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

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 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.

Entanglement and U(D)-spin squeezing in symmetric multi-quDit systems and applications to quantum phase transitions in Lipkin–Meshkov–Glick D-level atom models

“…The tensor product representation of U(D) is reducible and decomposes into a Clebsch-Gordan direct sum of mixed symmetry invariant subspaces. Here, we shall restrict ourselves to the N +D−1 N -dimensional fully symmetric sector (see [21] for the role of other mixed symmetry sectors), which means that our N atoms are indistinguishable (bosons). Denoting by a † i (resp.…”

“…written in terms of collective U(3)-spin operators S i j . Hamiltonians of this kind have already been proposed in the literature [38][39][40][41][42] (see also [21] for the role of mixed symmetry sectors in QPTs of multi-quDit LMG systems). We place levels symmetrically about i = 2, with intensive energy splitting per particle /N .…”

“…In general, parity-adapted CSs are a special set of "nonclassical" states with interesting statistical properties (see [18][19][20] for several seminal papers). Parity-adapted DSCSs arise as variational states reproducing the energy and structure of ground states in Lipkin-Meshkov-Glick (LMG) D-level atom models (see [21] and later in Sect. 5).…”

“…In this article, we want to explore squeezing and interparticle and interlevel quantum correlations in symmetric multi-quDit systems like the ones described by critical LMG models of identical D-level atoms (see, e.g., [21,[38][39][40][40][41][42] for D = 3 level atom models). The literature mainly concentrates on two-level atoms, displaying a U(2) symmetry, which is justified when we make atoms to interact with an external monochromatic electromagnetic field.…”

“…Collective operators generate a U(D) spin symmetry for the case of D-level identical atoms or D boson species (quDits). Recently [21], we have calculated the phase diagram of a three-level LMG atom model. Here, we want to explore the connection between entanglement and squeezing with QPTs for this symmetric multi-qutrit system.…”