Lifshitz formulas for finite-density Casimir effect

TL;DR

The paper addresses the lack of a finite-density treatment in the Lifshitz formalism by generalizing the Lifshitz formula to Dirac fermions at finite chemical potential μ. It derives a closed-form expression for the Casimir energy under periodic boundary conditions, shows how μ shifts the spectrum and induces oscillatory behavior when μ>M, and extends the framework to finite temperature, arbitrary spatial dimensions, and various boundary conditions. Key contributions include the μ-dependent Lifshitz expression, the separation of Dirac and Fermi sea contributions, and the beating Casimir effect arising from mismatched chemical potentials, with explicit connections to quark matter and Dirac/Weyl semimetals. This provides a versatile tool for density-controlled Casimir physics in both high-energy and condensed-matter contexts, and suggests pathways for lattice studies and experimental exploration of fermionic Casimir phenomena.

Abstract

The Lifshitz formula is well known as a theoretical approach to investigate the Casimir effect at finite temperature. In this Letter, we generalize the Lifshitz formula to the Casimir effect originating from quantum fields at finite chemical potential. To demonstrate the versatility of this formula, we discuss the typical phenomena of the Casimir effect at finite chemical potential in various systems, such as some boundary conditions, finite temperatures, arbitrary spatial dimensions, and mismatched chemical potentials. This formula can be applied to the Casimir effect in dense quark matter and Dirac/Weyl semimetals, where the chemical potential is regarded as a parameter to control the Casimir effect.

Figures (6)

• Figure 1: Typical behaviors of the Casimir energy $E_\mathrm{Cas}$ and its coefficient $C_\mathrm{Cas}^{[3]}$ at finite chemical potential larger than the mass ($\mu >M$). Solid lines: from the Lifshitz formula. Points: from the lattice regularization. The dotted line represents $E_\mathrm{Cas} = 4\pi^2/90L_z^3$ known for the massless field at $\mu=0$.
• Figure 2: Typical behaviors of the Casimir pressure $P_\mathrm{Cas}$ and its coefficient $P_\mathrm{Cas}L_z^4$ at finite chemical potential larger than the mass ($\mu >M$). The dotted line represents $P_\mathrm{Cas} = 2\pi^2/15L_z^4$ known for the massless field at $\mu=0$.
• Figure 3: Temperature dependence ($T/\mu=0,0.1,1.0$) of Casimir energy $E_\mathrm{Cas}$ and its coefficient $C_\mathrm{Cas}^{[3]}$ from massless Dirac fields at finite chemical potentials $\mu$.
• Figure 4: Casimir energy $E_\mathrm{Cas}$ and its coefficient $C_\mathrm{Cas}^{[d]}$ at spatial dimension $d=2$ and $d=1$. The dotted lines represent $E_\mathrm{Cas} = 2\zeta(3)/\pi L_z^2$ and $2\pi/3L_z$ known for the massless field at $\mu=0$.
• Figure 5: Contributions of Dirac and Fermi seas for Casimir energy $E_\mathrm{Cas}$ and its coefficient $C_\mathrm{Cas}^{[3]}$.