Role of mixed permutation symmetry sectors in the thermodynamic limit of critical three-level Lipkin-Meshkov-Glick atom models

TL;DR

This paper introduces the concept of Mixed Symmetry Quantum Phase Transitions (MSQPT) in multi-level atom models, demonstrating how permutation symmetry sectors influence phase behavior and ground state degeneracy in the thermodynamic limit.

Contribution

It defines MSQPTs within the three-level Lipkin-Meshkov-Glick model and analyzes their manifestation across permutation symmetry sectors, revealing complex phase diagrams and symmetry-breaking phenomena.

Findings

MSQPTs appear in all permutation sectors for finite N.

Four distinct quantum phases coexist at a quadruple point in the phase diagram.

Ground state exhibits four-fold degeneracy and Schrödinger cat states due to symmetry breaking.

Abstract

We introduce the notion of Mixed Symmetry Quantum Phase Transition (MSQPT) as singularities in the transformation of the lowest-energy state properties of a system of identical particles inside each permutation symmetry sector $\mu$, when some Hamiltonian control parameters $\lambda$ are varied. We use a three-level Lipkin-Meshkov-Glick (LMG) model, with $U(3)$ dynamical symmetry, to exemplify our construction. After reviewing the construction of $U(3)$ unirreps using Young tableaux and Gelfand basis, we firstly study the case of a finite number $N$ of three-level atoms, showing that some precursors (fidelity-susceptibility, level population, etc.) of MSQPTs appear in all permutation symmetry sectors. Using coherent (quasi-classical) states of $U(3)$ as variational states, we compute the lowest-energy density for each sector $\mu$ in the thermodynamic $N\to\infty$ limit. Extending the control parameter space by $\mu$, the phase diagram exhibits four distinct quantum phases in the $\lambda$-$\mu$ plane that coexist at a quadruple point. The ground state of the whole system belongs to the fully symmetric sector $\mu=1$ and shows a four-fold degeneracy, due to the spontaneous breakdown of the parity symmetry of the Hamiltonian. The restoration of this discrete symmetry leads to the formation of four-component Schr\"odinger cat states.