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arXiv:1805.05221 · Quantum Science and Technology

Quenches near criticality of the quantum Ising chain—power and limitations of the discrete truncated Wigner approximation

Stefanie Czischek, Martin Gärttner, Markus K. Oberthaler, Michael Kästner, Thomas Gasenzer

Published Sep 25, 2018• 18 citations
arXivHTMLPDF

Abstract

Abstract The semi-classical discrete truncated Wigner approximation has recently been proposed as a simulation method for spin-1/2 systems. While it appears to provide a powerful approach which shows promising results in higher dimensions and for systems with long-range interactions, its performance is still not well understood in general. Here we perform a systematic benchmarking on the one-dimensional transverse-field Ising model and point to limitations of the approximation arising after sudden quenches into the quantum critical regime. Our procedure allows to identify the limitations of the semi-classical simulations and with that to determine the regimes and questions where quantum simulators can provide information which is inaccessible to semi-classics.

DOI
10.1088/2058-9565/aae3f7

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Abstract The semi-classical discrete truncated Wigner approximation has recently been proposed as a simulation method for spin-1/2 systems. While it appears to provide a powerful approach which shows promising results in higher dimensions and for systems with long-range interactions, its performance is still not well understood in general. Here we perform a systematic benchmarking on the one-dimensional transverse-field Ising model and point to limitations of the approximation arising after sudden quenches into the quantum critical regime. Our procedure allows to identify the limitations of the semi-classical simulations and with that to determine the regimes and questions where quantum simulators can provide information which is inaccessible to semi-classics.
By means of the discrete truncated Wigner approximation, we study dynamical phase transitions arising in the steady state of transverse-field Ising models after a quantum quench. Starting from a fully polarized ferromagnetic initial condition, these transitions separate a phase with nonvanishing magnetization along the ordering direction from a disordered symmetric phase upon increasing the transverse field. We consider two paradigmatic cases, a one-dimensional long-range model with power-law interactions $\ensuremath{\propto}1/{r}^{\ensuremath{\alpha}}$ decaying algebraically as a function of distance rrr and a two-dimensional system with short-range nearest-neighbor interactions. In the former case, we identify dynamical phase transitions for $\ensuremath{\alpha}\ensuremath{\lesssim}2$ and we extract the critical exponents from a data collapse of the steady-state magnetization for up to 1200 lattice sites. We find identical exponents for $\ensuremath{\alpha}\ensuremath{\lesssim}0.5$, suggesting that the dynamical transitions in this regime fall into the same universality class as the nonergodic mean-field limit. The two-dimensional Ising model is believed to be thermalizing, which we also confirm using exact diagonalization for small system sizes. Thus, the dynamical transition is expected to correspond to the thermal phase transition, which is consistent with our data upon comparing to equilibrium quantum Monte Carlo simulations. We further test the accuracy of the discrete truncated Wigner approximation by comparing against numerically exact methods such as exact diagonalization, tensor network, as well as artificial neural network states and we find good quantitative agreement on the accessible time scales. Finally, our work provides an additional contribution to the understanding of the range and the limitations of qualitative and quantitative applicability of the discrete truncated Wigner approximation.
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.
Abstract We study the tractability of classically simulating critical phenomena in the quench dynamics of one-dimensional transverse field Ising models (TFIMs) using highly truncated matrix product states (MPSs). We focus on two paradigmatic examples: a dynamical quantum phase transition (DQPT) that occurs in nonintegrable long-range TFIMs, and the infinite-time correlation length of the integrable nearest-neighbor TFIM when quenched to the critical point, where the quantities of interest involve equal time one- and two- point correlation functions, which we associate with macroproperties. For the DQPT, we show that the order parameters can be efficiently simulated with heavy truncation of the MPS bond dimension. This can be used to reliably extract critical properties of the phase transition, including critical exponents, even when the full many-body state is not simulated with high fidelity. The long-time correlation length near the critical point is more sensitive to the full many-body state fidelity, and generally requires a large bond dimension MPS. Nonetheless, this can still be efficiently simulated with strongly truncated MPS because it can be extracted from the short-time behavior of the dynamics where entanglement is low. Our results provide illustrations of scenarios where accurate calculation of the full many-body state (microstate) is intractable due to the volume-law growth of entanglement, yet a precise specification of an exact microstate may not be required when simulating macroproperties that play a role in phases of matter of many-body systems. We also study the tractability of simulation using truncated MPS based on quantum chaos and equilibration in the models. We find a counterintuitive inverse relationship, whereby local expectation values are most easily approximated for chaotic systems whose exact many-body state is most intractable.
The near-critical unitary dynamics of quantum Ising spin chains in transversal and longitudinal magnetic fields is studied using an artificial neural network representation of the wave function. A focus is set on strong spatial correlations which build up in the system following a quench into the vicinity of the quantum critical point. We compare correlations obtained by optimizing the parameters of the network states with analytical solutions in integrable cases and time-dependent density matrix renormalization group (tDMRG) simulations, as well as with predictions from a semiclassical discrete truncated Wigner analysis. While the semiclassical approach yields quantitatively correct results only at very short times and near zero transverse fields, the neural-network representation is applicable in a much wider regime. However, for quenches close to the quantum critical point the representation becomes inefficient. For nonintegrable models we show that in regimes where tDMRG is limited to short times due to extensive entanglement growth, also the neural-network parametrization converges only at short times.
We develop a truncated Hamiltonian method to investigate the dynamics of the (1+1)(1+1)(1+1)-dimensional ${\ensuremath{\phi}}^{4}$ theory following quantum quenches. The results are compared to two different semiclassical approaches, the self-consistent Gaussian approximation and the truncated Wigner approximation, and used to determine the range of validity of these widely used approaches. We show that the self-consistent approximation is strongly limited in comparison to the truncated Hamiltonian method which for larger cutoffs is practically exact for the parameter range studied. We find that the self-consistent approximation is only valid when the effective mass is in the vicinity of the renormalized mass. Similarly to the self-consistent approximation, the truncated Wigner approximation (TWA) is not able to capture the correct mass renormalization, and breaks down for strong enough interactions where the bare mass becomes negative. We attribute the failure of TWA to the presence of a classical symmetry-broken fixed point. Aside from establishing the truncated Hamiltonian approach as a powerful tool for studying the dynamics of the ${\ensuremath{\phi}}^{4}$ model, our results on the limitation of semiclassical approximations are expected to be relevant for modeling the dynamics of other quantum field theories.
We present an approach to the numerical simulation of open quantum many-body systems based on the semiclassical framework of the discrete truncated Wigner approximation. We establish a quantum jump formalism to integrate the quantum master equation describing the dynamics of the system, which we find to be exact in both the noninteracting limit and the limit where the system is described by classical rate equations. We apply our method to simulation of the paradigmatic dissipative Ising model, where we are able to capture the critical fluctuations of the system beyond the level of mean-field theory.