Kinetically constrained freezing transition in a dipole-conserving system

We study a stochastic lattice gas of particles in one dimension with strictly finite-range interactions that respect the fractonlike conservation laws of total charge and dipole moment. As the charge density is varied, the connectivity of the system's charge configurations under the dynamics changes qualitatively. We find two distinct phases: Near half filling the system thermalizes subdiffusively, with almost all configurations belonging to a single dynamically connected sector. As the charge density is tuned away from half filling there is a phase transition to a frozen phase, where locally active finite bubbles cannot exchange particles and the system fails to thermalize. The two phases exemplify what has recently been referred to as weak and strong Hilbert space fragmentation, respectively. We study the static and dynamic scaling properties of this weak-to-strong fragmentation phase transition in a kinetically constrained classical Markov circuit model, obtaining some conjectured exact critical exponents.

Figure 1

The Markov circuit model. Markov gates (blue) can shuffle the local charge configuration in any way that conserves the total charge and dipole moment of those four sites. Hollow black circles represent the charge configuration before the action of the gates, and pink circles represent it afterwards. Each site is constrained to have 0, 1, or 2 particles. The black arrows show some possible actions of the gates; the only transitions that are allowed in this model are “pair hopping” moves where one (or two) particles hop to the left and one (or two) hop to the right. This model has a detailed balance so that in the long-time steady state all moves are equally likely to occur as the reverse of the same move.

Figure 2

Measures of the fragmentation transition in small systems. All plots correspond to zero total dipole: $N_1=0$. (a) One minus the fraction of states that belong to the largest Krylov sector. (b) The entropy of fragmentation, as defined in the main text. (c) The infinite-time fraction of frozen sites ${\rho }_F$ averaged over all possible initial states. All system sizes from $L=12$ (lightest blue) to $L=18$ (darkest blue) sites are represented, but only systems where the total charge and system size have the same parity have states with exactly zero total dipole. These data are symmetric about half filling ($\overline{n}=1$) so only $\overline{n}\ge 1$ are shown.

Figure 3

Ideal active bubble. (a) The vertical axis is the mean charge density of active sites over all histories and the horizontal axis is the number of time steps (layers in the Markov circuit). All histories were initialized with a block of length $L_0$ sites with $n_i=1$ embedded in an otherwise frozen infinite system of $n_i=2$. From left to right $L_0=12$ to 32 in steps of four. Each line represents an average over more than ${10}^3$ histories. (b) Histogram of the scaled time it takes for the active bubble to reach twice its initial size ($n_A=1.5$). The times are scaled by $L_0^z$, where $z=7$. There are at least ${10}^3$ histories for each curve, and the curves represent $L_0=12$ to 48 in steps of four (lightest purple to darkest purple). (c) Estimate of the dynamic exponent $z$ as a function of $L_0$ via $z\left(L_0\right)=log({\langle \tau \rangle }_{+}/{\langle \tau \rangle }_{-})/log(L_{+}/L_{-})$, where $L_{\pm }=L_0\pm 2$ and ${\langle \tau \rangle }_{\pm }$ are the corresponding mean values of $\tau$ from the data shown in (b). For example, the first point at $L_0=14$ is obtained by setting $L_{-}=12$ and $L_{+}=16$, then taking the corresponding mean values of $\tau$ from (b) and calling those mean values ${\langle \tau \rangle }_{-}$ and ${\langle \tau \rangle }_{+}$. Error bars represent the standard error.

Figure 4

Approximate model of the steady state. (a) Distributions (“normalized counts”) of the final lengths of the active blocks. Charge densities go from $\overline{n}=1.505$ (darkest blue) to 1.550 (lightest blue) in steps of $2.5\times {10}^{-3}$. Each line represents ${10}^2$ samples of length $L={10}^6$ drawn randomly from all configurations with the correct amount of total charge. The dashed line is $\sim L_A^{-\alpha }$ with $\alpha =3/2$, as is expected for the small $L_A$ regime. (b) The correlation length $\xi =\langle L_A^2\rangle /\langle L_A\rangle$. The dashed line represents the expected scaling $\xi \propto {(\overline{n}-{\overline{n}}_c)}^{-\nu }$ where $\nu =2$. (c) Scaled versions of the curves in part (a).

Figure 5

Time evolution of typical large systems. All plots represent the same simulations of the full Markov circuit dynamics of systems with $L={10}^4$ out to times $t={10}^8$. Each data point is averaged over $5–10$ samples. Quantities are shown at 40 times from $t=0$ to ${10}^8$ spaced evenly on a logarithmic scale (light to dark green). (a) The fraction of frozen sites. The dashed curve represents the simplified theoretical prediction detailed in the main text. (b) The charge density of the total area in the system that has been labeled active, i.e., $n_A={\sum }_{i\in A}{n}_i/(1-{\rho }_F)L$. (c), (d) The absolute difference in ${\rho }_F$ and $n_A$, respectively, from each of the 40 times at which ${\rho }_F$ and $n_A$ are shown to the next.

Figure 6

Autocorrelations of charge density fluctuations in half-filled and critical systems. (a) On-site charge autocorrelations at half filling $\overline{n}=1$. The dashed line represents the subdiffusive scaling $C(0,t)\propto t^{-1/4}$. Data for $C(0,t)$ (black crosses) was generated by simulating the dynamics of ${10}^3$ samples of systems with $L={10}^3$ sites and periodic boundaries out to time $t={10}^6$. Averaging was done over all sites, samples, and windows of time surrounding each data point shown above. (b) The scaled charge autocorrelation function for critical systems plotted as a function of $x$ for several times ${10}^4<t<{10}^8$. The scaling exponent was taken to be $z=7$. Data (black crosses) was generated by simulating the dynamics of $8\times {10}^2$ samples with $L={10}^3$ sites and periodic boundaries out to time $t={10}^8$. Averaging was done in the same way as in part (a). The curves are associated with times spaced evenly on a log scale (light to dark blue). (c) The distance at which charge autocorrelations crossover from positive to negative, i.e., $C(x_{C=0},t)=0$, in critical systems ($\overline{n}=1.5$). The dashed line represents the scaling $x_{C=0}\propto t^{1/z}$ again with $z=7$. The data is the same as for part (b). (d) Onsite charge autocorrelations at the critical density $\overline{n}=1.5$. Data and averaging is the same as in parts (b) and (c). The dashed line represents the scaling $C(0,t)\propto t^{-1/10}$.

Figure 7

Equilibrium of two merged active blocks. (a) Equilibrium charge distributions. The dashed line is at $\langle n_i\rangle =1.5$, the critical density. The purple curves correspond to the scenarios of two initial blocks of length $L_A=12$ sites with $n_i=1$ separated by an initial block of length $L_A+\{-1,0,1,2\}$ sites (light to dark) with $n_i=2$. Each system was time evolved for $t_f={10}^{10}$ time steps and the curves represent the steady-state charge density distributions over four samples during the final $2.5\times {10}^9$ time steps. (b) The standard deviation of the quadrupole moment fluctuations of the final active sites in equilibrium. The four rightmost data points correspond to the same simulations that generated the curves in part (a), with $L_A=12$. The center and left groups of four data points correspond to $L_A=10$ and $L_A=8$, respectively. The lines are fits of the form $\sigma \left(N_2\right)=A{L}_{A,\text{fin}}^{5/2}$, with $A=3\times {10}^{-2},3\times {10}^{-2},1\times {10}^{-2},8\times {10}^{-3}$ from top to bottom.