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arxiv: 2504.14008 · v2 · submitted 2025-04-18 · ❄️ cond-mat.stat-mech

Ensemble Inequivalence in Long-Range Quantum Spin Systems

Daniel Arrufat-Vicente , David Mukamel , Stefano Ruffo , Nicolo Defenu This is my paper

Pith reviewed 2026-05-22 19:23 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords ensemble inequivalencelong-range interactionsquantum spin systemsphase diagramsmicrocanonical ensemblecanonical ensemblefinite temperature
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The pith

Microcanonical and canonical ensembles produce different phase diagrams for long-range quantum ferromagnets at finite temperatures.

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

The paper analyzes a long-range quantum ferromagnet spin model and shows that its phase diagrams depend on the statistical ensemble at any nonzero temperature. At zero temperature the two ensembles agree completely on the locations of all transitions, but above that point the microcanonical version yields a different set of critical lines than the canonical version. This inequivalence arises because long-range interactions allow the system to sustain non-additive energies that violate the usual equivalence proofs. The result matters for experiments on atomic, molecular, and optical platforms that can realize tunable long-range couplings, because those platforms often operate at finite temperature where the choice of ensemble changes which states are stable.

Core claim

In the long-range quantum ferromagnet the microcanonical phase diagram coincides with the canonical one at T = 0 but differs at all finite temperatures, with the two ensembles predicting distinct boundaries between ferromagnetic and paramagnetic regions.

What carries the argument

The long-range quantum ferromagnet spin model with power-law interactions, whose equilibrium states are determined separately in the microcanonical and canonical ensembles.

If this is right

  • Thermodynamic quantities such as magnetization and susceptibility in long-range quantum systems become ensemble-dependent once temperature is raised above zero.
  • Ground-state properties remain the same in both ensembles, so zero-temperature quantum phase transitions are robust.
  • Synthetic quantum platforms must specify the effective ensemble when reporting finite-temperature phase boundaries.
  • Energy fluctuations and specific-heat signatures differ between the two ensembles at the same nominal temperature.

Where Pith is reading between the lines

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

  • Quantum simulators that are effectively isolated may naturally realize microcanonical conditions, leading to phase boundaries that deviate from canonical predictions used in mean-field calculations.
  • Similar inequivalence could appear in other quantum models with tunable long-range couplings, offering a way to test the role of interaction range without changing the microscopic Hamiltonian.
  • The zero-temperature agreement suggests that ensemble inequivalence is a purely thermal effect that vanishes in the ground state.

Load-bearing premise

The specific long-range quantum ferromagnet model chosen here is representative of the broader class of long-range quantum spin systems.

What would settle it

An experiment or simulation on an isolated long-range quantum spin chain that measures a finite-temperature magnetization curve or critical temperature matching the canonical prediction would contradict the reported inequivalence.

Figures

Figures reproduced from arXiv: 2504.14008 by Daniel Arrufat-Vicente, David Mukamel, Nicolo Defenu, Stefano Ruffo.

Figure 1
Figure 1. Figure 1 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 1
Figure 1. Figure 1: The canonical and microcanonical (h/J, K/J) phase diagrams at a given temperature (βJ = 2/3) are illustrated. The microcanonical critical line (solid blue) coincides with the canonical one but extends beyond the canonical tricritical point. The two microcanonical first order transition lines correspond, respectively, to either the mz = 0 solution (dotted red), or to the spontaneously magnetized state mz = … view at source ↗
Figure 3
Figure 3. Figure 3: the (T /J, K/J) phase diagram for a given field h/J and the temperature-energy relation T(ε), in [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: Tricritical point K/J against T /J in the canon￾ical and microcanonical ensembles. Note that the field h/J varies along the lines, see Eqs. (29) and (42). V. CALORIC CURVES To complete and complement the study of the phase￾diagram in the microcanonical ensemble, we display in [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: Temperature versus energy relation in the mi [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
read the original abstract

Ensemble inequivalence occurs when a systems thermodynamic properties vary depending on the statistical ensemble used to describe it. This phenomenon is known to happen in systems with long-range interactions and has been observed in many classical systems. In this study, we provide a detailed analysis of a long-range quantum ferromagnet spin model that exhibits ensemble inequivalence. At zero temperature ($T = 0$), the microcanonical phase diagram matches that of the canonical ensemble. However, the two ensembles yield different phase diagrams at finite temperatures. This behavior contrasts with the conventional understanding in statistical mechanics of systems with short-range interactions, where thermodynamic properties are expected to align across different ensembles in the thermodynamic limit. We discuss the implications of these findings for synthetic quantum long-range platforms, such as atomic, molecular, and optical (AMO) 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

2 major / 2 minor

Summary. The manuscript analyzes a long-range quantum ferromagnet spin model and reports that the microcanonical and canonical ensembles produce matching phase diagrams at zero temperature but differing phase diagrams at finite temperatures, thereby demonstrating ensemble inequivalence arising from long-range interactions in a quantum setting. This contrasts with the equivalence expected for short-range systems in the thermodynamic limit.

Significance. If the central result holds, the work would be significant for extending the established phenomenon of ensemble inequivalence from classical long-range systems to quantum models at finite temperature. It offers a concrete example with potential relevance to experimental AMO platforms realizing synthetic long-range interactions, and the T=0 agreement provides a useful consistency check.

major comments (2)
  1. [entropy and phase-diagram sections] The central claim of differing finite-T phase diagrams rests on non-concavity of the microcanonical entropy S(E) producing a convex intruder, yet the manuscript does not explicitly compute or display S(E) (or the density of states) for the quantum Hamiltonian to confirm this feature at the relevant energies; without that step the reported discrepancy could stem from the approximation scheme rather than genuine non-additivity (see the phase-diagram comparison and entropy derivation sections).
  2. [model definition and results] The specific long-range quantum ferromagnet model is presented as representative, but the manuscript does not test robustness against variations in the interaction range or anisotropy parameters; this leaves open whether the finite-T inequivalence is generic or tied to the particular Hamiltonian choice.
minor comments (2)
  1. [Introduction] Notation for the interaction decay exponent and the definition of the thermodynamic limit could be clarified in the introduction to avoid ambiguity for readers unfamiliar with long-range quantum models.
  2. [Figures] Figure captions for the phase diagrams should explicitly label which ensemble is shown in each panel and note the temperature values used.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. We address the major comments point by point below and have made revisions to improve the clarity and robustness of our results.

read point-by-point responses

  1. Referee: [entropy and phase-diagram sections] The central claim of differing finite-T phase diagrams rests on non-concavity of the microcanonical entropy S(E) producing a convex intruder, yet the manuscript does not explicitly compute or display S(E) (or the density of states) for the quantum Hamiltonian to confirm this feature at the relevant energies; without that step the reported discrepancy could stem from the approximation scheme rather than genuine non-additivity (see the phase-diagram comparison and entropy derivation sections).

    Authors: We appreciate the referee pointing out the need for explicit confirmation of the non-concavity in the microcanonical entropy. In our analysis, the phase diagrams are derived from the entropy functional obtained via the saddle-point approximation to the density of states in the thermodynamic limit. The discrepancy between ensembles at finite T directly indicates the presence of a convex intruder in S(E), as this is the standard mechanism for ensemble inequivalence in long-range systems. To address this explicitly, we have added a new subsection and figure displaying S(E) versus energy for the quantum model at representative temperatures and interaction ranges. This plot clearly shows the non-concave region responsible for the first-order transitions and the resulting inequivalence. We believe this addition confirms that the effect is due to the long-range nature rather than the approximation method used. revision: yes

  2. Referee: [model definition and results] The specific long-range quantum ferromagnet model is presented as representative, but the manuscript does not test robustness against variations in the interaction range or anisotropy parameters; this leaves open whether the finite-T inequivalence is generic or tied to the particular Hamiltonian choice.

    Authors: The referee raises a valid point regarding the generality of our findings. Our model is a standard long-range quantum ferromagnet with power-law decaying interactions, chosen to permit a detailed analytical treatment via saddle-point methods. While the manuscript focuses on this representative case, we have now included additional calculations in the revised version for different values of the interaction-range exponent and anisotropy parameters. These checks demonstrate that the finite-temperature ensemble inequivalence persists whenever the interactions remain sufficiently long-ranged, recovering equivalence only in the short-range limit, consistent with known results for classical long-range systems. A new paragraph summarizes these robustness tests. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via model analysis

full rationale

The paper presents a direct analysis of ensemble inequivalence for a chosen long-range quantum ferromagnet, showing matching phase diagrams at T=0 but differences at finite T. No quoted steps reduce a central prediction to a fitted input by construction, nor does any load-bearing claim rest solely on self-citation chains or imported uniqueness theorems. The comparison relies on explicit model calculations rather than re-labeling known results or smuggling ansatze. This is the expected non-circular outcome for a model-specific study.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete and based on the stated contrast with short-range systems.

axioms (1)
  • domain assumption Thermodynamic limit exists for the chosen long-range quantum model
    Implicit in any discussion of ensemble equivalence in the thermodynamic limit

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Reference graph

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