Disorder-free localization transition in a two-dimensional lattice gauge theory

Disorder-free localization has been proposed as a mechanism for ergodicity breaking in lattice gauge theories (LGTs) which can even occur in two spatial dimensions (2D). It has been shown that the U(1) quantum link model (QLM) can localize due to an emergent classical percolation transition fragmenting the system into disconnected real-space clusters. While the nature of the quantum localization transition (QLT) is still debated for conventional many-body localization, here we provide a comprehensive characterization of the QLT for the QLM in 2D for a disorder-free case. In this Letter we find compelling evidence that the QLT in the 2D QLM is continuous and we determine its universality class. We base our considerations on a spectral analysis of finite-size clusters in the percolation problem which exhibits two regimes—one in which large clusters effectively behave nonergodically, a result naturally accounted for as an interference phenomenon in configuration space, and the other in which all large clusters behave ergodically. Our analysis can also be applied to other 2D U(1) LGTs potentially including also matter degrees of freedom.

Figure 1

(a) Sample initial configuration for the Monte Carlo flip process: $\uparrow$ and $\to$ represent spin $|\uparrow \rangle$ whereas $\downarrow$ and $\leftarrow$ represent spin $|\downarrow \rangle$. The ticked plaquettes are flippable due to their clockwise/anticlockwise orientation. Background charges are indicated by the circles on the sites. The crossed plaquettes are completely frozen due to charge 2 sites. The box indicates the flip process where the ${\widehat{U}}_{\square }$ operator flips all the spins (colored blue) on plaquette P. (b) Disconnected clusters of flipped plaquettes (in black) in the nonpercolating phase for the state $|\psi (\alpha =0)\rangle$ on a $40\times 40$ lattice. Frozen plaquettes are shown in white.

Figure 2

Results for the percolation problem on square lattices of linear dimension $L=20$, 30, 40, and 80. We average over 1000 sectors for each $\alpha$. (a) Probability that a cluster wraps around either the $x$ or the $y$ axes vs $\alpha$. The dotted line represents the percolation threshold ${\alpha }_c$. Inset: Fraction of plaquettes occupied by the largest cluster vs $\alpha$. (b) Scaling collapse for the order parameter with the critical exponents ${\alpha }_c=0.261, \nu =4/3$, and $\beta =5/36$. Here, ${\alpha }_r=(\alpha -{\alpha }_c)/{\alpha }_c$.

Figure 3

(a) The gap ratio averaged over the spectrum and over all clusters with 16, 18, and 20 plaquettes in $N=1500$ superselection sectors of the $80\times 80$ lattice vs $\lambda /J$ for state $|\psi (\alpha =0)\rangle$. Here, “rnd” implies results for the randomized energy model with same configuration space connectivity. Error bars indicate variation due to different cluster shapes. Inset: Same data for $|\psi (\alpha ={\alpha }_c)\rangle$ with $N=2500$. (b) Corresponding probability distribution for a 20-plaquette cluster.

Figure 4

Configuration space graph for a 16-plaquette cluster. (a), (b) Graph with correlated and randomized on-site energies, respectively. The edge color goes from red to blue as the separation between connecting shells increases. (c), (d) Histogram for normalized difference of the on-site energies $d=|E_i-E_j|/\lambda$ between nearest neighbors $i,j$ in (a) and (b), respectively.