Local Reversibility and Divergent Markov Length in 1+1-D Directed Percolation

Abstract

Recent progress in open many-body quantum systems has highlighted the importance of the Markov length, the characteristic scale over which conditional correlations decay. It has been proposed that non-equilibrium phases of matter can be defined as equivalence classes of states connected by short-time evolution while maintaining a finite Markov length, a notion called local reversibility. A natural question is whether well-known classical models of non-equilibrium criticality fit within this framework. Here we investigate the Domany-Kinzel model -- which exhibits an active phase and an absorbing phase separated by a 1+1-D directed-percolation transition -- from this information-theoretic perspective. Using tensor network simulations, we provide evidence for local reversibility within the active phase. Notably, the Markov length diverges upon approaching the critical point, unlike classical equilibrium transitions where Markov length is zero due to their Gibbs character. Correspondingly, the conditional mutual information exhibits scaling consistent with directed percolation universality. Further, we analytically study the case of 1+1-D compact directed percolation, where the Markov length diverges throughout the phase diagram due to spontaneous breaking of domain-wall parity symmetry from strong to weak. Nevertheless, the conditional mutual information continues to faithfully detect the corresponding phase transition.

Figures (6)

• Figure 1: Summary of main results. (a) Starting from the non-absorbing steady state $\rho_0$ at $p=0$, the time-evolved state $\rho(t) = \mathcal{E}_p^t[\rho_0]$ in the active phase of the directed percolation ($p < p_c$) relaxes exponentially fast to the stationary state $\rho_p \equiv \mathcal{E}_p^\infty[\rho_0]$ ($L$ is the system size). Furthermore, the Markov length of $\rho(t)$ remains finite, implying local reversibility of the active phase. In contrast, at the critical point, $p = p_c$, the state $\rho(t) = \mathcal{E}_{p_c}^t[\rho_0]$ converges only polynomially to $\rho_{p_c} \equiv \mathcal{E}_{p_c}^\infty[\rho_0]$, and the associated Markov length diverges. (b) The Markov length $\xi_{\text{Markov}}$ is defined via the decay length of the conditional mutual information, $I(A\!:\!C \mid B) \sim e^{-|B|/\xi_{\text{Markov}}}$ with the buffer size $|B|$. Similar situation holds (trivially) for the absorbing phase of the DP ($p > p_c$).
• Figure 2: Schematic phase diagram of the Domany--Kinzel model. The critical behavior along the entire phase transition line (denoted as red in the figure) --- except for its lower terminal point --- is described by the DP universality class, while the critical behavior along the $q = 0$ line is described by the compact directed percolation (CDP). For the former, we focus on the diagonal line, $p = q$ (see main text).
• Figure 3: Local reversibility ((a), (b), (c)) and divergent Markov length ((d), (e)) in the Domany--Kinzel model along the $p=q$ line. (a) $|\rho_\Lambda(t) - \rho_\Lambda(t \approx \infty)|_1$ as a function of time $t$ with various $|\Lambda|$ at a representative $p < p_c$ where we choose $p = 0.24$. Here $t \approx \infty$ denotes the longest time (which is $310$ here) needed beyond which the magnetization does not change within the tolerance $10^{-10}$. The inset shows the $\log |\rho_\Lambda(t) - \rho_\Lambda(t\approx \infty)|_1$-$t$ plot. (b) $|\rho_\Lambda(t) - \rho_\Lambda(\infty)|_1$ as a function of $|\Lambda|$ with various $t$. (c) $\log I(A:C|B)$ as a function of $|B|$ for different times $t$, again at $p =0.24$. The slope determines the Markov length $\xi$. We find that $\xi$ increases from $\xi \approx 0.67$ at $t = 10$ and saturates to $\xi \approx 1.27$ for $t \gtrsim 50$. (d) The conditional mutual information $I(A:C|B)$ as a function of time at $p = p_c \approx 0.295$. The inset shows the data collapse with the scaling ansatz Eq.\ref{['Eq:I_collpase_z']}, with $z = 1.58$, $\alpha = 2$. (e) $I(A:C|B)$ as a function of the error rate $p$ for different size $|B|$ (the dashed line indicates $p_c$). The inset shows the data collapse of the scaling ansatz in Eq.\ref{['Eq:I_collpase_nu']} with $\nu_{\perp} = 1.1$ and $\alpha = 2$. Here we fix $t/|B|^z = 2$ with $z = 1.58$.
• Figure 4: Conditional mutual information (CMI) analysis of the compact directed percolation model with the initial state $|\rho_{\text{dw, sym}}\rangle \equiv \tfrac{1}{2} |\rho_{\text{dw}, \uparrow}\rangle + \tfrac{1}{2} |\rho_{\text{dw}, \downarrow}\rangle$, where one representative trajectory of $|\rho_{\text{dw}, \uparrow}\rangle$ and $|\rho_{\text{dw}, \downarrow}\rangle$ is shown in (a) and (b), respectively [see also Eq. \ref{['Eq:suppp_rho0_CDP']}]. If $b = |\uparrow\rangle^{|B|}$, there is an ambiguity in inferring whether the domain wall lies in region $C$ [as shown in (a)] or in region $A$ [as shown in (b)]. (c) $I(A:C|B)$ as a function of time at $p = p_c = 0.5$. (d) Finite-size, finite-time scaling collapse with $z = 2$ and $\alpha = 0$, compared with the analytical result derived in Eq. \ref{['Eq:supp_CMI_asymp']}. (e) $I(A:C|B)$ as a function of the error rate $p$ for different sizes $r$, where we fix $\tau/r^z = 6$ with $z = 2$. (f) Finite-size, finite-time scaling collapse with $\nu_{\perp} = 1$ and $\alpha = 0$, compared with the analytical result derived in Eq. \ref{['Eq:supp_CMI_asymp']}.
• Figure 5: Conditional mutual information (CMI) analysis of the compact directed percolation model with the initial state $|\rho_{-1}\rangle = |\cdots, \downarrow_{-4}, \downarrow_{-2}, \uparrow_{0}, \downarrow_{2}, \downarrow_{4}, \cdots\rangle$. One representative trajectory of $b = |\uparrow\rangle^{|B|}$ and $b = |\downarrow\rangle^{|B|}$ is shown in (a) and (b), respectively. (c) $I(A:C|B)$ as a function of time at $p = p_c = 0.5$. (d) Finite-size, finite-time scaling collapse with $z = 2$ and $\alpha = 1$. (e) $I(A:C|B)$ as a function of the error rate $p$ for different sizes $r$, where we fix $\tau/r^z = 6$ with $z = 2$. (f) Finite-size, finite-time scaling collapse with $\nu_{\perp} = 1$ and $\alpha = 1$.