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

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.

💡 Research Summary

The paper investigates the steady‑state phase transition of the one‑dimensional quantum contact process (QCP), a paradigmatic non‑equilibrium many‑body model that combines coherent branching/coagulation (rate Ω) with local incoherent decay (rate Γ). The dynamics are described by a Lindblad master equation, and the authors focus on the thermodynamic limit (L → ∞) where the order parameter is the average excitation density ⟨ \bar n ⟩.

First, a single‑site mean‑field (MF) treatment is performed by factorizing the density matrix into identical single‑site components. This yields three fixed points: the absorbing state (⟨ n ⟩=0) and two active states with ⟨ n ⟩>0 that exist for Ω/Γ>1/√2. The MF equations display a saddle‑node (fold) bifurcation: the two active branches emerge at S₁ (Ω=Γ/√2, ⟨ n ⟩≈0.25) and merge with the absorbing branch at S₂ (Ω→∞, ⟨ n ⟩→0). The coexistence of absorbing and active solutions between S₁ and S₂ resembles hysteresis and suggests a discontinuous transition, in contrast to the supercritical pitchfork bifurcation of continuous directed‑percolation‑type transitions.

To go beyond MF, the Liouvillian spectrum is computed for finite chains. The Liouvillian gap μ₀ (the magnitude of the largest negative real part of the eigenvalues) closes at Ω/Γ≈5.83 as L→∞, indicating critical slowing down. Moreover, the first non‑zero eigenvalue remains isolated from the rest of the spectrum, producing a clear separation of time scales: a long‑lived metastable plateau appears before the system finally relaxes to the absorbing state. This metastability is invisible in the single‑site MF picture.

The authors then introduce a cluster mean‑field (CMF) approach. The chain is partitioned into clusters of size L; intra‑cluster dynamics are treated exactly, while inter‑cluster couplings are replaced by self‑consistent effective fields (Fₙ, Fₓ). Because the x‑component of the Bloch vector vanishes for physical steady states, only the density field Fₙ is retained. The self‑consistency condition ⟨ n_j ⟩=Tr