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

The recently developed Truncated Wigner Approximation for spins (TWA, [1], [2]) is a semiclassical method to describe interacting spin-1/2-systems including dephasing and decay. Instead of finding exact solutions in the exponentially growing Hilbert space, the method employs a mapping from the equation of motion of many-body density matrix to stochastic differential equations of classical variables in a continuous phase space. The method, which improves on a mean-field description by including leading order quantum corrections, was successfully employed to simulate the real-time dynamics of several models.

We here further develop the TWA method to an imaginary-time evolution (iTWA), i.e. for the simulation of finite-temperature states and ground states of interacting spin systems. Specifically in this talk, we will derive the iTWA for spin-1/2-systems, highlight emerging problems and discuss how to deal with them. In order to assess the ability of the TWA method to faithfully describe quantum phase transitions, we show results of the one and two-dimensional transverse-field Ising model simulated via iTWA. Futhermore, we look at simulations of a random and in general frustrated anti-ferromagnetic Ising Hamiltonian on 3-regular-graphs and that the iTWA can provide very good approximations to the ground state of these systems.

[1] C. Mink et al., PhysRevResearch.4.043136

[2] J. Hartmann, T. Schlegel, C. Mink, M. Fleischhauer, Truncated Wigner approximation for unitary and open many-body spin systems (in preparation)