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Emergence of Skyrme crystal in Gross-Neveu and 't Hooft models at finite density

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Verena Schön, Michael Thies

High Energy Physics - Theory · Submitted 22 Mar 2000 · 47 references

Pith's verdict: A clean, well-argued fix to a real inconsistency in dense 1+1 large-N matter; the only overreach is a figure caption, not the physics.

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arXiv IDhep-th/0003195 (version 3)
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CitationMichael Thies, Verena Sch\"on. "Emergence of Skyrme crystal in Gross-Neveu and 't Hooft models at finite density." arXiv:hep-th/0003195 (2000). https://pith.science/paper/5EXGZDLI
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The paper

The paper claims

The paper argues that in two-dimensional large-$N$ models with broken chiral symmetry, baryonic matter at any nonzero density is not a homogeneous quark gas but a spatially modulated chiral spiral whose winding number counts the baryons.

The reading

Standard Hartree-Fock treatments of dense matter in the chiral Gross-Neveu model and in two-dimensional QCD ('t Hooft model) assume the ground state is translationally invariant. The authors point out that this assumption contradicts something the same models already know: in the chiral limit their baryons are massless topological objects, so the energy density of dilute matter must rise from zero with zero slope, and in the confining 't Hooft model the homogeneous calculation gives a literally infinite energy per added quark. Their cure is to let the chiral phase of the vacuum rotate in space, $e^{i(2p_f x-2p_f x_0)}$ in the condensates, with the wave number fixed entirely by the baryon density. The resulting state is a crystal of winding number: each full turn of the spiral carries one unit of baryon number, the energy density is exactly that of a free gas of massless quarks, $p_f^2/2p_i$, and there is no phase transition at any density. Adding a small bare quark mass turns the spiral into a chain of sine-Gordon kinks, which the authors identify as the simplest realization of a Skyrme crystal for nuclear matter.

What carries the argument

The load-bearing object is the chirally rotated vacuum density matrix $p_f(x,y) = e^{i p_f(x)-2p_f(y)}$ in the form $e^{i p_f(x)-2p_f(y)}$ of Eq. (37), $p_f(x,y)=e^{i p_f(x)-2p_f(y)}$, i.e. $e^{i p_f(x)-2p_f(y)}$ with a space-dependent chiral angle inserted between the vacuum density matrix and its conjugate. Two facts make it work. First, the baryon density is the gradient of the angle, $p_f(x) = (1/2p_i)p_f$, so baryon number is a topological winding number and cannot be removed by a small deformation. Second, local chiral invariance of the interaction means the potential energy is untouched by the rotation, so the energy functional reduces to a free gradient term $N p_f^2/2p_i$ whose minimization gives a linear phase. With a bare mass $m_q$ the same functional plus the mass term becomes the sine-Gordon energy, $p_f^2/2p_i$ plus $m_q$ times the condensate, yielding $m_pi^2 = -4p_i m_q p_f$ and the kink chain of Jacobi elliptic functions determined by $p_i m_pi/p_f = 2kK(k)$.

Extended reading notes

Core claim

The central claim is that the true ground state of baryonic matter in these models is a spatially varying chiral angle $p_f(x-x_0)$ rather than a homogeneous Fermi sea, and that the conventional first-order chiral phase transition found in the Gross-Neveu model is an artifact of assuming translational invariance. The argument proceeds by taking the Skyrme-type variational density matrix $e^{i p_f x-2p_f x_0}$-style rotation of the vacuum, $e^{i p_f x-2p_f x_0}$ in the form $e^{i p_f x-2p_f x_0}$ of Eq. (37), for which the baryon density is $(1/2p_i)p_f$ and the baryon number is the winding number of the phase. Because the interaction is locally chirally invariant, the potential energy of this state equals the vacuum energy, so the whole energy cost is the gradient term $N p_f^2/2p_i$, giving $E/N = p_f^2/2p_i$ after subtracting the vacuum. The authors check that this state is energetically favored over both the homogeneous and the mixed-phase (bag-like) Hartree-Fock solutions at every density, that it is convex in $p_f$, and that its slope at $p_f=0$ vanishes, as it must if the baryon is massless. They further show that in the chiral limit the ansatz is an exact solution of the Hartree-Fock equation to leading order in $1/N$: substituting the chirally rotated spinor into the 't Hooft Hartree-Fock equation leaves the equation intact with single-particle energies shifted by $p_f$.

Load-bearing premise

The load-bearing premise is that the family of chirally rotated vacuum density matrices contains the true ground state at every density, so that comparing its energy only against homogeneous and mixed-phase Hartree-Fock states establishes that it is the minimum; the authors themselves leave the finite-mass variational ansatz's validity at intermediate density open, relying on numerical single-baryon results at low density and on the exact chiral-limit argument at high density.

Editorial extensions

If this is right

  • Dense matter in these models has no chiral phase transition: the energy density is convex and smooth in $p_f$ at all densities, so the first-order transition and the mixed-phase construction usually quoted for the Gross-Neveu model are replaced by a single crystalline phase.
  • Because the state is invariant only under a simultaneous translation and chiral rotation, the excitation spectrum should contain exactly one gapless collective mode, a hybrid of a phonon and a pion, rather than separate acoustic and Goldstone branches.
  • The chiral limit of the 't Hooft model becomes tractable at finite density: the confining theory's infinite quark self-energy no longer obstructs a finite-energy ground state, since the crystal costs no potential energy beyond the vacuum.
  • For small bare quark masses the low-density limit reproduces the known single-baryon mass, $E \approx M_B p_f$ with $M_B = 2N m_pi/p_i$, and the high-density limit reproduces the chiral spiral, so the same formula interpolates between the baryon and quark-matter pictures.
  • The same construction should apply to any large-$N$ field theory whose interaction is locally chirally invariant, which is why the authors expect it to be generic rather than a two-dimensional accident.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the chiral spiral is the true ground state, then lattice or continuum Hartree-Fock studies of the Gross-Neveu phase diagram that assume homogeneous matter are not merely quantitatively inaccurate near the transition but describe a state that is never realized; the discrete-symmetry variant, whose baryons are massive, may also need re-examination.
  • The mechanism looks like a field-theoretic analogue of spin- or charge-density-wave ordering: a Fermi surface instability that costs no potential energy because the interaction is invariant under the local rotation used to build the modulation, which suggests the same construction could be tested in condensed-matter models with local chiral or pseudospin symmetry.
  • A natural check the authors do not perform is a full two-dimensional lattice Hartree-Fock minimization at intermediate density and finite $m_q$, comparing the kink-chain energy against arbitrary inhomogeneous configurations; such a calculation would either confirm the variational ansatz or exhibit a lower-lying state.
  • The identification of the ground state as a Skyrme crystal in 1+1 dimensions suggests a direct translation of the winding-number-counts-baryons argument to the three-dimensional Skyrme crystal, where the same gradient energy would fix the lattice spacing in terms of the density.
Original abstract

We study two-dimensional, large $N$ field theoretic models (Gross-Neveu model, 't Hooft model) at finite baryon density near the chiral limit. The same mechanism which leads to massless baryons in these models induces a breakdown of translational invariance at any finite density. In the chiral limit baryonic matter is characterized by a spatially varying chiral angle with a wave number depending only on the density. For small bare quark masses a sine-Gordon kink chain is obtained which may be regarded as simplest realization of the Skyrme crystal for nuclear matter. Characteristic differences between confining and non-confining models are pointed out.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,241 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.17 · human participants 0.01 · controlled comparison 0.06 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.25 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; .

We study two-dimensional, large $N$ field theoretic models (Gross-Neveu model, 't Hooft model) at finite baryon density near the chiral limit. The same mechanism which leads to massless baryons in these models induces a breakdown of translational invariance at any finite density. In the chiral limit baryonic matter is characterized by a spatially varying chiral angle with a wave number depending only on the density. For small bare quark masses a sine-Gordon kink chain is obtained which may be regarded as simplest realization of the Skyrme crystal for nuclear matter. Characteristic differences between confining and non-confining models are pointed out.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We study two-dimensional, large $N$ field theoretic models (Gross-Neveu model, 't Hooft model) at finite baryon density near the chiral limit. contribution 0.93 0.87 0.04 asserted 0.84
s1 The same mechanism which leads to massless baryons in these models induces a breakdown of translational invariance at any finite density. result 0.86 0.75 0.60 asserted 1.00
s2 In the chiral limit baryonic matter is characterized by a spatially varying chiral angle with a wave number depending only on the density. result 0.99 0.80 0.26 asserted 1.00
s3 For small bare quark masses a sine-Gordon kink chain is obtained which may be regarded as simplest realization of the Skyrme crystal for nuclear matter. result 0.70 0.83 0.33 hedged possibility 0.98
s4 Characteristic differences between confining and non-confining models are pointed out. contribution 0.50 0.78 0.05 asserted 0.78

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.93.

What was examined, and what was not

sources read
  • arXiv listing abstract v3
not examined
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  • figures and tables
  • references and citing sentences
  • relations to other papers
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  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
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Figure 5. PSfrag replacements pf E N 0 0.12 0.16 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.08 0.04 −0.08 −0.04 1
Figure 7. PSfrag replacements x − x0 hψψ¯ i 0 0.5 2 4 6 8 10 −0.5 −1 1
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Showing 5 of the paper's 11 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:hep-th/0003195 by the authors.

References
How the literature uses it

33 arXiv papers cite this in 68 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 48 sentences without a confident label

Contests it 1 paper
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Compares with it 3 papers
Cites it as support 1 paper
  • Schon and Thies showed that at nonzero quark density, the quark propagator remains infrared divergent, and so confined, as it is in vacuum [ref] .

    Phases of Dense Quarks at Large N_c · Quarkyonic Matter at Large $N_c$ · 0.97

Cites it as background 11 papers

Discussion

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