Using Matrix Product States to Study the Dynamical Large Deviations of Kinetically Constrained Models

Here we demonstrate that tensor network techniques—originally devised for the analysis of quantum many-body problems—are well suited for the detailed study of rare event statistics in kinetically constrained models (KCMs). As concrete examples, we consider the Fredrickson-Andersen and East models, two paradigmatic KCMs relevant to the modeling of glasses. We show how variational matrix product states allow us to numerically approximate—systematically and with high accuracy—the leading eigenstates of the tilted dynamical generators, which encode the large deviation statistics of the dynamics. Via this approach, we can study system sizes beyond what is possible with other methods, allowing us to characterize in detail the finite size scaling of the trajectory-space phase transition of these models, the behavior of spectral gaps, and the spatial structure and “entanglement” properties of dynamical phases. We discuss the broader implications of our results.

Figure 1

Finite size scaling of trajectory transition in the East model. (a) SCGF $\theta (s)/N$ as a function of $s$ at $c=0.2$ for system sizes $N=20–200$, showing the (extrapolated) crossing of the first two eigenvalues of $-H_s$. The dotted line corresponds to linear response and the dot-dashed lines correspond to the asymptotic values $\theta (s\to \infty )=-c$. (b) Dynamical susceptibilities $\chi (s)={\theta }^{\prime \prime }(s)$ exhibit a peak at $s_c(N)$ that gets sharper with $N$. For $s>s_c(N)$, we find an almost universal behavior $\chi \propto s^{-\gamma }$ with $\gamma \approx 1.4$. (c) $s_c(N)$ as a function of $N$ for $N\in [20,400]$ and various equilibrium concentrations $c$. The data are compatible with ${\lim }_{N\to \infty }{s}_c(N)\to 0$, but $s_c$ appears to scale as $s_c(N)\propto N^{-\alpha }$ with $\alpha >1$ (full lines are power-law fits; for comparison, we also show fits to $a/N+b/N^2$, dashed). (d) The scaling exponents $\alpha$ (blue diamonds) and fitting parameters $b/a$ (red squares) as a function of $c$. The departure from $1/N$ scaling (dotted-dashed) appears to be more pronounced the lower the $c$ is. (e) Rate functions $\phi (k)$ for $N\in [20,200]$ at $c=0.5$ (left) and $c=0.05$ (right). Dashed lines correspond to Poisson distributions with average $⟨k⟩=-{\theta }^{\prime }(0)/N$. For the analogous results for the FA model, see the Supplemental Material [63].

Figure 2

Structure of active phase. (a) Mean density $⟨n{⟩}_s$ for $s<0$ in the East model for $c=0.05$ (shown as a function of $-\nu =e^s-1$). The plateau structure of the density is evident as compared to $c=0.5$ in the inset. (b) Same for the FA model, where the plateaus are absent. (c) Density profile of the ground state of $H_s$ at $\nu =0.081$ ($s=-0.0845$) for the East model at $c=0.05$ for sizes $N=20$ (top) and 100 (bottom). (d) Density profiles across the active phase of the East model for $N=20$. (e) Extreme limit of the active phase $s\to -\infty$ in the East model. On the left, we show the rescaled $\tilde{\theta }(s=-\infty )/N≔e^s\theta (s=-\infty )/[N\sqrt{c(1-c)}]$ (black circles), $⟨n{⟩}_{s=-\infty }$ (blue squares), and $⟨n^x{⟩}_{s=-\infty }$ (green diamonds) for $N\in [20,400]$. The lines are fits to $a/N+b$ to extract the values in the thermodynamic limit: ${\lim }_{N\to \infty }\theta (s=-\infty )/N$, $⟨n^x{⟩}_{s=-\infty }$, $⟨n{⟩}_{s=-\infty }=0.67$, 0.82, 0.75. On the right is shown that the density profile at $s=-\infty$ is uniform, up to boundaries.

Figure 3

Entanglement. (a) Half-chain entanglement entropy $S_E$ of the ground state of $H_s$ as a function of $s$ for $c=0.5$, 0.1, 0.05 in the East model at $N=200$. (b) $S_E$ for $s<0$ for $c=0.1$ at various sizes $N$. The peak is correlated with the change in shape of the spectral gap $\mathrm{\Delta }E$ of $H_s$ shown in (c).