Fate of many-body localization in an Abelian lattice gauge theory

TL;DR

This work investigates whether many-body localization persists in a constrained $U(1)$ lattice gauge theory on ladder geometries in the mid-spectrum regime. Using exact diagonalization, it combines standard level-statistics diagnostics with a novel intensive observable $\mathcal{D}$ that probes both Fock-space localization and plaquette activity, analyzing its distribution $p(\mathcal{D})$ and its skewness to locate the ETH-MBL transition. The authors find a finite-size estimated transition for $L_y=2$ at $\alpha_c( L_y=2) = 31.04 \pm 0.54$, while wider ladders resist localization, implying absence of true MBL in 2D in this model. They also uncover strong dynamical heterogeneity: even before localization, certain high-energy Fock states exhibit robust oscillations of diagonal plaquette operators in connected real-space regions, a phenomenon tied to the constrained Hilbert space and the structure of mid-spectrum eigenstates. Overall, the study demonstrates that constrained lattice gauge theories challenge conventional MBL expectations and provides scalable diagnostics with potential experimental relevance.

Abstract

We address the fate of many-body localization (MBL) of mid-spectrum eigenstates of a matter-free $U(1)$ quantum-link gauge theory Hamiltonian with random couplings on ladder geometries. Apart from level spacing distribution indicators like disorder-averaged mean level spacing, we also consider an intensive estimator $\mathcal{D} \in [0,1/4]$, which acts as a measure of elementary plaquettes on the lattice that are active or inert in mid-spectrum eigenstates as well as the concentration of these eigenstates in Fock space, with $\mathcal{D}$ equal to its maximum value of $1/4$ for Fock states in the electric flux basis. We calculate its distribution, $p(\mathcal{D})$, for $L_x \times L_y$ lattices, with $L_y=2$ and $4$, as a function of (a dimensionless) disorder strength $α$ ($α=0$ implies zero disorder) using exact diagonalization in many disorder realizations. Although finite-size estimators based on level spacings do not give a reliable critical disorder strength, $α_c(L_y)$, beyond which MBL prevails as $L_x \rightarrow \infty$; a different estimator based on the skewness of $p(\mathcal{D})$ gives $α_c(L_y=2)=31.04 \pm 0.54$ using data for $L_x \leq 14$ due to faster convergence. $p(\mathcal{D})$ for wider ladders with $L_y=4$ show a lower tendency to localize, suggesting a lack of MBL in two dimensions. A remarkable observation is the resolution of the (monotonic) infinite-temperature autocorrelation function of single plaquette diagonal operators in typical high-energy Fock states into a plethora of emergent timescales of increasing spatio-temporal heterogeneity as the disorder is increased. At intermediate $α$ as well as for $α$ slightly below $α_c (L_y)$, a fraction of randomly selected initial Fock states display striking oscillatory temporal behavior of such plaquette operators in spatial regions formed out of connected plaquettes.

Figures (23)

• Figure 1: (Top panel) An electric flux configuration for a $L_x \times L_y = 6 \times 4$ lattice with periodic boundary conditions in both directions. The shading on the elementary plaquettes denote the different values of $- (1+\alpha R_\square)$ (see Eq. \ref{['eq:Hran']}) for one particular disorder realization where $\alpha=1$ and $R_\square$ is an independently chosen random number at each plaquette from the uniform distribution $[-1/2,1/2]$. (Bottom panel) Action of $\mathcal{O}_{\mathrm{kin},\square}$ and $\mathcal{O}_{\mathrm{pot},\square}$ shown for elementary flippable plaquettes. Here, clockwise (anti-clockwise) circulation of electric fluxes around a plaquette is marked in red (blue) inside the plaquette.
• Figure 2: Shannon entropy $S_1$ (Eq. \ref{['eq:Shannon']}) for the energy eigenstates of a single disorder realization of a $6 \times 4$ lattice with $\alpha=2$. The data for $C = \pm 1$ is shown together in the same plot and the density of states is indicated by a color map where warmer color corresponds to higher density of states. The sublattice scars are shown by a different point font and are enclosed by a box composed of dotted lines for clarity.
• Figure 3: (Top panel) Disorder-averaged distribution $P(r)$ shown for a ladder of dimension $12 \times 2$ for various values of $\alpha$. The universal distribution functions $P_{\mathrm{GOE}}(r)$ and $P_\mathrm{P}(r)$ (Eq. \ref{['eq:universalD']}) are also shown for comparison. (Bottom panel) Disorder-averaged mean level spacing $\bar{r}$ shown as a function of disorder strength $\alpha$ for various ladder dimensions. The dotted horizontal lines at $\bar{r} \approx 0.5307$ and $\bar{r} \approx 0.3863$ are the universal values for $P_{\mathrm{GOE}}(r)$ and $P_\mathrm{P}(r)$, respectively, and are shown here for comparison.
• Figure 4: (Top panel) Disorder-averaged $(\Delta r)_d$ shown for thin ladders with $L_y=2$ and $8 \leq L_x \leq 14$ for various values of $\alpha$. (Bottom panel) The peak location for each such curve has been extracted and this finite-size estimator for $\alpha_c$ is shown as a function of $L_x$. Another estimator, extracted from the skewness of $p(\mathcal{D})$ and discussed in Sec. \ref{['subsec:FSS']}, is shown in the same plot for comparison.
• Figure 5: (Top row) Shannon entropy $S_1$ (Eq. \ref{['eq:Shannon']}) as a function of energy for all the energy eigenstates of a particular disorder realization for a $12 \times 2$ ladder in the $C=-1$ sector at low ($\alpha=2$), intermediate ($\alpha=20$) and large ($\alpha=40$) disorder strengths. The top left panel displays a horizontal dotted line at the value of $\ln (\mathrm{HSD})$ for comparison. (Bottom row) $\mathcal{D}$ (Eq. \ref{['eq:formulaD']}) as a function of energy for all the energy eigenstates for the same system. In all panels, the density of states is indicated by a color map where warmer color corresponds to a higher density of states.