Finite-Size Scaling Exponents in the Dicke Model

TL;DR

This paper analyzes finite-size corrections at the critical point in the Dicke model, revealing singular scaling exponents for various quantities and establishing their relation to the Lipkin-Meshkov-Glick model.

Contribution

It introduces a method to determine finite-size scaling exponents in the Dicke model, showing their universality and connection to other models.

Findings

Finite-size corrections are singular at the phase transition.

Scaling exponents for atomic observables match those in the Lipkin-Meshkov-Glick model.

The order parameter's behavior is governed by the same scaling variable as atomic observables.

Abstract

We consider the finite-size corrections in the Dicke model and determine the scaling exponents at the critical point for several quantities such as the ground state energy or the gap. Therefore, we use the Holstein-Primakoff representation of the angular momentum and introduce a nonlinear transformation to diagonalize the Hamiltonian in the normal phase. As already observed in several systems, these corrections turn out to be singular at the transition point and thus lead to nontrivial exponents. We show that for the atomic observables, these exponents are the same as in the Lipkin-Meshkov-Glick model, in agreement with numerical results. We also investigate the behavior of the order parameter related to the radiation mode and show that it is driven by the same scaling variable as the atomic one.