Neural quantum states for non-Abelian lattice gauge theories with dynamical fermions

Research Approach

This paper addresses the challenge of simulating non-Abelian lattice gauge theories (LGTs) coupled to dynamical fermions, a task typically hindered by the sign problem in standard Monte Carlo methods. The authors propose a variational framework that operates in the Hamiltonian formulation, which is inherently sign-problem-free. They represent the gauge wavefunction using a neural network in the magnetic basis and describe the fermions through a gauge-covariant Gaussian correction. This correction is built upon a fixed Néel reference state and is parameterized by short Wilson lines and the eigenvectors of the mass-hopping Hamiltonian, ensuring the number of variational parameters scales polynomially with the system size.

Key Findings

By combining pure-gauge SU(2) variational Monte Carlo with a Gaussian fermionic state, the authors successfully model the interplay between gauge fields and matter. The framework allows for the analytical calculation of fermionic contributions to energy and observables. The researchers validated their approach against strong-coupling perturbation theory, recovering the expected effective antiferromagnetic spin Hamiltonian. Furthermore, they mapped a ground state phase diagram in the plane of electric and magnetic couplings, demonstrating that hysteresis analysis can effectively identify phase transitions in these gauge-matter systems.

Understanding the phase structure of non-Abelian gauge theories with dynamical matter is essential for studying confinement and the physics of the Standard Model at low energies. Traditional methods like lattice QCD often struggle with the sign problem when finite chemical potentials or topological terms are present. This variational approach provides a scalable, classical numerical tool that avoids these limitations, offering a promising pathway to study more complex matter content and higher-dimensional lattices without the hardware constraints currently faced by quantum simulators.