Landau functions for non-interacting bosons

TL;DR

This paper analyzes the Landau functions associated with Bose-Einstein condensation and spontaneous symmetry breaking in non-interacting bosons, revealing complex dependencies on system parameters and confirming expected phase transition behavior in 3D harmonic traps.

Contribution

It introduces and compares Landau functions for BEC and SSB in finite systems, highlighting their differences and dependencies on dimensionality and potential.

Findings

Landau functions describe BEC and SSB in finite bosonic systems.

For infinite N, BEC and SSB Landau functions coincide.

In 3D harmonic traps, the Landau function shows typical continuous phase transition behavior.

Abstract

We discuss the statistics of Bose-Einstein condensation (BEC) in a canonical ensemble of N non-interacting bosons in terms of a Landau function L_N^{BEC} (q) defined by the logarithm of the probability distribution of the order parameter q for BEC. We also discuss the corresponding Landau function for spontaneous symmetry breaking (SSB), which for finite N should be distinguished from L_N^{BEC}. Only for intinite N BEC and SSB can be described by the same Landau function which depends on the dimensionality and on the form of the external potential in a surprisingly complex manner. For bosons confined by a three-dimensional harmonic trap the Landau function exhibits the usual behavior expected for continuous phase transitions.