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

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.

Figure 1

Fermions hopping on the sites (black dots) of a 1D lattice and coupled to a ${\mathbb{Z}}_2$ gauge field (green arrows) living on the nearest neighbor bonds (links) of the lattice.

Figure 2

(a) Phase diagram and (b) gap at fixed $\mu$ of the ${\mathbb{Z}}_2$ gauge theory with fermions in (1+1)D, with Hamiltonian (1). The metal-insulator phase boundaries for the emergent gauge invariant $\tilde{c}$ fermions are given by $h/t={(h/t)}_{c,\pm }$, where we define ${(h/t)}_{c,\pm }\equiv \pm 1+\mu /2t$.

Figure 3

Fermions hopping on the sites (black dots) of a 2D square lattice and coupled to a ${\mathbb{Z}}_2$ gauge field (green arrows) living on the nearest neighbor bonds (links) of the lattice. The star of $i$, denoted ${+}_i$, corresponds to the four (orange) links $(i,i\pm \widehat{x}),(i,i\pm \widehat{y})$ connected to site $i$. The gauge invariant plaquette operator $P_i$ is defined as the product of the gauge fields on all four links bordering the blue square.

Figure 4

The flux of the background ${\mathbb{Z}}_2$ gauge field $B_{ij}$ through the plaquette at site $i$ (red square) is equal to the original gauge invariant flux $P_i$ [Eq. (25)], regardless of the choice of ${\sigma }^x$ string operator at the four sites of the plaquette (blue semi-infinite lines).

Figure 5

Ground state energy density of the 2D spinless fermion model for system size $L\times L$ at fermion fillings 1/4, 1/3, and 2/5. Open circles: Monte Carlo data, solid lines: linear fits, crosses: ground state energy densities in the thermodynamic limit for the inferred configurations in Fig. 6.

Figure 6

Optimal ground state flux configurations at fermion filling $\nu$ inferred from a visual inspection of the Monte Carlo configurations. Black squares: $\pi$ flux, white squares: zero flux.

Figure 7

Zero temperature phase diagram of the ${\mathbb{Z}}_2$ gauge theory on the 2D square lattice with spinful fermions [Eq. (47) and (23)] at half filling ($\mu =0$). For small values of the gauge coupling $h$ a gapless semimetallic (SM) phase with emergent Dirac fermions violating Luttinger's theorem is stabilized. For larger values one obtains either an AF insulator or a ground state with coexisting SC and CDW order.

Figure 8

Gauge invariant spectrum of emergent fermionic excitations in a 1D model of Majorana fermions interacting with a dynamical ${\mathbb{Z}}_2$ gauge field, Eq. (51), for $h/t=0.1$.

Figure 9

Single-particle spectrum in a 2D model of Majorana fermions interacting with a dynamical ${\mathbb{Z}}_2$ gauge field, for $h/t=0.5$. Owing to the particle-hole redundancy only the positive part of the spectrum is shown.

Figure 10

Diagrammatic representation of the gauge invariant $\tilde{\gamma }$ and $\tilde{f}$ fermion Green's functions (Matsubara frequencies are omitted for simplicity).

Figure 11

Infinite-temperature spectral function $A_{\tilde{f}}^{\infty }$ for the localized $\tilde{f}$ fermions obtained as a projection of the noninteracting density of states $\rho \left(\epsilon \right)$ for the itinerant $\tilde{\gamma }$ Majorana fermions.

Figure 12

Temperature dependence of the $\tilde{f}$ fermion spectral function in 1D for $h/t=0.5$ and $\delta {\mu }_{\tilde{f}}/t=1:T/t=0$ (blue), $T/t=0.5$ (red), $T/t=3$ (green).

Figure 13

Zero-temperature $\tilde{f}$ fermion spectral function in 2D for $h/t=0.5$ and $\delta {\mu }_{\tilde{f}}/t=1$. Inset: blowup of the spectral function showing (from left to right) the band edge discontinuity, logarithmic van Hove singularity, and linear vanishing of the spectral weight, that can be attributed to specific features of the $\tilde{\gamma }$ fermion dispersion in Fig. 9.