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arxiv: 2606.21246 · v1 · pith:TMG3Z5LVnew · submitted 2026-06-19 · ✦ hep-th

Phase diagram of the massless Gross-Neveu model with two components

Michael Thies This is my paper

Pith reviewed 2026-06-26 13:49 UTC · model grok-4.3

classification ✦ hep-th
keywords Gross-Neveu modelphase diagraminhomogeneous phasesfinite temperaturefinite densitystability analysislarge N limittwo-component fermions
0
0 comments X

The pith

Stability analyses determine the full large-N phase diagram of the two-component Gross-Neveu model without solving the thermal Hartree-Fock equations.

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

The paper extends prior work on the two-component Gross-Neveu model to finite temperature and constructs its complete large-N phase diagram, including regions with inhomogeneous condensates. It locates every phase boundary through multiple stability analyses rather than direct solution of the self-consistent thermal equations. The resulting diagram undergoes a qualitative change once the filling fraction passes a critical value. A reader would care because the approach shows how the phase structure of this solvable interacting-fermion model can be mapped efficiently at both zero and finite temperature.

Core claim

All phase boundaries, including those separating homogeneous and inhomogeneous phases, can be located solely by stability analyses of various kinds, so that the thermal Hartree-Fock problem need not be solved; this procedure produces a qualitative change in the phase diagram at a critical filling fraction.

What carries the argument

Various kinds of stability analyses applied to homogeneous and inhomogeneous candidate phases.

If this is right

  • The phase diagram is completely determined at large N for the two-flavor case at any temperature and chemical potential.
  • Inhomogeneous phases occupy definite regions whose boundaries are fixed by the stability criteria.
  • A critical filling fraction separates two qualitatively different arrangements of the phases.
  • Exact finite-N results from related work can be compared directly against the large-N boundaries obtained here.

Where Pith is reading between the lines

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

  • The same stability techniques might locate phase boundaries in other four-fermion models that share the same large-N structure.
  • The critical filling fraction could mark the density at which the Fermi surface first permits a new class of instabilities.
  • The predicted boundaries supply concrete targets for lattice simulations that include finite-N effects.
  • Adding a bare mass term would require checking whether the existing stability criteria continue to locate all transitions.

Load-bearing premise

Stability analyses of several kinds are enough to find every phase boundary and no extra phases or transitions exist that would only be seen by solving the full thermal Hartree-Fock equations.

What would settle it

Numerical solution of the thermal Hartree-Fock equations at a temperature and density near one of the predicted boundaries that reveals either a different transition point or an additional phase not captured by the stability criteria.

Figures

Figures reproduced from arXiv: 2606.21246 by Michael Thies.

Figure 1
Figure 1. Figure 1: FIG. 1: Perturbative phase boundaries of two-component GN m [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Determination of the tricritical point using the zer [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Full phase diagram of the two-component GN model with [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Phase diagram of the one-component GN model ( [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Generalizing the phase diagram from Fig. [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Wave number [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Homogeneous phase diagram of two-component GN model [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Full phase diagram of two-component GN model with [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Full phase diagram of two-component GN model with [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Like Fig [PITH_FULL_IMAGE:figures/full_fig_p010_10.png] view at source ↗
read the original abstract

Recently, Benini, Mamroud, Reis and Serone have presented exact results for the Gross-Neveu model at finite density, both at finite N and in the large N limit. Generalizing previous studies, they introduce a chemical potential acting only on a subset of fermion flavors - the two-component Gross-Neveu model. Here, we take up this idea and extend the semiclassical study to finite temperature. The full (large N) phase diagram including inhomogeneous phases is constructed without solving the thermal Hartree-Fock problem. All phase boundaries can be found on the basis of various kinds of stability analyses. As a result, we find a qualitative change of the phase diagram at some critical filling fraction.

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

1 major / 0 minor

Summary. The manuscript extends semiclassical studies of the two-component massless Gross-Neveu model to finite temperature. It constructs the full large-N phase diagram, including inhomogeneous phases, at finite T and density by means of various stability analyses of homogeneous and inhomogeneous reference configurations, without solving the thermal Hartree-Fock equations. The central result is a qualitative change in the phase diagram structure at a critical filling fraction.

Significance. If the central claim holds, the work provides a technically efficient route to complete phase diagrams in models with spatially modulated order parameters by replacing full minimization with targeted stability thresholds. This generalizes the recent exact results of Benini et al. and could be useful for related four-fermion models. The reported change at critical filling fraction would be a concrete, falsifiable prediction.

major comments (1)
  1. [Abstract / method paragraph] Abstract and the paragraph describing the method: the assertion that 'all phase boundaries can be found on the basis of various kinds of stability analyses' without solving the thermal Hartree-Fock problem is load-bearing for the completeness claim. Because the effective potential is non-convex and the order parameter can be modulated, an explicit argument is required showing that no first-order transitions or additional multi-critical points exist whose locations are invisible to linear stability around the homogeneous and previously identified inhomogeneous phases. Without this, the reported qualitative change at the critical filling fraction cannot be guaranteed to be exhaustive.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying this key point about the completeness of our phase-diagram construction. We address the major comment below.

read point-by-point responses

  1. Referee: Abstract and the paragraph describing the method: the assertion that 'all phase boundaries can be found on the basis of various kinds of stability analyses' without solving the thermal Hartree-Fock problem is load-bearing for the completeness claim. Because the effective potential is non-convex and the order parameter can be modulated, an explicit argument is required showing that no first-order transitions or additional multi-critical points exist whose locations are invisible to linear stability around the homogeneous and previously identified inhomogeneous phases. Without this, the reported qualitative change at the critical filling fraction cannot be guaranteed to be exhaustive.

    Authors: We agree that an explicit justification is required to support the claim that stability analyses capture every phase boundary. In the large-N Gross-Neveu model the inhomogeneous phases bifurcate continuously from the homogeneous ones at the linear-instability thresholds; likewise, the boundaries separating distinct inhomogeneous configurations are located by stability analyses of one inhomogeneous background against perturbations that connect it to the other. The non-convexity of the effective potential is handled by these global stability thresholds rather than local minima alone. This structure is inherited from the exact T=0 results of Benini et al., where all transitions remain continuous. We will add a short subsection in the revised manuscript that spells out this reasoning and confirms the absence of first-order lines or extra multi-critical points within the explored parameter range, thereby securing the reported qualitative change at the critical filling fraction. revision: yes

Circularity Check

0 steps flagged

No circularity; phase diagram constructed via external generalization and stability analyses

full rationale

The paper generalizes results from Benini et al. (distinct authors) to finite T using various stability analyses without solving the full thermal Hartree-Fock equations. No quoted equations or claims reduce by construction to fitted inputs, self-definitions, or self-citation chains. The central method statement is an explicit assumption about sufficiency of stability thresholds rather than a derived result that loops back to its own inputs. The work is self-contained against the cited external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities are identifiable from the provided text.

pith-pipeline@v0.9.1-grok · 5633 in / 1084 out tokens · 21627 ms · 2026-06-26T13:49:19.940347+00:00 · methodology

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Reference graph

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