Thermal Casimir effect in the spin-orbit coupled Bose gas

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 $D/\nu$ of the separation between the confining walls $D$ and the S-O coupling magnitude $\nu$. 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 $\nu$ 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 $\nu$