Phase space analysis of the 3-level Lipkin-Meshkov-Glick model using parity adapted U(D)-spin coherent states

Abstract

We study the phase-space properties of the 3-level Lipkin-Meshkov-Glick as paradigmatic case of critical, parity symmetric, N-quDit systems undergoing a quantum phase transition in the thermodynamic limit N → ∞. We generalize U(2) spin coherent states to U(D) (quDits), and define the coherent state representation Qψ (Husimi function) of a symmetric N- quDit state |ψ〉 in the phase space ℂP D −1 = U(D)/[U(1) × U(D − 1)]. This allows us to define parity adapted U(D) coherent states (𝕔-DCATs), which reproduce accurately the lowest energy Hamiltonian eigenstates obtained by numerical diagonalization. We visualize precursors of the QPTs by plotting localization measures (Husimi function and its moments) of the parity adapted U(D) coherent states and the numerical Hamiltonian eigenstates for a finite number of particles.