# Note on a Non Linear Perturbation of the Ideal Bose Gas

## TL;DR

This paper demonstrates that introducing a U(1) symmetry breaking field in an ideal Bose gas is equivalent to adding a nonlinear function of the zero mode number operator, leading to unconventional Bose-Einstein condensation at negative chemical potential.

## Contribution

It establishes the equivalence between a symmetry-breaking perturbation and a nonlinear modification of the Hamiltonian in the thermodynamic limit, extending to a class of diagonal models.

## Key findings

- Models exhibit non-conventional BEC at negative chemical potential.
- Limit pressures of the models coincide in the thermodynamic limit.
- Equivalence extends to a broader class of diagonal models.

## Abstract

In this work we show that the introduction of a U(1) symmetry breaking field in the energy operator of the boson-free gas, is equivalent, in the thermodynamic limit, to the inclusion, in the Hamiltonian of the ideal gas, of a non-linear function of the number operator associated with the zero mode. In other words, the limit pressures coincide. Moreover, both models undergo non conventional Bose-Einstein condensation (BEC) for strictly negative values of the chemical potential $\mu.$ Finally, the proof of equivalence of limit pressures is extended to a class of full-diagonal models.