We obtain a rigorous solution of universal Bose gases near resonance and offer an answer to one of the long-standing challenges of quantum gases at large scattering lengths, where the standard dilute theory breaks down. The solution was obtained by using an $\ensuremath{\epsilon}$ expansion near four spatial dimensions. In dimension $d=4\ensuremath{-}\ensuremath{\epsilon}$, the chemical potential of Bose gases near resonance is shown to approach the universal value ${\ensuremath{\epsilon}}^{\frac{2}{4\ensuremath{-}\ensuremath{\epsilon}}}{\ensuremath{\epsilon}}_{F}\sqrt{\frac{2}{3}}(1+0.474\ensuremath{\epsilon}\ensuremath{-}i1.217\ensuremath{\epsilon}+\ensuremath{\cdots})$, where ${\ensuremath{\epsilon}}_{F}$ is the Fermi energy defined for a Fermi gas of density $n$ , and the condensation fraction is equal to $\frac{2}{3}(1+0.0877\ensuremath{\epsilon}+\ensuremath{\cdots})$. We also discuss the implications on ultracold gases in physical dimensions.We obtain a rigorous solution of universal Bose gases near resonance and offer an answer to one of the long-standing challenges of quantum gases at large scattering lengths, where the standard dilute theory breaks down. The solution was obtained by using an $\ensuremath{\epsilon}$ expansion near four spatial dimensions. In dimension $d=4\ensuremath{-}\ensuremath{\epsilon}$, the chemical potential of Bose gases near resonance is shown to approach the universal value ${\ensuremath{\epsilon}}^{\frac{2}{4\ensuremath{-}\ensuremath{\epsilon}}}{\ensuremath{\epsilon}}_{F}\sqrt{\frac{2}{3}}(1+0.474\ensuremath{\epsilon}\ensuremath{-}i1.217\ensuremath{\epsilon}+\ensuremath{\cdots})$, where ${\ensuremath{\epsilon}}_{F}$ is the Fermi energy defined for a Fermi gas of density $n$, and the condensation fraction is equal to $\frac{2}{3}(1+0.0877\ensuremath{\epsilon}+\ensuremath{\cdots})$. We also discuss the implications on ultracold gases in physical dimensions.

arXiv:1307.4263Physical Review AMar 11, 201428 citations

ON THEORY OF NON-IDEAL BOSE GAS

Yu. V. Sanochkin

Soviet physics. DokladyApr 11, 19590 citations

Quantum ideal gases: Fermi-Dirac and Bose-Einstein statistics

Mark E. Tuckerman

Abstract Chapter 11 presents a statistical mechanical treatment of the quantum ideal gases, i.e., the ideal Boltzmann, fermion, and boson gases. The discussion begins with the microscopic description and the solution of the eigenvalue problem for a quantum ideal gas. It is argued that calculating the partition function is most readily accomplished in the grand canonical ensemble using a second-quantized formulation. When the particles are distinguishable, the equation of state is identical to that of a classical ideal gas. For fermions and bosons, however, the problem of computing thermodynamic properties is significantly more complex and can only be solved exactly in certain limits. Away from these limits approximations are needed and are discussed in detail. The relevant distributions - the Fermi-Dirac and Bose-Einstein distributions are derived. The local density approximation of density functional theory is derived for the ideal electron gas. For the ideal boson gas, the phenomenon of Bose-Einstein condensation is discussedAbstract Chapter 11 presents a statistical mechanical treatment of the quantum ideal gases, i.e., the ideal Boltzmann, fermion, and boson gases. The discussion begins with the microscopic description and the solution of the eigenvalue problem for a quantum ideal gas. It is argued that calculating the partition function is most readily accomplished in the grand canonical ensemble using a second-quantized formulation. When the particles are distinguishable, the equation of state is identical to that of a classical ideal gas. For fermions and bosons, however, the problem of computing thermodynamic properties is significantly more complex and can only be solved exactly in certain limits. Away from these limits approximations are needed and are discussed in detail. The relevant distributions - the Fermi-Dirac and Bose-Einstein distributions are derived. The local density approximation of density functional theory is derived for the ideal electron gas. For the ideal boson gas, the phenomenon of Bose-Einstein condensation is discussed

Aug 1, 20230 citations

Superfluid transition of homogeneous and trapped two-dimensional Bose gases

Markus Holzmann, Gordon Baym, Jean-Paul Blaizot +1 more

Current experiments on atomic gases in highly anisotropic traps present the opportunity to study in detail the low temperature phases of two-dimensional inhomogeneous systems. Although, in an ideal gas, the trapping potential favors Bose-Einstein condensation at finite temperature, interactions tend to destabilize the condensate, leading to a superfluid Kosterlitz-Thouless-Berezinskii phase with a finite superfluid mass density but no long-range order, as in homogeneous fluids. The transition in homogeneous systems is conveniently described in terms of dissociation of topological defects (vortex-antivortex pairs). However, trapped two-dimensional gases are more directly approached by generalizing the microscopic theory of the homogeneous gas. In this paper, we first derive, via a diagrammatic expansion, the scaling structure near the phase transition in a homogeneous system, and then study the effects of a trapping potential in the local density approximation. We find that a weakly interacting trapped gas undergoes a Kosterlitz-Thouless-Berezinskii transition from the normal state at a temperature slightly below the Bose-Einstein transition temperature of the ideal gas. The characteristic finite superfluid mass density of a homogeneous system just below the transition becomes strongly suppressed in a trapped gas.Current experiments on atomic gases in highly anisotropic traps present the opportunity to study in detail the low temperature phases of two-dimensional inhomogeneous systems. Although, in an ideal gas, the trapping potential favors Bose-Einstein condensation at finite temperature, interactions tend to destabilize the condensate, leading to a superfluid Kosterlitz-Thouless-Berezinskii phase with a finite superfluid mass density but no long-range order, as in homogeneous fluids. The transition in homogeneous systems is conveniently described in terms of dissociation of topological defects (vortex-antivortex pairs). However, trapped two-dimensional gases are more directly approached by generalizing the microscopic theory of the homogeneous gas. In this paper, we first derive, via a diagrammatic expansion, the scaling structure near the phase transition in a homogeneous system, and then study the effects of a trapping potential in the local density approximation. We find that a weakly interacting trapped gas undergoes a Kosterlitz-Thouless-Berezinskii transition from the normal state at a temperature slightly below the Bose-Einstein transition temperature of the ideal gas. The characteristic finite superfluid mass density of a homogeneous system just below the transition becomes strongly suppressed in a trapped gas.

Proceedings of the National Academy of SciencesJan 23, 200772 citations

n

, and the condensation fraction is equal to $\frac{2}{3}(1+0.0877\ensuremath{\epsilon}+\ensuremath{\cdots})$. We also discuss the implications on ultracold gases in physical dimensions.