Casimir effect in magnetic dual chiral density waves

TL;DR

The paper analyzes the Casimir energy of Dirac fields in finite-thickness MDCDW quark matter under a magnetic field, revealing a rich, Landau-level–dependent finite-size effect governed by the interplay of chemical potential $\mu$, magnetic field $B$, and the MDCDW order parameter $M$ and wavevector $b$. It develops a Lifshitz-type regularization for the MDCDW dispersion, decomposes the energy into lowest and higher Landau level contributions, and classifies the high-Landau-level dispersions into $(2,2)$, $(2,0)$, $(4,0)$ (island), and $(0,0)$ types, linking the presence of Fermi points to oscillatory Casimir patterns with periods set by $|k^{\rm FP}_z|=\sqrt{(\mu-b)^2-M^2}$. Across weak, intermediate, and strong magnetic fields, the authors demonstrate LLL-driven oscillations, flavor-beating phenomena in the two-flavor case, and sign-flips or transitions in the HLL contributions, providing a comprehensive framework for finite-size Casimir effects in magnetized Dirac matter. The findings have potential implications for the physics of thin quark matter in heavy-ion collisions and neutron stars, and offer routes to explore analogous Casimir phenomena in Dirac/Weyl semimetals and related systems.

Abstract

We theoretically investigate the Casimir effect originating from Dirac fields in finite-density matter under a magnetic field. In particular, we focus on quark fields in the magnetic dual chiral density wave (MDCDW) phase as a possible inhomogeneous ground state of interacting Dirac-fermion systems. In this system, the distance dependence of Casimir energy shows a complex oscillatory behavior by the interplay between the chemical potential, magnetic field, and inhomogeneous ground state. By decomposing the total Casimir energy into contributions of each Landau level, we elucidate what types of Casimir effects are realized from each Landau level: the lowest or some types of higher Landau levels lead to different behaviors of Casimir energies. Furthermore, we point out characteristic behaviors due to level splitting between different fermion flavors, i.e., up/down quarks. These findings provide new insights into Dirac-fermion (or quark) matter with a finite thickness.

Figures (8)

• Figure 1: Schematic picture of the Casimir effect in the two-flavor MDCDW phase, where the MDCDW phase is sandwiched by two boundary conditions at $z=0$ and $z=L_z$. The magnetic field $\vec{B}$ and the wave number $\vec{b}$ of density waves are parallel to the $z$ direction.
• Figure 2: Typical examples of dispersion relations for the LLLs (solid black line) and HLLs (red, blue, green, or dashed black line) of fermion fields in the MDCDW phase. Filled symbols stand for crossing points with the Fermi level, namely the FPs.
• Figure 3: (a) Dispersion relations of fermion fields in the one-flavor MDCDW phase. (b) Thickness dependence of Casimir coefficients $C^{[1]}_{\rm Cas}$ for each LL. Inset: the total Casimir coefficients $C^{[3]}_{\rm Cas}$.
• Figure 4: (a) Dispersion relations of fermion fields in the two-flavor MDCDW phase under a weak magnetic field $eB/\Lambda^2=(0.05)^2$. (b) Thickness dependence of Casimir coefficients $C^{[1]}_{\rm Cas}$ for each LL. Inset: the total Casimir coefficient $C^{[3]}_{\rm Cas}$.
• Figure 5: (a) Dispersion relations of fermion fields in the two-flavor MDCDW phase under an intermediate magnetic field $eB/\Lambda^2=(0.3)^2$. (b) Thickness dependence of Casimir coefficients $C^{[1]}_{\rm Cas}$ from each LL. Inset: the sum of Casimir coefficients $C^{[1]}_{\rm Cas}$ from $u$ and $d$ quarks at $l=1$.