Emergent fracton dynamics in a nonplanar dimer model

We study the late time relaxation dynamics of a pure $U\left(1\right)$ lattice gauge theory in the form of a dimer model on a bilayer geometry. To this end, we first develop a proper notion of hydrodynamic transport in such a system by constructing a global conservation law that can be attributed to the presence of topological solitons. The correlation functions of local objects charged under this conservation law can then be used to study the universal properties of the dynamics at late times, applicable to both quantum and classical systems. Performing the time evolution via classically simulable automata circuits unveils a rich phenomenology of the system's nonequilibrium properties: For a large class of relevant initial states, local charges are effectively restricted to move along one-dimensional “tubes” within the quasi-two-dimensional system, displaying fracton-like mobility constraints. The timescale on which these tubes are stable diverges with increasing systems size, yielding a novel mechanism for nonergodic behavior in the thermodynamic limit. We further explore the role of geometry by studying the system in a quasi-one-dimensional limit, where the Hilbert space is strongly fragmented due to the emergence of an extensive number of conserved quantities. This provides an instance of a recently introduced concept of “statistically localized integrals of motion,” whose universal anomalous hydrodynamics we determine by a mapping to a problem of classical tracer diffusion. We conclude by discussing how our approach might generalize to study transport in other lattice gauge theories.

Figure 1

Construction of transition graphs. (a) The dimer configuration on the bilayer geometry is seperated into upper and lower layer, with interlayer dimers connecting the two marked as dots. The two configurations are then projected on top of each other from a bird's eye view to give rise to a directed loop model as explained in the main text. The interlayer dimers are assigned charges corresponding to their sublattice. The shown configuration is also the simplest one hosting a single Hopfion. (b) Configurations with nontrivial fluxes $W_x$ and $W_y$ contain loops winding around the boundaries (other loops/interlayer charges not shown to avoid cluttering). (c) Loop-moves originating from the elementary dimer plaquette flips.

Figure 2

Loop interior and corner charges. (a) The interior $v_{\mathcal{L}}$ of a loop $\mathcal{L}$ contains all sites within the green shaded area, and turns into the complement upon inverting the chirality. In transition graphs, the loop chirality can be inverted via the inversion operator ${\widehat{I}}_z$ which exchanges top and bottom layer of the original dimer lattice. White and gray circles illustrate the corner charges ${\widehat{\mathcal{C}}}_{\mathit{r}}=\pm \frac{1}{4}$ from Eq. (8). The validity of Eq. (7) can directly be verified for this example. (b) Dictionary for the corner charges ${\widehat{C}}_{\mathit{r}}$, modulo lattice $C_4$ rotations.

Figure 3

Construction of chiral subcharges. An interlayer charge enclosed by a chiral loop cannot escape the loop under the dynamics of the Hamiltonian. We can then attach a string to an interlayer charge which extends all the way to the system boundary and determine the total chirality $\varphi$ of all loops enclosing the charge. The sum of all interlayer charges with a fixed chirality $\varphi$ then presents a conserved quantity.

Figure 4

Initial states. Local snapshots of typical example configurations within the transition graph picture, both for finite ($W_x/L_x=W_y/L_y>0$) and vanishing ($W_x/L_x=W_y/L_y=0$) flux densities.

Figure 5

Time evolution. A single time step of the deterministic unitary evolution, built on the elementary plaquette updates ${\widehat{U}}_p$ of Eq. (14). Within a fixed plaquette color, all associated local updates commute.

Figure 6

Relaxation dynamics of corner charge correlations. (a)–(c) The spatially resolved corner charge correlation function $G_{W/L}(\mathit{r},t)={\langle {\widehat{\mathcal{C}}}_0\left(0\right){\widehat{\mathcal{C}}}_{\mathit{r}}\left(t\right)\rangle }_{W/L}$ at different times $t$. The initial states in the average ${\langle ...\rangle }_{W/L}$ are sampled from random states within the fixed winding sector $W_x/L=W_y/L=0.4$. We observe a clear restriction of the charge dynamics to an effective one-dimensional ‘tube’ along the diagonal. (d) The dynamics along the diagonal is diffusive, demonstrated by the scaling collapse to a one-dimensional (Gaussian) diffusion kernel. (e) The return probability ${\langle {\widehat{\mathcal{C}}}_{\mathit{r}}\left(0\right){\widehat{\mathcal{C}}}_{\mathit{r}}\left(t\right)\rangle }_{W/L}$ is anomalously slow for a two-dimensional system with $W/L=0.4$. In contrast, for a vanishing winding density $W/L=0$, the correlations decay fast. The system sizes are $L_x=L_y=200$ and $L_x=L_y=1000$ for $W/L=0.4$ and $W/L=0$, respectively.

Figure 7

Thermalization process. The way to equilibrium, i.e., full delocalization of corner charges across the 2D system, can be split into four distinct stages that are reflected in local correlation functions: (I) A stage of diffusive dynamics along effective one-dimensional tubes, with $G\sim t^{-1/2}$. (II) A plateau where the charge is fully delocalized along the tube. (III) The delocalization along the second direction sets in. (IV) The charge is fully delocalized across the whole system. The system size in this example is $L_x=L_y=30$ and the green curve corresponds to the moving average. This demonstrates that the formation of 1D tubes is not due to disconnectivities, but rather bottlenecks in the Hilbert space structure.

Figure 8

Finite-size scaling. For increasing system size, the correlations follow the diffusive decay for increasingly long times, approximately scaling as $\sim L^2$ as shown in the inset. Therefore, the one-dimensional tubes are expected to persist up to infinite time in the thermodynamic limit. The displayed lines correspond to the moving average of the numerically sampled correlations.

Figure 9

Origin of reduced dimensional mobility. The numerical results presented in Sec. 3 can be understood intuitively in terms of bottlenecks in the Hilbert space structure: As the system size increases, the bottlenecks become narrower, and the time needed to eventually pass through the bottleneck to explore the full Hilbert space diverges.

Figure 10

Hilbert space fragmentation in a quasi-1D geometry. We display the probability distributions $P\left({\widehat{Q}}_{\varphi }\right)$ of the charges ${\widehat{Q}}_{\varphi }$ for randomly chosen states on a $L_x=100, L_y=6$ open-ended cylinder. (a) For vanishing flux $W_y=0$, almost all ${\widehat{Q}}_{\varphi }$ are statistically fixed to zero, i.e., $P\left(Q_{\varphi }\right)={\delta }_{Q_{\varphi },0}^{}$. Thus, a small number of large sectors dominates the Hilbert space, which is only weakly fragmented. (b) In contrast, for a finite winding density $W_y/L_x=0.4$, there is an extensive amount ($0\lesssim \varphi \lesssim W_y$) of charges ${\widehat{Q}}_{\varphi }$ with a finite width probability distribution. As a result, the Hilbert space is strongly fragmented.

Figure 11

Statistically localized integrals of motion (SLIOMs). (a) Probability distribution of the $x$-value $x_{\varphi }=r_x$ of the position of the chiral interlayer charges ${\widehat{q}}_{\mathit{r}}\left(\varphi \right)$ for fixed $\varphi$. The system size used for Monte Carlo simulations is $L_x=100, L_y=6$, as well as $W_y/L_x=0.4$. For a given $\varphi$ in the bulk, the ${\widehat{q}}_{\mathit{r}}\left(\varphi \right)$ are generally localized to a subextensive region of size $\sim \sqrt{L}$, see Inset. In addition, there exist modes localized exponentially on the boundary, shown here for $\varphi =0$. These results demonstrate that the ${\widehat{Q}}_{\varphi }$ are statistically localized integrals of motion (SLIOMs) according to Ref. [53]. (b) Probability of finding a chiral interlayer charge ${\widehat{q}}_{(x_{{\varphi }^{\prime }}^{\prime },0)}\left({\varphi }^{\prime }\right)$ to the right of another such charge ${\widehat{q}}_{(x_{\varphi },0)}\left(\varphi \right)$. The system-size-independent sharp step demonstrates the spatial ordering pattern of the ${\widehat{q}}_{\mathit{r}}\left(\varphi \right)$ along the cylinder. (c) Probability distribution of the distance between two interlayer charges ${\widehat{q}}_{\mathit{r}}\left(\varphi \right)$ of the same $\varphi$. The sharply peaked distribution shows that the SLIOMs ${\widehat{Q}}_{\varphi }$ can be assigned 1D positions along the cylinder. (d) The corner charge correlation function remains finite at the boundary due to the presence of edge modes, while decaying in the bulk.

Figure 12

Relaxation of corner charges in 1D. Local correlations of the corner charge relax subdiffusively in an effective 1D geometry (cylinder with finite circumference). The exponent of the decay is $1/4$ and the correlations assume a Gaussian form. This is in full agreement with the probabilty distributions of hard-core tracer particles, expected to describe the late time decay of SLIOM correlation functions.

Figure 13

Lattice magnetic field description. A given dimer configuration containing a certain loop in its transition graph in panel (a) can be mapped to a lattice magnetic field $B_{\alpha }\left(\mathit{r}\right)$ on the bonds of the bilayer lattice according to Eq. (25) in panel (b). The magnetic field can be derived from an associated vector potential $A_{\alpha }\left(\mathit{r}\right)$ living on the plaquettes of the lattice, see panel (c). The values of the vector potential for the example in panel (b) where chosen according to Eq. (32) and Eq. (33). It can directly be verified that this choice leads to the correct dimer occupation numbers, as well as a Hopf number $N_{\mathcal{H}}$ from Eq. (28) that agrees with the total chiral charge $\widehat{\mathcal{Q}}$ from Eq. (8), evaluated for the loop in panel (a).

Figure 14

Proof of Eq. (7). (a) A closed, directed loop $\mathcal{L}$ (black) encloses a region $V_{\mathcal{L}}$ (green shaded). The difference in the number of $A$ (blue) and $B$ (red) sublattice sites contained within $V_{\mathcal{L}}$ can be traced back to the corners of $\mathcal{L}$. (b) Changing the direction of $\mathcal{L}$ exchanges interior and exterior of $\mathcal{L}$. (c) For every corner, there exist two plaquettes $p_1$ and $p_2$ that potentially obtain a nontrivial difference of $A$ and $B$ sublattice sites. Which of these plaquettes is contained within the area $V_{\mathcal{L}}$ enclosed by $\mathcal{L}$ is determined by the chirality of $\mathcal{L}$.