Journal of High Energy Physics

Dense QC2D2 with uniform matrix product states

Kohei Fujikura, Yoshimasa Hidaka

Published Sep 24, 2026• 0 citations

Abstract

A bstract We study cold dense single-flavor 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 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 ≃ 1 in the dilute-baryon regime to K ≃ 1/2 at higher densities, suggesting a quarkyonic crossover. Furthermore, the quark momentum distribution reveals the coexistence of a quark Fermi sea with a baryonic infrared description, thereby realizing the quarkyonic picture from first principles.

DOI: 10.1007/jhep09(2026)257

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Journal of High Energy Physics

Dense QC2D2 with uniform matrix product states

Kohei Fujikura, Yoshimasa Hidaka

Published Sep 24, 2026• 0 citations

Publisher

Abstract

DOI: 10.1007/jhep09(2026)257

Related papers

Dense $\mathrm{QC_2D_2}$ with uniform matrix product statesDense $\mathrm{QC_2D_2}$ with uniform matrix product states

Kohei Fujikura, Yoshimasa Hidaka

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.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.

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We study equilibrium and nonequilibrium aspects of the normal state of cold and dense QCD and QED. The exchange of dynamically screened magnetic gluons (photons) leads to infrared singularities in the fermion propagator for excitations near the Fermi surface and the breakdown of the Fermi liquid description. We implement a resummation of these infrared divergences via the Euclidean renormalization group to obtain the spectral density, dispersion relation, widths, and wave function renormalization for single quasiparticles near the Fermi surface. We find that all features scale with anomalous dimensions: ${\ensuremath{\omega}}_{p}(k)\ensuremath{\propto}|k\ensuremath{-}{k}_{F}{|}^{1/(1\ensuremath{-}2\ensuremath{\lambda})},$ $\ensuremath{\Gamma}(k)\ensuremath{\propto}|k\ensuremath{-}{k}_{F}{|}^{1/(1\ensuremath{-}2\ensuremath{\lambda})};$ ${Z}_{p}(k)\ensuremath{\propto}|k\ensuremath{-}{k}_{F}{|}^{2\ensuremath{\lambda}/(1\ensuremath{-}2\ensuremath{\lambda})}$ with $\ensuremath{\lambda}=\ensuremath{\alpha}/6\ensuremath{\pi}$ for QED, $({\ensuremath{\alpha}}_{s}/6\ensuremath{\pi}{)(N}_{c}^{2}\ensuremath{-}{1)/2N}_{c}$ for QCD with ${N}_{c}$ colors and ${N}_{F}$ flavors. The discontinuity of the quasiparticle distribution at the Fermi surface vanishes. For $k\ensuremath{\approx}{k}_{F}$ we find ${n}_{k\ensuremath{\approx}{k}_{F}}=\mathrm{sin}[\ensuremath{\pi}\ensuremath{\lambda}]/2\ensuremath{\pi}\ensuremath{\lambda}\ensuremath{-}(k\ensuremath{-}{k}_{F})/\ensuremath{\pi}M(1\ensuremath{-}4\ensuremath{\lambda})+\mathcal{O}(k\ensuremath{-}{k}_{F}{)}^{2}$ with $M$ the dynamical screening scale of magnetic gluons (photons). The dynamical renormalization group is implemented to study nonequilibrium relaxation. The amplitude of single quasiparticle states with momentum near the Fermi surface falls off as $|{\ensuremath{\Psi}}_{k\ensuremath{\approx}{k}_{F}}(t)|\ensuremath{\approx}|{\ensuremath{\Psi}}_{k\ensuremath{\approx}{k}_{F}}{(t}_{0})|{e}^{\ensuremath{-}\ensuremath{\Gamma}(k)(t\ensuremath{-}{t}_{0})}[{t}_{0}/t{]}^{2\ensuremath{\lambda}}.$ Thus quasiparticle states with Fermi momentum have zero group velocity and relax with a power law with a coupling-dependent anomalous dimension.We study equilibrium and nonequilibrium aspects of the normal state of cold and dense QCD and QED. The exchange of dynamically screened magnetic gluons (photons) leads to infrared singularities in the fermion propagator for excitations near the Fermi surface and the breakdown of the Fermi liquid description. We implement a resummation of these infrared divergences via the Euclidean renormalization group to obtain the spectral density, dispersion relation, widths, and wave function renormalization for single quasiparticles near the Fermi surface. We find that all features scale with anomalous dimensions: ${\ensuremath{\omega}}_{p}(k)\ensuremath{\propto}|k\ensuremath{-}{k}_{F}{|}^{1/(1\ensuremath{-}2\ensuremath{\lambda})},$ $\ensuremath{\Gamma}(k)\ensuremath{\propto}|k\ensuremath{-}{k}_{F}{|}^{1/(1\ensuremath{-}2\ensuremath{\lambda})};$ ${Z}_{p}(k)\ensuremath{\propto}|k\ensuremath{-}{k}_{F}{|}^{2\ensuremath{\lambda}/(1\ensuremath{-}2\ensuremath{\lambda})}$ with $\ensuremath{\lambda}=\ensuremath{\alpha}/6\ensuremath{\pi}$ for QED, $({\ensuremath{\alpha}}_{s}/6\ensuremath{\pi}{)(N}_{c}^{2}\ensuremath{-}{1)/2N}_{c}$ for QCD with ${N}_{c}$ colors and ${N}_{F}$ flavors. The discontinuity of the quasiparticle distribution at the Fermi surface vanishes. For $k\ensuremath{\approx}{k}_{F}$ we find ${n}_{k\ensuremath{\approx}{k}_{F}}=\mathrm{sin}[\ensuremath{\pi}\ensuremath{\lambda}]/2\ensuremath{\pi}\ensuremath{\lambda}\ensuremath{-}(k\ensuremath{-}{k}_{F})/\ensuremath{\pi}M(1\ensuremath{-}4\ensuremath{\lambda})+\mathcal{O}(k\ensuremath{-}{k}_{F}{)}^{2}$ with $M$ the dynamical screening scale of magnetic gluons (photons). The dynamical renormalization group is implemented to study nonequilibrium relaxation. The amplitude of single quasiparticle states with momentum near the Fermi surface falls off as $|{\ensuremath{\Psi}}_{k\ensuremath{\approx}{k}_{F}}(t)|\ensuremath{\approx}|{\ensuremath{\Psi}}_{k\ensuremath{\approx}{k}_{F}}{(t}_{0})|{e}^{\ensuremath{-}\ensuremath{\Gamma}(k)(t\ensuremath{-}{t}_{0})}[{t}_{0}/t{]}^{2\ensuremath{\lambda}}.$ Thus quasiparticle states with Fermi momentum have zero group velocity and relax with a power law with a coupling-dependent anomalous dimension.

arXiv:hep-ph/0009172Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fieldsJan 11, 200141 citations

Particle formation and ordering in strongly correlated fermionic systems: Solving a model of quantum chromodynamics

P. Azaria, Robert Konik, P. Lecheminant +3 more

In this paper we study a ( $1+1$ )-dimensional version of the famous Nambu--Jona-Lasinio model of quantum chromodynamics (QCD2) both at zero and at finite baryon density. We use nonperturbative techniques (non-Abelian bosonization and the truncated conformal spectrum approach). When the baryon chemical potential, $\ensuremath{\mu}$, is zero, we describe the formation of fermion three-quark (nucleons and $\mathrm{\ensuremath{\Delta}}$ baryons) and boson (two-quark mesons, six-quark deuterons) bound states. We also study at $\ensuremath{\mu}=0$ the formation of a topologically nontrivial phase. When the chemical potential exceeds the critical value and a finite baryon density appears, the model has a rich phase diagram which includes phases with a density wave and superfluid quasi-long-range (QLR) order, as well as a phase of a baryon Tomonaga-Luttinger liquid (strange metal). The QLR order results in either a condensation of scalar mesons (the density wave) or six-quark bound states (deuterons).In this paper we study a ($1+1$)-dimensional version of the famous Nambu--Jona-Lasinio model of quantum chromodynamics (QCD2) both at zero and at finite baryon density. We use nonperturbative techniques (non-Abelian bosonization and the truncated conformal spectrum approach). When the baryon chemical potential, $\ensuremath{\mu}$, is zero, we describe the formation of fermion three-quark (nucleons and $\mathrm{\ensuremath{\Delta}}$ baryons) and boson (two-quark mesons, six-quark deuterons) bound states. We also study at $\ensuremath{\mu}=0$ the formation of a topologically nontrivial phase. When the chemical potential exceeds the critical value and a finite baryon density appears, the model has a rich phase diagram which includes phases with a density wave and superfluid quasi-long-range (QLR) order, as well as a phase of a baryon Tomonaga-Luttinger liquid (strange metal). The QLR order results in either a condensation of scalar mesons (the density wave) or six-quark bound states (deuterons).

arXiv:1601.02979Physical review. D/Physical review. D.Aug 3, 201622 citations

Dense

\mathrm{QC_2D_2}

with uniform matrix product states

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.

Dense

\textrm{QCD}_2

with matrix product states

We study one-flavor

\mathrm{SU}(2)

and

\mathrm{SU}(3)

lattice QCD in (

1+1

) dimensions at zero temperature and finite density using matrix product states and the density matrix renormalization group. We compute physical observables such as the equation of state, chiral condensate, and quark distribution function as functions of the baryon number density. As a physical implication, we discuss the inhomogeneous phase at nonzero baryon density, where the chiral condensate is inhomogeneous, and baryons form a crystal. We also discuss how the dynamical degrees of freedom change from hadrons to quarks through the formation of quark Fermi seas.

{N}_{c}

colors and

{N}_{F}

flavors. The discontinuity of the quasiparticle distribution at the Fermi surface vanishes. For $k\ensuremath{\approx}{k}_{F}$ we find ${n}_{k\ensuremath{\approx}{k}_{F}}=\mathrm{sin}[\ensuremath{\pi}\ensuremath{\lambda}]/2\ensuremath{\pi}\ensuremath{\lambda}\ensuremath{-}(k\ensuremath{-}{k}_{F})/\ensuremath{\pi}M(1\ensuremath{-}4\ensuremath{\lambda})+\mathcal{O}(k\ensuremath{-}{k}_{F}{)}^{2}$ with

M

the dynamical screening scale of magnetic gluons (photons). The dynamical renormalization group is implemented to study nonequilibrium relaxation. The amplitude of single quasiparticle states with momentum near the Fermi surface falls off as $|{\ensuremath{\Psi}}_{k\ensuremath{\approx}{k}_{F}}(t)|\ensuremath{\approx}|{\ensuremath{\Psi}}_{k\ensuremath{\approx}{k}_{F}}{(t}_{0})|{e}^{\ensuremath{-}\ensuremath{\Gamma}(k)(t\ensuremath{-}{t}_{0})}[{t}_{0}/t{]}^{2\ensuremath{\lambda}}.$ Thus quasiparticle states with Fermi momentum have zero group velocity and relax with a power law with a coupling-dependent anomalous dimension.

In this paper we study a (

1+1

)-dimensional version of the famous Nambu--Jona-Lasinio model of quantum chromodynamics (QCD2) both at zero and at finite baryon density. We use nonperturbative techniques (non-Abelian bosonization and the truncated conformal spectrum approach). When the baryon chemical potential, $\ensuremath{\mu}$, is zero, we describe the formation of fermion three-quark (nucleons and $\mathrm{\ensuremath{\Delta}}$ baryons) and boson (two-quark mesons, six-quark deuterons) bound states. We also study at $\ensuremath{\mu}=0$ the formation of a topologically nontrivial phase. When the chemical potential exceeds the critical value and a finite baryon density appears, the model has a rich phase diagram which includes phases with a density wave and superfluid quasi-long-range (QLR) order, as well as a phase of a baryon Tomonaga-Luttinger liquid (strange metal). The QLR order results in either a condensation of scalar mesons (the density wave) or six-quark bound states (deuterons).