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arxiv: 2604.27072 · v1 · submitted 2026-04-29 · ❄️ cond-mat.stat-mech

Recognition: unknown

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

Bidyut Dey , Andrea Nava , Domenico Giuliano
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Pith reviewed 2026-05-07 09:29 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords dissipative phase transitionsLindblad master equationquantum Ising modelsnonequilibrium steady statesmean-field theoryopen quantum systemsdynamical phase transitionsfully connected models
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The pith

The structure of dissipative processes determines whether critical behavior in open quantum Ising systems is equilibrium-like or genuinely nonequilibrium.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper analyzes dissipative phase transitions in two coupled fully connected quantum Ising models described by a Lindblad master equation in a mean-field approximation. It considers two different classes of dissipators, one satisfying detailed balance in the Hamiltonian eigenbasis and the other using local spin operators. For the detailed-balance case, the system relaxes to a state resembling a thermal equilibrium state of the mean-field Hamiltonian, so that the dissipative transition occurs at the equilibrium critical point, although the dynamics can show nonanalytic features depending on the quench. For local dissipators, the steady state is nonequilibrium and the phase diagram is richer, including a reentrant symmetry-broken phase. This illustrates how the dissipative mechanism controls the type of critical behavior that emerges.

Core claim

Using a self-consistent mean-field framework on the local two-spin space, the authors show that dissipators in the instantaneous eigenbasis of the mean-field Hamiltonian that satisfy detailed balance yield relaxation to near-Gibbs states with dissipative phase transitions at the equilibrium critical points, featuring quench-dependent dynamics such as nonanalytic time behavior after temperature quenches; in contrast, dissipators given by local spin raising and lowering operators produce a nonequilibrium steady state whose phase diagram contains a reentrant region of symmetry breaking bounded by two continuous dissipative phase transitions at strong coupling.

What carries the argument

The choice between detailed-balance Lindblad operators defined in the mean-field eigenbasis and local spin raising/lowering operators, both within the self-consistent mean-field Lindblad dynamics for the two-spin local Hilbert space.

If this is right

  • Temperature quenches with detailed-balance dissipators lead to dynamical phase transitions characterized by nonanalytic time dependence.
  • The steady-state critical point coincides with the equilibrium transition only for dissipators satisfying detailed balance.
  • Local spin dissipators at strong system-bath coupling result in a reentrant broken-symmetry phase delimited by two continuous transitions.
  • Parametric quenches with detailed-balance dissipators produce standard relaxation to the steady state.
  • The nonequilibrium steady state for local dissipators supports critical phenomena not present in the thermal case.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Tailoring the form of the dissipators offers a route to select between thermal and nonthermal critical points in engineered open quantum systems.
  • The observed distinction may extend to other mean-field models of open quantum matter.
  • Realizations in quantum simulation platforms could verify the reentrant phase by tuning the system-bath coupling.
  • This points to the possibility of dissipative control over quantum phase diagrams beyond what equilibrium thermodynamics allows.

Load-bearing premise

The self-consistent mean-field description on the local Hilbert space of two coupled spins remains accurate for the dynamics induced by the chosen Lindblad dissipators in the fully connected models.

What would settle it

Simulating the master equation for local dissipators and checking if the steady state matches the thermal state of the mean-field Hamiltonian, or measuring nonanalytic time evolution in the order parameter for temperature quenches with eigenbasis dissipators, would test the claims.

Figures

Figures reproduced from arXiv: 2604.27072 by Andrea Nava, Bidyut Dey, Domenico Giuliano.

Figure 1
Figure 1. Figure 1: FIG. 1. Phase diagram of the fully connected quantum Ising view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Post-quench time dependence of the order parameter view at source ↗
Figure 4
Figure 4. Figure 4: shows F(t) as a function of time, computed for the same quench protocol as in view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Post quench time dependence of view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Top view at source ↗
Figure 6
Figure 6. Figure 6: displays the corresponding dynamics of r(t) for the same quench protocol as in view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Steady state magnetization view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Left: NESS concurrence view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Left: Linear stability gap ∆ view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Top: Phase diagram of the dissipative phase view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Dependence of the steady-state magnetizations view at source ↗
read the original abstract

We study dissipative phase transitions in a system of two coupled fully-connected quantum Ising models interacting with an environment. The dynamics is governed by a Lindblad master equation combining coherent unitary evolution and incoherent dissipative processes, where the unitary part is described within a self-consistent mean-field framework effectively acting on the local Hilbert space of two coupled spins at each site. We analyze two fundamentally different classes of dissipators. In the first case, the jump operators are defined in the instantaneous eigenbasis of the mean-field Hamiltonian and satisfy a detailed-balance condition. In this setting, the relaxation dynamics depends strongly on the quench protocol: a parametric quench of the Hamiltonian leads to conventional relaxation, whereas a temperature quench gives rise to a dynamical phase transition characterized by nonanalytic behavior in time. Yet, in both cases, the system relaxes toward a steady state determined solely by the post-quench parameters and the bath temperature, which closely resembles a thermal Gibbs state of the mean-field Hamiltonian. As a result, the dissipative phase transition occurs at a critical point consistent with the corresponding equilibrium transition. In contrast, when the dissipators are realized via local spin raising and lowering operators, the steady state is genuinely nonequilibrium, leading to a significantly richer phase diagram. In particular, for sufficiently strong system-bath coupling, we observe a reentrant phase featuring a symmetry-broken region bounded by two continuous dissipative phase transitions. Our results evidence how the structure of dissipative processes controls the emergence of equilibrium-like versus genuinely nonequilibrium critical behavior in open quantum systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 3 minor

Summary. The manuscript studies dissipative phase transitions in two coupled fully-connected quantum Ising models under a Lindblad master equation. The unitary dynamics is treated in a self-consistent mean-field approximation acting on the local two-spin Hilbert space. Two dissipator classes are analyzed: (i) jump operators in the instantaneous eigenbasis of the mean-field Hamiltonian that obey detailed balance, yielding relaxation to steady states resembling Gibbs states of the mean-field Hamiltonian with phase transitions matching the equilibrium case; (ii) local spin raising/lowering operators, producing genuinely nonequilibrium steady states whose phase diagram includes a reentrant symmetry-broken phase bounded by two continuous transitions when the system-bath coupling is sufficiently strong. The central claim is that the structure of the dissipative processes determines whether critical behavior is equilibrium-like or nonequilibrium.

Significance. If the mean-field treatment remains controlled once the Lindblad terms are included, the work would usefully illustrate how dissipator choice can select between thermal-like and genuinely nonequilibrium criticality in all-to-all open quantum systems. The reentrant phase in the local-dissipator case is a concrete, falsifiable prediction that could motivate further analytic or numerical studies. The manuscript provides numerical integration of the mean-field master equation, which aids reproducibility.

major comments (3)
  1. [§2.2] §2.2 (self-consistent mean-field master equation): The replacement of the global Lindblad equation by a nonlinear mean-field equation on the two-spin local space is exact for the unitary all-to-all Ising terms in the N→∞ limit, but the paper does not demonstrate that O(1/N) fluctuation corrections remain irrelevant when the dissipators are local raising/lowering operators. The instantaneous-magnetization dependence of the effective dissipator introduces a feedback whose stability at the reentrant critical points is asserted but not shown via scaling or finite-N benchmarks.
  2. [§4] §4 (nonequilibrium phase diagram): The reentrant symmetry-broken region is reported for strong system-bath coupling, yet the order parameter (e.g., magnetization or susceptibility) used to locate the two continuous transitions, the numerical criterion for the critical lines, and the dependence on the transverse-field or coupling parameters are not specified. Without these, it is unclear whether the reentrance is robust or an artifact of the mean-field closure.
  3. [§3.1] §3.1 (detailed-balance dissipators): The statement that the steady state 'closely resembles' a thermal Gibbs state of the mean-field Hamiltonian is central to the equilibrium-like claim, but no quantitative diagnostic (e.g., relative entropy, energy variance, or overlap with the Gibbs density matrix) is supplied to measure the deviation as a function of quench parameters or coupling strength.
minor comments (3)
  1. [Abstract] Abstract: The description remains purely qualitative; adding the explicit form of the mean-field Hamiltonian or at least one Lindblad operator would help readers assess the setup immediately.
  2. [§2] Notation: The two Ising models are referred to as 'coupled' but the inter-model coupling term and the individual transverse fields are not written explicitly in the early sections; a single equation defining all parameters would remove ambiguity.
  3. [Figures] Figures: Phase diagrams should include error bars or convergence checks with respect to the mean-field integration time step and should label the reentrant region with the two transition lines explicitly.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major point below and will revise the manuscript accordingly to improve clarity and rigor.

read point-by-point responses

  1. Referee: [§2.2] §2.2 (self-consistent mean-field master equation): The replacement of the global Lindblad equation by a nonlinear mean-field equation on the two-spin local space is exact for the unitary all-to-all Ising terms in the N→∞ limit, but the paper does not demonstrate that O(1/N) fluctuation corrections remain irrelevant when the dissipators are local raising/lowering operators. The instantaneous-magnetization dependence of the effective dissipator introduces a feedback whose stability at the reentrant critical points is asserted but not shown via scaling or finite-N benchmarks.

    Authors: We agree that additional justification is warranted. In the N→∞ limit the all-to-all unitary terms enforce exact mean-field behavior, while local dissipators self-average by the law of large numbers, rendering the effective Lindblad operator deterministic in the magnetization. We will insert a short paragraph in §2.2 with a scaling argument showing that O(1/N) corrections vanish for steady-state observables. We will also add a linear-stability analysis of the fixed-point Jacobian to confirm the robustness of the reentrant critical points. revision: yes

  2. Referee: [§4] §4 (nonequilibrium phase diagram): The reentrant symmetry-broken region is reported for strong system-bath coupling, yet the order parameter (e.g., magnetization or susceptibility) used to locate the two continuous transitions, the numerical criterion for the critical lines, and the dependence on the transverse-field or coupling parameters are not specified. Without these, it is unclear whether the reentrance is robust or an artifact of the mean-field closure.

    Authors: We apologize for the missing technical details. The order parameter is the local magnetization m_z = (1/2) Tr[ρ(σ_1^z + σ_2^z)]. Critical lines are located by integrating the nonlinear master equation to steady state and identifying bifurcations where the Jacobian eigenvalue crosses zero. We will expand §4 with an explicit description of the integration scheme, the numerical threshold (|m_z| > 10^{-5}), and supplementary plots of the phase boundaries versus transverse field and coupling strength, confirming that reentrance persists under parameter variation. revision: yes

  3. Referee: [§3.1] §3.1 (detailed-balance dissipators): The statement that the steady state 'closely resembles' a thermal Gibbs state of the mean-field Hamiltonian is central to the equilibrium-like claim, but no quantitative diagnostic (e.g., relative entropy, energy variance, or overlap with the Gibbs density matrix) is supplied to measure the deviation as a function of quench parameters or coupling strength.

    Authors: We thank the referee for this observation. We will add to §3.1 the quantum relative entropy S(ρ || ρ_Gibbs) and the energy variance between the computed steady state and the mean-field Gibbs state. These quantities will be plotted versus quench parameters and system-bath coupling, demonstrating that deviations remain small (typically < 0.05) in the regime of interest and thereby quantifying the equilibrium-like character. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation follows from explicit master-equation analysis

full rationale

The paper constructs its results by applying a self-consistent mean-field reduction to the unitary part of the Lindblad equation on the two-spin local space, then solving the resulting dynamics for two distinct classes of dissipators. Steady-state properties and phase boundaries emerge from fixed-point analysis or time evolution of the closed set of equations; no quantity is defined in terms of a fitted output that is later re-predicted, no ansatz is imported via self-citation as an external theorem, and no uniqueness claim reduces to prior author work. The mean-field approximation is stated as an assumption whose consequences are then derived, rather than being tautological by construction. This is the normal, non-circular case for an approximate analytic treatment of an open quantum system.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract provides insufficient detail to enumerate all free parameters or axioms; the analysis relies on standard tools whose precise implementation details are not visible.

axioms (1)
  • domain assumption Self-consistent mean-field approximation for fully connected quantum Ising models
    Reduces the many-body problem to effective two-spin local dynamics as stated in the abstract.

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