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REVIEW 3 major objections 7 minor 94 references

Temperature beyond equilibrium defined by energy coherence, not entropy

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2026-07-10 03:31 UTC pith:OMABEYQ3

load-bearing objection Genuine conceptual advance in nonequilibrium quantum temperature, but the thermodynamic-limit argument has a gap that matters. the 3 major comments →

arxiv 2607.08655 v1 pith:OMABEYQ3 submitted 2026-07-09 quant-ph cond-mat.stat-mechmath-phmath.MP

Temperature Beyond Equilibrium in Isolated Quantum Many-Body Systems and Their Subsystems

Maurizio Fagotti This is my paper
classification quant-ph cond-mat.stat-mechmath-phmath.MP PACS 05.30.-d05.70.Ln03.65.Yz
keywords temperatureenergythermodynamicequilibriumquantumstatestatescoherence
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper proposes that temperature for an isolated quantum many-body system far from equilibrium should be defined not by analogy with equilibrium, but by identifying where the state sits within the family of states sharing its coherent energy content. The key insight is that a non-stationary quantum state carries two distinct kinds of energy fluctuations: classical population fluctuations, which have the usual thermodynamic meaning, and coherent fluctuations arising from quantum superpositions between energy sectors, which drive time evolution and have no classical analogue. The author shows that states can be organized into leaves of a foliation, where each leaf fixes the coherent energy content. On each leaf, a canonical flow parameterized by an inverse temperature can be defined. Crucially, this inverse temperature is obtained through a principle of minimum discrimination information (relative entropy minimization), not through the usual principle of maximum entropy. The result is that, out of equilibrium, the inverse temperature is generally not the derivative of thermodynamic entropy with respect to energy. The framework extends to subsystems, where the temperature cannot be read off from the reduced density matrix alone but requires knowledge of the dynamical trajectory, with the rate of change of subsystem temperature being a genuinely local quantity.

Core claim

The central object is the minimum-variance foliation of quantum state space, which groups states by their coherent energy content. On each leaf of this foliation, the author defines a canonical pseudolocal flow whose coordinate plays the role of inverse temperature. This coordinate is fixed by solving an equation (Eq. 45) that identifies the reference point invariant under rescaling of the Hamiltonian, analogous to the infinite-temperature state. The construction replaces the maximum entropy principle with the principle of minimum discrimination information, yielding an inverse temperature that is generally different from the entropy derivative dS/dE. Numerical evidence on spin chains with 8

What carries the argument

The construction rests on the minimum-variance foliation, defined via the solution of a Lyapunov equation (Eq. 7) that separates population-like from coherent energy fluctuations. In the thermodynamic limit, the foliation is characterized by the harmonic conjugate of the Hamiltonian (Eq. 19), and leaf membership is tested through the vanishing of a transverse functional (Eq. 29). The canonical flow on a leaf is generated by a pseudolocal charge (Eq. 47) combining the symmetrized connected correlation of the Hamiltonian with the commutator of its harmonic conjugate. The leaf canonical ensemble (Eq. 46) is built by dressing with the reference state's square root, and inverse temperature is the

Load-bearing premise

The construction assumes that the canonical pseudolocal flow preserves exponential clustering of correlations in a neighborhood of the reference state, which is proven only for a finite interval of inverse temperatures. Extending this to all finite temperatures is stated as an expectation by analogy with equilibrium systems, and if it fails, the leaf canonical states may not be well-defined in the thermodynamic limit.

What would settle it

If numerical or analytical evidence shows that the canonical flow on non-commuting leaves does not preserve exponential clustering beyond a finite temperature window, or that the leaf-typicality hypothesis fails (meaning local observables on a leaf do not concentrate around their leaf-canonical values), then the thermodynamic-limit states defining temperature would not be regular, and the temperature assignment would lose its physical meaning.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If the framework is correct, nonequilibrium temperature becomes a well-defined intrinsic property of isolated quantum states rather than a vestige of equilibrium, enabling thermodynamic descriptions of states that retain significant energy coherence.
  • The replacement of maximum entropy by minimum discrimination information as the governing principle suggests that standard thermodynamic relations (e.g., dS = βdE) fail generically out of equilibrium, requiring reformulation of thermodynamic identities for coherent quantum states.
  • The subsystem temperature equation (Eq. 64) implies that local temperature evolves on macroscopic time scales proportional to subsystem size, providing a dynamical law for thermalization that is distinct from and transverse to the equilibration mechanism itself.
  • The classification of leaves into generic and non-generic types predicts that integrable-like behavior can appear on individual leaves even when the full Hamiltonian is generic, and vice versa, suggesting a leaf-resolved notion of thermalization distinct from the usual Hamiltonian-based one.
  • The framework provides a principled resolution to the ambiguity of temperature in integrable post-quench systems: temperature is defined by continuity from the generic (integrability-broken) case, making it stable under infinitesimal perturbations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 7 minor

Summary. This paper proposes a definition of temperature for isolated quantum many-body systems far from equilibrium, building on the author's earlier minimum-variance foliation framework (Ref. [27]). The key idea is to decompose energy fluctuations into a population-like (classical) part and a coherent (quantum) part, and to define temperature as the coordinate of a canonical pseudolocal flow on the leaf of fixed energy coherence. The inverse temperature is defined via a principle of minimum discrimination information (relative entropy minimization) rather than maximum entropy, and the paper argues that the two principles generically disagree on non-commuting leaves. The construction is extended to subsystems, where the temperature is shown to require a reference point along the dynamical trajectory and cannot be inferred from the reduced state at a single time. The framework is illustrated with explicit computations for the XY model with Dzyaloshinskii-Moriya interactions.

Significance. The problem of defining temperature out of equilibrium is longstanding and important. The paper's approach is conceptually novel: rather than defining nonequilibrium temperature by reference to equilibrium, it treats equilibrium as a special case of a more general construction organized by energy coherence. The minimum-variance foliation provides a mathematically grounded structure, and the reformulation in terms of pseudolocal flows in the thermodynamic limit is a nontrivial technical achievement. The explicit computation of dressed Hamiltonians and QFI densities for the XY model (Eqs. 69-72) provides concrete, falsifiable predictions. The subsystem theory, including the local inverse-temperature rate equation (Eq. 64) and the identification of the O(1/|A|) boundary ambiguity, is a physically transparent result. The claim that the maximum entropy principle fails on non-commuting leaves, if confirmed, would be a significant conceptual finding with implications for quantum thermodynamics.

major comments (3)
  1. The existence of β^h_ω within the proven analyticity domain is not established. Appendix A proves analyticity of F^{(I)}_O(Δβ) only for |Δβ| < Δβ* with Δβ* > 0 (Eq. 87). The temperature is defined as the solution to Eq. (45), but the text explicitly conditions on β^h_ω lying within this domain ('If the negative of the inverse temperature β^h_ω of ω lies in the domain of analyticity...'). Footnote 7 establishes monotonicity of the energy density in Δβ, but existence of a zero within |Δβ| < Δβ* is not proven. For KMS states at large |β_0|, the energy density can be O(1) while Δβ* depends on clustering constants and ||h^-_{(ω)}||_{loc}, which could be small. The paper should either prove that the zero lies within the proven domain for a nontrivial class of states, or clarify that the definition is conditional and discuss what happens when the condition fails.
  2. The claim that β^h_ω ≠ dS_th/dE on non-commuting leaves (a central distinctive prediction of the framework) rests on numerical evidence from systems with L ≤ 12 (Fig. 5, Eq. 48). The bias between the minimum-discrimination-information temperature and the maximum-entropy temperature could vanish as L → ∞. The paper argues structurally that the two principles differ because the barycenter may not be a regular thermodynamic state (§4.1), but does not prove that the bias survives the thermodynamic limit. Given that this is the framework's key physical prediction, the authors should either provide larger-system numerics, a scaling analysis of the bias with L, or a more rigorous argument for why the bias cannot vanish.
  3. The leaf-typicality hypothesis (generalizing ETH to non-commuting leaves) is stated as an expectation but is load-bearing for the physical interpretation of the temperature. If leaf-typicality fails, the leaf canonical ensemble ω_β (Eq. 46) may not describe typical states on the leaf, undermining the thermodynamic meaning of β. The paper acknowledges this in §4.1 ('the leaf typicality hypothesis... is expected to have the same status as ETH') but does not discuss what evidence or tests could distinguish leaf-typicality holding from failing. A brief discussion of falsifiable consequences or known obstructions would strengthen the paper.
minor comments (7)
  1. §1.1.2: The statement that β_0 'no longer has an intrinsic meaning' after the quench could be misread as implying that the pre-quench temperature is physically meaningless. Consider clarifying that it is the coordinate on the commutant foliation that loses thermodynamic meaning, not the pre-quench state itself.
  2. Eq. (48): The notation β^{(I)}_{bare}(β) is introduced without sufficient explanation of how it is computed in practice for the numerics in Fig. 5. A brief description of the finite-size procedure would help readers reproduce the results.
  3. Fig. 5: The curves for different L (8, 10, 12) are visually nearly indistinguishable, which is cited as evidence of fast convergence. However, the bias being measured is itself small for the parameters shown. A log-scale plot of |β - β_{bare}| vs L would make the convergence claim more convincing.
  4. §5.3, Eq. (66): The expression β_A(t_0) = β^h_{ω∘σ_{t_0}} - β_{τ,A}(t_0) involves β_{τ,A}(t_0), which is defined implicitly by Eq. (65). The relationship between these quantities and the physical interpretation is not immediately transparent; a sentence expanding on why this decomposition is natural would help.
  5. The term 'pseudo-Hermitian statistical mechanics' is mentioned in passing (after Eq. 23) with references [50-52] but not developed. If this connection is not explored further, it could be trimmed to avoid distracting from the main thread.
  6. Typo in §6 (Bipartitioning protocol): 'two systems associates with disjoint sets' should read 'two systems associated with disjoint sets.'
  7. The incoherent inner product (Eq. 52) and its norm (Eq. 53) are introduced in §4.2 but not used elsewhere in the paper. If they are included for completeness, a forward reference to where they become relevant (or a statement that their applications are deferred) would help.

Circularity Check

0 steps flagged

No significant circularity found; the temperature definition (Eq. 45) is anchored to an external zero-energy condition, and the max-entropy failure claim rests on independent (if small-system) numerics rather than a definitional identity.

full rationale

The paper's central derivation chain is largely self-contained against circularity. (1) The temperature β^h_ω is defined by Eq. (45) as the solution to lim F^{(I)}_H(-β^h_ω)/|I| = 0, which is the zero-energy-density condition analogous to the equilibrium infinite-temperature anchor (Eq. 41). This is an external constraint, not a self-referential definition. (2) The minimum discrimination information principle (Eqs. 39–42) is a standard variational principle (minimize DKL under energy and normalization constraints), applied to the leaf structure. The resulting leaf canonical ensemble (Eq. 42) is distinct from the maximum-entropy leaf canonical ensemble (Eq. 38) by construction, and the claim that they yield different temperatures is supported by numerical computation of two independently defined quantities (Fig. 5), not by a definitional identity. (3) The main self-citation is to Ref. [27] (same author), which introduced the minimum-variance foliation and leaf canonical ensembles. However, the foliation itself rests on the Lyapunov equation from external Ref. [38] (Yu, 2013) and the QFI–variance connection from external Ref. [39] (Tóth–Petz, 2013). The present paper's key results—thermodynamic-limit reformulation, the minimum-discrimination-information temperature definition, the subsystem theory—are developed here, not imported from Ref. [27]. (4) The canonical charge (Eq. 47) is obtained by straightforward differentiation of Eq. (43). (5) The subsystem temperature rate (Eq. 64) follows from restricting the global canonical functional to subsystem observables (Appendix B, Eq. 93), which is a construction with independent content. The leaf-typicality hypothesis from Ref. [27] is an unverified assumption, but the paper is transparent about this; it is not a circular step. The small-system numerics (L ≤ 12) raise a correctness concern about whether the bias survives the thermodynamic limit, but this is a question of empirical validity, not circularity. Score 1 reflects the minor self-citation dependency on Ref. [27] for the foliation framework, which is not load-bearing for the specific results claimed in this paper.

Axiom & Free-Parameter Ledger

2 free parameters · 5 axioms · 4 invented entities

The paper introduces several new mathematical objects (foliation leaves, canonical flow, leaf modular Hamiltonian, incoherent inner product) that are well-motivated by the finite-volume construction but whose thermodynamic-limit properties rest on unproven clustering and typicality assumptions. The free parameters are model-specific (§6) rather than theory-level. The key axioms are the leaf-typicality hypothesis and the higher-order clustering estimates, both of which are standard-type assumptions in this field but are not proven for the specific dressed operators used here.

free parameters (2)
  • Δβ* (radius of analyticity) = not specified; existence only
    Appendix A proves existence of Δβ* > 0 but does not compute it; the extension to all finite β is assumed by analogy with equilibrium.
  • Model parameters in §6 (J, γ, h, D, J₀, β₀, δ_y) = J=1, γ=-2, h=0.5, D=0.75, δ_y=0.3, various β₀
    These are illustrative choices for the examples, not fitted parameters; they are chosen to demonstrate generic and non-generic leaves.
axioms (5)
  • domain assumption Higher-order clustering estimates for truncated correlations of the dressed Hamiltonian H^-[ω] (Eq. 84)
    Appendix A states: 'In the following we assume the corresponding higher-order clustering estimate for the truncated correlations entering the cumulants.' This is standard for equilibrium KMS states but not proven for the dressed operators.
  • ad hoc to paper Leaf-typicality hypothesis: states on generic leaves are typical in the sense that local observables are captured by the leaf canonical ensemble
    Introduced in Ref. [27] as a generalization of ETH; assumed throughout but not proven. Section 4.1 states 'the leaf typicality hypothesis proposed in Ref. [27] is expected to have the same status as ETH.'
  • domain assumption The canonical pseudolocal flow preserves exponential clustering in a neighborhood of the reference state
    Appendix A sketches a proof of analyticity for |Δβ| < Δβ*, and Eq. (91) suggests clustering inheritance, but the extension beyond the local neighborhood is stated as expectation.
  • domain assumption The ω-dressed Hamiltonian H^(ω) has simple spectrum (finite volume) or generic pseudolocal structure (thermodynamic limit)
    Section 3 defines M_H as the set of full-rank density matrices with simple spectrum for H^(ω); the thermodynamic limit replaces this with genericity of the leaf (§4.1).
  • standard math KMS condition characterizes equilibrium states for 1D quantum spin chains with local interactions
    Standard result cited from Araki, Kishimoto (Refs. 28-34); used to define the initial state.
invented entities (4)
  • Minimum-variance foliation leaves (thermodynamic limit) independent evidence
    purpose: Organize nonequilibrium state space by energy-coherence content
    The foliation is defined via the Lyapunov equation (Eq. 7) and characterized by leaf conditions (A), (B), (C) in §3. The QFI density (Eq. 33) and continuity equation (Eq. 36) provide operational handles. Numerical evidence in §6 shows concrete dressed Hamiltonians.
  • Canonical pseudolocal flow on leaves independent evidence
    purpose: Defines the temperature coordinate on non-commuting leaves
    The flow is defined by Eq. (47) and its analyticity is established in Appendix A. The numerical solutions of Eq. (45) in §6 provide concrete values of β for specific models.
  • Leaf subsystem modular Hamiltonian K^leaf_A no independent evidence
    purpose: Determines the subsystem temperature rate via Eq. (64)
    Defined in Eq. (58-60) via the Lyapunov equation on the subsystem. No independent verification is provided; the bipartitioning protocol in §6 uses chains too small to test the anchoring assumptions.
  • Incoherent inner product (·,·)^inc_ω independent evidence
    purpose: Hilbert-space structure for pseudolocal charges on a leaf
    Defined in Eq. (52); related to quantum covariance and the difference between variance and QFI/4 (Eq. 53). This is a mathematical construction with a clear interpretation.

pith-pipeline@v1.1.0-glm · 36200 in / 3693 out tokens · 505641 ms · 2026-07-10T03:31:55.509119+00:00 · methodology

0 comments
read the original abstract

Temperature is one of the central concepts of thermodynamics, yet its meaning away from equilibrium remains elusive. This problem is particularly acute in isolated quantum many-body systems: their states evolve unitarily, need not be close to equilibrium, and can retain energy coherence, a feature with no classical thermodynamic analogue. A non-stationary quantum state contains two kinds of energy fluctuations. One is associated with energy populations and has the usual thermodynamic interpretation; the other arises from coherence between energy sectors and drives time dependence. We propose that temperature, also out of equilibrium, locates the state within the family of regular states compatible with its energy-coherence structure. This leads to a natural definition of temperature for regular nonequilibrium states. The resulting inverse temperature is not generally the derivative of thermodynamic entropy with respect to energy. Indeed the principle of maximum entropy does not extend in its usual form; it is replaced by a principle of minimum discrimination information. We also develop the corresponding theory for subsystems, where temperature cannot in general be inferred from the reduced state alone. Instead, it is determined by the induced local thermodynamic structure, with boundary ambiguities removed in the thermodynamic limit.

Figures

Figures reproduced from arXiv: 2607.08655 by Maurizio Fagotti.

Figure 1
Figure 1. Figure 1: Two foliations of state space. The cartoon compares the commutant foliation and the minimum-variance foliation of the state space of a two-dimensional Hilbert space. The state space is represented by the Bloch ball. The commutant foliation is defined on the state space with the tracial state 1 2 I removed; its leaves are half-open radial segments. By contrast, the leaves of the minimum-variance foliation a… view at source ↗
Figure 2
Figure 2. Figure 2: From foliations to thermodynamics. Left: The six leaves of the commutant foliation that contain stationary states of the Hamiltonian H3 = P3 i=1 Ei|φi⟩⟨φi| on a three-dimensional Hilbert space. Since each such leaf is two-dimensional, a generic state on it requires two coordinates. For local observables, however, the eigenstate thermalization hypothesis suggests an effective reduction of the relevant state… view at source ↗
Figure 3
Figure 3. Figure 3: Schematic illustration of four possible types of leaves. [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Geometric construction of subsystem temperature. [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Temperature vs local temperature vs bare temperature. [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Bipartitioning protocol. The half-chain inverse temperature after having prepared the subchains at inverse temperature (β0,L, β0,R) (shown on the upper-left corner of the plots in units of J −1 0 ) for the Hamiltonian H (I) 0,p in (67). The Hamiltonian is the same as in (5) with the additional interaction (73) with δy = 0.3. The symbols correspond to different chain’s lengths (in the legend). The inverse t… view at source ↗

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