Entropies and IPR as Markers for a Phase Transition in a Two-Level Model for Atom-Diatomic Molecule Coexistence.

Ignacio Baena,Pedro Pérez-Fernández,Manuela Rodríguez-Gallardo,José Miguel Arias

doi:10.3390/e24010113

Ignacio Baena, Pedro Pérez-Fernández + Show 2 more

Open Access

https://doi.org/10.3390/e24010113

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Journal: Entropy	Publication Date: Jan 12, 2022
Citations: 2	License type: CC BY 4.0

Affiliation: Universidad de Sevilla, University of Granada

Abstract

A quantum phase transition (QPT) in a simple model that describes the coexistence of atoms and diatomic molecules is studied. The model, which is briefly discussed, presents a second-order ground state phase transition in the thermodynamic (or large particle number) limit, changing from a molecular condensate in one phase to an equilibrium of diatomic molecules–atoms in coexistence in the other one. The usual markers for this phase transition are the ground state energy and the expected value of the number of atoms (alternatively, the number of molecules) in the ground state. In this work, other markers for the QPT, such as the inverse participation ratio (IPR), and particularly, the Rényi entropy, are analyzed and proposed as QPT markers. Both magnitudes present abrupt changes at the critical point of the QPT.

Highlights

The study of phase transitions in quantum systems is a topic of present interest, usually referred to as quantum phase transitions (QPT) [1–3]
We propose the use of the inverse participation ratio (IPR) and the Rényi entropy as other good markers for the critical point
We have studied a two-level model for the coexistence of atoms and diatomic molecules

Summary

Introduction

The study of phase transitions in quantum systems is a topic of present interest, usually referred to as quantum phase transitions (QPT) [1–3]. Particular Hamiltonians based on algebraic structures that are exactly solvable have been proposed, such as the Lipkin [10], the Bose-Hubbard [11], the Jaynes–Cummings, the Tavis–Cummings and the Dicke models [12–14], just to cite a few of them. These models present specific dynamical symmetries that correspond to different equilibrium configurations of the system in the ground state. We propose the use of the inverse participation ratio (IPR) and the Rényi entropy as other good markers for the critical point

The Model for the Atom–Diatomic Molecule Coexistence

Exact Solution of the Eigenvalue Problem

Mean Field for the Model Hamiltonian

Inverse Participation Ratio

Renyi Entropy

Conclusions