A quantum Monte Carlo method on asymptotic Lefschetz thimbles for quantum spin systems: An application to the Kitaev model in a magnetic field

TL;DR

This paper introduces a quantum Monte Carlo method on asymptotic Lefschetz thimbles to mitigate the sign problem in quantum spin systems, demonstrated on the Kitaev model in a magnetic field, enabling more accurate simulations of topological quantum states.

Contribution

The authors extend the asymptotic Lefschetz thimble Monte Carlo method to quantum spin models using Hubbard-Stratonovich and Popov-Fedotov transformations, improving sign problem mitigation.

Findings

Sign problem is significantly alleviated in the Kitaev model.

The method produces unbiased, precise results at low temperatures.

Visualization of thimbles and saddle points confirms the effectiveness.

Abstract

The quantum Monte Carlo method on asymptotic Lefschetz thimbles is a numerical algorithm devised specifically for alleviation of the sign problem appearing in the simulations of quantum many-body systems. In this method, the sign problem is alleviated by shifting the integration domain for the auxiliary fields, appearing for example in the conventional determinant quantum Monte Carlo method, from real space to an appropriate manifold in complex space. Here we extend this method to quantum spin models with generic two-spin interactions, by using the Hubbard-Stratonovich transformation to decouple the exchange interactions and the Popov-Fedotov transformation to map the quantum spins to complex fermions. As a demonstration, we apply the method to the Kitaev model in a magnetic field whose ground state is predicted to deliver a topological quantum spin liquid with non-Abelian anyonic excitations. To illustrate how the sign problem is alleviated in this method, we visualize the asymptotic Lefschetz thimbles in complex space, together with the saddle points and the zeros of the fermion determinant. We benchmark our method in the low-temperature region in a magnetic field and show that the sign of the action is recovered considerably and unbiased numerical results are obtained with sufficient precision.