Dissipation Mechanisms and Dissipative Phase Transitions of two coupled Fully Connected Quantum Ising models

This paper investigates dissipative phase transitions in two coupled fully-connected quantum Ising models, demonstrating that while jump operators satisfying detailed balance lead to equilibrium-like steady states and conventional critical behavior, local dissipators generate genuinely nonequilibrium steady states with a richer phase diagram featuring reentrant symmetry-broken phases.

The Gist

Imagine you have two giant, perfectly synchronized crowds of people (representing quantum spins) standing in a field. These crowds are connected to each other, and they are constantly interacting with a noisy, windy environment (the "bath").

The paper asks a simple question: How does the way the wind blows affect the behavior of these crowds? Specifically, does the wind just calm them down to a normal state, or does it create weird, new patterns that never happen in a calm room?

The researchers studied two different ways the "wind" (dissipation) interacts with the crowds. Here is the breakdown of their findings using everyday analogies.

The Setup: Two Crowds and a Windy Field

The system consists of two "Quantum Ising Models." Think of these as two crowds of people who want to agree on a direction to face (like all facing North or all facing South).

• The Crowd: They are "fully connected," meaning every person in the crowd can hear every other person. This makes them act like a single giant organism rather than individuals.
• The Wind (Dissipation): This is the environment trying to push the crowd around. In the real world, friction slows things down; in quantum physics, this "friction" is the environment stealing energy or adding noise.

The researchers looked at two different types of "wind":

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Scenario 1: The "Smart Thermostat" Wind (Self-Consistent Dissipation)

In this scenario, the wind is very intelligent. It knows exactly what the crowd is doing at every single moment. It adjusts its blowing direction based on the crowd's current "energy state."

• The Analogy: Imagine a thermostat that doesn't just blow cold air; it senses the exact temperature of the room and blows just enough cold air to bring the room to a specific, comfortable temperature. It follows the rules of thermodynamics perfectly (this is called "detailed balance").
• What Happens:
  • The Result: No matter how you start the crowd, this "smart wind" eventually cools them down into a state that looks exactly like a normal, calm equilibrium. It's as if the wind forces the crowd to settle into a predictable, thermal state (like a Gibbs state).
  • The "Quench" Experiments: The researchers tried two ways to disturb the system:
    1. Changing the Rules (Parametric Quench): They suddenly changed the rules of the game (like telling the crowd to face East instead of North). The crowd just slowly relaxed to the new rule. No drama.
    2. Changing the Temperature (Temperature Quench): They suddenly made the "wind" much hotter. Here, they saw something interesting: a Dynamical Phase Transition. For a brief moment, the crowd's reaction was sharp and jagged (non-analytic), like a sudden snap. But as the wind got stronger, this "snap" smoothed out into a gentle curve.
  • The Bottom Line: Even though the crowd was being pushed by the wind, the final result was just a standard, predictable state. The "smart wind" forced the system to behave like it was in a normal, closed room. The critical point (where the crowd changes from ordered to disordered) was exactly the same as if there were no wind at all.

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Scenario 2: The "Chaotic Fan" Wind (Local Pump-Loss Dissipation)

In this scenario, the wind is dumb and local. It doesn't know the crowd's overall state. It just randomly pushes people up or pulls them down based on simple local rules (like a fan blowing on individual people).

• The Analogy: Imagine a chaotic fan blowing on the crowd. It doesn't care about the group's temperature or energy. It just randomly shoves people up (pumping) or lets them fall (loss). It ignores the "smart rules" of the thermostat.
• What Happens:
  • The Result: This creates a Genuine Nonequilibrium State. The crowd never settles into a normal, calm state. They are stuck in a constant tug-of-war between the crowd's desire to agree and the fan's random shoving.
  • The Surprise (The Reentrant Phase): This is the most creative part of the discovery.
    • When the fan is weak, the crowd behaves normally (ordered).
    • When the fan gets stronger, it usually destroys the order (disordered).
    • But then, something weird happens: If the fan gets very strong, the crowd actually re-forms an ordered pattern!
  • The "Reentrant" Metaphor: Imagine a crowd trying to march in step.
    1. Quiet: They march in perfect step.
    2. Moderate Noise: The noise is loud enough to break their rhythm; they stumble and lose order.
    3. Extreme Noise: The noise becomes so chaotic and rhythmic in its own way that it accidentally forces them back into a synchronized march, but a different kind of march than before.
  • The Bottom Line: The wind didn't just destroy order; it created a new, strange window where order could exist again. This "reentrant phase" is bounded by two critical points. It proves that when you use "dumb" local dissipation, the system creates entirely new physics that doesn't exist in the real world.

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The Big Takeaway

The paper's main message is about how you define the "wind" (dissipation).

1. If the dissipation is "smart" (aligned with the system's energy levels), the system acts like a normal, closed system. It eventually forgets the chaos and settles into a standard thermal state. The phase transitions look exactly like they do in a quiet room.
2. If the dissipation is "local" and "dumb" (just pushing and pulling without regard for the system's energy), the system enters a genuine nonequilibrium state. This leads to rich, complex behaviors, like the "reentrant phase" where order returns only under specific, strong conditions.

In short: The nature of the noise determines whether the system behaves like a calm, predictable object or a chaotic, creative one that invents new phases of matter. The researchers showed that by changing how the environment interacts with the system, you can switch between "boring" equilibrium physics and "exciting" new nonequilibrium physics.

Technical Summary

1. Problem Statement

The paper investigates how the specific structure of dissipative processes (coupling to an environment) influences the emergence of equilibrium-like versus genuinely nonequilibrium critical behavior in open quantum systems.

• Context: Open quantum systems are governed by the competition between coherent unitary dynamics and incoherent dissipation. While Dissipative Phase Transitions (DPTs) are well-studied, it remains unclear how the nature of the dissipation (e.g., thermalizing vs. local pumping) dictates whether the system settles into a state resembling thermal equilibrium or a distinct nonequilibrium steady state (NESS).
• Model: The authors study a system of two coupled, fully connected (infinite-range interaction) one-dimensional Quantum Ising Models (QIMs). In the thermodynamic limit, this system is exactly solvable via a Self-Consistent Mean-Field (SCMF) approach, where the many-body density matrix factorizes into single-site density matrices acting on a local 4-dimensional Hilbert space (two spins per site).

2. Methodology

The authors employ the Lindblad Master Equation (LME) framework to describe the time evolution of the density matrix $\rho$. They compare two fundamentally different classes of dissipators (jump operators):

Case A: Self-Consistent Thermalization (Detailed Balance)

• Jump Operators: Defined in the instantaneous eigenbasis of the mean-field Hamiltonian ($H_{MF}$).
• Condition: Transition rates satisfy the detailed balance condition relative to the bath temperature.
• Dynamics: The LME is nonlinear and explicitly time-dependent because $H_{MF}$ depends self-consistently on $\rho$.
• Goal: To determine if this "thermal" dissipation drives the system to an equilibrium-like state and how it responds to quenches.

Case B: Local Pump-Loss Dynamics (Nonequilibrium)

• Jump Operators: Defined as local spin raising and lowering operators ($\sigma^{\pm}$) in the fixed computational basis, independent of the instantaneous eigenstates of $H_{MF}$.
• Condition: These operators do not satisfy detailed balance with respect to the mean-field Hamiltonian.
• Dynamics: The system is driven into a genuinely nonequilibrium steady state (NESS).
• Analysis Tool: Instead of integrating the full LME, the authors derive a closed set of Bloch-type equations for local magnetizations and correlation functions. Stability is analyzed via the Jacobian matrix of these coupled equations.

Quench Protocols

The authors analyze post-quench relaxation dynamics for both cases:

1. Parametric Quench: Sudden change in the transverse field ($h_1$).
2. Temperature Quench: Sudden change in the bath temperature ($T$).

3. Key Contributions and Results

A. Relaxation Dynamics and Quenches (Case A)

• Parametric Quench ($h_1$): The system exhibits conventional relaxation with damped oscillations toward a steady state. No dynamical phase transitions (DPTs) are observed; the fidelity and observables evolve smoothly.
• Temperature Quench ($T$): This protocol triggers a Dynamical Phase Transition (DPT).
  • For weak dissipation, observables (magnetization, inter-model correlator) and the fidelity rate function exhibit nonanalytic cusps at a critical time $t^*$.
  • As dissipation strength increases, these nonanalytic features are smoothed out, transitioning to a crossover behavior.
• Steady State: In both cases, the asymptotic steady state is a thermal Gibbs state of the mean-field Hamiltonian. Consequently, the critical point of the dissipative phase transition coincides exactly with the equilibrium quantum phase transition point.

B. Steady-State Phase Diagram (Case A)

• The transition is continuous (second-order). The order parameter (magnetization) vanishes continuously at the critical point $h_{1,c}$.
• Stability Analysis: Unlike linear dissipative systems where the Liouvillian gap closes at criticality, here the linear stability gap (derived from the Jacobian of the nonlinear LME) remains finite and positive. However, it exhibits a pronounced minimum (cusp) near the critical point, indicating "softening" of the relaxation mode without true critical slowing down (divergence).
• Diagnosis: The transition is confirmed via fidelity susceptibility (which peaks at $h_{1,c}$) and concurrence (entanglement), both showing signatures of criticality.

C. Steady-State Phase Diagram (Case B: Local Dissipators)

• Genuinely Nonequilibrium: The steady state is not a thermal Gibbs state. The critical point of the DPT deviates from the equilibrium critical point and depends on the dissipation strength.
• Reentrant Phase: For sufficiently strong system-bath coupling ($\gamma$), the phase diagram exhibits a reentrant behavior:
  • At low $h_1$, the system is in a symmetric (disordered) phase.
  • At intermediate $h_1$, a symmetry-broken (ordered) phase emerges.
  • At high $h_1$, the system returns to the symmetric phase.
• Mechanism: This reentrance arises from the competition between coherent dynamics (which requires a finite transverse field to generate coherences for ordering) and dissipation (which suppresses order at low fields and polarizes spins along $z$ at high fields).
• Transition Type: The transition is characterized by a pitchfork bifurcation of the steady-state solutions, confirming a continuous DPT.

4. Significance and Conclusions

1. Control of Criticality: The paper demonstrates that the structure of the dissipative channel is the primary determinant of whether an open quantum system exhibits equilibrium-like critical behavior or novel nonequilibrium phenomena.
   • Detailed balance $\rightarrow$ Equilibrium-like steady state, critical point pinned to equilibrium.
   • Local pump-loss $\rightarrow$ Genuinely nonequilibrium steady state, rich phase diagram with reentrant phases.
2. Beyond Liouvillian Gap: The study highlights limitations of the standard Liouvillian gap as a universal diagnostic for DPTs in nonlinear, self-consistent mean-field systems. In Case A, the gap does not close, yet a phase transition occurs. The authors advocate for linear stability analysis of the steady state and information-theoretic quantities (fidelity susceptibility) as robust alternatives.
3. Reentrant Phenomena: The discovery of a dissipation-driven reentrant phase in a mean-field model suggests that environmental coupling can stabilize ordered phases within specific parameter windows, a phenomenon absent in isolated equilibrium systems.
4. Generalizability: While based on a mean-field model, the authors argue these mechanisms are generic for systems with long-range interactions and provide a framework for understanding driven-dissipative criticality in more complex, finite-dimensional systems.

In summary, the work provides a rigorous comparison of how "thermal" versus "non-thermal" dissipation shapes the phase diagram and dynamical response of open quantum systems, revealing that dissipation is not merely a source of noise but a tunable resource that can fundamentally reshape quantum phases.