Steady-state phase transition in one-dimensional quantum contact process

Abstract

We investigate the steady-state phases of the one-dimensional quantum contact process model. We present the Liouvillian gap in the thermodynamic limit and uncover the metastability of the system. Exploiting the mean-field approximations with a novel self-consistent condition based on the effective field, we capture the avoid the interference of the metastable state. We show the feature of saddle-node bifurcation of the order parameter revealing the discontinuous phase transition of the steady state and extract the transition point for infinite-size system. We show the monotonic decreasing of the steady-state magnetic susceptibility by the linked-cluster expansion, which does not support the divergence of the correlation length at the vicinity of the transition point. The present results may be tested in the quantum simulator of Rydberg atoms.

Figures (7)

• Figure 1: (a) The schematic diagram of the QCP in one dimension. The spin on each site coherently flips with a rate $\Omega$ between the occupied (spin-up) and empty (spin-down) state conditioned on its neighboring sites being occupied. Each spin incoherently decays to the empty state at a rate $\Gamma$. (b) The distribution of the steady states in the $y$-$z$ plane of the Bloch sphere. The blue (green) circles represent the stable (unstable) steady states and the red circles represent the absorbing phase $|\downarrow\rangle$. (c) The three branches of steady-state population $\langle n \rangle_{\text{ss}}$ as functions of $\Omega$ within the single-site mean-field approximation.
• Figure 2: (a) Liouvillian gap. The continuous lines and discrete symbols denote the Liouvillian gap for finite-size system with $L=5,6,$.., and $12$ (from top to bottom). The connected black circles denote the extrapolated Liouvillian gap in the limit of $L\rightarrow\infty$. The gap closes at $\Omega/\Gamma\approx5.83$. (b) The real part of the Liouvillian eigenvalue with different $L$ for $\Omega/\Gamma=5.8$. The black, blue, and red circles represent the zero, first largest negative, and second largest negative real parts of the eigenvalues. The triangles represent the real parts of the rest eigenvalues which are bounded by $\text{Re}(\mu)=-0.5$ (red circles). The inset shows the time-evolution of the averaged population $\langle \bar{n}(t) \rangle$ with the CMF approximation for $\Omega/\Gamma=5.03$ (red dotted-dashed line) and $5.035$ (blue dashed line). The initial state is the full active state $|\uparrow\uparrow...\uparrow\rangle$ and $L=11$.
• Figure 3: (a) The $\langle \hat{n}_1 \rangle_\text{ss}$ as a function of the effective field $F^n$ for the CMF simulation with $L=7$. The slope of the thin solid line is one. The intersection of the thin solid line and the curve of $\langle n_1 \rangle_{\text{ss}}$ denotes the steady-state solution. The intersection denoted by full (empty) magenta diamond indicates the stable (unstable) steady-state solution. (b) The time-evolution trajectories of the reduced state of the boundary site originating from different initial states $|\psi\rangle=\bigotimes_j{|\psi\rangle_j}$ where the identical $|\psi_j\rangle$s locate at the surface of the Bloch sphere. The diamonds and the circle correspond to the intersections in (a). (c) The branches of $\langle \bar{n} \rangle_\text{ss}$ as a function of $\Omega/\Gamma$ in the CMF approximation with $L=7,9$ and $11$. The upper solid (lower dashed) curves denote the stable (unstable) active states. The red solid line denotes the absorbing state. (d) The resulting CMF transition point $\Omega_c(L) / \Gamma$ for different $L$. The green and blue symbols denote the odd- and even-size clusters, respectively. The square in magenta marks the transition point $\Omega_c/\Gamma = 5.83$ through the Liouvillian gap analysis in Fig. \ref{['fig2_Lgap']}(a).
• Figure 4: (a) The weight $w_{\chi}(L)$ of the cluster with size $L$ (in logarithmic scale) as a function of $\Omega/\Gamma$. The curves above (blow) the $x$ axis denote the $w_{\chi}(L)$ with odd (even) $L=1,2,3,...,$ and $12$. The two arrows points to the directions along which $L$ increases. (b) The log-log plot of the magnitude $|w_{\chi}(L)|$ scaling with $L$ for various $\Omega/\Gamma$, indicated by the three vertical dashed lines in (a). For $\Omega/\Gamma=2$ the weight scales as $w_{\chi}(L)\approx 8.18e^{-0.34L}$, for $\Omega/\Gamma=6$ the weight scales as $w_{\chi}(L)\approx46.39L^{-1.77}$, and for $\Omega/\Gamma=8$ the weight scales as $w_{\chi}(L)\approx30.13L^{-1.55}$. The dashed line in black represents a power-low function with power $n=-2$ and is plotted as a guide to the eye. (c) The LCE results of $\chi$ with the sum up to various orders of $R$. The connected circles in black represent the extrapolated result with $R\rightarrow\infty$.
• Figure 5: The maximal real part of the eigenvalue of Jacobian Matrix $M_p$ as a function of $\Omega/\Gamma$. The magenta pentagram marks the saddle-node point $S_1$ in Fig. 1 of the main text. Since $\text{Re}(\lambda_0)=-0.5$, the $\langle \hat{\hat{\sigma}} \rangle_{\text{ss},0}$ is stable for all $\Omega$. The $\langle \hat{\hat{\sigma}} \rangle_{\text{ss},+}$ becomes stable after $\Omega$ crosses the saddle-node point. The inset shows the details of $\text{Re}(\lambda_\pm)$ near the saddle-node point bifurcation.