Stability of the $\pi$-Flux Phase for $\mathbb{Z}_{2}$ Lattice Gauge Theory Coupled to Fermionic Matter

TL;DR

This paper proves that in a 2D $bZ_2$ lattice gauge theory coupled to fermions, the ground state at half-filling is a $bpi$-flux phase with semimetallic properties, supported by rigorous mathematical methods.

Contribution

It provides a rigorous proof that the $bpi$-flux phase is the ground state under certain conditions and confirms the magnetic susceptibility matches that of massless Dirac fermions.

Findings

Ground state at half-filling is the $bpi$-flux phase.

Magnetic susceptibility matches that of massless 2D Dirac fermions.

Ground state exhibits semimetallic behavior.

Abstract

We consider the two-dimensional $\mathbb{Z}_{2}$ Ising gauge theory coupled to fermionic matter. In absence of electric fields, we prove that, at half-filling, the ground state of the gauge theory coincides with the $\pi$-flux phase, associated with magnetic flux equal to $\pi$ in every elementary lattice plaquette, provided the fermionic hopping is large enough. This proves in particular the semimetallic behavior of the ground state of the model. Furthermore, we compute the magnetic susceptibility of the gauge theory, and we prove that it is given by the one of massless $2d$ Dirac fermions, thus rigorously justifying recent numerical computations. The proof is based on reflection positivity and chessboard estimates, and on lattice conservation laws for the computation of the transport coefficient.