Baryonic Vortex Lattice in Dense QCD

• Baryonic Vortex Lattice is a dense QCD phase where baryon number is carried by topologically linked pion vortices rather than conventional baryon fields.
• It features a unique interplay between local charged (ANO-like) vortices and global neutral-pion vortices connected through domain walls, yielding Skyrmion-like objects.
• Energy calculations indicate that above a critical baryon chemical potential, a triangular lattice of vortices becomes energetically favorable compared to homogeneous pion condensation.

Searching arXiv for the cited works and closely related papers. A baryonic vortex lattice is a proposed phase of low-energy dense QCD in which baryon number is carried by topological vortex configurations rather than by conventional localized baryon fields. In two-flavor chiral perturbation theory with finite isospin chemical potential, finite baryon chemical potential, finite pion mass, and dynamical electromagnetism, the relevant ground state for $\mu_I>m_\pi$ contains pion condensation and supports vortices in both charged and neutral pion sectors. The recent linked-vortex formulation identifies each baryon with a topological linking between a local $\pi^\pm$ Abrikosov–Nielsen–Olesen-like vortex and a closed $\pi^0$ global vortex line attached to a domain wall, while earlier work emphasized a vortex-Skyrmion state in which neutral-pion modulation inside a charged vortex core endows the vortex with baryon number. When the baryon chemical potential exceeds a critical value, these objects become energetically favorable and can organize into a triangular Abrikosov-type lattice, yielding a candidate dense baryonic phase of hadronic matter (Hamada et al., 25 Sep 2025, Qiu et al., 2024).

1. Effective-field-theory setting

The construction is formulated in two-flavor Chiral Perturbation Theory at leading order in the derivative expansion. The pion field is parametrized by an $SU(2)$ matrix

$$U(x)=\exp\Big(i\,\tau^a\pi^a(x)/f_\pi\Big)\in SU(2),$$

with Pauli matrices $\tau^a$. The effective Lagrangian is written as

$L_{\rm eff}=L_{\rm chiral}+L_{\rm EM}+L_{\rm WZW},$

where

$$L_{\rm chiral} =\frac{f_\pi^2}{4}\,\mathrm{Tr}\!\left[D_\mu U^\dagger D^\mu U\right] +\frac{f_\pi^2 m_\pi^2}{4}\,\mathrm{Tr}\!\left[U+U^\dagger-2\right],$$

$$L_{\rm EM}=-\frac{1}{4e^2}F_{\mu\nu}F^{\mu\nu}, \qquad L_{\rm WZW}=\bigl(A_\mu^B+q\,A_\mu^{\rm EM}\bigr)j_B^\mu, \qquad q=\tfrac12.$$

The covariant derivative is

$$D_\mu U=\partial_\mu U-iA_\mu[Q,U], \qquad Q=\frac{1}{6}\mathbf1+\frac{\tau^3}{2},$$

with the combination $\pi^\pm$0 including the static isospin chemical potential as $\pi^\pm$1 plus the dynamical photon. A fictitious baryon gauge field implements the baryon chemical potential through $\pi^\pm$2 (Hamada et al., 25 Sep 2025).

The topological input is the conserved baryon current. In the absence of gauge fields it is

$\pi^\pm$3

up to covariant gauge-field corrections. Integrating $\pi^\pm$4 over space yields an integer Skyrmion number. In the complementary notation used for the vortex-Skyrmion formulation, $\pi^\pm$5 with $\pi^\pm$6, and the same low-energy description couples chiral fields, electromagnetism, and the Goldstone–Wilczek current to finite $\pi^\pm$7 and $\pi^\pm$8 (Qiu et al., 2024).

2. Homogeneous pion-condensed background

The existence of the baryonic vortex lattice relies on the structure of the homogeneous ground state at finite isospin density. For $\pi^\pm$9, the minimum of the chiral Lagrangian is the trivial vacuum,

$\pi^0$0

Once $\pi^0$1, charged pions condense in the direction $\pi^0$2, and one may write the uniform condensate as

$\pi^0$3

In the basis

$\pi^0$4

the condensates satisfy

$\pi^0$5

Thus both neutral and charged pions have nonzero expectation values in the bulk once $\pi^0$6 exceeds $\pi^0$7 (Hamada et al., 25 Sep 2025).

This coexistence is structurally important because it allows two different topological sectors to coexist in the same medium. The charged-pion condensate breaks $\pi^0$8 spontaneously and supports local flux-carrying vortices. Simultaneously, the nonzero neutral-pion condensate leaves a phase modulus that supports global vortices, with explicit breaking by the pion mass converting the neutral sector into a vortex–domain-wall system. A plausible implication is that the baryonic lattice is not an auxiliary add-on to the pion-condensed phase but an instability of that phase once $\pi^0$9 becomes sufficiently large.

3. Local, global, and linked vortex configurations

In the phase $SU(2)$0, the charged condensate supports an Abrikosov–Nielsen–Olesen-like local vortex. For a straight unit-winding vortex along the $SU(2)$1-axis, the standard cylindrical ansatz is

$SU(2)$2

with boundary conditions

$SU(2)$3

In the simplest pure ANO solution one holds the neutral-pion phase $SU(2)$4 constant, so that the azimuthal baryon current vanishes, $SU(2)$5 (Hamada et al., 25 Sep 2025).

Because $SU(2)$6 remains nonzero in the bulk, the neutral sector supports a global vortex of winding $SU(2)$7,

$SU(2)$8

At finite pion mass, the global vortex line is attached to a two-dimensional domain wall, the chiral soliton, across which $SU(2)$9 jumps by $$U(x)=\exp\Big(i\,\tau^a\pi^a(x)/f_\pi\Big)\in SU(2),$$0. Locally the wall is a sine-Gordon soliton solving

$$U(x)=\exp\Big(i\,\tau^a\pi^a(x)/f_\pi\Big)\in SU(2),$$1

with a kink interpolating $$U(x)=\exp\Big(i\,\tau^a\pi^a(x)/f_\pi\Big)\in SU(2),$$2 (Hamada et al., 25 Sep 2025).

The baryonic configuration of the linked-vortex proposal is obtained by superposing the local charged vortex with a closed neutral-pion vortex ring that encircles it and is capped by a domain wall. Earlier work described a related baryonic vortex phase from a different angle: a vortex with the same quantized magnetic flux as the conventional Abrikosov vortex carries baryon number once one takes into account a modulation of the neutral pion inside the vortex core, and Qiu and Nitta therefore dubbed the resulting object the baryonic vortex (Qiu et al., 2024).

4. Topology, baryon number, and Skyrmion interpretation

The topological classification follows from $$U(x)=\exp\Big(i\,\tau^a\pi^a(x)/f_\pi\Big)\in SU(2),$$3 in the chiral formulation and, equivalently, from $$U(x)=\exp\Big(i\,\tau^a\pi^a(x)/f_\pi\Big)\in SU(2),$$4 in the $$U(x)=\exp\Big(i\,\tau^a\pi^a(x)/f_\pi\Big)\in SU(2),$$5 description. For any smooth configuration approaching a constant at spatial infinity, the baryon number is

$$U(x)=\exp\Big(i\,\tau^a\pi^a(x)/f_\pi\Big)\in SU(2),$$6

In the linked-vortex construction, this integer is identified directly with the linking number,

$$U(x)=\exp\Big(i\,\tau^a\pi^a(x)/f_\pi\Big)\in SU(2),$$7

Because each linked pair carries $$U(x)=\exp\Big(i\,\tau^a\pi^a(x)/f_\pi\Big)\in SU(2),$$8, it is interpreted as a single Skyrmion realized without any Skyrme term; the stability is attributed to the Wess–Zumino–Witten-induced coupling to $$U(x)=\exp\Big(i\,\tau^a\pi^a(x)/f_\pi\Big)\in SU(2),$$9 (Hamada et al., 25 Sep 2025).

In the vortex-Skyrmion formulation, a periodic modulation of the neutral pion along the vortex axis plays an analogous role. For the ansatz

$\tau^a$0

imposing $\tau^a$1, $\tau^a$2, and $\tau^a$3, $\tau^a$4 yields one unit of baryon number per period $\tau^a$5 (Qiu et al., 2024).

Two points are often conflated. First, the baryonic vortex lattice is not merely an Abrikosov lattice of ordinary $\tau^a$6 flux tubes; the baryon number arises from nontrivial neutral-pion structure. Second, the 2025 linked-vortex construction explicitly emphasizes that the Skyrmion-type object is realized without the Skyrme term, whereas the 2024 effective-theory exposition notes that beyond $\tau^a$7 one may add the usual Skyrme term for stability. This suggests that the central topological mechanism is not identical to a conventional Skyrme-model stabilization, even though the resulting objects carry integer baryon charge (Hamada et al., 25 Sep 2025, Qiu et al., 2024).

5. Energetics, lattice formation, and phase structure

The energetics are organized around the competition between the cost of forming a vortex and the gain from the baryon chemical potential. In the linked-vortex formulation, the energy of one baryonic vortex is

$\tau^a$8

and becomes negative when $\tau^a$9 exceeds a critical value $L_{\rm eff}=L_{\rm chiral}+L_{\rm EM}+L_{\rm WZW},$0. Physically, this means that it becomes favorable to nucleate the linked pairs once $L_{\rm eff}=L_{\rm chiral}+L_{\rm EM}+L_{\rm WZW},$1 is large enough. To construct a dense phase, one arranges many identical baryonic vortices on a two-dimensional lattice in the transverse plane and extends them uniformly in $L_{\rm eff}=L_{\rm chiral}+L_{\rm EM}+L_{\rm WZW},$2. The unit-cell energy density is then computed as a function of $L_{\rm eff}=L_{\rm chiral}+L_{\rm EM}+L_{\rm WZW},$3, $L_{\rm eff}=L_{\rm chiral}+L_{\rm EM}+L_{\rm WZW},$4, $L_{\rm eff}=L_{\rm chiral}+L_{\rm EM}+L_{\rm WZW},$5, and the lattice spacing $L_{\rm eff}=L_{\rm chiral}+L_{\rm EM}+L_{\rm WZW},$6 or vortex density. Above $L_{\rm eff}=L_{\rm chiral}+L_{\rm EM}+L_{\rm WZW},$7, the unit-cell energy density is lower than that of a homogeneous charged-pion condensate or an Abrikosov lattice of pure $L_{\rm eff}=L_{\rm chiral}+L_{\rm EM}+L_{\rm WZW},$8 vortices. Varying the lattice geometry, the lowest unit-cell energy is typically achieved by a triangular (hexagonal) lattice, and the optimal spacing $L_{\rm eff}=L_{\rm chiral}+L_{\rm EM}+L_{\rm WZW},$9 decreases as $$L_{\rm chiral} =\frac{f_\pi^2}{4}\,\mathrm{Tr}\!\left[D_\mu U^\dagger D^\mu U\right] +\frac{f_\pi^2 m_\pi^2}{4}\,\mathrm{Tr}\!\left[U+U^\dagger-2\right],$$0 grows (Hamada et al., 25 Sep 2025).

The earlier baryonic-vortex-phase analysis expresses the same instability in terms of the string tension of a single vortex on top of the homogeneous $$L_{\rm chiral} =\frac{f_\pi^2}{4}\,\mathrm{Tr}\!\left[D_\mu U^\dagger D^\mu U\right] +\frac{f_\pi^2 m_\pi^2}{4}\,\mathrm{Tr}\!\left[U+U^\dagger-2\right],$$1 condensate,

$$L_{\rm chiral} =\frac{f_\pi^2}{4}\,\mathrm{Tr}\!\left[D_\mu U^\dagger D^\mu U\right] +\frac{f_\pi^2 m_\pi^2}{4}\,\mathrm{Tr}\!\left[U+U^\dagger-2\right],$$2

with the term $$L_{\rm chiral} =\frac{f_\pi^2}{4}\,\mathrm{Tr}\!\left[D_\mu U^\dagger D^\mu U\right] +\frac{f_\pi^2 m_\pi^2}{4}\,\mathrm{Tr}\!\left[U+U^\dagger-2\right],$$3 arising from $$L_{\rm chiral} =\frac{f_\pi^2}{4}\,\mathrm{Tr}\!\left[D_\mu U^\dagger D^\mu U\right] +\frac{f_\pi^2 m_\pi^2}{4}\,\mathrm{Tr}\!\left[U+U^\dagger-2\right],$$4. For $$L_{\rm chiral} =\frac{f_\pi^2}{4}\,\mathrm{Tr}\!\left[D_\mu U^\dagger D^\mu U\right] +\frac{f_\pi^2 m_\pi^2}{4}\,\mathrm{Tr}\!\left[U+U^\dagger-2\right],$$5 below a certain $$L_{\rm chiral} =\frac{f_\pi^2}{4}\,\mathrm{Tr}\!\left[D_\mu U^\dagger D^\mu U\right] +\frac{f_\pi^2 m_\pi^2}{4}\,\mathrm{Tr}\!\left[U+U^\dagger-2\right],$$6 the minimum tension remains positive, while above

$$L_{\rm chiral} =\frac{f_\pi^2}{4}\,\mathrm{Tr}\!\left[D_\mu U^\dagger D^\mu U\right] +\frac{f_\pi^2 m_\pi^2}{4}\,\mathrm{Tr}\!\left[U+U^\dagger-2\right],$$7

one finds negative tension, so the homogeneous condensate is unstable. In the London approximation the free energy per unit volume of a periodic Abrikosov lattice with areal density $$L_{\rm chiral} =\frac{f_\pi^2}{4}\,\mathrm{Tr}\!\left[D_\mu U^\dagger D^\mu U\right] +\frac{f_\pi^2 m_\pi^2}{4}\,\mathrm{Tr}\!\left[U+U^\dagger-2\right],$$8 is

$$L_{\rm chiral} =\frac{f_\pi^2}{4}\,\mathrm{Tr}\!\left[D_\mu U^\dagger D^\mu U\right] +\frac{f_\pi^2 m_\pi^2}{4}\,\mathrm{Tr}\!\left[U+U^\dagger-2\right],$$9

and minimization yields an optimal vortex density. For a triangular lattice the inter-vortex spacing is

$$L_{\rm EM}=-\frac{1}{4e^2}F_{\mu\nu}F^{\mu\nu}, \qquad L_{\rm WZW}=\bigl(A_\mu^B+q\,A_\mu^{\rm EM}\bigr)j_B^\mu, \qquad q=\tfrac12.$$0

while a square lattice is slightly higher in energy by the standard factor $$L_{\rm EM}=-\frac{1}{4e^2}F_{\mu\nu}F^{\mu\nu}, \qquad L_{\rm WZW}=\bigl(A_\mu^B+q\,A_\mu^{\rm EM}\bigr)j_B^\mu, \qquad q=\tfrac12.$$1. The triangular-array parameter is $$L_{\rm EM}=-\frac{1}{4e^2}F_{\mu\nu}F^{\mu\nu}, \qquad L_{\rm WZW}=\bigl(A_\mu^B+q\,A_\mu^{\rm EM}\bigr)j_B^\mu, \qquad q=\tfrac12.$$2 (Qiu et al., 2024).

These energetics lead to a simple low-temperature, zero-magnetic-field phase diagram. For $$L_{\rm EM}=-\frac{1}{4e^2}F_{\mu\nu}F^{\mu\nu}, \qquad L_{\rm WZW}=\bigl(A_\mu^B+q\,A_\mu^{\rm EM}\bigr)j_B^\mu, \qquad q=\tfrac12.$$3, the ground state is the trivial vacuum. For $$L_{\rm EM}=-\frac{1}{4e^2}F_{\mu\nu}F^{\mu\nu}, \qquad L_{\rm WZW}=\bigl(A_\mu^B+q\,A_\mu^{\rm EM}\bigr)j_B^\mu, \qquad q=\tfrac12.$$4 but $$L_{\rm EM}=-\frac{1}{4e^2}F_{\mu\nu}F^{\mu\nu}, \qquad L_{\rm WZW}=\bigl(A_\mu^B+q\,A_\mu^{\rm EM}\bigr)j_B^\mu, \qquad q=\tfrac12.$$5, the ground state is the homogeneous charged-pion condensate, possibly interrupted by an Abrikosov lattice of pure $$L_{\rm EM}=-\frac{1}{4e^2}F_{\mu\nu}F^{\mu\nu}, \qquad L_{\rm WZW}=\bigl(A_\mu^B+q\,A_\mu^{\rm EM}\bigr)j_B^\mu, \qquad q=\tfrac12.$$6 vortices if an external magnetic field is present. Once $$L_{\rm EM}=-\frac{1}{4e^2}F_{\mu\nu}F^{\mu\nu}, \qquad L_{\rm WZW}=\bigl(A_\mu^B+q\,A_\mu^{\rm EM}\bigr)j_B^\mu, \qquad q=\tfrac12.$$7 crosses the critical line $$L_{\rm EM}=-\frac{1}{4e^2}F_{\mu\nu}F^{\mu\nu}, \qquad L_{\rm WZW}=\bigl(A_\mu^B+q\,A_\mu^{\rm EM}\bigr)j_B^\mu, \qquad q=\tfrac12.$$8, it becomes energetically favorable to form a two-dimensional lattice of linked vortices, with each unit cell carrying one unit of baryon number (Hamada et al., 25 Sep 2025).

6. Magnetic fields, rotation, and physical interpretation

One of the striking consequences of the baryonic vortex phase is that magnetic flux is intrinsic to the vortex lattice rather than externally imposed. In the 2024 analysis, when the tension becomes negative the system lowers its energy by creating vortices spontaneously without an external magnetic field. The magnetic field in the lattice phase is a superposition of single-vortex profiles,

$$L_{\rm EM}=-\frac{1}{4e^2}F_{\mu\nu}F^{\mu\nu}, \qquad L_{\rm WZW}=\bigl(A_\mu^B+q\,A_\mu^{\rm EM}\bigr)j_B^\mu, \qquad q=\tfrac12.$$9

with maximal magnitude near the core reaching

$$D_\mu U=\partial_\mu U-iA_\mu[Q,U], \qquad Q=\frac{1}{6}\mathbf1+\frac{\tau^3}{2},$$0

in the quoted numerics, while the spatial average

$$D_\mu U=\partial_\mu U-iA_\mu[Q,U], \qquad Q=\frac{1}{6}\mathbf1+\frac{\tau^3}{2},$$1

may be of order $$D_\mu U=\partial_\mu U-iA_\mu[Q,U], \qquad Q=\frac{1}{6}\mathbf1+\frac{\tau^3}{2},$$2–$$D_\mu U=\partial_\mu U-iA_\mu[Q,U], \qquad Q=\frac{1}{6}\mathbf1+\frac{\tau^3}{2},$$3 for astrophysical densities. For $$D_\mu U=\partial_\mu U-iA_\mu[Q,U], \qquad Q=\frac{1}{6}\mathbf1+\frac{\tau^3}{2},$$4 and $$D_\mu U=\partial_\mu U-iA_\mu[Q,U], \qquad Q=\frac{1}{6}\mathbf1+\frac{\tau^3}{2},$$5 in the range $$D_\mu U=\partial_\mu U-iA_\mu[Q,U], \qquad Q=\frac{1}{6}\mathbf1+\frac{\tau^3}{2},$$6–$$D_\mu U=\partial_\mu U-iA_\mu[Q,U], \qquad Q=\frac{1}{6}\mathbf1+\frac{\tau^3}{2},$$7, the ground state is described there as this vortex lattice rather than uniform nuclear matter or pion condensate (Qiu et al., 2024).

A distinct extension concerns rotating nuclear matter. In a co-rotating frame with angular velocity $$D_\mu U=\partial_\mu U-iA_\mu[Q,U], \qquad Q=\frac{1}{6}\mathbf1+\frac{\tau^3}{2},$$8 about the $$D_\mu U=\partial_\mu U-iA_\mu[Q,U], \qquad Q=\frac{1}{6}\mathbf1+\frac{\tau^3}{2},$$9-axis, the chiral action is formulated with the rotating metric

$\pi^\pm$00

Within this framework two configurations are identified: a local vortex with charged-pion winding and a global vortex with neutral-pion winding, both carrying baryon number. The analysis emphasizes a point often treated as an objection to global vortices: in an infinite system their energy diverges logarithmically, but the finite-size constraint dictated by causality in a rotating frame regularizes the divergence physically, rendering the global vortex a viable excitation. The critical angular velocity is defined by

$\pi^\pm$01

and numerically one finds

$\pi^\pm$02

for $\pi^\pm$03 and $\pi^\pm$04, corresponding to $\pi^\pm$05 for $\pi^\pm$06. Once $\pi^\pm$07, the vortex areal density obeys the Feynman relation

$\pi^\pm$08

and the equilibrium lattice is triangular, with

$\pi^\pm$09

For heavy-ion-collision parameters $\pi^\pm$10 and $\pi^\pm$11, this gives $\pi^\pm$12 (Mameda et al., 31 Mar 2026).

The present literature therefore places the baryonic vortex lattice at the intersection of several low-energy QCD mechanisms: pion condensation at finite isospin density, anomaly-induced baryon number from the Wess–Zumino–Witten or Goldstone–Wilczek current, Skyrmion topology, Abrikosov lattice energetics, and, in rotating systems, causality-limited finite-size regularization of global vortices. The astrophysical interpretation advanced in these works is that such a phase may be relevant to neutron-star interiors, neutron-star crusts, or phases of heavy-ion collisions where both isospin and baryon asymmetries are large; the rotational analysis further suggests that the previously overlooked global vortex can play a significant role in the topological structure of rotating dense QCD matter (Hamada et al., 25 Sep 2025, Mameda et al., 31 Mar 2026).