of quantum-enhanced machine-learning models but also advance the classical machine-learning technique by introducing the quantum-inspired activation function., Comment: 12 pages, 6 figures

Analog quantum simulation based on ultracold atoms in optical lattices has catalyzed significant breakthroughs in the study of quantum many-body systems. These simulations rely on the statistical sampling of electronic Fock states, which are not easily accessible in classical algorithms. In this work, we modify the determinantal quantum Monte Carlo by integrating a Fock-state update mechanism alongside the auxiliary field. This method enables efficient sampling of Fock-state configurations. The Fock-state restrictive sampling scheme further enables the pre-selection of multiple ensembles at no additional computational cost, thereby broadening the scope of simulation to more general systems and models. Employing this method, we analyze static correlations of the Hubbard model up to the fourth order and achieve quantitative agreement with cold-atom experiments. The simulations of dynamical spectroscopies of the Hubbard and Kondo-lattice models further demonstrate the reliability and advantage of this method., Comment: 11 pages, 6 figures

Recent resonant x-ray scattering (RXS) experiments revealed a novel charge order in highly overdoped La$_{2-x}$Sr$_x$CuO$_4$ (LSCO). The observed charge order appears around the $(\pi/3,0)$ wavevector and remains robust from cryogenic temperatures to room temperature. To investigate the origin of this charge order in the overdoped region, we use determinant quantum Monte Carlo (DQMC) simulations to examine models with various interactions. We demonstrate that this CDW originates from remnant correlations in overdoped cuprates. The doping-independent wavevector $(\pi/3,0)$ further reflects the presence of nonlocal electron-phonon couplings. Our study reveals the importance of phonons in the cuprates, which assist correlated electrons in the formation of exotic phases., Comment: 7 pages, 4 figures

Motivated by the potential to suppress the antiferromagnetic long-range order in favor of the long-sought-after Kitaev quantum spin liquid state, we study the effect of spin vacancies in the extended Kitaev-Heisenberg model. In particular, we focus on a realistic model obtained from fitting inelastic neutron scattering on $\alpha$-RuCl$_3$. We observe that the long-range zigzag magnetic ordered state only survives when the doping concentration is smaller than 5\%. Upon further increasing the spin vacancy concentration, the ground state becomes a short-range ordered state at low temperatures. Compared with experiments, our classical solution over-stabilizes the zigzag correlation in the presence of spin vacancies. Our theoretical results provide guidance toward interpreting inelastic neutron scattering experiments on magnetically diluted Kitaev candidate materials., Comment: 9 figures

The Hubbard model on an anisotropic triangular lattice in two dimensions, a fundamental model for frustrated electron physics, displays a wide variety of phases and phase transitions. This work investigates the model using the ladder dual fermion approximation which captures local correlations non-perturbatively but approximates non-local correlations. We find metallic, one-dimensional antiferromagnetic, non-collinear antiferromagnetic, square-lattice antiferromagnetic, and spiral phases but no evidence of collinear antiferromagnetic order in different parts of the phase diagram. Analyzing the spin susceptibility in detail, we see both regions of agreement and of discrepancy with previous work. The case of Cs$_2$CuCl$_4$ is discussed in detail., Comment: 13 pages, 6 figures

We propose a new Rashba-free mechanism to realize topological superconductivity with electron-phonon interaction. In the presence of a magnetic field, electron-phonon interaction with small momentum transfer is found to favor spin-triplet Cooper pairing. This process facilitates the formation of chiral topological superconductivity even when Rashba spin-orbital coupling is absent. As a proof of concept, we propose an experimentally feasible heterostructure to systematically study the entangled relationship among forward-phonon scatterings, Rashba spin-orbital couplings, pairing symmetries, and superconducting topology. Our theory sheds light on the important role of electron-phonon coupled materials in the pursuit of non-Abelian Majorana quasiparticles., Comment: comments are welcome. 6 pages, 4 figures. The supplemental material is available upon reasonable request

In crystalline materials, electron-phonon coupling (EPC) is a ubiquitous many-body interaction that drives conventional Bardeen-Cooper-Schrieffer superconductivity. Recently, in a new kagome metal $\rm{CsV_3Sb_5}$, superconductivity that possibly intertwines with time-reversal and spatial symmetry-breaking orders is observed. Density functional theory calculations predicted weak EPC strength,$\lambda$, supporting an unconventional pairing mechanism in $\rm{CsV_3Sb_5}$. However, experimental determination of $\lambda$ is still missing, hindering a microscopic understanding of the intertwined ground state of $\rm{CsV_3Sb_5}$. Here, using 7-eV laser-based angle-resolved photoemission spectroscopy and Eliashberg function analysis, we determine an intermediate $\lambda$=0.45~0.6 at T=6 K for both Sb 5p and V 3d electronic bands, which can support a conventional superconducting transition temperature on the same magnitude of experimental value in $\rm{CsV_3Sb_5}$. Remarkably, the EPC on the V 3d-band enhances to $\lambda$~0.75 as the superconducting transition temperature elevated to 4.4 K in $\rm{Cs(V_{0.93}Nb_{0.07})_3Sb_5}$. Our results provide an important clue to understand the pairing mechanism in the Kagome superconductor $\rm{CsV_3Sb_5}$., Comment: To appear in Nature Communications

We study a multi-orbital Hubbard--Su-Schrieffer-Heeger model for the one-dimensional (1D) corner-shared cuprates in the adiabatic and nonadiabatic limits using the exact diagonalization and determinant quantum Monte Carlo methods. Our results demonstrate that lattice dimerization can be achieved only over a narrow range of couplings slightly below a critical coupling $g_c$ at half-filling. Beyond this critical coupling, the sign of the effective hopping changes, and the lattice becomes unstable. We also examine the model's temperature-dependent uniform magnetic susceptibility and the dynamical magnetic susceptibility and compare them to the results of an effective spin-$1/2$ Heisenberg model. In doing so, we numerically demonstrate that the lattice fluctuation induced by the $e$-ph interaction suppresses the effective superexchange interaction. Our results elucidate the effect of bond-stretching phonons in the parent cuprate compounds in general and are particularly relevant to 1D cuprates, where strong $e$-ph interactions have recently been inferred., Comment: 6 pages, 4 figures

We investigate the thermal Hall effect in a Kitaev-Heisenberg system, a model for Kitaev candidate $\alpha$-RuCl$_3$, in the presence of the coupling between spin and phonon arising from chlorine atoms' vibration. We observe that the coupling modifies the relative stability between different magnetic states under a magnetic field, especially stabilizing a canted zigzag antiferromagnetic state. Remarkably, the spin-phonon interaction has distinct effects on the thermal Hall conductivity in different magnetically ordered states. For the canted zigzag state, which is relevant to $\alpha\text{-RuCl}_3$, the spin-phonon interaction enhances the energy gap that is induced by a magnetic field and suppresses the thermal Hall conductivity at low temperatures. Importantly, we find that the spin-phonon interaction destabilizes the quantized thermal Hall effect in the spin-liquid state. Our results demonstrate a crucial role of phonon degrees of freedom to the thermal Hall effect in Kitaev materials., Comment: 6 pages, 4 figures

The collective spin and charge excitations of doped cuprates and their relationship to superconductivity are not yet fully understood, particularly in the case of the charge excitations. Here, we study the doping-dependent dynamical spin and charge structure factors of single and multi-orbital models for the one-dimensional corner shared spin-chain cuprates using several numerically exact methods. We find that the singleband Hubbard model can describe the spin and charge excitations of the $pd$-model in the low-energy region, including the particle-hole asymmetry in the spin response. However, our results also reveal that the weight of the interorbital spin excitations between Cu and O orbitals is comparable to the weight of the spin excitations between two Cu orbitals. This finding elucidates the microscopic nature of the spin excitations in the 1D cuprates and sheds light on the spin properties of other oxides. Importantly, we find a particle-hole asymmetry in the orbital-resolved charge excitations, which cannot be described by the singleband Hubbard model and is relevant to resonant inelastic x-ray scattering experiments. Our results imply that the explicit inclusion of the oxygen degrees of freedom may be required to understand experimental observations., Comment: 10 figures

Due to the strong coupling between magnetism and ferroelectricity, $(\mathrm{ND}_4)_2\mathrm{FeCl}_5\cdot\mathrm{D}_2\mathrm{O}$ exhibits several intriguing magnetic and electric phases. In this letter, we include high-order onsite spin anisotropic interactions in a spin model that successfully captures the ferroelectric phase transitions of $(\mathrm{ND}_4)_2\mathrm{FeCl}_5\cdot\mathrm{D}_2\mathrm{O}$ under a magnetic field and produces the large weights of high-order harmonic components in the cycloid structure that are observed from neutron diffraction experiments. Moreover, we predict a new ferroelectric phase sandwiched between the FE II and FE III phases in a magnetic field. Our results emphasize the importance of the high-order spin anisotropic interactions and provide a guideline to understand multiferroic materials with rich phase diagrams., Comment: 6 pages, 5 figures

Motivated by a recent inelastic neutron scattering experiment on $\mathrm{YbMgGaO}_4$ \cite{William2019}, we reinvestigate the homogeneous spin model on the triangular lattice. Using the cluster mean-field theory, we study the phase diagram and the magnetic-field-induced phase transition. We find that the phase boundary between the stripe state and the $120^{\circ}$ antiferromagnetic state is broadened by the magnetic field, leading to a field-induced phase transition. This phase transition is suppressed by the next-nearest neighbor exchange interaction $J_2/J_1$ and vanishes as $J_2/J_1>0.13$. We find a parameter space at $J_2/J_1=0.1$, in which the field-induce transition can be achieved and the deviation of theoretical spin excitation energies from experimental data is only $5.4\%$. Our results imply that an effective homogeneous spin model still works in $\mathrm{YbMgGaO}_4$., Comment: 7 pages, 7 figures. Accepted by Physics Review B

We study magnetic and charge susceptibilities in the half-filled two-dimensional triangular Hubbard model within the dual fermion approximation in the metallic, Mott insulating, and crossover regions of parameter space. In the \textcolor{black}{insulating state}, we find strong spin fluctuations at the K point at low energy corresponding to the \textcolor{black}{120$^{\circ}$} antiferromagnetic order. These spin fluctuations persist into the metallic phase and move to higher energy. We also present data for simulated neutron spectroscopy and \textcolor{black}{spin-lattice} relaxation times, and perform direct comparisons to inelastic neutron spectroscopy experiments on the triangular material Ba$_8$CoNb$_6$O$_{24}$ and to the relaxation times on $\kappa$-(ET)$_2$Cu$_2$(CN)$_3$. Finally, we present charge susceptibilities in different areas of parameter space, which should correspond to momentum-resolved electron-loss spectroscopy measurements on triangular compounds., Comment: 6 pages, 5 figures

Monte Carlo (MC) simulations are essential computational approaches with widespread use throughout all areas of science. We present a method for accelerating lattice MC simulations using fully connected and convolutional artificial neural networks that are trained to perform local and global moves in configuration space, respectively. Both networks take local spacetime MC configurations as input features and can, therefore, be trained using samples generated by conventional MC runs on smaller lattices before being utilized for simulations on larger systems. This new approach is benchmarked for the case of determinant quantum Monte Carlo (DQMC) studies of the two-dimensional Holstein model. We find that both artificial neural networks are capable of learning an unspecified effective model that accurately reproduces the MC configuration weights of the original Hamiltonian and achieve an order of magnitude speedup over the conventional DQMC algorithm. Our approach is broadly applicable to many classical and quantum lattice MC algorithms., Comment: 6 pages, 3 figures

We compute dynamic spin susceptibilities in the two-dimensional Hubbard model using the method of Dual Fermions and provide comparison to lattice Monte Carlo and cluster dynamical mean field theory. We examine the energy dispersion identified by peaks in ${\rm Im}\chi(\omega,q)$ which define spin modes and compare the exchange scale and magnon dispersion to neutron experiments on the parent La$_2$CuO$_4$ cuprate. We present the evolution of the spin excitations as a function of Hubbard interaction strengths and doping and explore the particle-hole asymmetry of the spin excitations. We also study the correlation lengths and the spin excitation dispersion peak structure and find a `Y'-shaped dispersion similar to neutron results on doped HgBa$_2$CuO$_{4+\delta}$.

We study a three-orbital Su-Schrieffer-Heeger model defined on a two-dimensional Lieb lattice and in the negative charge transfer regime using determinant quantum Monte Carlo. At half-filling (1 hole/unit cell), we observe a bipolaron insulating phase, where the ligand oxygen atoms collapse and expand about alternating cation atoms to produce a bond-disproportionated state. This phase is robust against moderate hole doping but is eventually suppressed at large hole concentrations, leading to a metallic polaron-liquid-like state with fluctuating patches of local distortions. Our results suggest that the polarons are highly disordered in the metallic state and freeze into a periodic array across the metal-to-insulator transition. We also find an $s$-wave superconducting state at finite doping that primarily appears on the oxygen sublattices. Our approach provides an efficient, non-perturbative way to treat bond phonons in higher dimensions and our results have implications for many materials where coupling to bond phonons is the dominant interaction., Comment: 4 figures, 1 supplementary

We study the interplay between the electron-phonon (e-ph) and on-site electron-electron (e-e) interactions in a three-orbital Hubbard-Holstein model on an extended one-dimensional lattice using determinant quantum Monte Carlo. For weak e-e and e-ph interactions, we observe a competition between an orbital-selective Mott phase (OSMP) and a (multicomponent) charge-density-wave (CDW) insulating phase, with an intermediate metallic phase located between them. For large e-e and e-ph couplings, the OSMP and CDW phases persist, while the metallic phase develops short-range orbital correlations and becomes insulating when both the e-e and e-ph interactions are large but comparable. Many of our conclusions are in line with those drawn from a prior dynamical mean field theory study of the two-orbital Hubbard-Holstein model [Phys. Rev. B 95, 12112(R) (2017)] in infinite dimension, suggesting that the competition between the e-ph and e-e interactions in multiorbital Hubbard-Holstein models leads to rich physics, regardless of the dimension of the system., Comment: 12 pages and 11 figures

We study the interplay between the electron-electron (e-e) and the electron-phonon (e-ph) interactions in the two-orbital Hubbard-Holstein model at half filling using the dynamical mean field theory. We find that the e-ph interaction, even at weak couplings, strongly modifies the phase diagram of this model and introduces an orbital-selective Peierls insulating phase (OSPI) that is analogous to the widely studied orbital-selective Mott phase (OSMP). At small e-e and e-ph coupling, we find a competition between the OSMP and the OSPI, while at large couplings, a competition occurs between Mott and charge-density-wave (CDW) insulating phases. We further demonstrate that the Hund's coupling influences the OSPI transition by lowering the energy associated with the CDW. Our results explicitly show that one must be cautious when neglecting the e-ph interaction in multiorbital systems, where multiple electronic interactions create states that are readily influenced by perturbing interactions., Comment: 5 pages, 4 figures

We study non-local correlations in a three-orbital Hubbard model defined on an extended one-dimensional chain using determinant quantum Monte Carlo and density matrix renormalization group methods. We focus on a parameter with robust Hund's coupling, which produces an orbital selective Mott phase (OSMP) at intermediate values of the Hubbard U, as well as an orbitally ordered ferromagnetic insulating state at stronger coupling. An examination of the orbital and spin-correlation functions indicates that the orbital ordering occurs before the onset of magnetic correlations in this parameter regime as a function of temperature. In the OSMP, we find that the self-energy for the itinerant electrons is momentum dependent, indicating a degree of non-local correlations while the localized electrons have largely momentum independent self-energies. These non-local correlations also produce relative shifts of the hole-like and electron-like bands within our model. The overall momentum dependence of these quantities is strongly suppressed in the orbitally-ordered insulating phase., Comment: we have 12 pages and 9 figures

A recently introduced one-dimensional three-orbital Hubbard model displays orbital-selective Mott phases with exotic spin arrangements such as spin block states [J. Rinc\'on {\em et al.}, Phys. Rev. Lett. \textbf{112}, 106405 (2014)]. In this publication we show that the Constrained-Path Quantum Monte Carlo (CPQMC) technique can accurately reproduce the phase diagram of this multiorbital one-dimensional model, paving the way to future CPQMC studies in systems with more challenging geometries, such as ladders and planes. The success of this approach relies on using the Hartree-Fock technique to prepare the trial states needed in CPQMC. We also study a simplified version of the model where the pair-hopping term is neglected and the Hund coupling is restricted to its Ising component. The corresponding phase diagrams are shown to be only mildly affected by the absence of these technically difficult-to-implement terms. This is confirmed by additional Density Matrix Renormalization Group and Determinant Quantum Monte Carlo calculations carried out for the same simplified model, with the latter displaying only mild Fermion sign problems. We conclude that these methods are able to capture quantitatively the rich physics of the several orbital-selective Mott phases (OSMP) displayed by this model, thus enabling computational studies of the OSMP regime in higher dimensions, beyond static or dynamic mean field approximations., Comment: 13 pages, 14 figures

We use determinant quantum Monte Carlo to study the single particle properties of quasiparticles and phonons in a variant of the two-dimensional Holstein model that includes an additional non-linear electron-phonon (e-ph) interaction. We find that a small positive non-linear interaction reduces the effective coupling between the electrons and the lattice, suppresses charge-density wave (CDW) correlations, and hardens the effective phonon frequency. Conversely, a small negative non-linear interaction can enhance the e-ph coupling resulting in heavier quasiparticles, an increased tendency towards a CDW phase at all fillings, and a softened phonon frequency. An effective linear model with a renormalized interaction strength and phonon frequency can qualitatively capture this physics; however, the quantitative effects of the non-linearity on both the electronic and phononic degrees of freedom cannot be captured by such a model. These results are significant for typical non-linear coupling strengths found in real materials, indicating that non-linearity can have a significant influence on the physics of many e-ph coupled systems., Comment: We have 12 pages and 13 figures

Determinant quantum Monte Carlo (DQMC) simulations are used to study non-linear electron-phonon interactions in a two-dimensional Holstein-like model on a square lattice. We examine the impact of non-linear electron-lattice interactions on superconductivity and on Peierls charge-density-wave (CDW) correlations at finite temperatures and carrier concentrations. We find that the CDW correlations are dramatically suppressed with the inclusion of even a small non-linear interaction. Conversely, the effect of the non-linearity on superconductivity is found to be less dramatic at high temperatures; however, we find evidence that the non-linearity is ultimately detrimental to superconductivity. These effects are attributed to the combined hardening of the phonon frequency and a renormalization of the effective linear electron-phonon coupling towards weaker values. These results demonstrate the importance of non-linear interactions at finite carrier concentrations when one is addressing CDW and superconducting order and have implications for experiments that drive the lattice far from equilibrium., Comment: 5 Pages, 5 Figures