Simulating 2+1D Quantum Electrodynamics at Finite Density with Neural Flow Wavefunctions

We variationally simulate the 2+1D quantum electrodynamics with finite density dynamical fermions using a newly developed neural flow wavefunction, Gauge-Fermion FlowNet. The Gauge-Fermion FlowNet allows us to study the U(1) gauge theory with no truncation. It obeys Gauss's law by construction, performs autoregressive sampling without equilibration time, and simulates gauge-fermion systems with sign problems. Using our approach, we first study the string breaking phenomena in the U(1) gauge theory. Then we investigate the phase transition from the vacuum phase to the charge crystal phase due to the effect the electric field energy and the fermion mass at both the zero and finite density regime. In addition, we find a phase transition in the magnetic field driven by the competition between the kinetic energy of fermions and the magnetic energy of the gauge field. Our neural network methods provide state-of-the-art tools for studying various lattice gauge theories coupled to matter.