Entropy Governed by the Absorbing State of Directed Percolation

We investigate the informational aspect of ($1+1$)-dimensional directed percolation, a canonical model of a nonequilibrium continuous transition to a phase dominated by a single special state called the “absorbing” state. Using a tensor network scheme, we numerically calculate the time evolution of state probability distribution of directed percolation. We find a universal relaxation of Rényi entropy at the absorbing phase transition point as well as a new singularity in the active phase, slightly but distinctly away from the absorbing transition point. At the new singular point, the second-order Rényi entropy has a clear cusp. There we also detect a singular behavior of “entanglement entropy,” defined by regarding the probability distribution as a wave function. The entanglement entropy vanishes below the singular point and stays finite above. We confirm that the absorbing state, though its occurrence is exponentially rare in the active phase, is responsible for these phenomena. This interpretation provides us with a unified understanding of time evolution of the Rényi entropy at the critical point as well as in the active phase.

Figure 1

Time evolution of the second-order Rényi entropy per site $h_2$ (solid) and the density of active sites (dashed) at critical points of bond DP and site DP.

Figure 2

The second-order Rényi entropy per site $h_2$ of the steady-state distribution of the bond DP in the thermodynamic limit. There is a clear cusp slightly above the bond-DP threshold, $p_c=0.644700185(5)$ [28]. The inset is the enlarged view near $p_2^{*}=0.6785(5)$.

Figure 3

Time evolution of entanglement entropy $H_E$ between the left and the right halves of the system from system size 128 to 8192. Solid and dashed lines represent $H_E$ at $p=0.678$ and 0.679, respectively. Solid and dashed lines in the inset represent singular values ${\xi }_0$, ${\xi }_1$, and ${\xi }_2$ at $p=0.678$ and 0.679 for 512 sites, respectively.

Figure 4

Crossover time ${\tau }_0$ between two levels ${\xi }_0$ and ${\xi }_1$ from $p=0.676$ to 0.681. $N$ denotes a system size. The inset shows a gap ${\mathrm{\Delta }}_0$ between ${\xi }_0({\tau }_0)$ and ${\xi }_1({\tau }_0)$.

Figure 5

Comparison between the second-order Rényi entropy per site $h_2$ (solid lines) and the absorbing-state entropy $-2\log [P_N(0)]/N$ (symbols).