Fluctuation-induced forces in confined ideal and imperfect Bose gases

Abstract

Fluctuation-induced forces are investigated for ideal and imperfect Bose gases confined to $d$-dimensional films of size ${\infty }^{d-1}\times D$ under periodic (P), antiperiodic (A), Dirichlet-Dirichlet (DD), Neumann-Neumann (NN), and Robin (R) boundary conditions (BCs). The full scaling functions ${{\rm Y}}_d^{\text{BC}}(x_{\lambda }=D/{\lambda }_{th},x_{\xi }=D/\xi )$ of the residual reduced grand potential per area, ${\phi }_{\text{res},d}^{\text{BC}}(T,\mu ,D)=D^{-(d-1)}{{\rm Y}}_d^{\text{BC}}(x_{\lambda },x_{\xi })$, are determined for the ideal gas case with these BCs, where ${\lambda }_{th}$ and $\xi$ are the thermal de-Broglie wavelength and the bulk correlation length, respectively. The scaling functions ${\Theta }_d^{\text{BC}}(x_{\xi })\equiv {{\rm Y}}_d^{\text{BC}}(\infty ,x_{\xi })$ describing the critical behavior at the bulk condensation transition are shown to agree with those previously determined from a massive free $O(2)$ theory for $\text{BC}=\text{P},\text{A},\text{DD},\text{DN},\text{NN}$. For $d=3$, they are expressed in closed analytical form. The analogous functions ${{\rm Y}}_d^{\text{BC}}(x_{\lambda },x_{\xi },c_{1}D,c_{2}D)$ and ${\Theta }_d^{\text{R}}(x_{\xi },c_{1}D,c_{2}D)$ under the RBCs $({\partial }_z-c_1)\varphi {|}_{z=0}=({\partial }_z+c_2)\varphi {|}_{z=D}=0$ with $c_1\ge 0$ and $c_2\ge 0$ are also determined. The functions ${{\rm Y}}_{\infty ,d}^{\text{P}}(x_{\lambda },x_{\xi })$ and ${\Phi }_{\infty ,d}^{\text{P}}(x_{\xi })$ for the imperfect Bose gas are shown to agree with those of the interacting Bose gas with $n\to \infty$ internal degrees of freedom. Hence for $d=3$, ${\Phi }_{\infty ,d}^{\text{P}}(x_{\xi })$ is known exactly in closed analytic form. A modified imperfect Bose-gas model with free BC is introduced that corresponds to the limit $n\to \infty$ of this interacting Bose gas. Exact results for the function ${\Theta }_{\infty ,3}^{\mathbb{DD}}(x_{\xi })$ therefore follow from those of the $O(2n)$ ${\varphi }^4$ model for $n\to \infty$.

pdf-LaTeX, 24 pages; minor additions, cosmetic modifications, and typos corrected

References in corpus (16)

• Many-Body Physics with Ultracold Gases
• The Casimir effect: from quantum to critical fluctuations
• Casimir Forces between Compact Objects: I. The Scalar Case
• Bose-Einstein Condensation Temperature of Homogenous Weakly Interacting Bose Gas in Variational Perturbation Theory Through Seven Loops
• Crossover from Attractive to Repulsive Casimir Forces and Vice Versa
• Superfluid Bose gas in two dimensions
• Relation between the thermodynamic Casimir effect in Bose-gas slabs and critical Casimir forces
• Exact thermodynamic Casimir forces for an interacting three-dimensional model system in film geometry with free surfaces
• Fluctuation-induced forces in strongly anisotropic critical systems
• Dynamic critical behavior of model A in films: Zero-mode boundary conditions and expansion near four dimensions
• Casimir force in O(n) lattice models with a diffuse interface
• Finite-size effects in the spherical model of finite thickness
• Comment on "Casimir force in the $O(n\to \infty )$ model with free boundary conditions"
• Inverse scattering-theory approach to the exact large-$n$ solutions of $O(n)$ ${\varphi }^4$ models on films and semi-infinite systems bounded by free surfaces
• Inverse scattering theory and trace formulae for one-dimensional Schrödinger problems with singular potentials
• The three-dimensional $O(n)$ ${\varphi }^4$ model on a strip with free boundary conditions: exact results for a nontrivial dimensional crossover in the limit $n\to \infty$

Cited by in corpus (13)

• Critical Casimir Effect: Exact Results
• Dimensional crossovers and Casimir forces for the Bose gas in anisotropic optical lattices
• Repulsive Casimir force in Bose-Einstein Condensate
• Thermodynamic equivalence of two-dimensional imperfect attractive Fermi and repulsive Bose gases
• Thermodynamic Casimir forces in strongly anisotropic systems within the $N\to \infty$ class
• First-order, continuous, and multicritical Bose-Einstein condensation in Bose mixtures
• Large-$N$ properties of a non-ideal Bose gas
• Variance of the Casimir force in an ideal Bose gas
• Condensation and superfluidity of $SU(N)$ Bose gas
• Exact results for the Casimir force of a three-dimensional model of relativistic Bose gas in a film geometry
• Spin-orbit coupled mean-field Bose gas at finite temperature
• Phase diagram of two-component mean-field Bose mixtures
• Thermal Casimir effect in the spin-orbit coupled Bose gas