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arXiv:2510.26240 · Journal of Statistical Mechanics Theory and Experiment

Thermal Casimir effect in the spin–orbit coupled Bose gas

Marek Napiórkowski, Paweł Jakubczyk

Published Mar 1, 2026• 0 citations
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Abstract

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 in both 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 a singular behavior in the limit of vanishing ν .

DOI
10.1088/1742-5468/ae4fa9

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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 in both 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 a singular behavior in the limit of vanishing ν .
The Casimir effect of an ideal Bose gas trapped in a generic power-law potential and confined between two slabs with Dirichlet, Neumann, and periodic boundary conditions is investigated systematically, based on the grand potential of the ideal Bose gas, the Casimir potential and force are calculated. The scaling function is obtained and discussed. The special cases of free and harmonic potentials are also discussed. It is found that when T ⩽ T c (where T c is the critical temperature of Bose-Einstein condensation), the Casimir force is a power-law decay function; when T > T c , the Casimir force is an exponential decay function; and when T ≫ T c , the Casimir force vanishes.
Abstract We consider an ideal Bose gas enclosed in a d -dimensional slab of thickness D . Using the grand canonical ensemble we calculate the variance of the thermal Casimir force acting on the slab’s walls. The variance evaluated per unit wall area is shown to decay like Δ var / D for large D . The amplitude Δ var is a non-universal function of two scaling variables λ / ξ and D / ξ , where λ is the thermal de Broglie wavelength and ξ is the bulk correlation length. It can be expressed via the bulk pressure, the Casimir force per unit wall area, and its derivative with respect to chemical potential. For thermodynamic states corresponding to the presence of the Bose–Einstein condensate the amplitude Δ var retains its non-universal character while the ratio of the mean standard deviation and the Casimir force takes the scaling form ( D / L ) d − 1 2 ( D / λ ) d / 2 , where L is the linear size of the wall.
Abstract We study the Casimir effect for a three dimensional system of ideal free massive Bose gas in a slab geometry with Zaremba and anti-periodic boundary conditions. It is found that for these type of boundary conditions the resulting Casimir force is repulsive in nature, in contrast with usual periodic, Dirichlet or Neumann boundary condition where the Casimir force is attractive (Martin and Zagrebnov 2006 Europhys. Lett. 73 15). Casimir forces in these boundary conditions also maintain a power law decay function below condensation temperature and exponential decay function above the condensation temperature albeit with a positive sign, identifying the repulsive nature of the force.
We consider the Bose gas on a ddd-dimensional anisotropic lattice employing the imperfect (mean-field) gas as a prototype example. We study the dimensional crossover arising as a result of varying the dispersion relation at finite temperature TTT. We analyze in particular situations where one of the relevant effective dimensionalities is located at or below the lower critical dimension, so that the Bose-Einstein condensate becomes expelled from the system by anisotropically modifying the lattice parameters controlling the kinetic term in the Hamiltonian. We clarify the mechanism governing this phenomenon. Subsequently we study the thermodynamic Casimir effect occurring in this system. We compute the exact profile of the scaling function for the Casimir energy. As an effect of strongly anisotropic scale invariance, the Casimir force below or at the critical temperature Tc{T}_{c}Tc​ may be repulsive even for periodic boundary conditions. The corresponding Casimir amplitude is universal only in a restricted sense, and the power law governing the decay of the Casimir interaction becomes modified. We also demonstrate that, under certain circumstances, the scaling function is constant for sufficiently large values of the scaling variable, and in consequence is not an analytical function. At T>TcT>{T}_{c}T>Tc​ the Casimir-like interactions reflect the structure of the correlation function, and, for certain orientations of the confining walls, show exponentially damped oscillatory behavior so that the corresponding force is attractive or repulsive depending on the distance.
In this study, we have derived close form of the Casimir force for the non-interacting ideal Bose gas between two slabs in harmonic-optical lattice potential by using Ketterle and van Druten approximation. We find that Bose-Einstein condensation temperature TcT_{c}Tc​ is a critical point for different physical behavior of the Casimir force. We have shown that Casimir force of confined Bose gas in the presence of the harmonic-optical potential decays with inversely proportional to d5d^{5}d5 when T≤TcT\leq T_{c}T≤Tc​. However, in the case of T>TcT>T_{c}T>Tc​, it decays exponentially depends on separation ddd of the slabs. Additionally we have discussed temperature dependence of Casimir force and importance of the harmonic-optical lattice potential on quantum critical systems, quantum phase transition and nano-devices.
We investigate theoretically Bose-Einstein condensation of an ideal, trapped Bose gas in the presence of Rashba spin-orbit coupling. Analytic results for the critical temperature and condensate fraction are derived based on a semiclassical approximation to the single-particle-energy spectrum and density of states and are compared with exact results obtained by explicitly summing discrete energy levels for a small number of particles. We find a significant decrease of the critical temperature and of the condensate fraction due to finite spin-orbit coupling. For a large coupling strength and a finite number of particles NNN, the critical temperature scales as N2/5{N}^{2/5}N2/5 and N2/3{N}^{2/3}N2/3 in three and two dimensions, respectively, contrasted to the predictions of N1/3{N}^{1/3}N1/3 and N1/2{N}^{1/2}N1/2 in the absence of spin-orbit coupling. Finite-size corrections in three dimensions are also discussed.