---
title: Quarkyonic Model
url: https://www.emergentmind.com/topics/quarkyonic-model
type: topic
---

# Quarkyonic Model

Searching arXiv for recent quarkyonic-model papers to support the article.
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{"cmd":"bash -lc \"python - <<'PY'\nimport urllib.parse, urllib.request, ssl\nquery='all:quarkyonic matter'\nurl='https://export.arxiv.org/api/query?search_query=%s&start=0&max_results=10&sortBy=submittedDate&sortOrder=descending' % urllib.parse.quote(query)\nprint(url)\nctx=ssl.create_default_context()\nprint(urllib.request.urlopen(url, timeout=20, context=ctx).read().decode('utf-8')[:5000])\nPY\""}ართული to=commentary ็ตทรู code  on იძლिच्छ?
{"cmd":"bash -lc \"python - <<'PY'\nimport requests, urllib.parse\nquery='all:quarkyonic matter'\nurl='https://export.arxiv.org/api/query?search_query=%s&start=0&max_results=5&sortBy=submittedDate&sortOrder=descending' % urllib.parse.quote(query)\nprint(url)\nr=requests.get(url, timeout=20)\nprint(r.status_code)\nprint(r.text[:3000])\nPY\""}
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Quarkyonic model denotes a class of dense-QCD descriptions in which the bulk thermodynamics becomes quark-like while low-energy excitations remain confined or baryonic. In these constructions, matter at low density is hadronic, but beyond an onset density the low-momentum region of phase space is increasingly occupied by quarks, whereas baryons persist near the Fermi surface as a shell or surface layer. The framework was originally motivated by large-\(N_c\) reasoning and is commonly used as a crossover alternative to both purely hadronic matter and sharply deconfined quark matter in the density range relevant to neutron-star interiors [1811.12503][2606.00943].

## 1. Large-\(N_c\) origin and defining idea

The standard large-\(N_c\) motivation is that confinement need not disappear immediately at high baryon density. One formulation states that the color Debye mass scales as \(m_D \simeq g\,\mu_Q\), so confinement can persist up to \(\mu_Q \lesssim \sqrt{N_c}\,\Lambda_{\mathrm{QCD}}\); this supports a regime in which dense matter is quark-filled in the bulk yet still confined near the Fermi surface [1811.12503]. In this sense, quarkyonic matter is not simply “hadronic matter plus quark corrections,” but a distinct momentum-space organization of dense matter.

A central clarification in the literature is that the quarkyonic phase is a momentum-space mixed phase rather than a spatial mixed phase. Deep inside the Fermi sea, quarks occupy low-momentum states; near the Fermi surface, excitations are more naturally baryonic because confining dynamics remain important there. This distinction matters thermodynamically: because the quarkyonic construction is not a constant-pressure spatial coexistence region, it need not produce the soft equation of state usually associated with a first-order Maxwell construction [1908.04799].

Some formulations define quarkyonic matter through the separation of confinement and chiral restoration. In a holographic Einstein–Maxwell–Dilaton construction, the deconfinement line from the Polyakov loop remains a crossover and depends weakly on \(\mu\), while the chiral line turns from crossover to first order at sufficiently large density, generating an intermediate region that is chirally restored but still confined; that region is identified there as quarkyonic [1908.02000]. This use of the term is narrower than the momentum-shell definition, but both emphasize dense matter that is not conventionally deconfined.

## 2. Momentum-space shell structure and dual quark–baryon description

The canonical quarkyonic construction splits phase space into an inner quark region and an outer baryonic shell. A widely used zero-temperature parameterization writes the baryon density as
\[
n_B=\frac{2}{3\pi^2}\left[F_B^3-(F_B-\Delta)^3+F_Q^3\right],
\qquad
F_Q=\frac{F_B-\Delta}{N_c}\,\Theta(F_B-\Delta),
\]
where \(F_B\) is the baryon Fermi momentum, \(F_Q\) is the quark Fermi momentum, and \(\Delta\) is the shell thickness [1811.12503]. In related constituent-quark and excluded-volume constructions, the occupied region is expressed as \(k_Q<k_F/N_c\) for quarks and \(k_F<k_N<k_F+\Delta\) for nucleons [1908.04799].

A more explicit dual description relates quark and baryon occupation probabilities through a convolution with the quark momentum distribution inside a baryon,
\[
f_Q(q)=\int_k \varphi\!\left(\mathbf q-\frac{\mathbf k}{N_c}\right) f_B(k).
\]
This relation is central to the ideal dual quarkyonic and QQMC constructions, where quark Pauli blocking is imposed directly at the level of occupation numbers rather than introduced as a purely phenomenological onset criterion [2410.22758][2603.19839].

When low-momentum quark states saturate, the baryon distribution reorganizes into a depleted bulk plus a shell,
\[
f_B(k)=\frac{1}{N_c^3}\,\Theta(k_{\rm bu}-k)
+\Theta(k_{\rm sh}-k)\,\Theta(k-k_{\rm bu}).
\]
In the ideal construction, quark saturation \(f_Q(q)=1\) implies \(f_B(N_c q)=1/N_c^3\) in the saturated region, so the low-momentum baryon occupation is reduced by the color factor while the shell remains fully occupied [2606.00943]. This is the standard quarkyonic shell picture in its most explicit form: quarks fill the bulk, baryons live at the surface.

## 3. Major model realizations

One important realization derives the shell structure dynamically from hard-core nucleon repulsion treated in excluded volume. In that framework,
\[
n_{\rm ex}=\frac{n}{1-n/n_0},
\qquad
\epsilon=\left(1-\frac{n}{n_0}\right)\epsilon_{\rm ex},
\]
and minimization of the total energy at fixed density yields the appearance of a nucleonic shell near the hard-core density scale. At large density the shell becomes thin, with \(\Delta \sim n_{\rm ex}^N/k_F^2\), which reproduces the quarkyonic expectation that quarks dominate most of phase space while baryons survive only in a narrow surface layer [1908.04799].

The finite-temperature excluded-volume extension keeps the shell geometry while introducing thermal smearing through Fermi–Dirac factors. In that formulation, the quark onset density decreases only slightly with temperature, by about \(\sim 2\%\) up to \(T\sim 50\) MeV, and for \(n_0=2.5\,n_{\text{sat}}\) the onset remains well approximated by \(n_{\text{onset}}\approx 0.863\,n_0\) with about \(5\%\) accuracy over \(0\le T\le 50\) MeV [2002.11133]. The weak thermal sensitivity is attributed there to large-\(N_c\) suppression.

A different realization is field-theoretical rather than kinematic. By combining the Walecka model with the quark–meson model, a mean-field quarkyonic Lagrangian with simultaneous quark and nucleon degrees of freedom produces a continuous transition from nucleon-dominance to quark-dominance while reproducing symmetric nuclear matter saturation. In that calibration the best agreement is obtained for \(m_q^v=370.4\pm 0.8\) MeV, with \(g_{\rm Nv}\approx 7.2\), \(g_{\rm sv}\approx 81\), and a nucleon fraction \(R_N\approx 85.8\%\) at saturation [2007.02028].

More recent quark-substructure realizations combine quarkyonic phase-space rearrangement with the quark–meson coupling model. In the QQMC construction, confined quarks in the nucleon are described by relativistic Gaussian wavefunctions derived from
\[
U(r)=\frac{c}{2}(1+\gamma_0)r^2,
\]
and in-medium quark parameters are shifted by scalar and vector mean fields,
\[
m^\ast=m-g_\sigma^q\bar\sigma,\qquad
\epsilon^\ast=\epsilon_j-g_\omega^q\bar\omega \mp g_\rho^q\bar\rho.
\]
This makes the onset of quark saturation sensitive not only to kinematics but also to the internal quark structure of the nucleon and to nuclear mean fields [2603.19839].

## 4. Thermodynamics, quark saturation, and sound speed

A recurrent result across quarkyonic models is rapid pressure growth once the quark-filled interior appears. In the original shell model, the sound speed is written as
\[
c_s^2=\frac{\partial P}{\partial \epsilon}
=\frac{n_B}{\mu_B\, dn_B/d\mu_B},
\]
so any suppression of the density response \(dn_B/d\mu_B\) around onset produces a sharp increase in \(c_s^2\) [1811.12503]. In one representative fit, the onset occurs near \(n_B\simeq 0.24\,\mathrm{fm}^{-3}\) and the sound speed reaches \(c_s\simeq 0.94\) at \(n_B\simeq 0.64\,\mathrm{fm}^{-3}\) before decreasing and then tending toward the asymptotic value \(1/\sqrt{3}\) [1811.12503].

In quark-substructure models, the key control quantity is the quark saturation density \(\rho_{\rm sat}\), defined as the density at which the lowest quark state becomes fully occupied. The QQMC calculation finds a strong dependence on the proton size parameter \(r_p\): in symmetric nuclear matter,
\[
\rho_{\rm sat}/\rho_0 = 3.59,\ 2.21,\ 1.46
\]
for \(r_p=0.6,\ 0.7,\ 0.8\) fm, respectively [2603.19839]. The same work reports that nuclear interactions lower the onset relative to the noninteracting Gaussian quarkyonic model; for \(r_p=0.7\) fm, \(\rho_{\rm sat}/\rho_0\) is \(2.21\) in QQMC versus \(2.53\) in GQ, indicating that interactions enhance stiffening in the quarkyonic regime [2603.19839].

The ideal Gaussian quarkyonic construction also reveals a technical difficulty: while the energy density remains continuous at \(\rho_{\rm sat}\), the chemical potential and pressure are discontinuous and \(v_s^2\) diverges there. An infrared regulator can remove this singular behavior, and for \(\frac32<\nu<3\) the regulated model makes \(\mu\), \(P\), and \(v_s^2\) continuous at \(\rho_{\rm sat}\) [2512.04505]. This singular-onset problem is one of the clearest indicators that many quarkyonic implementations are effective descriptions of a crossover, not microscopic derivations from QCD.

The same calculations also show a systematic asymmetry between symmetric nuclear matter and pure neutron matter. Pure neutron matter reaches quark saturation earlier and stiffens more strongly because \(d\)-quark states fill more rapidly than \(u\)-quark states, so the quarkyonic rearrangement has a larger impact on neutron-rich matter [2603.19839].

## 5. Beta equilibrium, magnetism, hyperons, and strangeness

For neutron-star applications, a major development is the beta-equilibrated quarkyonic equation of state in which neutrons, protons, \(u\) and \(d\) quarks, electrons, and muons are treated self-consistently. In one such formulation, the EOS is controlled by only three free parameters—\(L\), \(n_t\), and \(\Lambda\)—yet still reproduces nuclear saturation and pure neutron matter constraints while allowing \(M_{\max}\gtrsim 2M_\odot\) and \(R_{1.4}\lesssim 13.5\) km over a wide region of parameter space [2004.08293]. This construction is explicitly presented as an alternative to first-order hybrid-star models that often require fine tuning near nuclear saturation density.

Spin-polarized extensions show that quarkyonic matter can have qualitatively different magnetic response from conventional nuclear matter. In a model where quarks in the deep Fermi sea remain unpolarized but nucleons in the shell can polarize, the magnetic susceptibility
\[
\chi=\frac{\partial^2 \varepsilon_{\rm total}}{\partial \xi^2}\bigg|_{\xi=0}
\]
can become negative in pure neutron matter if the spin-dependent interaction coefficient \(\tilde p\) is sufficiently attractive. For \(\tilde p=-0.002\ {\rm MeV\cdot fm^6}\), the system returns from a ferromagnetic to a paramagnetic regime around \(n_B\sim 5.5\,n_0\) as Pauli pressure overtakes spin alignment [2507.06577].

Hyperonic matter provides a distinct use of quarkyonic reasoning. In the three-flavor IdylliQ extension, neutron occupation of down-quark states delays the onset of \(S=-1\) hyperons, shifting the threshold from the naive hadronic value \(M_Y\) to
\[
\mu_B^{\rm onset}=2M_Y-M_N.
\]
The same work argues that the equation of state softens only mildly above threshold because low-energy hyperon phase space is strongly restricted by quark Pauli blocking [2410.22758]. A related ideal model with hyperons and charge-neutral matter states that statistical quark-level constraints can delay \(\Lambda_0\) onset from around \(\sim 2n_0\) to around \(\sim 5n_0\), thereby mitigating the hyperon puzzle without inserting strong hyperon repulsion by hand [2606.00943].

Extended RMF–equivparticle models that include \(\Lambda\), \(\Xi\), \(\Sigma\), and strange quarks draw a more conservative conclusion: hyperons inevitably soften the EOS around \(n_{\rm b}\approx 2n_0\), and the quark-hadron transition softens it further at high density, yielding a maximum sound speed \(v_{\rm max}\approx 0.6\,c\), close to the ultrarelativistic limit \(0.58\,c\) [2601.17300]. Taken together, these results show that the treatment of strangeness is model-dependent even within the quarkyonic paradigm.

## 6. Astrophysical realizations, phenomenology, and open questions

Relativistic mean-field implementations of quarkyonic matter have been used extensively for neutron stars. One RMF-based quarkyonic EOS with transition density \(n_t\) and confinement-scale parameter \(\Lambda\) yields stars with maximum masses of order \(\sim 2.8\,M_\odot\), while its tidal deformabilities are reported to be compatible with GW170817 and GW190425 constraints [2304.08223]. The same work notes that quarkyonic merger remnants in numerical-relativity simulations tend to merge earlier by about \(\sim 1\) ms, show reduced post-merger \(f_2\), and are more resistant to prompt collapse than their purely hadronic counterparts [2304.08223].

Oscillation spectroscopy has also been proposed as a discriminant. In a multicomponent star with a hadronic mantle and a quarkyonic core, spacetime \(\omega\)-mode frequencies are reported in the \(5\)–\(8\) kHz range with damping times of order \(10^2\ \mu\)s for the fundamental branch, and approximate universal relations are found after suitable rescaling by compactness or central pressure [2602.22641]. This treatment emphasizes smooth crossover rather than Maxwell or Gibbs interfaces.

Not all hadronic baselines favor a quarkyonic transition equally. In an extended RMF plus equivparticle construction, the TW99 functional already gives satisfactory nuclear and neutron-star properties and makes quarkyonic transition unfavorable, whereas the stiffer PKDD and DD-ME2 functionals become more acceptable only after quarkyonic softening is included [2307.03032]. This is a reminder that quarkyonic matter is often introduced not as a unique prediction but as a controlled modification of otherwise over-stiff or over-soft high-density matter.

The phenomenological domain of the model is wider than neutron stars. A percolation reinterpretation describes quarkyonic matter as a confined but percolating medium in which quark wavefunctions propagate through dense baryonic matter, leaving a possible confined-percolating sliver even for \(N_c=3,\ N_f=2\) at densities above roughly one baryon per baryonic volume [1211.2433]. Another line of work asks whether ordinary nuclear matter may already be close to quarkyonic onset: in the IdylliQ sigma model, nuclear matter forms at a density close to but slightly above quarkyonic onset, and the predicted depletion of low-momentum nucleons for \(k\lesssim 120\) MeV is argued to be consistent with electron scattering data [2403.15375].

The principal unresolved issue across quarkyonic models is not whether they can produce stiff intermediate-density matter—they often can—but how smooth the crossover should be and how the shell structure should emerge microscopically. Singular behavior at \(\rho_{\rm sat}\), model dependence of shell thickness \(\Delta\), sensitivity to nucleon size parameters, and differing treatments of hyperons and strange quarks all indicate that “the quarkyonic model” is best understood as a family of related dense-matter constructions rather than a single unique theory. What unifies that family is the same organizing principle: dense matter can be quark-dominated in bulk occupancy while remaining baryonic in its low-energy excitations.

Source: https://www.emergentmind.com/topics/quarkyonic-model