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arXiv:2011.07924 · Physical review. B./Physical review. B

Time-evolving Weiss fields in the stochastic approach to quantum spins

Samuel E. Begg, Andrew G. Green, M. J. Bhaseen

Published Jul 6, 2021• 4 citations
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Abstract

We investigate nonequilibrium quantum spin systems via an exact mapping to stochastic differential equations. This description is invariant under a shift in the mean of the Gaussian noise. We show that one can extend the simulation time for real-time dynamics in one and two dimensions by a judicious choice of this shift. This can be updated dynamically in order to reduce the impact of stochastic fluctuations. We discuss the connection to drift gauges in the gauge-$P$ literature.

DOI
10.1103/physrevb.104.024408

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We investigate nonequilibrium quantum spin systems via an exact mapping to stochastic differential equations. This description is invariant under a shift in the mean of the Gaussian noise. We show that one can extend the simulation time for real-time dynamics in one and two dimensions by a judicious choice of this shift. This can be updated dynamically in order to reduce the impact of stochastic fluctuations. We discuss the connection to drift gauges in the gauge-PPP literature.
We explore the dynamics of quantum spin systems in two and three dimensions using an exact mapping to classical stochastic processes. In recent work, we explored the effectiveness of sampling around the mean-field evolution as determined by a stochastically averaged Weiss field. Here, we show that this approach can be significantly extended by sampling around the instantaneous Weiss field associated with each stochastic trajectory taken separately. This trajectory-resolved approach incorporates sample to sample fluctuations and allows for longer simulation times. We demonstrate the utility of this approach for quenches in the two-dimensional and three-dimensional quantum Ising model. We show that the method is particularly advantageous in situations where the average Weiss field vanishes, but the trajectory-resolved Weiss fields are nonzero. We discuss the connection to the gauge-P phase-space approach, where the trajectory-resolved Weiss field can be interpreted as a gauge degree of freedom.
We explore the dynamics of quantum spin systems in two and three dimensions using an exact mapping to classical stochastic processes. In recent work we explored the effectiveness of sampling around the mean field evolution as determined by a stochastically averaged Weiss field. Here, we show that this approach can be significantly extended by sampling around the instantaneous Weiss field associated with each stochastic trajectory taken separately. This trajectory-resolved approach incorporates sample to sample fluctuations and allows for longer simulation times. We demonstrate the utility of this approach for quenches in the two-dimensional and three-dimensional quantum Ising model. We show that the method is particularly advantageous in situations where the average Weiss-field vanishes, but the trajectory-resolved Weiss fields are non-zero. We discuss the connection to the gauge-P phase space approach, where the trajectory-resolved Weiss field can be interpreted as a gauge degree of freedom.
We use the Gaussian Phase-Space Representation to solve the real-time dynamic of interacting fermions in 1D, 2D, and 3D systems. The method is exact up to a spiking point, which represents a limit on the practical simulation time. The spiking can be delayed, and the practical simulation time extended, by adjusting the gauges of the representation, resulting in different equivalent stochastic differential equations. Here, we work on the so-called diffusion gauge and propose an algorithm to find efficiently new implementations of the noise terms. Compared with our initial results [F. Rousse \textit{et al.} 2024, J. Phys. A: Math. Theor. \textbf{57}, 015303], the new method achieves a significantly longer practical simulation time and can be applied to significantly larger systems.
Abstract We investigate the non-equilibrium dynamics of isolated quantum spin systems via an exact mapping to classical stochastic differential equations. We show that one can address significantly larger system sizes than recently obtained, including two-dimensional systems with up to 49 spins. We demonstrate that the results for physical observables are in excellent agreement with exact results and alternative numerical techniques where available. We further develop a hybrid stochastic approach involving matrix product states. In the presence of finite numerical sampling, we show that the non-Hermitian character of the stochastic representation leads to the growth of the norm of the time-evolving quantum state and to departures for physical observables at late times. We demonstrate approaches that correct for this and discuss the prospects for further development.
We reformulate the full quantum dynamics of spin systems using a phase-space representation based on SU(2) coherent states which generates an exact mapping of the dynamics of any spin system onto a set of stochastic differential equations. This representation is superior in practice to an earlier phase-space approach based on Schwinger bosons, with the numerical effort scaling only linearly with system size. By also implementing extrapolation techniques from quasiclassical equations to the full quantum limit, we are able to extend useful simulation times severalfold. This approach is applicable in any dimension including cases where frustration is present in the spin system. The method is demonstrated by simulating quenches in the transverse-field Ising model in one and two dimensions.
We discuss a scheme for simulating the real-time quantum quench dynamics of interacting quantum spin systems within the positive-P formalism. As model systems we study the transverse field Ising model as well as the Heisenberg model undergoing a quench away from the classical ferromagnetic ordered state and the antiferromagnetic Néel state, depending on the sign of the Heisenberg exchange interaction. The connection to the positive-P formalism as it is used in quantum optics is established by mapping the spin operators on to Schwinger bosons. In doing so, the dynamics of the interacting quantum spin system is mapped onto a set of Ito stochastic differential equations the number of which scales linearly with the number of spins, N , compared to an exact solution through diagonalization that, in the case of the Heisenberg model, would require matrices exponentially large in N . This mapping is exact and can be extended to higher dimensional interacting systems as well as to systems with an explicit coupling to the environment.