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arXiv:cond-mat/0411133 · Physical Review A

Nonperturbative effects on Tc of interacting Bose gases in power-law traps

O. Zobay, Georgios Metikas, Hagen M. Kleinert

Published Apr 27, 2005• 13 citations
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Abstract

The critical temperature ${T}_{c}$ of an interacting Bose gas trapped in a general power-law potential $V(\mathbf{x})={\ensuremath{\sum}}_{i}{U}_{i}{\ensuremath{\mid}{x}_{i}\ensuremath{\mid}}^{{p}_{i}}$ is calculated with the help of variational perturbation theory. It is shown that the interaction-induced shift in ${T}_{c}$ fulfills the relation $({T}_{c}\ensuremath{-}{T}_{c}^{0})∕{T}_{c}^{0}={D}_{1}(\ensuremath{\eta})\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}+{D}^{\ensuremath{'}}(\ensuremath{\eta}){\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}^{2\ensuremath{\eta}}+\mathcal{O}({\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}^{2})$ with ${T}_{c}^{0}$ the critical temperature of the trapped ideal gas, $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}$ the $s$-wave scattering length divided by the thermal wavelength at ${T}_{c}$, and $\ensuremath{\eta}=1∕2+{\ensuremath{\sum}}_{i}{p}_{i}^{\ensuremath{-}1}$ the potential-shape parameter. The terms ${D}_{1}(\ensuremath{\eta})\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}$ and ${D}^{\ensuremath{'}}(\ensuremath{\eta}){\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}^{2\ensuremath{\eta}}$ describe the leading-order perturbative and nonperturbative contributions to the critical temperature, respectively. This result quantitatively shows how an increasingly inhomogeneous potential suppresses the influence of critical fluctuations. The appearance of the ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}^{2\ensuremath{\eta}}$ contribution is qualitatively explained in terms of the Ginzburg criterion.

DOI
10.1103/physreva.71.043614

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The critical temperature Tc{T}_{c}Tc​ of an interacting Bose gas trapped in a general power-law potential V(x)=\ensuremath∑iUi\ensuremath∣xi\ensuremath∣piV(\mathbf{x})={\ensuremath{\sum}}_{i}{U}_{i}{\ensuremath{\mid}{x}_{i}\ensuremath{\mid}}^{{p}_{i}}V(x)=\ensuremath∑i​Ui​\ensuremath∣xi​\ensuremath∣pi​ is calculated with the help of variational perturbation theory. It is shown that the interaction-induced shift in Tc{T}_{c}Tc​ fulfills the relation (Tc\ensuremath−Tc0)∕Tc0=D1(\ensuremathη)a\ifmmode^\elseˆ\fi+D\ensuremath′(\ensuremathη)a\ifmmode^\elseˆ\fi2\ensuremathη+O(a\ifmmode^\elseˆ\fi2)({T}_{c}\ensuremath{-}{T}_{c}^{0})∕{T}_{c}^{0}={D}_{1}(\ensuremath{\eta})\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}+{D}^{\ensuremath{'}}(\ensuremath{\eta}){\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}^{2\ensuremath{\eta}}+\mathcal{O}({\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}^{2})(Tc​\ensuremath−Tc0​)∕Tc0​=D1​(\ensuremathη)a\ifmmode^\elseˆ\fi+D\ensuremath′(\ensuremathη)a\ifmmode^\elseˆ\fi2\ensuremathη+O(a\ifmmode^\elseˆ\fi2) with Tc0{T}_{c}^{0}Tc0​ the critical temperature of the trapped ideal gas, a\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}a\ifmmode^\elseˆ\fi the sss-wave scattering length divided by the thermal wavelength at Tc{T}_{c}Tc​, and \ensuremathη=1∕2+\ensuremath∑ipi\ensuremath−1\ensuremath{\eta}=1∕2+{\ensuremath{\sum}}_{i}{p}_{i}^{\ensuremath{-}1}\ensuremathη=1∕2+\ensuremath∑i​pi\ensuremath−1​ the potential-shape parameter. The terms D1(\ensuremathη)a\ifmmode^\elseˆ\fi{D}_{1}(\ensuremath{\eta})\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}D1​(\ensuremathη)a\ifmmode^\elseˆ\fi and D\ensuremath′(\ensuremathη)a\ifmmode^\elseˆ\fi2\ensuremathη{D}^{\ensuremath{'}}(\ensuremath{\eta}){\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}^{2\ensuremath{\eta}}D\ensuremath′(\ensuremathη)a\ifmmode^\elseˆ\fi2\ensuremathη describe the leading-order perturbative and nonperturbative contributions to the critical temperature, respectively. This result quantitatively shows how an increasingly inhomogeneous potential suppresses the influence of critical fluctuations. The appearance of the a\ifmmode^\elseˆ\fi2\ensuremathη{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}^{2\ensuremath{\eta}}a\ifmmode^\elseˆ\fi2\ensuremathη contribution is qualitatively explained in terms of the Ginzburg criterion.
We investigate the critical temperature of an interacting Bose gas confined in a trap described by a generic isotropic power-law potential. We compare the results with respect to the non-interacting case. In particular, we derive an analytical formula for the shift of the critical temperature holding to first order in the scattering length. We show that this shift scales as Nn/3(n+2), where N is the number of Bosons and n is the exponent of the power-law potential. Moreover, the sign of the shift critically depends on the power-law exponent n. Finally, we find that the shift of the critical temperature due to finite-size effects vanishes as N-2n/3(n+2).
Bose-Einstein condensation (BEC) of an ideal gas is investigated, beyond the thermodynamic limit, for a finite number NNN of particles trapped in a generic three-dimensional power-law potential. We derive an analytical expression for the condensation temperature Tc{T}_{c}Tc​ in terms of a power series in ${x}_{0}={\ensuremath{\varepsilon}}_{0}/{k}_{B}{T}_{c}$, where ${\ensuremath{\varepsilon}}_{0}$ denotes the zero-point energy of the trapping potential. This expression, which applies in Cartesian, cylindrical, and spherical power-law traps, is given analytically at infinite order. It is also given numerically for specific potential shapes as an expansion in powers of x0{x}_{0}x0​ up to the second order. We show that, for a harmonic trap, the well-known first-order shift of the critical temperature $\ensuremath{\Delta}{T}_{c}/{T}_{c}\ensuremath{\propto}{N}^{\ensuremath{-}1/3}$ is inaccurate when $N\ensuremath{\leqslant}{10}^{5}$, the next order (proportional to ${N}^{\ensuremath{-}1/2}$) being significant. We also show that finite-size effects on the condensation temperature cancel out in a cubic trapping potential, e.g., $V(r)\ensuremath{\propto}{r}^{3}$. Finally, we show that in a generic power-law potential of higher order, e.g., $V(r)\ensuremath{\propto}{r}^{\ensuremath{\alpha}}$ with $\ensuremath{\alpha}>3$, the shift of the critical temperature becomes positive. This effect provides a large increase of Tc{T}_{c}Tc​ for relatively small atom numbers. For instance, an increase of about $+$40% is expected with 104{10}^{4}104 atoms in a $V(r)\ensuremath{\propto}{r}^{12}$ trapping potential.
We review some recent theoretical work on the phase transition of interacting Bose gases in the presence of external trapping potentials. A general framework for the study of such questions is presented which is based on the application of perturbative momentum-shell renormalization group methods to the trapped gas in the uncondensed phase. After giving an overview of this approach, we first establish its validity by comparing to previous results for homogeneous and harmonically trapped gases. Using this theoretical framework, we then examine various aspects of how external potentials influence the physics of condensation. (i) By studying the case of general power-law potentials and complemented by arguments from variational perturbation theory, it is quantitatively worked out how a growing inhomogeneity of the trapping potential diminishes nonperturbative effects at the transition. (ii) It is shown how by superimposing a weak periodic potentials on the homogeneous system, the characteristic nonperturbative momentum scale of critical interacting Bose gases can be probed. (iii) For a gas in a random potential, it is studied how condensation is affected by the combined influence of disorder effects and particle interactions.
For some time, the theoretical result for the transition temperature of a dilute three-dimensional Bose gas in an arbitrarily wide harmonic trap has been known to first order in the interaction strength. We extend that result to second order. The first-order result for a gas trapped in a harmonic potential can be computed in mean-field theory (in contrast to the first-order result for a uniform gas, which cannot). We show that, at second order, perturbation theory suffices for relating the transition temperature to the chemical potential at the transition, but the chemical potential is nonperturbative at the desired order. The necessary information about the chemical potential can be extracted, however, from recent lattice simulations of uniform Bose gases.
We discuss certain specific features of the calculation of the critical temperature of a dilute repulsive Bose gas. Interactions modify the critical temperature in two different ways. First, for gases in traps, temperature shifts are introduced by a change of the density profile, arising itself from a modification of the equation of state of the gas (reduced compressibility); these shifts can be calculated simply within mean field theory. Second, even in the absence of a trapping potential (homogeneous gas in a box), temperature shifts are introduced by the interactions; they arise from the correlations introduced in the gas, and thus lie inherently beyond mean field theory – in fact, their evaluation requires more elaborate, non-perturbative, calculations. One illustration of this non-perturbative character is provided by the solution of self-consistent equations, which relate together non-linearly the various energy shifts of the single particle levels k . These equations predict that repulsive interactions shift the critical temperature (at constant density) by an amount which is positive, and simply proportional to the scattering length a ; nevertheless, the numerical coefficient is difficult to compute. Physically, the increase of the temperature can be interpreted in terms of the reduced density fluctuations introduced by the repulsive interactions, which facilitate the propagation of large exchange cycles across the sample.
The transition temperature for a dilute, homogeneous, three-dimensional Bose gas has the expansion ${T}_{\mathrm{c}}{=T}_{0}{{1+c}_{1}{\mathrm{an}}^{1/3}+[{c}_{2}^{\ensuremath{'}}\mathrm{ln}{(an}^{1/3}{)+c}_{2}^{\ensuremath{''}}]{a}^{2}{n}^{2/3}{+O(a}^{3}n)},$ where a is the scattering length, n the number density, and T0{T}_{0}T0​ the ideal gas result. The first-order coefficient c1{c}_{1}c1​ depends on nonperturbative physics. In this paper, we show that the coefficient ${c}_{2}^{\ensuremath{'}}$ may be computed perturbatively. We also show that the remaining second-order coefficient ${c}_{2}^{\ensuremath{''}}$ depends on non-perturbative physics but may be related, by a perturbative calculation, to quantities that have previously been measured using lattice simulations of three-dimensional O(2)O(2)O(2) scalar field theory. Making use of those simulation results, we find ${T}_{\mathrm{c}}\ensuremath{\simeq}{T}_{0}{1+(1.32\ifmmode\pm\else\textpm\fi{}0.02){\mathrm{an}}^{1/3}+[19.7518\mathrm{ln}{(an}^{1/3})+(75.7\ifmmode\pm\else\textpm\fi{}0.4)]{a}^{2}{n}^{2/3}{+O(a}^{3}n)}.$