Searching Repulsive Casimir Forces Between Magneto-Electric Materials

TL;DR

The paper investigates how discrete symmetry breaking—specifically parity and time-reversal symmetry breaking—affects the sign of the Casimir force between magneto-electric bi-isotropic plates. It develops a Lifshitz-type formalism for BIMs, deriving a phase diagram of the force sign as a function of the magneto-electric parameters χ and κ and showing how the diagram evolves with plate separation due to dispersion. The results indicate that time-reversal symmetry breaking plays a more fundamental role than parity breaking in enabling repulsion, with larger separations favoring repulsion because low-frequency material responses dominate. The study provides a unified theoretical framework for calculating Casimir forces in chiral and nonreciprocal media and suggests metamaterial designs to realize repulsive forces, potentially mitigating stiction in nano-devices.

Abstract

The Casimir effect, arising from vacuum quantum fluctuations, plays a fundamental role in the development of modern quantum electrodynamics. In parallel, the field of condensed matter has flourished through the discovery of various materials exhibiting broken symmetries, often connected to topology and characterized by magneto-electric coupling. To enhance the comprehension of the role of parity symmetry and time-reversal symmetry in determining the sign of the Casimir force, we calculate the Casimir forces between magneto-electric materials and obtain a phase diagram governing the sign of symmetry-breaking-induced Casimir forces. We also investigate how the force phase diagram varies with the separation distances between the objects. Our results contribute to a better understanding of the sign of the Casimir force, a subject bearing both theoretical interest and practical significance.

Figures (5)

• Figure 1: Schematic diagram of Casimir effect between two bi-isotropic plates which are separated with distance $d$. Each BIM plate is characterized by four parameters: $\epsilon_i, \mu_i, \chi_i, \kappa_i(i = 1,2)$. The permittivity and permeability for two plates are assumed to be the same: $\epsilon_1 = \epsilon_2 = \epsilon, \mu_1 = \mu_2 = \mu$.
• Figure 2: The phase diagram of the Casimir force $F_c/F_0$ between two BIM plates with separation $d = 1\mu m$. The reference force $F_0 = \frac{\pi^2\hbar c A}{240d^4}$ is the magnitude of the Casimir force between two parallel perfect metallic plates. The red (blue) region represents attractive (repulsive) force. $\omega_{\kappa_1} = 0.2\omega_R, \omega_{\chi_1} = \omega_R$ is fixed and the Casimir force varies with different $\omega_{\kappa_2}, \omega_{\chi_2}$. The sign of the $\omega_{\kappa_i}, \omega_{\chi_i}$ represents the sign in the dispersion model of $\kappa_i, \chi_i$. The parameters for the permittivity and permeability are: $\omega_{R} = \omega_p = \omega_m = 10^{15} rad/s, B = 0.04, \gamma = 0.05\omega_R, \lambda = 10^{-5}\omega_R^{-1}$, which impose the constraints $|\omega_{\kappa_i}/\omega_R|\leq 0.2, |\omega_{\chi_i}/\omega_R|\leq 1$.
• Figure 3: Phase diagram of the Casimir force $F_c(d)/F_0(d)$ at varying distances: (a) $d = 0.01\mu m$, (b) $d = 0.1\mu m$, (c) $d = 1\mu m$, (d) $d = 10\mu m$. The reference force $F_0(d) = \frac{\pi^2 \hbar c A}{240d^4}$ is the magnitude of the force between two perfect conducting plates at each distance. The parameters are chosen such that $\omega_{\chi1} = -\omega_{\chi2} = \omega_{\chi}, \omega_{\kappa1}= \omega_{\kappa2} = \omega_{\kappa}$. For fixed $\omega_{\kappa}, \omega_{\chi}$, Casimir repulsion emerges more readily at larger separations. Parameters in the permittivity and permeability are the same as those in Fig.\ref{['fig:fix1side']}.
• Figure 4: The response functions and ($-r^2_{ss}-r^2_{pp}+r^2_{ps}+r^2_{sp}$) for $\omega_\kappa = 0.2\omega_R, \omega_\chi = \omega_R$. (a) Dispersion of the material response functions $\epsilon(i\xi)$, $\mu(i\xi)$, $\chi(i\xi)$ and $\kappa(i\xi)$. The vertical gray lines mark the characteristic frequency $\xi = \frac{c}{d}$ for separations $d = 10, 1, 0.1, 0.01 \mu m$ (left to right). $\epsilon(i\xi)$, $\mu(i\xi)$, $\chi(i\xi)$ are real and decrease monotonically. $\kappa(i\xi)$ is purely imaginary. (b) The reflection coefficient combination ($-r^2_{ss}-r^2_{pp}+r^2_{ps}+r^2_{sp}$) as a function of $k_\parallel$ and $\xi$. The horizontal and vertical gray lines corresponds to the characteristic frequency $\xi = \frac{c}{d}$ and cut-off momentum $k_\parallel = \frac{1}{d}$ for the same set of separations. Red (blue) regions indicate negative (positive) values, corresponding to attractive (repulsive) contributions to the Casimir force. Parameters for permittivity and permeability are the same as those in Fig.\ref{['fig:fix1side']}.
• Figure 5: Distance dependence of the normalized Casimir force $F_c(d)/F_0(d)$ between two BIM plates. The magneto-electric coupling parameters are chosen such that $\omega_{\chi_1}=-\omega_{\chi_2}=\omega_{\chi}, \omega_{\kappa_1}=\omega_{\kappa_2}=\omega_{\kappa}$, with other parameters matching Fig.\ref{['fig:fix1side']}. When $\omega_\kappa/\omega_R=0.2,\omega_\chi/\omega_R=1$ (red curve), the force is attractive at short distance but repulsive at large distance. When $\omega_\kappa/\omega_R=\omega_\chi/\omega_R=0$ (black curve) or $\omega_\kappa/\omega_R=0.1,\omega_\chi/\omega_R=0.5$ (blue curve), the force is always attractive. The inset figure shows the total force per unit area: $F_c(d)/A$. In the case of $\omega_\kappa/\omega_R = \omega_\chi/\omega_R = 0$ (black dots) and $\omega_\kappa/\omega_R = 0.1, \omega_\chi/\omega_R = 0.5$ (blue curve), the attractive force are nearly identical and decay with the distance.