Critical Point of a Weakly Interacting Two-Dimensional Bose Gas

Abstract

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 |ψ| 4 -model on a lattice. The critical density and chemical potential are given by relations nc = (mT /2πh2 ) ln(ξh 2 /mU ) and µc = (mT U/πh 2 ) ln(ξµh 2 /mU ), where T is the temperature, m is the mass, and U is the effective interaction. The dimensionless constant ξ = 380 ± 3 is very large and thus any quantitative analysis of the experimental data crucially depends on its value. For ξµ our result is ξµ = 13.2 ± 0.4. We also report the study of the quasi-condensate correlations at the critical point.An accurate microscopic expression for the critical temperature of the Berezinskii-Kosterlitz-Thouless (BKT) transition [1] has been a weak point of the theory of weakly interacting two-dimensional Bose gas. The theory of Ref.[2] (see also [3,4] and analysis below), suggests that the critical density of the BKT transition in the weakly interacting system reads (we seth = 1)However, the value of ξ cannot be obtained within standard analytical treatments since ξ is related to the system behavior in the fluctuation region where the perturbative expansion in powers of U does not work. With unknown ξ, one finds Eq. (1) rather inaccurate unless mU is exponentially small. Moreover, as we will find in this Letter, the value of ξ is very large: ξ ≈ 380. This means that for all experimentally available up to date (quasi-)2D weakly interacting Bose gases [5,6] the quantitative analysis of the data for the critical ratio n c /T c requires a precise value of ξ. In the system of spin-polarized atomic hydrogen on helium film [5], the value of mU is of order unity [3]; in the recently created quasi-2D system of sodium atoms [6], mU is of order 10 −2 , according to the formula of Ref. [7].To quantitatively describe the BKT transition in a weakly interacting Bose gas, it is sufficient to solve a classical-field |ψ| 4 -model with the effective long-wave Hamiltonian [2]where µ ′ is the chemical potential, and ψ is the classical complex field.In this Letter, we first discuss the origin of the relation (1) in the limit of small U , and how quantum and classical models relate to each other. Then we present our numeric results (for the critical density, critical chemical potential, and quasi-condensate correlations at the BKT point) obtained by simulating the critical behavior of the 2D |ψ| 4 -model on a lattice using recently developed Worm algorithm [8] for classical statistical models. In particular, we show that the quasi-condensate correlations are very strong at T c , in agreement with the experimental observation in the spin-polarized atomic hydrogen [5] and quantum Monte Carlo simulations [9].A simple dimensional analysis of the Hamiltonian (2) allows to write a generic formula for the critical point in a weakly interacting 2D |ψ| 4 -model. The routine itself is completely analogous to that in the 3D case (see, e.g., [10,11]), but final results naturally ...

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