Slow dynamics and large deviations in classical stochastic Fredkin chains

The Fredkin spin chain serves as an interesting theoretical example of a quantum Hamiltonian whose ground state exhibits a phase transition between three distinct phases, one of which violates the area law. Here we consider a classical stochastic version of the Fredkin model, which can be thought of as a simple exclusion process subject to additional kinetic constraints, and study its classical stochastic dynamics. The ground-state phase transition of the quantum chain implies an equilibrium phase transition in the stochastic problem, whose properties we quantify in terms of numerical matrix product states (MPSs). The stochastic model displays slow dynamics, including power-law decaying autocorrelation functions and hierarchical relaxation processes due to exponential localization. Like in other kinetically constrained models, the Fredkin chain has a rich structure in its dynamical large deviations—which we compute accurately via numerical MPSs—including an active-inactive phase transition and a hierarchy of trajectory phases connected to particular equilibrium states of the model. We also propose, via its height field representation, a generalization of the Fredkin model to two dimensions in terms of constrained dimer coverings of the honeycomb lattice.

Figure 1

Fredkin spin chains. (a) The local stochastic transition rates for neighboring occupied and unoccupied sites, given by all choices of their neighbors. The fourth transition is not allowed. (b) The disallowed configuration change in the height representation. The troughs ($\cdots 0011\cdots$) are locally immobile. (c) An example configuration in the chosen symmetry sector. The top shows the RW representation of the height field, which must always satisfy $h>0$. The middle is the corresponding particle representation. The bottom is in terms of Dyck words, where opening “(“ must always be matched with a closing ”)”.

Figure 2

Equilibrium properties of the Fredkin model. (a) The average area (scaled by maximum area) $\langle A\rangle /A_m$ as a function of $c$ for various systems sizes $N\in [20,400]$. The dashed line shows the extrapolated value for $N\to \infty$. (b) The average area (symbols) for $c=0.4$ (red/dark grey), $c=0.5$ (blue/medium grey) and $c=0.6$ (green/light grey). The lines show the power laws $\langle N\rangle \sim N^{-\beta }$ with $\beta =1,1.5,2,$ respectively. (c), (d) The spin occupation ${\langle n\rangle }_i$ and height profiles $\langle h_i\rangle$ for each equilibrium phase with a system size $N=60$. (e) The average dynamical activity (per unit time and system size) as a function of $c$ for various systems sizes $N\in [20,400]$. The dashed line shows the extrapolated value for $N\to \infty$. All results are calculated using numerical DMRG.

Figure 3

Localization in the Fredkin chain. (a) The occupation profile $\langle n_i\rangle$ of the steady state for $c>0.5$ and $N=20$. The occupations exhibit an exponential decay for $i>N/2$. (b) The average domain-wall occupations $\langle n_{N/2-i}^{\mathrm{DW}}\rangle$ for $c=0.75$ and $N=20,40,60$. We see the same exponential decay of domain-wall density as we move away from the center of the lattice. (c) The localization length $\lambda$ as a function of $c$. The line shows the result from the theory, Eq. (20), and the blue circles and red crosses the numerically extracted lengths from the occupation and DW profiles, respectively. The numerical data is from DMRG with $N=100$.

Figure 4

Stochastic trajectories and dynamics. (a) Representative trajectories with initial states sampled from equilibrium for $c=0.4$ (top), $c=0.5$ (middle), and, $c=0.6$ (bottom), respectively, for system size $N=100$ and time $t={10}^3$. (b) The autocorrelation functions Eq. (22) for each of the three distinct equilibrium phases. At large times, the autocorrelator for $c=0.5$ decays as the power law $t^{-0.464}$ (from size $N=100$). (c) The same autocorrelation functions plotted on a double-log ordinate scale. For small times, they show exponential decay in the three phases. For large times, they take a stretched-exponential form for $c<1/2$ and $c>1/2$. (d) The numerically estimated timescales Eq. (23) as a function of $c$ (for $N=40,100$, and 400). (e) Example trajectories after a quench from the initial state $1010\cdots 1010$ for $c=0.5$ (left) and $c=0.6$ (on a logarithmic time scale). The former relaxes to equilibrium quickly, while the latter shows hierarchical relaxation (both panels for $N=100$ and $t={10}^5).$ (f) The area (scaled by system size) $\langle A\rangle /N$ after the same quench, for various system sizes $N\in [20,100]$ and $c=0.6$. The dashed line shows $logt$. All data is obtained using continuous-time Monte Carlo.

Figure 5

Finite-size scaling of dynamical LD transitions. The dynamical LD statistics for each equilibrium phase. The top (middle) row of (a)–(d) shows $c=0.4$ ($c=0.5$) with $N\in [20,400]$ obtained via DMRG and the bottom row shows $c=0.9$ obtained through ED. (a) The SCGF $\theta \left(s\right)$ as a function of $s$. The upper and middle panels are scaled by system size, with dotted lines showing the value for $s\to \infty$ and the dashed line showing the linear response prediction (see the main text). (b) The average dynamic activity $k\left(s\right)$ as a function of $s$. The top and bottom panels are shown on log-log scales, whereas the middle one is shown in linear scale. The dashed lines in the bottom panel correspond to integer multiples of $k\left(0\right)$. (c) The dynamical susceptibility $\chi \left(s\right)={\theta }^{\prime \prime }\left(s\right)$ as a function of $s$. (d) The LD rate function scaled by system size $\phi \left(k\right)/N$ as a function of activity $k$. The black dashed lines show a Poisson distribution with mean $k\left(0\right)/N$ in the thermodynamic limit $N\to \infty$, extrapolated from finite-size DMRG data. (e) We estimate the critical point as a function of system size from the peaks of the dynamical susceptibility for $c=0.4,0.5$ in the top panel. The dashed lines shows a fitted power law $s_c\sim N^{-\alpha }$, with the bottom panel showing the obtained $\alpha$ for various $c$.

Figure 6

Structural properties of the LDs. We show observables for each equilibrium phase with (a) $c=0.2$, (b) $c=0.5$, and (c) $c=0.9$. The top row shows the average occupations ${\langle n_i\rangle }_s$ for site $i$ (with differing system sizes and ranges of $s$). The middle row shows the area scaled by system size ${\langle A\rangle }_s/N$ for $s<0$. Finally, we show the area scaled by system size squared ${\langle A\rangle }_s/N^2$ for $s>0$ in the bottom row.

Figure 7

Entanglement entropy of the LD eigenvectors. The bipartite entanglement entropy $S_E\left(s\right)$ for $c=0.4$ (top) and $c=0.5$ (bottom) and system sizes $N\in [20,200]$. For $c=0.5$, the entanglement entropy scales as approximately $S_E\sim logN$.

Figure 8

Extreme active limit. (a) The rescaled SCGF $\tilde{\theta }/N$ (top) and the area ${\langle A\rangle }_{-\infty }/N$ (bottom) as functions of $N$ measured via DMRG. We fit the SCGF as $a+b{N}^{-1}$, allowing us to extrapolate the value in the thermodynamic limit $N\to \infty$ (see main text), shown by the dashed line. The value for the area quickly settles with increasing system size, indicated by the dashed line. (b) The occupation profiles ${\langle n_i\rangle }_{-\infty }$ for $N=40$.

Figure 9

Two-dimensional generalization of the Fredkin model. (a) Dimer covering of the honeycomb lattice. (b) Equivalent rhombus tiling of the plane. (c) Definition of the height field: Given a tiling, moving along the edges of the rhombi the height increases of decreases by one unit as shown. For example, the path in (b) shows that the start and end points have a height difference of $+2$. (d) The elementary local moves that preserve the perfect tiling character (i.e., no tiling defects or no monomers in the dimer representation) are rotations of a triplet of tiles forming an elementary hexagon. These are the dimer/tiler equivalent of the particle-hole exchange in the SEP. These moves change the height field of the central site by three units. (e)–(g) Constrained moves: Requiring the presence of the extra tile guarantees that the height of the central site (indicated by the filled circle) never goes below the lowest height of the arrangement (indicated by the open circle). These are the two-dimensional equivalents of the allowed moves in the Fredkin chain, see Fig. 1.