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 µ, when some Hamiltonian control parameters λ 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 µ in the thermodynamic N → ∞ limit. Extending the control parameter space by µ, the phase diagram exhibits four distinct quantum phases in the λ-µ plane that coexist at a quadruple point. The ground state of the whole system belongs to the fully symmetric sector µ = 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 Schrödinger cat states.

Collective spin operators for symmetric multi-quDit (namely identical D-level atom) systems generate a U(D) symmetry. We explore generalizations to arbitrary D of SU(2)-spin coherent states and their adaptation to parity (multi-component Schrödinger cats), together with multi-mode extensions of NOON states. We write level, one- and two-quDit reduced density matrices of symmetric N-quDit states, expressed in the last two cases in terms of collective U(D)-spin operator expectation values. Then, we evaluate level and particle entanglement for symmetric multi-quDit states with linear and von Neumann entropies of the corresponding reduced density matrices. In particular, we analyze the numerical and variational ground state of Lipkin–Meshkov–Glick models of 3-level identical atoms. We also propose an extension of the concept of SU(2)-spin squeezing to SU(D) and relate it to pairwise D-level atom entanglement. Squeezing parameters and entanglement entropies are good markers that characterize the different quantum phases, and their corresponding critical points, that take place in these interacting D-level atom models.

We study phase space properties of critical, parity symmetric, N-qudit systems undergoing a quantum phase transition (QPT) in the thermodynamic N → ∞ limit. The D = 3 level (qutrit) Lipkin-Meshkov-Glick model is eventually examined as a particular example. For this purpose, we consider U(D)-spin coherent states (DSCS), generalizing the standard D = 2 atomic coherent states, to define the coherent state representation Q ψ (Husimi function) of a symmetric N-qudit state |ψ in the phase space CP D−1 (complex projective manifold). DSCS are good variational approximations to the ground state of an N-qudit system, especially in the N → ∞ limit, where the discrete parity symmetry Z D−1 2 is spontaneously broken. For finite N, parity can be restored by projecting DSCS onto 2 D−1 different parity invariant subspaces, which define generalized "Schrödinger cat states" reproducing quite faithfully low-lying Hamiltonian eigenstates obtained by numerical diagonalization. Precursors of the QPT are then visualized for finite N by plotting the Husimi function of these parity projected DSCS in phase space, together with their Husimi moments and Wehrl entropy, in the neighborhood of the critical points. These are good localization measures and markers of the QPT.

In this paper we pursue the use of information measures (in particular, information diagrams) for the study of entanglement in symmetric multi-quDit systems. We use generalizations to $${U}(D)$$ U ( D ) of spin $${U}(2)$$ U ( 2 ) coherent states and their adaptation to parity (multicomponent Schrödinger cats), and we analyse one- and two-quDit reduced density matrices. We use these correlation measures to characterize quantum phase transitions occurring in Lipkin–Meshkov–Glick models of $$D=3$$ D = 3 -level identical atoms, and we propose the rank of the corresponding reduced density matrix as a discrete order parameter.

In this paper we pursue the use of information measures (in particular, information diagrams) for the study of entanglement in symmetric multi-quDit systems. We use generalizations to U(D) of spin U(2) coherent states and their adaptation to parity (multicomponent Schrödinger cats) and we analyse one-and two-quDit reduced density matrices. We use these correlation measures to characterize quantum phase transitions occurring in Lipkin-Meshkov-Glick models of D = 3-level identical atoms and we propose the rank of the corresponding reduced density matrix as a discrete order parameter.