Holographic QCD Matter: Chiral Soliton Lattices in Strong Magnetic Field

Abstract

We investigate the chiral soliton lattice (CSL) in the framework of holographic QCD in magnetic field. Under appropriate boundary conditions for the gauge field and the quark mass deformation, we demonstrate that the ground state in the gravitational dual of QCD is given by the CSL in the background magnetic field and the baryon number density. In the presence of the background magnetic field, we show that the CSL is interpreted as a uniformly distributed D4-branes in the holographic setup, where the chiral soliton is identified with a non-self-dual instanton vortex or a center vortex in the five dimensional bulk gauge theory. While the baryon numbers are given to chiral solitons as well as Skyrmions due to the different terms in the Wess-Zumino-Witten (WZW) term in the chiral perturbation theory, these baryon numbers with different origins are unified in terms of the instanton charge density in five dimensions. With bulk analysis of the WZW term, we find that the pion decay constant becomes dependent on the magnetic field. For the massless pion case, we obtain an analytical form that is in qualitative agreement with lattice QCD results for strong magnetic fields.

Figures (6)

• Figure 1: A schematic picture of the brane configuration for the localized D4-brane (the red dot in the left figure) and the partially dissolved, periodic D4-branes (the dotted lines in the right figure). The former is identified with a Skyrmion while the latter corresponds to the CSL in the background field. The blue rectangle is the D8-branes that fill the whole $(x^1,x^2,x^3, z)$-plane. The (partially) dissolved D4-branes are periodically localized in the $(x^3, z)$-plane but uniformly distributed along the $(x^1,x^2)$-directions. Note that the worldvolume of the D4-branes is extended along the $(x^0, x^6,x^7,x^8,x^9)$-directions, which are omitted in the figure.
• Figure 2: The instanton density $\text{Tr} [F \wedge F]$ in the $(x^3,z)$-plane. The single soliton (left) and the CSL (middle). The parameters are set to $f_{\pi} = B_3 = U_{\text{KK}} = m_\pi = 1$. The $m_\pi \to 0$ behavior of the kink is also shown (right).
• Figure 3: The gradient of the ground state for the massless case.
• Figure 4: Normalized cross sections of $\text{Tr}\left(F\wedge F\right)$ along the $z$-direction. The quantities are normalized such that at $z=0$, $\text{Tr}\left(F\wedge F\right) = 0$. For sufficiently large magnetic fields, one can see the onset of two peaks. The peaks grow in size relative to the value at the origin. If the normalization were removed, the peaks would be larger because of the hyperbolic scaling of $F_{z3}$.
• Figure 5: The instanton density magnitude, $\left|\text{Tr} [F \wedge F]\right|$, in the $(x^3,z)$-plane with back reaction for two $m_\pi=0$ kink configurations. Plots are made with $\mu_B = f_\pi = 1$. Left plot: $\mathcal{B}_3 = 1$; right plot: $\mathcal{B}_3 = 3$.