Exact solution for the filling-induced thermalization transition in a one-dimensional fracton system

We study a random circuit model of constrained fracton dynamics, in which particles on a one-dimensional lattice undergo random local motion subject to both charge and dipole moment conservation. The configuration space of this system exhibits a continuous phase transition between a weakly fragmented (“thermalizing”) phase and a strongly fragmented (“nonthermalizing”) phase as a function of the number density of particles. Here, by mapping to two different problems in combinatorics, we identify an exact solution for the critical density $n_c$. Specifically, when evolution proceeds by operators that act on $\ell$ contiguous sites, the critical density is given by $n_c=1/(\ell -2)$. We identify the critical scaling near the transition, and we show that there is a universal value of the correlation length exponent $\nu =2$. We confirm our theoretical results with numeric simulations. In the thermalizing phase the dynamical exponent is subdiffusive, $z=4$, while at the critical point it increases to $z_c\gtrsim 6$.

Figure 1

An illustration of the dynamics with charge- and dipole-conserving three-site gates $U_{3,\pm }$. The circuit (above) shows the sequence of random operations, while the balls (below) illustrate the occupation numbers of the state. Yellow balls represent the starting positions of particles involved in the first two applied operators, blue balls represent the final positions of these particles, and gray balls show particles that remain in place. These two operations are the only three-site gates for our model.

Figure 2

Analogy between nondecreasing integer lattice paths and symmetry sector states. The $x$ axis corresponds to position and each particle corresponds to a one unit move in the $y$ direction. This construction guarantees that the area bounded between the curve and the $y$ axis is equal to the dipole moment of the state.

Figure 3

Analogy between tournament scoring sequences and states in the apparent LKS. On the left are tournament graphs for $N=5$ teams, with the score of each team (the number of outgoing edges) labeled. A given scoring sequence corresponds to a particle distribution, with the number of wins for each team corresponding to the position $x$ of a particle. In this analogy, it is clear that $U_{3,\pm }$ corresponds to flipping the result of a game between two teams that that have either the same number of wins ($U_{3,+}$) or a number of wins that differ by 2 ($U_{3,-}$). Note, particles are indistinguishable in our dynamics and in this figure are only labeled for clarity.

Figure 4

The relative size of the apparent LKS as a function of filling $n$ and system size $L$. (a) For $n\le 1$, the relative size $D$ of the LKS approaches the scaling suggested by Eq. (5) in the limit $L\to \infty$ with no fitting parameters. Increasingly dark symbols correspond to progressively larger system size. (b) Exactly at the critical filling $n=1$, $D$ decays as $\sim L^{-1/2}$. This transition from exponential decay at $n<1$ to power-law decay at $n=1$ is indicative of the thermalization transition.

Figure 5

The relative size $D$ of the LKS for systems with periodic boundary conditions, as measured by numeric simulations. The value of $D$ is plotted as a function of the filling $n$ for different values of the gate size $\ell$. Data shown here correspond to system size $L=241$ and is averaged over 1000 random choices of the initial state (see Appendix pp6 for a full description of the simulation protocol). The vertical dashed lines show the predicted critical filling $1/(\ell -2)$.

Figure 6

Scaling of the correlation function $C(x,t)$ at the critical point $n=1$ for dynamics with three-site gates. We scale the value of the correlation function and the position coordinate by the value $x_0\left(t\right)$ at which the correlation function is first equal to zero. Curves correspond to different values of the time, logarithmically spaced from $t={10}^2$ (light blue) to ${10}^7$ (dark blue). The inset shows that the growth of $x_0$ with $t$ can be fit to a power law with exponent $z$ larger than 5.

Figure 7

Allowed game flips that correspond to three-site dipole-conserving operations. The game flips on the left correspond to $U_{3,\pm }$ and the game flips on the right leave the scoring sequence unchanged and thus correspond to the identity operation.

Figure 8

Numerically estimated probability that a segment of length $L_0$ segment has particle density $n\ge n_c$. We generate 375 random states with $L=10000$ with values of the average density $n$ ranging from 0.6 (bottom curve) to 0.8 (top curve). For each curve, we sample 100 random segments of length $L_0$ and calculate the probability that the segment is locally thermalized (its density exceeds $n_c=1$). Our numerical results closely match the theoretical result of Eq. (G1), and they display exponential decay with $L_0$ in the range $1\ll L_0\ll L$.