Normal and generalized Bose condensation in traps: One dimensional examples

TL;DR

This paper proves that one-dimensional Bose gases in certain traps exhibit complete generalized Bose-Einstein condensation at all temperatures, with detailed phase transition behavior and effects of interactions.

Contribution

It establishes the occurrence of complete generalized BEC in one-dimensional traps with unscaled interactions and analyzes phase transitions in harmonic traps with varying parameters.

Findings

Complete generalized BEC occurs at all temperatures in certain 1D Bose gases.

Phase transition at a=1 in harmonic traps with specific frequency scaling.

Interactions of order o(N log N) do not prevent generalized BEC.

Abstract

We prove the following results. (i) One-dimensional Bose gases which interact via unscaled integrable pair interactions and are confined in an external potential increasing faster than quadratically undergo a complete generalized Bose-Einstein condensation (BEC) at any temperature, in the sense that a macroscopic number of particles are distributed on a o(N)number of one-particle states. (ii) In a one dimensional harmonic trap the replacement of the oscillator frequency \omega by \omega\ln N/N gives rise to a phase transition at a=\hbar\omega\beta=1 in the noninteracting gas. For a<1 the limit distribution of n_0/N^a is exponential and <n_0>/N^a tends to 1. For a>1 there is BEC with a condensate density <n_0>/N going to 1-1/a. For a>=1, (\ln N/N)(n_0-<n_0>) is asymptotically distributed following Gumbel's law. For any a>0 the free energy is -(\pi^2/6a\beta)N/\ln N+o(N/\ln N), with no singularity at a=1. (iii) In Model (ii) both above and below the critical temperature the the gas undergoes a complete generalized BEC, thus providing a coexistence of ordinary and generalized condensates below the critical point. (iv) Adding an interaction <U_N>=o(N\ln N) to Model (ii) we prove that a complete generalized BEC occurs at all temperatures.