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arxiv: 2605.17183 · v1 · pith:HRKY5ENInew · submitted 2026-05-16 · ✦ hep-lat · hep-ph· hep-th

Dense QC₂D₂ with uniform matrix product states

Kohei Fujikura , Yoshimasa Hidaka This is my paper

Pith reviewed 2026-05-20 13:48 UTC · model grok-4.3

classification ✦ hep-lat hep-phhep-th
keywords dense QCDlattice gauge theorymatrix product statesTomonaga-Luttinger liquidquarkyonic matterfinite baryon densitySU(2) gauge theoryone-dimensional systems
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The pith

At finite baryon density, single-flavor SU(2) gauge theory in two dimensions shows Tomonaga-Luttinger liquid infrared behavior with central charge 1 and smoothly varying Luttinger parameter.

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

The paper applies a gauge-invariant variational uniform matrix product state ansatz to single-flavor SU(2) Yang-Mills theory in one spatial dimension. This sign-problem-free method works directly in the thermodynamic limit at nonzero baryon density. It establishes that the low-energy excitations match those of a Tomonaga-Luttinger liquid, with central charge fixed at one and baryon-density correlations oscillating at the wave number required by the theory. The Luttinger parameter decreases continuously from near one in the dilute regime to one-half at higher densities. The momentum distribution of quarks simultaneously shows a filled Fermi sea, indicating that baryonic infrared physics and quark degrees of freedom coexist.

Core claim

At finite baryon density the infrared behavior is consistent with a Tomonaga-Luttinger liquid: the central charge is determined to be c=1, and the two-point function of the baryon-number density exhibits spatial modulation with the wavenumber predicted by Tomonaga-Luttinger liquid theory. The Luttinger parameter varies smoothly from K≃1 in the dilute-baryon regime to K≃1/2 at higher densities, suggesting a quarkyonic crossover. The quark distribution reveals the coexistence of a quark Fermi sea with a baryonic infrared description, thereby realizing the quarkyonic picture from first principles.

What carries the argument

Gauge-invariant variational uniform matrix product state ansatz optimized in the thermodynamic limit, which supplies a sign-problem-free variational representation of the ground state at finite density.

If this is right

  • The low-energy theory is a Tomonaga-Luttinger liquid whose central charge equals one.
  • Baryon-number density correlations oscillate at the wave number fixed by the baryon density.
  • The Luttinger parameter K changes continuously from approximately 1 at low density to 1/2 at higher density.
  • Quark and baryon descriptions coexist inside the same ground state.

Where Pith is reading between the lines

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

  • The continuous change in K provides a first-principles benchmark for how the quarkyonic crossover might appear in other numerical approaches.
  • The same modulation pattern in density correlations offers a concrete observable for larger-volume lattice simulations.
  • The method supplies a controlled setting in which to test whether similar Luttinger-liquid signatures persist when the gauge group or spatial dimension is altered.

Load-bearing premise

The gauge-invariant variational uniform matrix product state ansatz, when optimized in the thermodynamic limit, faithfully represents the true ground state without significant truncation or bias that would alter the extracted central charge, Luttinger parameter, or density correlations.

What would settle it

An independent calculation that measures a central charge different from 1 or finds baryon-density correlations lacking spatial modulation at the wave number set by the baryon density would falsify the Tomonaga-Luttinger liquid identification.

Figures

Figures reproduced from arXiv: 2605.17183 by Kohei Fujikura, Yoshimasa Hidaka.

Figure 1
Figure 1. Figure 1: FIG. 1. Dependence of the Schmidt spectrum on the representation [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Dependence of the entanglement entropy [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Dependence of the baryon number density on the baryon chemical potential (left), and dependence of the energy [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Dependence of the pressure [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The baryon-number susceptibilities [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Dependence of the entanglement entropy [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: shows the dependence of the speed of sound and the Luttinger parameter on the baryon number density. At m/g = 0, we find agreement with the analytic result for both cs = 1 and K = 1/2. The small deviations in the high￾density region can be understood as lattice artifacts. For a nonvanishing quark mass m/g ̸= 0, the overall behavior changes drastically. In particular, cs → 0 is found at the transition point… view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Eigenvalues of the transfer matrix [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. The solid and dashed curves correspond to the cases without and with gauge interactions for the same baryon chemical [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
read the original abstract

We study cold dense single-flavor $\mathrm{SU}(2)$ gauge theory in $(1+1)$ dimensions in the thermodynamic limit using a gauge-invariant variational uniform matrix product state ansatz. This formulation provides a sign-problem-free, first-principles approach to dense QCD. We show that, at finite baryon density, the infrared behavior is consistent with a Tomonaga--Luttinger liquid: the central charge is determined to be $c=1$, and the two-point function of the baryon-number density exhibits spatial modulation with the wavenumber predicted by Tomonaga--Luttinger liquid theory. The Luttinger parameter varies smoothly from $K\simeq 1$ in the dilute-baryon regime to $K\simeq 1/2$ at higher densities, suggesting a quarkyonic crossover. Furthermore, the quark distribution reveals the coexistence of a quark Fermi sea with a baryonic infrared description, thereby realizing the quarkyonic picture from first principles.

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

3 major / 2 minor

Summary. The manuscript studies dense single-flavor SU(2) gauge theory in (1+1) dimensions using a gauge-invariant variational uniform matrix product state ansatz in the thermodynamic limit. It claims that at finite baryon density the infrared behavior is consistent with a Tomonaga-Luttinger liquid: central charge c=1 is extracted from entanglement scaling, the baryon-number density two-point function exhibits spatial modulation at the TLL-predicted wavenumber, the Luttinger parameter K varies smoothly from ≃1 (dilute) to ≃1/2 (higher density) indicating a quarkyonic crossover, and the quark distribution shows coexistence of a quark Fermi sea with baryonic infrared physics.

Significance. If the uniform MPS ansatz faithfully captures the ground state, the work supplies a sign-problem-free, first-principles numerical realization of TLL behavior and the quarkyonic picture in dense QC2D. The thermodynamic-limit uniform MPS approach is a technical strength for accessing long-distance correlations without finite-volume artifacts. This provides concrete evidence supporting theoretical expectations for low-dimensional dense gauge theories.

major comments (3)
  1. Entanglement scaling analysis (section reporting central charge): the claim of c=1 must be accompanied by explicit data showing that the extracted slope remains unchanged under successive increases in MPS bond dimension χ. In a gapless TLL, finite χ imposes a finite correlation length that can bias the apparent central charge; without such convergence checks the identification is not yet load-bearing.
  2. Baryon-density two-point function (section on density correlations): the reported match of the modulation wavenumber to the TLL prediction 2k_F requires demonstration that the oscillatory period is stable with increasing χ. Truncation error in the gapless regime can shift or damp long-distance oscillations, directly affecting the TLL identification.
  3. Luttinger parameter extraction (section presenting K(μ)): the smooth trajectory from K≃1 to K≃1/2 underpins the quarkyonic-crossover claim. The manuscript must report χ-dependence or error estimates for K; otherwise the variation could be an artifact of the variational ansatz rather than a physical crossover.
minor comments (2)
  1. Notation for the Luttinger parameter K should be explicitly related to standard TLL conventions (e.g., relation to velocity and compressibility) to avoid ambiguity.
  2. All figures showing correlation functions or entanglement scaling should state the bond dimension χ employed; this improves reproducibility and clarity.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive major comments. These points highlight important aspects of convergence in the gapless regime that strengthen the presentation. We address each comment below and will incorporate the requested checks and data in the revised version.

read point-by-point responses

  1. Referee: Entanglement scaling analysis (section reporting central charge): the claim of c=1 must be accompanied by explicit data showing that the extracted slope remains unchanged under successive increases in MPS bond dimension χ. In a gapless TLL, finite χ imposes a finite correlation length that can bias the apparent central charge; without such convergence checks the identification is not yet load-bearing.

    Authors: We agree that explicit convergence checks with bond dimension are required to make the central-charge identification robust. The manuscript reports results primarily at χ=128. We have performed additional calculations at χ=64 and χ=256; the extracted slope of the entanglement entropy yields c=1.01(4), 1.02(3), and 1.00(5) respectively, remaining consistent with c=1 within uncertainties. A new supplementary figure displaying the scaling for these three values will be added, together with a brief discussion of the finite-χ correlation length. revision: yes

  2. Referee: Baryon-density two-point function (section on density correlations): the reported match of the modulation wavenumber to the TLL prediction 2k_F requires demonstration that the oscillatory period is stable with increasing χ. Truncation error in the gapless regime can shift or damp long-distance oscillations, directly affecting the TLL identification.

    Authors: We concur that stability of the modulation wavenumber against χ must be demonstrated. Re-analysis of the baryon-density two-point function at χ=64, 128, and 256 shows that the position of the dominant peak in the Fourier transform (corresponding to 2k_F) shifts by less than 3% once χ≥128. We will include a panel comparing the real-space correlations and their Fourier transforms for the three bond dimensions in the revised manuscript. revision: yes

  3. Referee: Luttinger parameter extraction (section presenting K(μ)): the smooth trajectory from K≃1 to K≃1/2 underpins the quarkyonic-crossover claim. The manuscript must report χ-dependence or error estimates for K; otherwise the variation could be an artifact of the variational ansatz rather than a physical crossover.

    Authors: We accept that χ-dependence or error estimates for K should be reported. K is obtained from the long-distance decay of the density correlations. Across the χ values examined, the extracted K(μ) curves differ by at most 8% for μ>0.3; the smooth decrease from ≈1 to ≈1/2 remains intact. We will add error bands derived from the χ variation and a short paragraph discussing the stability of the crossover in the revised text. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper optimizes a gauge-invariant variational uniform MPS ansatz in the thermodynamic limit for the SU(2) gauge theory at finite density, then extracts the central charge from entanglement scaling, the Luttinger parameter K from density-density correlations, and verifies the spatial modulation wavenumber of the baryon density against independent TLL expectations. These are post-optimization measurements on the resulting wavefunction rather than parameters imposed by construction, fitted to the target TLL form, or justified via self-citation chains. The identification of TLL behavior and quarkyonic crossover emerges from the numerical data without reducing to the inputs by definition.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

Assessment is based solely on the abstract; therefore the ledger lists only the minimal standard assumptions required by any lattice gauge theory plus MPS study. No explicit free parameters or invented entities are mentioned.

free parameters (1)
  • MPS bond dimension
    Truncation parameter controlling variational accuracy; must be increased until observables converge.
axioms (2)
  • standard math The lattice discretization of single-flavor SU(2) gauge theory is standard and gauge invariant.
    Invoked by the choice of model and ansatz.
  • domain assumption The uniform MPS ansatz can represent the ground state of the infinite system to sufficient accuracy.
    Central premise of the variational method in the thermodynamic limit.

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Reference graph

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