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arXiv:2404.18594 · Physical Review A

Two-dimensional correlation propagation dynamics with a cluster discrete phase-space method

Kazuma Nagao, Seiji Yunoki

Published Feb 12, 2025• 4 citations
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Abstract

Nonequilibrium dynamics of highly controlled quantum systems is a challenging issue in statistical physics and quantum many-body physics, relevant to recent experimental developments of analog and digital quantum simulations. In this work, we develop a discrete phase-space approach for general $\mathrm{SU}(N$) spin systems that utilizes cluster mean-field equations, which capture nontrivial quantum correlations inside each cluster, beyond the capability of the standard discrete truncated Wigner approximation for individual classical spins. Our formalism, based on a cluster phase-point operator, enables efficient numerical samplings of cluster phase-space variables, where the total number of noise variables for a direct product state is independent of the specific way in which the entire system is divided into multiple equally sized finite clusters. We numerically demonstrate that the cluster discrete truncated Wigner approximation (C-dTWA) method can reproduce key results in a recent experiment on correlation propagation dynamics in a two-dimensional Bose-Hubbard system. We further compare the results of C-dTWA for clusters of $2\ifmmode\times\else\texttimes\fi{}2$ sites with those from a two-dimensional tensor network method and discuss that both approaches agree very well in the short-time region, where the energy conservation is well maintained in the tensor network simulations. Since we formulate the C-dTWA method in a general form, it has the potential for application to various dynamical problems in isolated and open quantum systems, even in higher dimensions.

DOI
10.1103/physreva.111.023315

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Nonequilibrium dynamics of highly controlled quantum systems is a challenging issue in statistical physics and quantum many-body physics, relevant to recent experimental developments of analog and digital quantum simulations. In this work, we develop a discrete phase-space approach for general SU(N\mathrm{SU}(NSU(N) spin systems that utilizes cluster mean-field equations, which capture nontrivial quantum correlations inside each cluster, beyond the capability of the standard discrete truncated Wigner approximation for individual classical spins. Our formalism, based on a cluster phase-point operator, enables efficient numerical samplings of cluster phase-space variables, where the total number of noise variables for a direct product state is independent of the specific way in which the entire system is divided into multiple equally sized finite clusters. We numerically demonstrate that the cluster discrete truncated Wigner approximation (C-dTWA) method can reproduce key results in a recent experiment on correlation propagation dynamics in a two-dimensional Bose-Hubbard system. We further compare the results of C-dTWA for clusters of 2\ifmmode×\else\texttimes\fi22\ifmmode\times\else\texttimes\fi{}22\ifmmode×\else\texttimes\fi2 sites with those from a two-dimensional tensor network method and discuss that both approaches agree very well in the short-time region, where the energy conservation is well maintained in the tensor network simulations. Since we formulate the C-dTWA method in a general form, it has the potential for application to various dynamical problems in isolated and open quantum systems, even in higher dimensions.
A phase-space method, suitable to study the many-body spin dynamics in generic spin lattice models is introduced, discussed, and tested. This technique, which we refer to as the dTWA, is based on the discrete sampling of the Wigner function instead of the continuous sampling in traditional Truncated Wigner Approximation (TWA) methods. The power of the method is demonstrated by comparisons with analytical and numerically exact calculations, which show a significant improvement compared to the TWA, e.g. by capturing revivals of spin-observables. The dTWA is demonstrated to not only correctly reproduce dynamics of one-particle observables but also two-point correlations and spin squeezing, a property relying on entanglement. This method thus opens the possibility to study the quantum dynamics in long-range spin models, accessible to recent state-of-the-art experiments, in regimes where other numerical techniques are inapplicable.
Interacting quantum spin models are remarkably useful for describing different types of physical, chemical, and biological systems.Significant understanding of their equilibrium properties has been achieved to date, especially for the case of spin models with short-range couplings.However, progress toward the development of a comparable understanding in long-range interacting models, in particular out-of-equilibrium, remains limited.In a recent work, we proposed a semiclassical numerical method to study spin models, the discrete truncated Wigner approximation (DTWA), and demonstrated its capability to correctly capture the dynamics of one-and two-point correlations in one-dimensional (1D) systems.Here we go one step forward and use the DTWA method to study the dynamics of correlations in two-dimensional (2D) systems with many spins and different types of long-range couplings, in regimes where other numerical methods are generally unreliable.We compute spatial and time-dependent correlations for spin-couplings that decay with distance as a power-law and determine the velocity at which correlations propagate through the system.Sharp changes in the behavior of those velocities are found as a function of the power-law decay exponent.Our predictions are relevant for a broad range of systems including solid state materials, atom-photon systems and ultracold gases of polar molecules, trapped ions, Rydberg, and magnetic atoms.We validate the DTWA predictions for small 2D systems and 1D systems, but ultimately, in the spirt of quantum simulation, experiments will be needed to confirm our predictions for large 2D systems.
Abstract Numerical techniques to efficiently model out-of-equilibrium dynamics in interacting quantum many-body systems are key for advancing our capability to harness and understand complex quantum matter. Here we propose a new numerical approach which we refer to as generalized discrete truncated Wigner approximation (GDTWA). It is based on a discrete semi-classical phase space sampling and allows to investigate quantum dynamics in lattice spin systems with arbitrary S ≥ 1/2. We show that the GDTWA can accurately simulate dynamics of large ensembles in arbitrary dimensions. We apply it for S > 1/2 spin-models with dipolar long-range interactions, a scenario arising in recent experiments with magnetic atoms. We show that the method can capture beyond mean-field effects, not only at short times, but it also can correctly reproduce long time quantum-thermalization dynamics. We benchmark the method with exact diagonalization in small systems, with perturbation theory for short times, and with analytical predictions made for models which feature quantum-thermalization at long times. We apply our method to study dynamics in large S > 1/2 spin-models and compute experimentally accessible observables such as Zeeman level populations, contrast of spin coherence, spin squeezing, and entanglement quantified by single-spin Renyi entropies. We reveal that large S systems can feature larger entanglement than corresponding S = 1/2 systems. Our analyses demonstrate that the GDTWA can be a powerful tool for modeling complex spin dynamics in regimes where other state-of-the art numerical methods fail.
An accurate description of the nonequilibrium dynamics of systems with coupled spin and bosonic degrees of freedom remains theoretically challenging, especially for large system sizes and in higher than one dimension. Phase-space methods such as the truncated Wigner approximation (TWA) have the advantage of being easily scalable and applicable to arbitrary dimensions. In this work we adapt the TWA to generic spin-boson models by making use of recently developed algorithms for discrete phase spaces [J. Schachenmayer, A. Pikovski, and A. M. Rey, Phys. Rev. X 5, 011022 (2015)]. Furthermore we go beyond the standard TWA approximation by applying a scheme based on the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy of equations to our coupled spin-boson model. This allows us, in principle, to study how systematically adding higher-order corrections improves the convergence of the method. To test various levels of approximation we study an exactly solvable spin-boson model, which is particularly relevant for trapped-ion arrays. Using TWA and its BBGKY extension we accurately reproduce the time evolution of a number of one- and two-point correlation functions in several dimensions and for an arbitrary number of bosonic modes.
We develop and utilize the SU(3) truncated Wigner approximation (TWA) in order to analyze far-from-equilibrium quantum dynamics of strongly interacting Bose gases in an optical lattice. Specifically, we explicitly represent the corresponding Bose-Hubbard model at an arbitrary filling factor with restricted local Hilbert spaces in terms of SU(3) matrices. Moreover, we introduce a discrete Wigner sampling technique for the SU(3) TWA and examine its performance as well as that of the SU(3) TWA with the Gaussian approximation for the continuous Wigner function. We directly compare outputs of these two approaches with exact computations regarding dynamics of the Bose-Hubbard model at unit filling with a small size and that of a fully connected spin-1 model with a large size. We show that both approaches can quantitatively capture quantum dynamics on a timescale of $\ensuremath{\hbar}/(Jz)$, where JJJ and zzz denote the hopping energy and the coordination number. We apply the two kinds of SU(3) TWA to dynamical spreading of a two-point correlation function of the Bose-Hubbard model on a square lattice with a large system size, which has been measured in recent experiments. Noticeable deviations between the theories and experiments indicate that proper inclusion of effects of the spatial inhomogeneity, which is not straightforward in our formulation of the SU(3) TWA, may be necessary.
Connecting short-time microscopic dynamics with long-time hydrodynamics in strongly correlated quantum systems is one of the outstanding questions. In particular, it is hard to determine various hydrodynamic coefficients such as the diffusion constant or viscosity starting from a microscopic model: exact quantum simulations are limited to either small system sizes or to short times, which are insufficient to reach asymptotic behavior and so various approximations must be applied. We show that these difficulties, at least for particular models, can be circumvented by using the cluster truncated Wigner approximation (CTWA), which maps quantum Hamiltonian dynamics into classical Hamiltonian dynamics in auxiliary high-dimensional phase space. We apply CTWA to XXZ next-nearest-neighbor spin-1/2 chains and XY spin ladders, and find behavior consisting of short-time spin relaxation which gradually crosses over to emergent diffusive behavior at long times. For a random initial state, we show that CTWA correctly reproduces the whole spin spectral function. Necessary in this construction is sampling from properly fluctuating initial conditions: the Dirac mean-field (variational) ansatz, which neglects such fluctuations, leads to incorrect predictions.