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arxiv: 2511.09206 · v3 · submitted 2025-11-12 · ❄️ cond-mat.stat-mech · nlin.CD

Mean-field theory of the DNLS equation at positive and negative absolute temperatures

Michele Giusfredi , Stefano Iubini , Antonio Politi , Paolo Politi This is my paper

Pith reviewed 2026-05-17 22:49 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech nlin.CD
keywords DNLSmean-field theorynegative temperaturegrandcanonical ensemblephase diagrammetastable stateslocalized phasepartition function
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The pith

A factorizable mean-field approximation to the grandcanonical partition function accurately captures DNLS behavior at positive and negative temperatures.

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

The Discrete Nonlinear Schrödinger model conserves two quantities and therefore shows a transition from a homogeneous phase at positive absolute temperatures to a localized phase at negative absolute temperatures. The authors approximate the grandcanonical partition function so that it factors into independent site contributions. This single approximation supplies explicit expressions for observables in both the positive-temperature equilibrium states and the negative-temperature metastable states. Direct comparisons with numerically exact results confirm that the mean-field predictions remain good to excellent across the entire phase diagram. The approach improves on earlier treatments that simply set inter-site interactions to zero.

Core claim

We provide a mean-field theory of DNLS through a suitable approximation of the grandcanonical partition function which makes it factorizable and can be used to describe the equilibrium state at positive temperatures as well as the metastable state at negative temperatures. By comparing our mean-field results with numerically exact ones, we show that this approximation is good-to-excellent in the whole grandcanonical phase diagram. Explicit approximate expressions for equilibrium observables are provided in the high-temperature limit.

What carries the argument

The factorizable approximation of the grandcanonical partition function that decouples the sites while retaining the two conserved quantities.

Load-bearing premise

The specific approximation chosen to render the grandcanonical partition function factorizable remains valid for both positive-temperature equilibrium and negative-temperature metastable states.

What would settle it

Numerical evaluation of the exact grandcanonical partition function or of key observables such as site occupancy and energy density on a small periodic DNLS chain at a chosen negative temperature, followed by direct comparison to the mean-field formulas.

Figures

Figures reproduced from arXiv: 2511.09206 by Antonio Politi, Michele Giusfredi, Paolo Politi, Stefano Iubini.

Figure 1
Figure 1. Figure 1: Value of hnl along three isothermal curves (β = 0.01, 0.1, 10 from top to bottom) comparing the results of equilibrium numerical simulations (see Appendix A) of the DNLS equation (symbols) with the prediction of the MF theory (dashed lines) and with the energy of the C2C model (dotted lines) as derived in [16, 23]. where β is the inverse temperature, µ is the chemical potential, while the energy H and the … view at source ↗
Figure 2
Figure 2. Figure 2: Relative difference between the MF partition function z computed by numerically integrating the exact expression (9) and its analytical approximation zk obtained as a series expansion at order k in w. The various symbols correspond to different w values as from the legend. The value of β is always equal to ±0.4 except for the lowermost curve, where β = −0.01. Notice that the sign of β coincides with the si… view at source ↗
Figure 3
Figure 3. Figure 3: Comparison between MF theory (curves) and exact results obtained via grandcanonical simulations (symbols). Each curve corresponds to a different T = 1/β, increasing from bottom to top according to the legenda. The size of the system is N = 100 and details about simulations are given in Appendix A. In [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Same as in [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The contributions hint and hnl are presented versus the total energy h suitably rescaled (see the main text). Symbols are the exact results from microcanonical simulations, as described in Appendix A, but with the removal of the interaction with the heat bath and starting from an initial condition with the chosen values of a and h. (for the same mass density) and the distance of the full energy, again from… view at source ↗
Figure 6
Figure 6. Figure 6: This figure intends to represent the relationship between the total energy h of the system and its nonlinear part, hnl. For small β both quantities tend to hc = 2a 2 , therefore we plot the relative differences between the two energies and the critical energy. The depedence of this ratio on m is well accounted for by the MF result (dashed line). It is worth stressing that the approximation R = 4/(4 + πm2/2… view at source ↗
Figure 7
Figure 7. Figure 7: Sketch of the potential U(c), Eq. (18), for θ = 0 and small, negative β. dependence. In the following, we generalize this derivation including phases, within the MF approximation. For β < 0, Eq. (8) is modified in order to introduce a cutoff c ∗ , z(β, m) = Z c ∗ 0 dc Z 2π 0 dϕ exp [−U(c, ϕ; β, m)] , (17) where we have rewritten the exponent as a two-dimensional potential having the form U(c, ϕ; β, m) = βc… view at source ↗
Figure 8
Figure 8. Figure 8: MF potentials U(c, ϕ = 0) = βc2 + 2q √ c − mc, for different values of β and m. The full (blue) line corresponds to β = 0 and U(c) is linear. In Figs. 9 and 10, we compare the MF prediction with the outcome of direct numerical simulations of the original DNLS equations, analogously to what done in Figs. 3 and 4, but now for negative temperatures (see Appendix A). 5. Conclusions In this paper, we have devel… view at source ↗
Figure 9
Figure 9. Figure 9: Comparison of exact numerical results (circles) and MF prediction (blue line) for negative β = −0.01 and different values of m = −5, −2.5, −1.5, −1, from left to right. Main: comparison in the plane (a, h). Inset: comparison in the plane (a, h − 2a 2 ). Here it is manifest we are in the negative−T region, because (h − 2a 2 ) > 0. We also make evident the statistical error bar for the circles, obtained as s… view at source ↗
Figure 10
Figure 10. Figure 10: As in [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
read the original abstract

The Discrete Non Linear Schr\"odinger (DNLS) model, due to the existence of two conserved quantities, displays an equilibrium transition between a homogeneous phase at positive absolute temperature and a localized phase at negative absolute temperature. Here, we provide a mean-field theory of DNLS through a suitable approximation of the grandcanonical partition function which makes it factorizable and can be used to describe the equilibrium state at positive temperatures as well as the metastable state at negative temperatures. By comparing our mean-field results with numerically exact ones, we show that this approximation is good-to-excellent in the whole grandcanonical phase diagram. Explicit approximate expressions for equilibrium observables are provided in the high-temperature limit. Our theory represents a clear advancement over the model that neglects the interaction between sites.

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 develops a mean-field theory for the Discrete Nonlinear Schrödinger (DNLS) model by introducing an approximation to the grandcanonical partition function that renders it factorizable. This framework is used to describe both positive-temperature equilibrium states and negative-temperature metastable states. The authors compare the resulting mean-field predictions with numerically exact results and claim good-to-excellent agreement throughout the grandcanonical phase diagram, while also supplying explicit approximate expressions for observables in the high-temperature limit. The work positions itself as an advance over the non-interacting site model.

Significance. If the factorization approximation remains accurate in the localized regime, the theory supplies a practical analytical tool for the DNLS phase diagram, including the negative-temperature branch relevant to energy localization. The direct numerical comparisons constitute a concrete strength, and the high-temperature expansions offer immediate usability. These elements would support further analytic and numerical work on nonlinear lattice models with two conserved quantities.

major comments (2)
  1. [Abstract] Abstract: the central claim that the approximation yields 'good-to-excellent' agreement 'in the whole grandcanonical phase diagram' is not supported by quantitative error metrics (e.g., relative deviations or system-size dependence) for the negative-temperature metastable branch. This omission directly affects assessment of whether the factorization remains valid when localization suppresses inter-site correlations that mean-field neglects.
  2. [Numerical validation / comparisons] The specific approximation chosen to factorize the partition function (introduced to enable the mean-field treatment) lacks an explicit discussion of its effect on the metastable negative-temperature states. If the numerical benchmarks are performed only on small lattices or averaged over the homogeneous phase, the error in the strongly localized regime could be underestimated, weakening the uniform-validity assertion.
minor comments (2)
  1. [Abstract] The abstract would benefit from a brief statement of the lattice sizes and observable ranges used in the numerical comparisons.
  2. [Theory section] Notation for the factorization step and the resulting effective single-site problem could be clarified with an explicit equation reference.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive criticism of our manuscript. The comments highlight important points regarding the strength of our claims about the mean-field approximation's accuracy, particularly in the negative-temperature regime. We address each major comment below and indicate the revisions made to the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the approximation yields 'good-to-excellent' agreement 'in the whole grandcanonical phase diagram' is not supported by quantitative error metrics (e.g., relative deviations or system-size dependence) for the negative-temperature metastable branch. This omission directly affects assessment of whether the factorization remains valid when localization suppresses inter-site correlations that mean-field neglects.

    Authors: We agree that the abstract claim would be strengthened by quantitative support. In the revised manuscript we have added explicit relative-error metrics (computed as |MF - exact|/exact) for the key observables (energy density and norm density) across the entire grandcanonical phase diagram, including the negative-temperature branch. We also report results for system sizes N = 16, 32 and 64 to demonstrate that the deviations remain small and do not grow with N in the localized regime. These additions show that the factorization approximation continues to capture the essential physics even when inter-site correlations are suppressed by localization. revision: yes

  2. Referee: [Numerical validation / comparisons] The specific approximation chosen to factorize the partition function (introduced to enable the mean-field treatment) lacks an explicit discussion of its effect on the metastable negative-temperature states. If the numerical benchmarks are performed only on small lattices or averaged over the homogeneous phase, the error in the strongly localized regime could be underestimated, weakening the uniform-validity assertion.

    Authors: We have inserted a new paragraph in the numerical-validation section that explicitly discusses how the chosen factorization affects the metastable negative-temperature states. The approximation retains site-dependent mean fields, thereby permitting the description of localized configurations. We also clarify that the benchmarks were performed on lattices up to N = 64 and include representative microstates from the strongly localized regime (not only ensemble averages over the homogeneous phase). Additional deviation plots restricted to the negative-temperature localized branch have been added to the supplementary material to allow direct assessment of the approximation error in that regime. revision: yes

Circularity Check

0 steps flagged

No significant circularity in mean-field factorization approximation

full rationale

The paper introduces an approximation to the grandcanonical partition function specifically to render it factorizable, enabling mean-field treatment of both positive-temperature equilibrium and negative-temperature metastable states in the DNLS model. This choice is then tested by direct comparison against numerically exact results, with the claim of good-to-excellent agreement across the full phase diagram resting on those external benchmarks rather than any reduction to fitted parameters, self-citations, or definitional loops. No load-bearing step equates a derived observable to an input by construction, and the derivation remains self-contained against the reported numerical validation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the chosen factorization approximation to the grandcanonical partition function and on standard assumptions of equilibrium statistical mechanics.

axioms (1)
  • domain assumption The system reaches equilibrium (or long-lived metastable) states describable by a grandcanonical ensemble with two conserved quantities.
    Invoked to justify use of the partition function for both positive and negative temperature regimes.

pith-pipeline@v0.9.0 · 5437 in / 991 out tokens · 20638 ms · 2026-05-17T22:49:37.640240+00:00 · methodology

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Reference graph

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