Fermionic domain-wall Skyrmions of QCD in a magnetic field

TL;DR

The paper investigates the ground-state structure of low-energy QCD in strong magnetic fields, focusing on the chiral soliton lattice (CSL) and domain-wall Skyrmion (DWSk) phases. Using chiral perturbation theory with the Wess-Zumino-Witten term and a half-period moduli approximation, it shows that the minimal DWSk is fermionic with baryon number $1$, and that a bosonic $N_B=2$ DWSk can split into two fermionic $N_B=1$ DWSks on opposite sides of the CSL without energy cost, leaving the phase boundary intact. The analysis in the chiral limit reveals a linear CSL profile and symmetric half-density DWSks, with a clarified scaling of the critical chemical potential and magnetic-field threshold. By employing a 5D WZW embedding, the work establishes the spin-statistics of half DWSks and demonstrates the decoupling of the two halves, contributing to a coherent picture of baryon-number localization on CSLs and their topological nature in QCD under strong magnetic fields.

Abstract

The ground state of low-energy QCD matter in strong magenetic fields is either a chiral soliton lattice (CSL), a periodic array of neutral pion domain walls (chiral solitons) perpendicular to the magnetic field, or domain-wall Skyrmion phase, in which Skyrmions are induced on top of the CSL. Previously found domain-wall Skyrmions are bosons with the baryon number two. In this paper, we show that the minimum domain-wall Skyrmions are fermions with the baryon number one; a bosonic domain-wall Skyrmion can be separated without cost of energy into two fermionic domain-wall Skyrmions attached on the opposite sides of a chiral soliton. The phase boundary between the CSL and domain-wall Skyrmion phases is unchanged. In the chiral limit, the CSL reduces to a linearly dependent neutral pion on the direction of the magnetic field, while fermionic domain-wall Skyrmions sit in an equal distance of a half period.

Figures (5)

• Figure 1: Isosurfaces of $\mathcal{B}=1/(50\pi^2)$ for both effective DWSk theories on (a) $0<z<\ell/2$ shown in amber and (b) $\ell/2<z<\ell$ shown in green. Both are for the single soliton case, $\kappa=1$. The CSL in blue extending into both halves of (a) and (b) is shown for $\pi/2<\chi_3^{\textrm{CSL}}<3\pi/2$ with blue plane intersecting each half. Notice the separation of each macaron across each intersecting plane depicting the separation of baryon topological charges $k=1$ and $k=1$ for each half.
• Figure 2: The separation of the macaron into two halves by a smooth step function (left) and by a step function (right).
• Figure 3: $\chi_3^{\text{CSL}}$ of the CSL shown for both $\kappa=1000/1001$ (blue) given by eq. \ref{['eq:CSL_solution']} and $\kappa\to0$ (yellow) in the chiral limit given by eq. \ref{['eq:CSL_chiral_limit']}. For comparison both are shown against their respective period of $K(\kappa)z$. Notice the flattening linear profile in the chiral limit.
• Figure 4: Skyrion charge density isosurface at $\mathcal{B}=(50\pi^{2})^{-1}$ for 'half' $n\ell<z<\ell/2+n\ell$ shown in amber and 'half' $\ell/2+n\ell<z<(n+1)\ell$ shown in green $\forall n\in \mathbb{Z}$. Both are shown for $\kappa\to 0$ in the chiral limit. In between each Skyrmion sits alternating vacuua at $\Sigma=\pm 1$ (not shown in figure). The baby Skyrmion charge density has been evaluated to eq. \ref{['eq:baby_Skyrmion']} for the single lump with $\omega=x+iy\pm d$ for offset constant $d$.
• Figure 5: General configurations of fermionic DWSks and topological nature of the baryon number. The $SU(2)$ flavor symmetry is spontaneously broken to $U(1)$ and $\text{SU}(2)_V/\text{U}(1)_3\cong \mathbb{C}P^1\simeq S^2$ moduli appear between the two planes $\Sigma=-\boldsymbol{I}_2$ (black) and $\Sigma=+\boldsymbol{I}_2$ (blue), while on the boundary planes the $SU(2)$ is recovered. (a) A fermionic DWSk is described by an open string (orange line) interpolating between the two with baryon topological charge density $N_B=1$. (b) An open string parallel to the planes carries no baryon number. (c) Here, a DWSk's interpolating path has been changed due to the effective moduli parameter promotion. Notice that paths in which the two planes are not connected give zero baryon topological charge (in red).