Split orthogonal group: A guiding principle for sign-problem-free fermionic simulations

TL;DR

This paper introduces a mathematical principle based on split orthogonal groups that guides the design of fermionic Hamiltonians and QMC methods to avoid the sign problem, enabling more efficient simulations of complex quantum systems.

Contribution

It establishes a unifying mathematical framework using Lie groups to identify sign-problem-free fermionic models and suggests new algorithms for their simulation.

Findings

Provides a rigorous criterion for sign-free fermionic models on bipartite lattices.

Unifies existing solutions based on continuous-time QMC and Majorana representation.

Proposes new algorithms for simulating previously intractable systems.

Abstract

We present a guiding principle for designing fermionic Hamiltonians and quantum Monte Carlo (QMC) methods that are free from the infamous sign problem by exploiting the Lie groups and Lie algebras that appear naturally in the Monte Carlo weight of fermionic QMC simulations. Specifically, rigorous mathematical constraints on the determinants involving matrices that lie in the split orthogonal group provide a guideline for sign-free simulations of fermionic models on bipartite lattices. This guiding principle not only unifies the recent solutions of the sign problem based on the continuous-time quantum Monte Carlo methods and the Majorana representation, but also suggests new efficient algorithms to simulate physical systems that were previously prohibitive because of the sign problem.