Revisiting the Hybrid Quantum Monte Carlo Method for Hubbard and Electron-Phonon Models

TL;DR

This paper critically revisits the hybrid quantum Monte Carlo method, identifying its limitations with the Hubbard model and highlighting its efficiency for specific models like the Su-Schrieffer-Heeger Hamiltonian, achieving sub-cubic scaling.

Contribution

It analyzes the limitations of HQMC for the Hubbard model and proposes that models with singularity-free effective actions, like the SSH model, can be efficiently simulated with HQMC.

Findings

HQMC faces ergodicity issues and unbounded forces in the Hubbard model.

Complexification of auxiliary fields can address some issues.

Models like the SSH Hamiltonian allow sub-cubic scaling with HQMC.

Abstract

A unique feature of the hybrid quantum Monte Carlo (HQMC) method is the potential to simulate negative sign free lattice fermion models with subcubic scaling in system size. Here we will revisit the algorithm for various models. We will show that for the Hubbard model the HQMC suffers from ergodicity issues and unbounded forces in the effective action. Solutions to these issues can be found in terms of a complexification of the auxiliary fields. This implementation of the HQMC that does not attempt to regularize the fermionic matrix so as to circumvent the aforementioned singularities does not outperform single spin flip determinantal methods with cubic scaling. On the other hand we will argue that there is a set of models for which the HQMC is very efficient. This class is characterized by effective actions free of singularities. Using the Majorana representation, we show that models such as the Su-Schrieffer-Heeger Hamiltonian at half filling and on a bipartite lattice belong to this class. For this specific model sub-cubic scaling is achieved.