Scalable hybrid quantum Monte Carlo simulation of U(1) gauge field coupled to fermions on GPU

TL;DR

This paper introduces a GPU-accelerated hybrid quantum Monte Carlo algorithm for simulating U(1) gauge fields coupled to fermions, enabling large-scale studies of quantum spin liquids with improved efficiency and accuracy.

Contribution

The paper presents novel GPU-based computational techniques that significantly improve the scalability and efficiency of quantum Monte Carlo simulations for U(1) gauge theories coupled to fermions.

Findings

Achieved nearly linear scaling in computational complexity.

Simulated large system sizes up to 660×66×66.

Confirmed conformal nature of the U(1) Dirac spin liquid.

Abstract

We develop a GPU-accelerated hybrid quantum Monte Carlo (QMC) algorithm to solve the fundamental yet difficult problem of $U(1)$ gauge field coupled to fermions, which gives rise to a $U(1)$ Dirac spin liquid state under the description of (2+1)d quantum electrodynamics QED$_3$. The algorithm renders a good acceptance rate and, more importantly, nearly linear space-time volume scaling in computational complexity $O(N_{\tau} V_s)$, where $N_\tau$ is the imaginary time dimension and $V_s$ is spatial volume, which is much more efficient than determinant QMC with scaling behavior of $O(N_\tau V_s^3)$. Such acceleration is achieved via a collection of technical improvements, including (i) the design of the efficient problem-specific preconditioner, (ii) customized CUDA kernel for matrix-vector multiplication, and (iii) CUDA Graph implementation on the GPU. These advances allow us to simulate the $U(1)$ Dirac spin liquid state with unprecedentedly large system sizes, which is up to $N_\tau\times L\times L = 660\times66\times66$, and reveal its novel properties. With these technical improvements, we see the asymptotic convergence in the scaling dimensions of various fermion bilinear operators and the conserved current operator when approaching the thermodynamic limit. The scaling dimensions find good agreement with field-theoretical expectation, which provides supporting evidence for the conformal nature of the $U(1)$ Dirac spin liquid state in the QED$_3$. Our technical advancements open an avenue to study the Dirac spin liquid state and its transition towards symmetry-breaking phases at larger system sizes and with less computational burden.