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

Recently developed quantum Monte Carlo (QMC) method on asymptotic Lefschetz thimbles is a numerical algorithm capable of alleviating the sign problem mostly inevitable in the simulations of quantum many-body systems [1]. In this method, the sign problem is alleviated by shifting the integration domain for the auxiliary fields, appearing for example in the conventional determinant QMC method, from real space to an appropriate manifold in complex space. In this talk, we describe a way to extend this method to quantum spin models with generic two-spin interactions. In particular, we utilize the Hubbard-Stratonovich transformation to decouple the exchange interactions and the Popov-Fedotov transformation to map the quantum spins to complex fermions [2]. As a demonstration, we apply the method to the Kitaev model in a magnetic field whose ground state is predicted to deliver a quantum spin liquid [3]. Specifically, we visualize the asymptotic Lefschetz thimbles in complex space, as well as show that in the low-temperature region the sign of the action is recovered considerably and unbiased numerical results are obtained with sufficient precision [4].

[1] A. Alexandru et al., JHEP 05, 53 (2016).

[2] V. N. Popov and S. A. Fedotov, Sov. Phys. - JETP 67, 535 (1988); Proc. Steklov Inst. Math. 184, 177 (1991).

[3] A. Kitaev, Ann. Phys. 321, 2 (2006).

[4] P. A. Mishchenko et al., Phys. Rev. D 104, 074517 (2021).