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arXiv:cond-mat/9811096 · Physical Review A

Application of renormalization-group techniques to a homogeneous Bose gas at finite temperature

Jens O. Andersen, Michael T. Strickland

Published Aug 1, 1999• 39 citations
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Abstract

A homogeneous Bose gas is investigated at finite temperature using renormalization-group techniques. A nonperturbative flow equation for the effective potential is derived using sharp and smooth cutoff functions. Numerical solutions of these equations show that the system undergoes a second-order phase transition in accordance with universality arguments. We obtain the critical exponent $\ensuremath{\nu}=0.73.$

DOI
10.1103/physreva.60.1442

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A homogeneous Bose gas is investigated at finite temperature using renormalization-group techniques. A nonperturbative flow equation for the effective potential is derived using sharp and smooth cutoff functions. Numerical solutions of these equations show that the system undergoes a second-order phase transition in accordance with universality arguments. We obtain the critical exponent \ensuremathν=0.73.\ensuremath{\nu}=0.73.\ensuremathν=0.73.
A homogeneous non-relativistic Bose gas is investigated at finite temperature using renormalization group methods. The phase transition is shown to be second order and the effective chemical potential and the effective quartic coupling vanish at the critical temperature. We obtain the critical exponent ν=0.73ν=0.73ν=0.73 at leading order in the derivative expansion.
We apply perturbative renormalization-group theory to the symmetric phase of a dilute interacting Bose gas which is trapped in a three-dimensional harmonic potential. Using Wilson's energy-shell renormalization and the $\ensuremath{\epsilon}$ expansion, we derive the flow equations for the system. We relate these equations to the flow for the homogeneous Bose gas. In the thermodynamic limit, we apply our approach to examine the transition temperature of the harmonically trapped Bose gas as a function of the scattering length. The results are compared to previous studies of the problem.
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Using renormalization-group arguments we show that the low-temperature thermodynamics of a three- or two-dimensional dilute Bose gas is fully determined by a universal scaling function ${\mathcal{F}}_{d}(\ensuremath{\mu}/{k}_{B}T,\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}(T))$ once the mass mmm and the sss-wave scattering length ad{a}_{d}ad​ of the bosons are known (ddd is the space dimension). Here $\ensuremath{\mu}$ and TTT denote the chemical potential and temperature of the gas, and the temperature-dependent dimensionless interaction constant $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}(T)$ is a function of $m{a}_{d}^{2}{k}_{B}T/{\ensuremath{\hbar}}^{2}$. We compute the scaling function F2{\mathcal{F}}_{2}F2​ using a nonperturbative renormalization-group approach and find that both the $\ensuremath{\mu}/{k}_{B}T$ and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}(T)$ dependencies are in very good agreement with recent experimental data obtained for a quasi-two-dimensional Bose gas with or without optical lattice. We also show that the nonperturbative renormalization-group estimate of the Berezinskii-Kosterlitz-Thouless transition temperature compares well with the result obtained from a quantum Monte Carlo simulation of an effective classical field theory.
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