Thermal Casimir effect in the spin-orbit coupled Bose gas

TL;DR

This work addresses how Rashba spin-orbit coupling modifies the thermal Casimir effect in an ideal Bose gas below the condensation temperature. By diagonalizing the spin-orbit coupled Hamiltonian into two branches and focusing on the condensed plus branch, the authors derive scaling forms for the Casimir free energy and compute the resulting forces in $d=2$ and $d=3$, including two distinct orientations in 3D. They show that the Casimir force remains attractive but its decay exponent and amplitude are altered by $x_0=ig(mν^2/(2k_B T)ig)^{1/2}$ and by wall orientation, with notable singular behavior as $ν\to0$ in 2D. The results reveal nontrivial scaling with $D$, $λ$, and $x_0$, and demonstrate that universality is restricted by spin-orbit coupling; open questions include $T\to0$ behavior and other forms of S-O coupling.

Abstract

We study the thermal Casimir effect in ideal Bose gases with spin-orbit (S-O) coupling of Rashba type below the critical temperature for Bose-Einstein condensation. In contrast to the standard situation involving no S-O coupling, the system exhibits long-ranged Casimir forces both in two and three dimensions ($d=2$ and $d=3$). We identify the relevant scaling variable involving the ratio $D/ν$ of the separation between the confining walls $D$ and the S-O coupling magnitude $ν$. We derive and discuss the corresponding scaling functions for the Casimir energy. In all the considered cases the resulting Casimir force is attractive and the S-O coupling $ν$ has impact on its magnitude. In $d=3$ the exponent governing the decay of the Casimir force becomes modified by the presence of the S-O coupling, and its value depends on the orientation of the confining walls relative to the plane defined by the Rashba coupling. In $d=2$ the obtained Casimir force displays singular behavior in the limit of vanishing $ν$

Figures (3)

• Figure 1: The scaling function $\phi=\beta\omega_{s,+}D$ implied by Eq. (\ref{['omegas2']}) plotted as a function of $x_0$ at fixed $D/\lambda=100$. For $x_0$ small the function $\phi$ is well approximated by the asymptotic form given by $\phi_1$ (dotted line), while for $x_0$ large, it coincides with the function $\phi_2$ (dashed line). The limit of vanishing $x_0$ is not resolved in this plot scale - compare Fig. 2.
• Figure 2: The scaling function $\phi=\beta\omega_{s,+}D$ implied by Eq. (\ref{['omegas2']}) plotted for a sequence of values of $D/\lambda$ for $x_0$ approaching zero. The dotted line represents the asymptotic form given by $\phi_1$ to which the sequence of plotted functions converges nonuniformly. The red star represents the value $-2\zeta(2)/\pi$ -see the main text.
• Figure 3: Numerical evaluation of the Casimir energy as a function of $D/\lambda$ from Eq. (\ref{['oms02']}) (red crosses). The value of $x_0=10$ is fixed. The fit to the data (blue solid line) indicates a power law with the exponent -1.50 - see the main text and Eq. (\ref{['oms081']}).