Simple $\mathbb{Z}_2$ lattice gauge theories at finite fermion density

TL;DR

This paper explores exactly solvable or reducible $ obreak ext{Z}_2$ lattice gauge theories with gapless fermions at finite density in one and two dimensions, revealing diverse phases including metals, insulators, and flux phases with emergent Dirac fermions.

Contribution

It introduces new methods using duality and slave-spin representations to analyze $ obreak ext{Z}_2$ gauge theories with gapless fermions, expanding understanding of their phase diagrams.

Findings

Identification of free-fermion, insulating, and superconducting phases in 1D.

Discovery of flux phases with emergent Dirac fermions in 2D.

Violation of Luttinger's theorem in certain flux phases.

Abstract

Lattice gauge theories are a powerful language to theoretically describe a variety of strongly correlated systems, including frustrated magnets, high-$T_c$ 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 $\mathbb{Z}_2$ 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 $\mathbb{Z}_2$ 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.