Pfaffian quantum Monte Carlo: solution to Majorana sign ambiguity and applications

TL;DR

This paper introduces Pfaffian quantum Monte Carlo (PfQMC), a novel method that resolves sign ambiguity in Majorana-based QMC, enabling efficient simulations of interacting fermion systems and exploring Majorana edge modes.

Contribution

The paper derives a closed-form Pfaffian formula for the Boltzmann weight in Majorana QMC, overcoming sign ambiguity and broadening applicability to generic interacting fermion models.

Findings

Successfully applied PfQMC to study Majorana edge modes in Kitaev chains.

Resolved sign ambiguity in Majorana-based QMC with a polynomial-time Pfaffian formula.

Potential to improve sign mitigation and study doped Hubbard models.

Abstract

Determinant quantum Monte Carlo (DQMC), formulated in complex-fermion representation, has played a key role in studying strongly-correlated fermion systems. However, its applicability is limited due to the requirement of particle-number conservation after Hubbard-Stratonovich transformation. In going beyond the conventional DQMC, one encouraging development occurred when Majorana fermions were introduced for QMC [1,2]. But in previous Majorana-based QMC, Boltzmann weight is determined often with a sign ambiguity. Here we successfully resolved this ambiguity by deriving a closed-form Pfaffian formula for the weight, enabling efficient calculation of the weight with its sign in polynomial time. We call it ''Pfaffian quantum Monte Carlo'' (PfQMC), which can be applied to generic interacting fermion models. We have successfully employed PfQMC to explore how robust Majorana edge modes in Kitaev chain are against strong interactions. By offering greater flexibility, PfQMC can potentially enhance existing sign-mitigating and approximation methods and help address challenging issues such as the ground-state properties of the doped Hubbard model.