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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Kosterlitz-Thouless transition in a dilute Bose gas

Henk T. C. Stoof, M. Bijlsma

Published Feb 1, 1993• 44 citations
Publisher

Abstract

We study the two-dimensional dilute Bose gas using the ladder or T-matrix approximation, which is valid if ln(1/${\mathit{na}}^{2}$)\ensuremath{\gg}1, and show that this approximation can describe also the critical region of the Kosterlitz-Thouless transition accurately. In particular, we consider the superfluid properties of the gas and find that the superfluid density shows the square-root cusp together with the universal jump at the critical temperature. The theory is applied to spin-polarized atomic hydrogen adsorbed on a superfluid helium film, since this appears to be one of the most promising experimental setups to achieve the Kosterlitz-Thouless transition in a weakly interacting Bose gas.

DOI
10.1103/physreve.47.939

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We study the two-dimensional dilute Bose gas using the ladder or T-matrix approximation, which is valid if ln(1/na2{\mathit{na}}^{2}na2)\ensuremath{\gg}1, and show that this approximation can describe also the critical region of the Kosterlitz-Thouless transition accurately. In particular, we consider the superfluid properties of the gas and find that the superfluid density shows the square-root cusp together with the universal jump at the critical temperature. The theory is applied to spin-polarized atomic hydrogen adsorbed on a superfluid helium film, since this appears to be one of the most promising experimental setups to achieve the Kosterlitz-Thouless transition in a weakly interacting Bose gas.
Abstract Using a formalism based on the two-body S-matrix we study two-dimensional Bose and Fermi gases with both attractive and repulsive interactions. Approximate analytic expressions, valid at weak coupling and beyond, are developed and applied to the Berezinskii–Kosterlitz–Thouless (BKT) transition. We successfully recover the correct logarithmic functional form of the critical chemical potential and density for the Bose gas. For fermions, the BKT critical temperature is calculated in BCS and BEC regimes through consideration of Tan’s contact.
We study the Berezinskii-Kosterlitz-Thouless transition in a weakly interacting 2D quantum Bose gas using the concept of universality and numerical simulations of the classical $|\ensuremath{\psi}{|}^{4}$ model on a lattice. The critical density and chemical potential are given by relations ${n}_{c}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}(mT/2\ensuremath{\pi}{\ensuremath{\Elzxh}}^{2})\mathrm{ln}(\ensuremath{\xi}{\ensuremath{\Elzxh}}^{2}/mU)$ and ${\ensuremath{\mu}}_{c}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}(mTU/\ensuremath{\pi}{\ensuremath{\Elzxh}}^{2})\mathrm{ln}({\ensuremath{\xi}}_{\ensuremath{\mu}}{\ensuremath{\Elzxh}}^{2}/mU)$, where TTT is the temperature, mmm is the mass, and UUU is the effective interaction. The dimensionless constant $\ensuremath{\xi}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}380\ifmmode\pm\else\textpm\fi{}3$ is very large and thus any quantitative analysis of the experimental data crucially depends on its value. For ${\ensuremath{\xi}}_{\ensuremath{\mu}}$ our result is ${\ensuremath{\xi}}_{\ensuremath{\mu}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}13.2\ifmmode\pm\else\textpm\fi{}0.4$. We also report the study of the quasicondensate correlations at the critical point.
The weakly interacting Bose gas in two dimensions is considered in the dilute limit n1/2{n}^{1/2}n1/2a\ensuremath{\ll}1, where n is the particle density and a is the range of the potential. The standard many-body perturbation theory for this system has two separate divergences: the first, associated with classical phase fluctuations, is responsible for the vanishing of the long-range order; the second is quantum mechanical and is connected with the vanishing of the scattering t matrix at long wavelengths and low energies. An earlier diagrammatic theory of Popov, which provides a consistent description of the system in the dilute limit, is rederived heuristically from a quasiparticle picture, and also using the renormalization group. It is shown that the superfluid transition temperature is Tc{T}_{c}Tc​\ensuremath{\approxeq}4\ensuremath{\pi}(${\ensuremath{\Elzxh}}^{2}$/2m)n/[ln ln(1/na2{\mathrm{na}}^{2}na2)], and the condition of validity of the dilute limit is ln ln(1/na2{\mathrm{na}}^{2}na2)\ensuremath{\gg}1. The connection to the dilute Bose gas in dimensions d>2 and the universal behavior beyond the extreme asymptotic low-density domain are also discussed.
We study the crossover between the mean-field and critical behavior of the two-dimensional Bose gas throughout the fluctuation region of the Berezinskii-Kosterlitz-Thouless phase transition point. We argue that this crossover is described by universal (for all weakly interacting $|\ensuremath{\psi}{|}^{4}$ models) relations between thermodynamic parameters of the system, including superfluid and quasicondensate densities. We establish these relations with high-precision Monte Carlo simulations of the classical $|\ensuremath{\psi}{|}^{4}$ model on a lattice, and check their asymptotic forms against analytic expressions derived on the basis of the mean-field theory.
We present an improved many-body T-matrix theory for partially Bose-Einstein condensed atomic gases by treating the phase fluctuations exactly. The resulting mean-field theory is valid in arbitrary dimensions and able to describe the low-temperature crossover between three-, two-, and one-dimensional Bose gases. When applied to a degenerate two-dimensional atomic hydrogen gas, we obtain a reduction of the three-body recombination rate, which compares favorably with experiment. Supplementing the mean-field theory with a renormalization-group approach to treat the critical fluctuations, we also incorporate into the theory the Kosterlitz-Thouless transition that occurs in a homogeneous Bose gas in two dimensions. In particular, we calculate the critical conditions for the Kosterlitz-Thouless phase transition as a function of the microscopic parameters of the theory. The proposed theory is further applied to a trapped one-dimensional Bose gas, where we find good agreement with exact numerical results obtained by solving a nonlinear Langevin field equation.
We present an improved many-body T-matrix theory for partially Bose-Einstein condensed atomic gases by treating the phase fluctuations exactly. The resulting mean-field theory is valid in arbitrary dimensions and able to describe the low-temperature crossover between three-, two-, and one-dimensional Bose gases. When applied to a degenerate two-dimensional atomic hydrogen gas, we obtain a reduction of the three-body recombination rate, which compares favorably with experiment. Supplementing the mean-field theory with a renormalization-group approach to treat the critical fluctuations, we also incorporate into the theory the Kosterlitz-Thouless transition that occurs in a homogeneous Bose gas in two dimensions. In particular, we calculate the critical conditions for the Kosterlitz-Thouless phase transition as a function of the microscopic parameters of the theory. The proposed theory is further applied to a trapped one-dimensional Bose gas, where we find good agreement with exact numerical results obtained by solving a nonlinear Langevin field equation.