Emergent superconformal symmetry in the phase diagram of a 1D $\mathbb{Z}_{2}$ lattice gauge theory

Abstract

We investigate the phase diagram and critical properties of a one-dimensional $\mathbb{Z}_{2}$ lattice gauge theory describing an orthogonal metal, where spinless fermions and Ising spins are minimally coupled to a deconfined $\mathbb{Z}_{2}$ gauge field. Working at half-filling of fermions, we derive an exact gauge-invariant formulation that maps the model onto decoupled XXZ and transverse-field Ising chains. This mapping enables a controlled low-energy field-theory description in terms of a perturbed Luttinger liquid and Ising conformal field theories. Combining analytical arguments with numerical simulations, we determine the full phase diagram and identify various critical and multi-critical regimes. Along a specific multi-critical line, where the fermionic and bosonic velocities coincide, we find strong evidence for an emergent superconformal symmetry. Our results establish a minimal lattice realization of emergent superconformal criticalities in a gauge-matter system and provide a route toward its exploration in quantum simulators.

Figures (9)

• Figure 1: (color online) Phase diagram of the Hamiltonian Eq. \ref{['eq:H-in-Gauge-Invariant-Fields']}. For any finite $\kappa>0$ and $|\Delta|<1$, conventional and gauged Luttinger liquid phases occur for $g<1$ (LL) and $g>1$ (LL$^{*}$), respectively. For $|\Delta|>1$, symmetry-broken phases appear: for $g<1$ (SSB) the $\mathbb{Z}_{2}^{C}$ symmetry is broken, while for $g>1$ (SSB$^{*}$) both $\mathbb{Z}_{2}^{C}$ and $\mathbb{Z}_{2}^{W}$ are spontaneously broken. Transitions along solid and dashed red lines are of BKT type, whereas those along solid and dashed blue lines are in the Ising universality class. The SSB–SSB$^{*}$ transition has central charge $c=\flatfrac{1}{2}$, and the LL–LL$^{*}$ line has $c=\flatfrac{3}{2}$ due to an additional gapless Luttinger liquid mode. Superconformal field theories (SUSY CFTs) appear on the dashed blue line, for $\kappa$ given by Eq. \ref{['eq:SUSY-Condition']}.
• Figure 2: (color online) Schematic illustration of the ground states in the $\mathbb{Z}_{2}^{C}$-breaking SSB phase ($\Delta > 1$, $g < 1$) and $\mathbb{Z}_{2}^{C}\times\mathbb{Z}_{2}^{W}$-breaking SSB$^{*}$ ($\Delta > 1$, $g > 1$) phase.
• Figure 3: (color online) Numerically obtained $M(q)$ for $L=256$, for several values of $0 \leq \Delta \leq 2$ at fixed $\kappa = 1$. Here we set $g = 0.5$, although the results are identical for any $g$, as $M(q)$ is insensitive to the properties of the QI sector of the model. For small anisotropies $0< \Delta < 1$, $M(q)$ increases linearly with $q$. In contrast, for larger anisotropies $1 < \Delta$ we observe a pronounced peak at $q = \pi = 2k_{F}$, signaling the emergence of CDW order in the XXZ sector.
• Figure 4: (color online) Numerically obtained values of $K$ and $\alpha$ over a wide range of $\Delta$ at fixed system size $L=256$. The Luttinger parameter $K$ is extracted by fitting the structure factor at momenta $q/\pi < 0.1$ to the form $f(q) = Kq/2\pi$, yielding $K_{\text{DMRG}}$ in excellent agreement with the exact result. The exponent $\alpha$ is obtained from a fit of $M(\pi)$ to $f(L) = AL^{\alpha}$ for $L\in[32,64,128,256]$. Over this interval of system sizes, $\alpha$ gradually converges towards the expected value $\alpha=1$, signaling a tendency towards true long-range CDW order in the corresponding parameter regime.
• Figure 5: (color online) Numerical results for the expectation values of the order and disorder parameters. We fix the system size to $L=256$ and $\Delta=0.5$. Since the definitions in Eqs. \ref{['eq:O_P,O_xx']} depend on bulk indices $i$ and $r$, we choose $i=L/4$ and $r=L/2$ to minimize finite-size effects, i.e., we evaluate $\mathcal{O}_{P}$ and $\mathcal{O}_{xx}$ in the central part of the chain. As $g$ varies, these order and disorder parameters clearly signal the expected Ising transition at $g=1$.