# Scaling Properties of the Lipkin Model at the Critical Point

## TL;DR

This paper analyzes the scaling behavior of the Lipkin model at the critical point, highlighting differences between first and second order quantum phase transitions and potential applications to the interacting boson model.

## Contribution

It provides a detailed analysis of the Lipkin model's spectral scaling at phase transitions, emphasizing distinctions between transition orders and applicability to other models.

## Key findings

- Identifies distinct scaling properties at the critical point for different transition orders.
- Shows how results can be applied to the interacting boson model.
- Enhances understanding of quantum phase transition characteristics.

## Abstract

The influence of Franco Iachello in Physics during the last 50 years and, in particular, in the use of algebraic methods applied to very different physical problems has been broad, extense and have permeated most branches of Physics, from Nuclear and Molecular to Particle and Condensed Matter physics. Apart of many other contributions, at the beginning of the 2000's he introduced the concept of critical point symmetry and triggered the study of the many faces of quantum phase transitions in nuclei and other mesoscopic systems. In this contribution, we present the analysis of the scaling properties of the Lipkin model spectrum at the phase transition point and we focus on the differences between first and second order quantum phase transitions. Moreover, we explain how the obtained results can be also of application for the interacting boson model.