No singularities at the phase transition in the Dicke model

TL;DR

This paper analyzes the phase transition in the Dicke model, demonstrating that divergences at the critical point are artifacts of infinite atom number and remain finite for finite systems, supported by analytical and numerical evidence.

Contribution

It provides analytical expressions for observables in the Dicke model for any atom number and clarifies the nature of divergences at the phase transition.

Findings

Divergences at the critical point are limits as atom number approaches infinity.

Finite systems do not exhibit true singularities at the phase transition.

Analytical results agree with numerical solutions across different atom numbers.

Abstract

The Dicke Hamiltonian describes the simplest quantum system with atoms interacting with photons: N two level atoms inside a perfectly reflecting cavity which allows only one electromagnetic mode. It has also been successfully employed to describe superconducting circuits which behave as artificial atoms coupled to a resonator. The system exhibits a transition to a superradiant phase at zero temperature. When the interaction strength reaches its critical value, both the number of photons and of atoms in excited states in the cavity, together with their fluctuations, exhibit a sudden increase from zero. Employing symmetry-adapted coherent states it is shown that these properties scale with the number of atoms, that their reported divergences at the critical point represent the limit when this number goes to infinity, and that in this limit they remain divergent in the superradiant phase. Analytical expressions are presented for all observables of interest, for any number of atoms. Comparisons with exact numerical solutions strongly support the results.