Imaginary-time evolution of interacting spin systems in the truncated Wigner approximation

TL;DR

This paper introduces an imaginary-time extension of the truncated Wigner approximation for spins (iTWA), enabling efficient semiclassical simulations of large interacting spin systems' ground states and thermal properties.

Contribution

The authors develop and validate the iTWA method, extending TWA to imaginary time for improved ground state and thermal state calculations in complex spin systems.

Findings

iTWA accurately approximates ground states of frustrated Ising models.

The method captures quantum phase transition behavior in 1D and 2D models.

Results show good agreement with exact solutions and quantum-classical correspondence.

Abstract

We present a semiclassical phase-space method to calculate thermal and ground states of large interacting spin systems. To this end, we extend the recently developed truncated Wigner approximation for spins (TWA) to the imaginary time, termed iTWA. The evolution of the canonical density matrix in imaginary time is mapped to a partial differential equation of its Wigner function. Truncation at the Fokker-Planck level leads to a set of stochastic differential equations, which can be efficiently simulated. We show that the iTWA can provide very good approximations to the ground state of a random and in general frustrated anti-ferromagnetic Ising Hamiltonian on a 3-regular graph, for which finding the exact ground state and approximations to it beyond a certain accuracy is NP hard. Furthermore in order to assess the ability of the method to properly account for leading-order quantum effects, we analyze the ground-state quantum phase transition of the nearest-neighbor, transverse-field Ising model in one and two spatial dimensions, finding very good agreement with the exact behaviour. The critical behavior obtained in iTWA follows the quantum-classical correspondence.