Cluster truncated Wigner approximation in strongly interacting systems

Abstract

We present a general method by which linear quantum Hamiltonian dynamics with exponentially many degrees of freedom is replaced by approximate classical nonlinear dynamics with the number of degrees of freedom (phase space dimensionality) scaling polynomially in the system size. This method is based on generalization of the truncated Wigner approximation (TWA) to a higher dimensional phase space, where phase space variables are associated with a complete set of quantum operators spanning finite size clusters. The method becomes asymptotically exact with the increasing cluster size. The crucial feature of TWA is fluctuating initial conditions, which we approximate by a Gaussian distribution. We show that such fluctuations dramatically increase accuracy of TWA over traditional cluster mean field approximations. In this way we can treat on equal footing quantum and thermal fluctuations as well as compute entanglement and various equal and non-equal time correlation functions. The main limitation of the method is exponential scaling of the phase space dimensionality with the cluster size, which can be significantly reduced by using the language of Schwinger bosons and can likely be further reduced by truncating the local Hilbert space variables. We demonstrate the power of this method analyzing dynamics in various spin chains with and without disorder and show that we can capture such phenomena as long time hydrodynamic relaxation, many-body localization and the ballistic spread of entanglement.

Self-consistent Hartree-Fock approach to many-body localization

“…The fact that the performance of meanfield approaches is sensitive to the fluctuations encoded in the initial state-as we see here-was recently pointed out in Ref. [66].…”

A generalized phase space approach for solving quantum spin dynamics

“…By comparing the dynamics obtained for increasing cluster sizes s c (the simulation would hypothetically become exact in the limit N=s c ) and different cluster choices, one could benchmark the convergence of the phase space method in regimes where no other methods are available for verification. For example, such an approach was recently suggested and analyzed in [42], however without utilizing initial discrete distributions.…”

“…This can be achieved by applying the same unitary transformations to the equations of motion (see appendix A). Note that, alternatively, one may also use a Gaussian approximation for the distribution of the initial l m i [42,43], i.e.for all  -1 generalized Bloch sphere variables. However, we point out that in contrast to the discrete distribution given by equation (15), the Gaussian sampling not only does not reproduce all initial intra-spin correlations correctly, but also can lead to worse longer-time predictions.…”

Emergent Hydrodynamics in Nonequilibrium Quantum Systems

“…Until now, such questions have remained largely unexplored owing to the fact that they sit in a region of phase space where neither theoretical techniques nor numerical methods easily apply. However, a number of recently proposed numerical methods [41][42][43][44][45] promise to bridge this gap and directly connect microscopic models to emergent macroscopic hydrodynamics. Here, we will focus on one such method-density matrix truncation (DMT) [41]-which modifies time-evolving block decimation (TEBD) by representing states as matrix product density operators (MPDOs) and prioritizing short-range (over long-range) correlations.…”