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Cluster phase space and variational subspace approaches to the quantum many-body problem

Jonathan Wurtz

Published Jan 1, 2020• 1 citation
OpenAlex

Abstract

Simulating the nonequilibrium behavior of interacting quantum systems is an important way to understand results of experimental quantum simulators, motivate new materials, and refine new quantum algorithms. However, this is a challenging task due to the exponential difficulty of such systems, which motivates dimensional reduction methods, such as semiclassical limits. This work extends semiclassical phase space methods to spin systems with no clear classical limit with the cluster truncated Wigner approximation (cTWA), and improves on Schrieffer-Wolff low energy effective dynamics with variational adiabatic generators. The cTWA was used to compute nonequilibrium dynamics in spin chains, finding behavior such as signatures of many body localization; rapid thermalization and preservation of fluctuations; effective thermodynamic classical behaviors; and signatures of quantum chaos and butterfly velocities, in 1d spin 1/2 chains. Variational Schrieffer-Wolff methods were used to find efficient non-perturbative dressings for the Hubbard model and find effective quasiparticle dynamics and nonthermal states in quantum chaotic spin chains. These methods are potentially effective tools to separate essential quantum behavior from classical behavior, and can be used to diagnose quantum thermalization behavior in interacting quantum systems.

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Simulating the nonequilibrium behavior of interacting quantum systems is an important way to understand results of experimental quantum simulators, motivate new materials, and refine new quantum algorithms. However, this is a challenging task due to the exponential difficulty of such systems, which motivates dimensional reduction methods, such as semiclassical limits. This work extends semiclassical phase space methods to spin systems with no clear classical limit with the cluster truncated Wigner approximation (cTWA), and improves on Schrieffer-Wolff low energy effective dynamics with variational adiabatic generators. The cTWA was used to compute nonequilibrium dynamics in spin chains, finding behavior such as signatures of many body localization; rapid thermalization and preservation of fluctuations; effective thermodynamic classical behaviors; and signatures of quantum chaos and butterfly velocities, in 1d spin 1/2 chains. Variational Schrieffer-Wolff methods were used to find efficient non-perturbative dressings for the Hubbard model and find effective quasiparticle dynamics and nonthermal states in quantum chaotic spin chains. These methods are potentially effective tools to separate essential quantum behavior from classical behavior, and can be used to diagnose quantum thermalization behavior in interacting quantum systems.
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.
The description of nonequilibrium dynamics in interacting quantum systems is a long-standing challenge within condensed matter physics. In this theoretical work, the authors introduce a method to block-diagonalize low-energy sectors of strongly interacting Hamiltonians by variationally computing generators of rotations, equivalent to a nonperturbative dressing of quasiparticle excitations. This procedure allows for the calculation of effective dynamics including quenches and response functions, as is demonstrated in a Fermi-Hubbard model and an integrability-broken XY model, both of which demonstrate excellent convergence.
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 $2\ifmmode\times\else\texttimes\fi{}2$ 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.
The discrete truncated Wigner approximation (DTWA) is a semiclassical phase-space method useful for the exploration of many-body quantum dynamics. In this work we investigate many-body localization (MBL) and thermalization using DTWA and compare its performance to exact numerical solutions. By taking as a benchmark case a one-dimensional random field Heisenberg spin chain with short-range interactions, and by comparing to numerically exact techniques, we show that DTWA is able to reproduce dynamical signatures that characterize both the thermal and the MBL phases. It exhibits the best quantitative agreement at short times deep in each of the phases and larger mismatches close to the phase transition. The DTWA captures the logarithmic growth of entanglement in the MBL phase, even though a pure classical mean-field analysis would lead to no dynamics at all. Our results suggest the DTWA can become a useful method to investigate MBL and thermalization in experimentally relevant settings intractable with exact numerical techniques, such as systems with long-range interactions and/or systems in higher dimensions.
Many-body quantum-optical systems, where a collection of emitters interacts through a common electromagnetic reservoir, exhibit rich out-of-equilibrium behavior and hold promise for applications in quantum technologies. However, exact numerical simulations of their dynamics quickly become unfeasible due to the exponential growth of the Hilbert space with system size. Semiclassical, phase-space approaches -- such as the Truncated Wigner approximation (TWA) -- provide computationally efficient alternatives by capturing leading-order quantum fluctuations. In this paper, we present a comprehensive overview of how to tackle problems in many-body quantum optics using phase-space methods. We derive the exact partial differential equation governing many-body dissipative evolution in any phase-space representation and discuss the approximations that yield the dissipative TWA proposed by Mink and Fleischhauer [SciPost Phys. 15, 233 (2023)]. We find that PPP and QQQ distributions are generally suboptimal for many-body quantum optics. Additionally, we extend the formalism to calculate multi-time correlation functions, thereby broadening the scope of phase-space simulations of open spin systems to include coherence and spectral properties, as well as directional correlations of collectively radiating emitters. These developments provide valuable tools for investigating exotic light sources driven by collective dissipation, driven-dissipative phase transitions, and a wealth of many-body phenomena arising in state-of-the-art experimental platforms.
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.