🔬 condensed matter

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

This paper investigates the steady-state phase transitions in a one-dimensional hybrid contact process, revealing both continuous and discontinuous transitions through mean-field approximations and the coherent anomaly method, which characterizes the system's nonclassical scaling behavior.

Original authors: Lin Shang, Shuai Geng, Xingli Li, Jiasen Jin

Published 2026-08-12

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CC BY 4.0

Original authors: Lin Shang, Shuai Geng, Xingli Li, Jiasen Jin

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a crowded dance floor where people can either stand still or start dancing. In the world of physics, this is similar to how particles behave in a system. Sometimes, a system settles into a quiet, "sleeping" state where nothing happens (called an absorbing state), and sometimes it stays lively and active forever (the active state). The big question scientists ask is: what makes the system switch from sleeping to dancing? In the classical world, this switch is usually smooth and predictable, like a dimmer switch slowly turning up the lights. But when we enter the quantum world—where particles can be in two places at once or influence each other instantly through "entanglement"—things get weird. Quantum rules can sometimes make the switch sudden and jumpy, or create a confusing middle ground where the system doesn't know whether to sleep or dance. Understanding these switches is crucial because they help us model everything from how diseases spread to how information travels through networks, and figuring out how quantum mechanics changes these rules could unlock new ways to control future technologies.

Now, let's look at what Lin Shang, Shuai Geng, and their team at Dalian University of Technology and other institutions discovered. They studied a specific setup called the one-dimensional hybrid contact process (HCP). Think of this as a long line of light switches (qubits) where each switch can be either "on" (occupied) or "off" (empty). In this hybrid model, the switches have two ways to interact with their neighbors: a "classical" way and a "quantum" way. The classical way is like a neighbor nudging you to turn on your light if they are already on, or turning off if they are off, but it happens randomly and messily. The quantum way is like a synchronized dance move where the neighbors flip their states together in perfect rhythm, without the random mess. Additionally, every switch has a chance to spontaneously turn off on its own, like a battery running out.

The researchers wanted to see what happens when they mix these two types of interactions. They used a series of mathematical "approximations"—basically, smart guesses that simplify the complex math—to map out the different states the system could be in. Their simulations revealed a fascinating landscape. They found that if they tweaked the strength of the quantum "dance moves" (the coherent interaction), the system could jump suddenly from a sleeping state to a lively one. This is a discontinuous phase transition, like a light switch snapping on rather than fading up. They even found a "bistable" region, a sort of limbo where the system could end up either sleeping or dancing depending on how it started, much like a ball balanced on a hilltop that could roll either way with the slightest nudge.

However, the story changes when they tweaked the "classical nudging" (the correlated decay rate). In this scenario, the transition from sleeping to dancing was smooth and continuous, just like the classical dimmer switch. This is a key finding: even though quantum mechanics is present, it doesn't always force the system to be jumpy. The team used advanced techniques to pinpoint exactly where these switches happen. They calculated that the critical point for the smooth transition occurs when the ratio of the classical nudging strength to the spontaneous decay rate is roughly 0.988 (specifically, $\kappa_c/\Gamma \approx 0.9884$ for one setting and 0.986 for another).

One of the most exciting parts of their work is that they didn't just find where the switch happens, but also how it behaves right at the edge. They discovered that the "critical exponent"—a number that describes how the system behaves right at the tipping point—is about 0.796 and 0.787 for their different settings. This suggests that even with quantum mechanics mixed in, the smooth transition shares a similar "personality" to the classical version, though the authors note they need even larger simulations to be absolutely sure.

Crucially, the paper rules out a few things. They explicitly show that this smooth transition is not caused by a "pitchfork bifurcation," which is a common mathematical reason for symmetry breaking in other systems. They also found that, unlike some other quantum models, this hybrid system doesn't show signs of "metastability" (staying in a fake, temporary state for a long time) in the way some pure quantum models do. Instead, the system's behavior is governed by a "Liouvillian gap" that slowly closes as the system gets bigger, indicating a slowing down of the dynamics right before the transition.

In short, this paper suggests that by mixing quantum and classical rules, we can create a system that exhibits both sudden, jumpy switches and smooth, gradual ones, depending on which knob we turn. The authors' simulations provide a detailed map of these behaviors, showing that while quantum coherence can create complex bistable regions and sudden jumps, it doesn't necessarily destroy the smooth, continuous nature of the transition when the classical interactions are strong enough. It's a reminder that in the quantum world, the outcome depends heavily on the balance between the orderly quantum dance and the chaotic classical nudges.

Technical Summary: Steady-state phase transition in one-dimensional hybrid contact process

Problem Statement
The paper investigates the nonequilibrium steady-state phase transitions and critical behavior of a one-dimensional hybrid contact process (HCP). This model serves as an interpolation between the classical contact process (CP) and the quantum contact process (QCP). In the HCP, the dynamics involve local decay, branching, and coagulation, where the branching and coagulation mechanisms can occur via both coherent (quantum) and incoherent (classical) channels. The study aims to determine how the introduction of quantum coherence affects the nature of the phase transition (continuous vs. discontinuous) and the critical exponents compared to the purely classical or purely quantum limits. Specifically, the authors seek to characterize the steady-state phases, identify the bifurcation mechanisms driving transitions, and extract true critical exponents for the continuous transition, which is known to belong to the directed percolation universality class in the classical limit.

Methodology
The authors employ a multi-faceted theoretical approach to analyze the system's dynamics, governed by a Lindblad master equation:

Single-Site Mean-Field (MF) Approximation: The system is initially analyzed using the Gutzwiller factorization ansatz, assuming the total density matrix is a product of identical single-site density matrices. This reduces the many-body problem to a set of Bloch equations, allowing for the identification of steady-state solutions and their stability via Jacobian eigenvalue analysis.
Cluster Mean-Field (CMF) Approximation: To incorporate short-range correlations neglected in the single-site MF approach, the authors utilize the CMF method. The density matrix is factorized into identical clusters of size $L$ . Interactions within the cluster are treated exactly, while interactions between clusters are approximated via effective fields. This allows for the calculation of critical points that converge toward the thermodynamic limit as cluster size increases.
Liouvillian Spectral Analysis: The relaxation dynamics are characterized by diagonalizing the Liouvillian superoperator for finite-size systems. The Liouvillian gap (the magnitude of the largest non-zero real part of the eigenvalues) is analyzed to detect critical slowing down and potential metastability.
Coherent Anomaly Method (CAM): To extract the true critical exponents of the continuous phase transition, the authors apply the CAM. This method systematically analyzes the scaling of the coherent anomaly (the amplitude of the order parameter) across different cluster sizes in the CMF approximation to extrapolate the non-classical critical exponent and the true critical point in the thermodynamic limit.

Key Results

Phase Diagram and Bifurcations: The single-site MF analysis reveals a phase diagram with three regions: an absorbing phase (zero particle density), an active phase (non-zero particle density), and a bistable region.
- Discontinuous Transition: As the coherent interaction strength ( $\Omega$ ) increases, the transition from the absorbing phase to the bistable region is found to be discontinuous. This is driven by a saddle-node bifurcation where an unstable active branch emerges, leading to a jump in the order parameter.
- Continuous Transition: As the incoherent correlated decay rate ( $\kappa$ ) increases, the transition from the absorbing phase to the active phase is continuous. This transition does not arise from a pitchfork bifurcation associated with spontaneous symmetry breaking (unlike in some dissipative spin models) but rather from the intersection of the absorbing solution with the stable active branch.
Liouvillian Spectrum: The analysis of the Liouvillian gap shows a progressive closing as system size increases, indicating critical slowing down. The spectrum exhibits a non-monotonic behavior with respect to $\kappa$ due to the crossing of two low-lying spectral branches. Unlike the QCP, the HCP does not exhibit isolated slow modes or clear signatures of metastability in the accessible system sizes.
Critical Points and Exponents:
- Using CMF with cluster sizes up to $L=10$ , the critical points ( $\kappa_c$ ) for the continuous transition were extrapolated to the thermodynamic limit, yielding values of $\kappa_c/\Gamma \approx 0.9881$ (for $\Omega/\Gamma=0.1$ ) and $0.9857$ (for $\Omega/\Gamma=0.4$ ).
- Applying the CAM to these results, the true critical exponents ( $\beta^*$ ) for the order parameter were estimated. For $\Omega/\Gamma=0.1$ , $\beta^* \approx 0.7962$ , and for $\Omega/\Gamma=0.4$ , $\beta^* \approx 0.7865$ .
- The authors note that these exponents are distinct from the mean-field value ( $\beta_{mf}=1$ ) and appear consistent across different $\Omega$ values, though larger cluster sizes are required for higher precision.

Significance and Claims
The paper claims to provide a comprehensive characterization of the steady-state phase transitions in a hybrid quantum-classical system. The primary significance lies in demonstrating that the continuous nature of the absorbing-to-active transition in the classical contact process is preserved in the hybrid model, despite the presence of quantum coherence. The study highlights that this continuous transition is fundamentally different from those driven by spontaneous symmetry breaking (pitchfork bifurcations). Furthermore, by successfully applying the Coherent Anomaly Method to a dissipative quantum system, the authors extract non-classical critical exponents, suggesting that the HCP may share a universality class with the classical directed percolation, albeit modified by quantum effects. The work underscores the subtle role of quantum coherence in modifying dynamical relaxation and critical behavior without necessarily driving the system into discontinuous behavior or bistability in the parameter regime studied.

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