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arXiv:2501.10065 · Journal of Statistical Physics

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

Leonardo Goller, Marcello Porta

Published May 22, 2026• 0 citations
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Abstract

Abstract We consider the two-dimensional $$\mathbbm {Z}_{2}$$ Z 2 lattice 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 $$\pi $$ π -flux phase, associated with magnetic flux equal to $$\pi $$ π 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 2 d Dirac fermions, thus rigorously justifying recent numerical simulations. The proof is based on reflection positivity and chessboard estimates, and on lattice conservation laws for the computation of the transport coefficient.

DOI
10.1007/s10955-026-03617-y

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A study of spinless matter fermions coupled to a constrained Z2{\mathbb{Z}}_{2}Z2​ lattice gauge theory on a triangular ladder is presented. The triangular unit cell and the ladder geometry strongly modify the physics, as compared with a previous analysis on the square lattice. In the static case, the even and odd gauge theories for the empty and filled ladder are identical. The gauge field dynamics due to the electric coupling is drastically influenced by the absence of periodic boundary conditions, rendering the deconfinement-confinement process a crossover in general and a quantum phase transition (QPT) only for decorated couplings. At finite doping and in the static case, a staggered flux insulator at half-filling and vanishing magnetic energy competes with a uniform flux metal at elevated magnetic energy. As for the square lattice, a single QPT into a confined fermionic dimer gas is found vs electric coupling. Dimer resonances in the confined phase are, however, a second-order process only, likely reducing the tendency to phase separate for large magnetic energy. The results obtained employ a mapping to a pure spin model of Z2{\mathbb{Z}}_{2}Z2​ gauge-invariant moments, adapted from the square lattice, and density matrix renormalization group calculations thereof for numerical purposes. Global scans of the quantum phases in the intermediate coupling regime are provided.
Lattice gauge theories are a powerful language to theoretically describe a variety of strongly correlated systems, including frustrated magnets, high-Tc{T}_{c}Tc​ superconductors, and topological phases. However, in many cases gauge fields couple to gapless matter degrees of freedom, and such theories become notoriously difficult to analyze quantitatively. In this paper we study several examples of Z2{\mathbb{Z}}_{2}Z2​ lattice gauge theories with gapless fermions at finite density, in one and two spatial dimensions, that are either exactly soluble or whose solution reduces to that of a known problem. We consider complex fermions (spinless and spinful) as well as Majorana fermions and study both theories where Gauss' law is strictly imposed and those where all background charge sectors are kept in the physical Hilbert space. We use a combination of duality mappings and the Z2{\mathbb{Z}}_{2}Z2​ slave-spin representation to map our gauge theories to models of gauge-invariant fermions that are either free, or with on-site interactions of the Hubbard or Falicov-Kimball type that are amenable to further analysis. In 1D, the phase diagrams of these theories include free-fermion metals, insulators, and superconductors, Luttinger liquids, and correlated insulators. In 2D, we find a variety of gapped and gapless phases, the latter including uniform and spatially modulated flux phases featuring emergent Dirac fermions, some violating Luttinger's theorem.