Mean-field theory of 1+1D $\mathbb{Z}_2$ lattice gauge theory with matter

TL;DR

We develop a mean-field theory for a $1+1$D $\mathbb{Z}_2$ lattice gauge theory with dynamical matter and a superconducting term, revealing a gauged Kitaev-chain structure and confinement physics. The authors solve the full LGT by DMRG and construct a self-consistent mean-field decoupling that enforces Gauss-law constraints, then compare phase diagrams, Green's functions, string-length distributions, and ground-state energies/polarizations. The MF theory reproduces the main confinement/deconfinement features and captures the qualitative phase structure, though it omits the $U(1)$-symmetric Luttinger-liquid line and exact critical lines. The results highlight a close connection between Kitaev-chain topology and confinement in LGTs, and point to feasible cold-atom implementations to probe string-length and polarization observables. Overall, the MF framework offers a tractable, accurate description of gauged Kitaev chains and motivates extensions to higher dimensions.

Abstract

Lattice gauge theories (LGTs) provide valuable insights into problems in strongly correlated many-body systems. Confinement which arises when matter is coupled to gauge fields is just one of the open problems, where LGT formalism can explain the underlying mechanism. However, coupling gauge fields to dynamical charges complicates the theoretical and experimental treatment of the problem. Developing a simplified mean-field theory is thus one of the ways to gain new insights into these complicated systems. Here we develop a mean-field theory of a paradigmatic 1+1D $\mathbb{Z}_2$ lattice gauge theory with superconducting pairing term, the gauged Kitaev chain, by decoupling charge and $\mathbb{Z}_2$ fields while enforcing the Gauss law on the mean-field level. We first determine the phase diagram of the original model in the context of confinement, which allows us to identify the symmetry-protected topological transition in the Kitaev chain as a confinement transition. We then compute the phase diagram of the effective mean-field theory, which correctly captures the main features of the original LGT. This is furthermore confirmed by the Green's function results and a direct comparison of the ground state energy. This simple LGT can be implemented in state-of-the art cold atom experiments. We thus also consider string-length histograms and the electric field polarization, which are easily accessible quantities in experimental setups and show that they reliably capture the various phases.

Figures (14)

• Figure 1: One-dimensional $\mathbb{Z}_2$ lattice gauge theory with superconducting term Eq. \ref{['eqDefLGTModelSupercond']}. (a) The Gauss law constraint Eq. \ref{['eqDefGauss']} ensures that the $\mathbb{Z}_2$ electric field changes its prefactor across an occupied lattice site, which allows us to define the $\mathbb{Z}_2$ strings (orange lines) labeling $\tau^{x} = -1$ and anti-strings (no line) which labels $\tau^{x} = +1$. (b) Different regimes of the 1+1D $\mathbb{Z}_2$ LGT Eq. \ref{['eqDefLGTModelSupercond']}. In the first row (from above) we sketch the free parton regime, in the second row we sketch the regime where partons are confined into mesons, in the third row we illustrate deconfined partons with SC term, where the U(1) symmetry is explicitly broken (meson creation and annihilation), and in the last row we illustrate the confined regime with the SC term. (c) A qualitative sketch of a phase diagram in the deconfined regime $h = 0$, which exhibits a deconfined parton Luttinger liquid (LL) on the $\lambda = 0$ line, a deconfined symmetry protected state (SPT) for intermediate fillings which correspond to $t < 2 | \mu |$, ferromagnetic (FM) symmetry broken phase at low filling, and antiferromagnetic (AFM) symmetry broken phase at high filling. (d) A qualitative sketch of a phase diagram in the regime $h \neq 0$, which exhibits a confined meson LL on the $\lambda = 0$ line, confined Higgs phase for $\lambda \neq 0$ up to moderate doping, and a symmetry broken AFM phase for high doping.
• Figure 2: Entanglement entropy of the $\mathbb{Z}_2$ LGT Eq. \ref{['eqDefLGTModelSupercond']}, after integrating out matter fields, as a function of filling $n$ and value of the SC term $\lambda$. (a) Symmetric behavior $(n = 0.5 + \Delta n) \leftrightarrow (n = 0.5 - \Delta n)$ is observed for $h = 0$. The entanglement entropy is larger for positive values of the SC term $\lambda > 0$ than for their negative values. We notice a trapezoid shape where the entanglement entropy drastically decreases at low filling $n \lesssim 0.15$ and high filling $n \gtrsim 0.85$, which corresponds to the transition between the topological phase and trivial phase in the Kitaev chain. (b) The entanglement entropy gradually increases with increasing filling $n$ at finite field term $h/t = 1.0$ . The white dots represent the data points, obtained from DMRG, from which the software triangulates the heat map values.
• Figure 3: Central charge as a function of filling $n$ and value of the SC term $\lambda$. (a) On the free parton line $h = 0$ and $\lambda = 0$ we extract the central charge $c = 1$, which indicates a gapless Luttinger liquid. In the case when $\lambda \neq 0$ we observe non-zero value of the central charge on the lines which correspond to the transition between the trivial and topological regime, in agreement with the results in Fig. \ref{['figMiddleEntropyValuesLGT']}. (b) In the confined regime $h/t = 1$ the $\lambda = 0$ line, where $c = 1$, remains which signals a meson Luttinger liquid. For the values $\lambda \neq 0$ we only observe one $c$ non-zero line at high filling $n \gtrsim 0.85$. The white dots represent the data points, obtained from DMRG, from which the software triangulates the heat map values.
• Figure 4: Gauge-invariant Green's function Eq. \ref{['eqGreensFct']} for the $\mathbb{Z}_2$ LGT with SC terms. (a) Free parton regime $h = 0$, $\lambda = 0$, where the Green's function decays with a power law for every filling. (b) The Green's function remains nearly constant at $h = 0$ when the SC terms are included $\lambda / t = -1$ for fillings $0.2 \lesssim n \lesssim 0.8$, which is a topologically non-trivial regime. For lower $n \lesssim 0.2$ and higher fillings $n \gtrsim 0.8$ the Green's function decays exponentially which is signaling a confined state. (c) For $h / t = 1$ and without the SC terms, $\lambda = 0$, the Green's function has an exponential decay which gets weaker with higher filling. (d) When $\lambda / t = -1$ and $h/t=1$ we observe exponential decay which is getting weaker with increasing filling until $n \approx 0.8$, when it starts to decay stronger with increasing filling. This again signals a transition to a different state. In all plots, in order to decrease the boundary effects we start at $x_0 = 30$ in the chain length of $L = 96$. On the right side we highlighted the parameter regime in the phase diagrams where we considered the Green's functions.
• Figure 5: String and anti-string length distributions in the 1+1D $\mathbb{Z}_2$ LGT with SC terms Eq. \ref{['eqDefLGTModelSupercond']}. (a) In the regime $h = \lambda = 0$ we obtain the same distributions for the string and anti-string length histograms with peaks at approximately $\ell = 3$ for quarter filling $n_s = 1/4$. (b) By including the SC terms $\lambda / t= -1$ in the deconfined SPT phase $h = 0$ at filling $n_s = 0.234$, we also obtain identical distributions of strings and anti-strings, however with a peak at $\ell = 1$. (c) For lower filling at $n_s = 0.128$ in the FM phase ($h = 0$ and $\lambda / t = -1$) we see a stronger peak at $\ell = 1$ for the strings which indicates confinement of partons into mesons. (d) In the U(1) confined phase $h / t = 1$, $\lambda = 0$, we observe two distinct distributions for strings and anti-strings, which is an indication of confinement. (e) For $h / t = 1$ with SC terms $\lambda / t = -1$ and at filling $n_s = 0.338$, both distributions peak at $\ell =1$, however the string length distribution has a significantly higher peak than the broader anti-string length distribution, which signals confinement. (f) No qualitative change of behavior can be seen for the results in the same parameter regime as in (e) but with lower filling $n_s = 0.172$. The yellow "x" in all insets indicates the parameter regime in the corresponding phase diagram, and $n_s$ denotes the filling.