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

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 model, with U(3) dynamical symmetry, to exemplify our construction. After reviewing the construction of U(3) unitary irreducible representations using Young tableaux and the Gelfand basis, we first 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 (quasiclassical) states of U(3) as variational states, we compute the lowest-energy density for each sector mu. in the thermodynamic N -> infinity 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 fourfold 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 Schrodinger cat states.

Automated summary

This paper introduces the concept of mixed symmetry quantum phase transition, using a three-level Lipkin-Meshkov-Glick model with U(3) dynamical symmetry to illustrate it. The study shows that precursors of MSQPTs appear in all permutation symmetry sectors, and a fourfold degeneracy is observed in the ground state due to the spontaneous breakdown of the parity symmetry.