Confined Phases of One-Dimensional Spinless Fermions Coupled to $Z_2$ Gauge Theory

We investigate a quantum many-body lattice system of one-dimensional spinless fermions interacting with a dynamical $Z_2$ gauge field. The gauge field mediates long-range attraction between fermions resulting in their confinement into bosonic dimers. At strong coupling we develop an exactly solvable effective theory of such dimers with emergent constraints. Even at generic coupling and fermion density, the model can be rewritten as a local spin chain. Using the density matrix renormalization group the system is shown to form a Luttinger liquid, indicating the emergence of fractionalized excitations despite the confinement of lattice fermions. In a finite chain we observe the doubling of the period of Friedel oscillations which paves the way towards experimental detection of confinement in this system. We discuss the possibility of a Mott phase at the commensurate filling $2/3$.

Figure 1

Fermions (red) that occupy sites interact with the $Z_2$ gauge field (blue) defined on links. Any physical configuration satisfies the Gauss law.

Figure 2

Effective dimer model from the second-order perturbation theory: (a) Effective hopping of a dimer and (b) dimer length fluctuation that decreases the energy of a dimer.

Figure 3

(a) Friedel oscillations in a chain of length $L=160$ at filling ${\rho }_F=1/8$. At $h=0$ the oscillations have the wave vector $2k_F$, but for $h>0$ the wave vector is reduced to $k_F$ resulting in the doubling of the period. (b) Absolute value of the fermion correlator $⟨f_0^{†}{f}_i⟩$ for different values of the gauge coupling $h$. At $h=0$ the correlators decay algebraically. For $h>0$ the decay is exponential, indicating confinement of fermions $f_i$. The black lines are exponentials, obtained by fitting the data. (c) Exponent of the power-law decay of the pair-pair correlator $⟨b_i^{†}{b}_{j^{*}}⟩$. The numerical results at $h=10$ are in a good agreement with the predictions from the exactly solvable bosonic model (5). For figures (a) and (b) the ground state is obtained by applying finite DMRG to the original Hamiltonian (1). For (c), iDMRG on the dual spin model (4) was used.