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SciPost Physics Proceedings

Mixed permutation symmetry quantum phase transitions of critical three-level atom models

Alberto Mayorgas, Julio Guerrero, Manuel Calixto

Published Nov 24, 2023• 0 citations
Publisher

Abstract

We define the concept of Mixed Symmetry Quantum Phase Transition (MSQPT), considering each permutation symmetry sector \mu μ of an identical particles system, as singularities in the properties of the lowest-energy state into each \mu μ when shifting a Hamiltonian control parameter \lambda λ . A three-level Lipkin-Meshkov-Glick (LMG) model is chosen to typify our construction. Firstly, we analyze the finite number N N of particles case, proving the presence of MSQPT precursors. Then, in the thermodynamic limit N\to∞ N → ∞ , we calculate the lowest-energy density inside each sector \mu μ , augmenting the control parameter space by \mu μ , and showing a phase diagram with four different quantum phases.

DOI
10.21468/scipostphysproc.14.036

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We define the concept of Mixed Symmetry Quantum Phase Transition (MSQPT), considering each permutation symmetry sector \mu μ of an identical particles system, as singularities in the properties of the lowest-energy state into each \mu μ when shifting a Hamiltonian control parameter \lambda λ . A three-level Lipkin-Meshkov-Glick (LMG) model is chosen to typify our construction. Firstly, we analyze the finite number N N of particles case, proving the presence of MSQPT precursors. Then, in the thermodynamic limit N\to∞ N → ∞ , we calculate the lowest-energy density inside each sector \mu μ , augmenting the control parameter space by \mu μ , and showing a phase diagram with four different quantum phases.
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 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 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 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 Schrödinger cat 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.
Quantum phase transitions are currently studied in several fields of physics, as examples one finds: in nuclear physics the shape phase transitions within the Interacting Boson Model; in quantum optics two level atoms interacting with a one mode electromagnetic radiation in a cavity; and in condensed matter in the analysis of the behavior of spin systems. In this contribution we employ the fidelity and its susceptibility, two concepts widely used in quantum information theory, to determine the quantum phase transitions of the Lipkin-Meshkov-Glick (LMG) model together with its scaling properties. By means of these concepts, we propose a quantum method to determine the crossings and anti-crossings present in a model Hamiltonian as a function of the control parameters of the model. A review of the separatrix of the LMG model is done to compare the results obtained by means of quantum information concepts with those related with the singular behavior of the energy surface of the model, which is the expectation value of the Hamiltonian with respect to spin coherent states.
Excited-state quantum phase transitions (ESQPTs) have been extensively studied in two-level models, but their characterization remains challenging in systems displaying mixed regular and chaotic dynamics. In this work, we investigate ESQPTs within the three-level Lipkin-Meshkov-Glick model, where an enlarged Hilbert space and multiple separatrices give rise to rich spectral structures strongly influenced by chaos. To investigate the different dynamical regions, we have calculated Poincaré sections and Peres lattices. In addition, by combining chaos-sensitive measures with standard ESQPT diagnostics, we provide a static analysis of ESQPT signatures in this model and establish a robust framework for future studies of its dynamical behavior. The degree of chaos and the Kullback-Leibler divergence are found to be very effective chaos-sensitive measures, which are complementary to ESQPT diagnostics such as the mean field limit and the participation ratio. Hence we provide a standard framework to work with ESQPTs in chaotic three-level systems.
The basic Lipkin-Meshkov-Glick model displays a second-order ground-state quantum phase transition and an excited-state quantum phase transition (ESQPT). The inclusion of an anharmonic term in the Hamiltonian implies a second ESQPT of a different nature. We characterize this ESQPT using the mean field limit of the model. The alternative ESQPT, associated with the changes in the boundary of the finite Hilbert space of the system, can be properly described using the order parameter of the ground-state quantum phase transition, the energy gap between adjacent states, the participation ratio, and the quantum fidelity susceptibility.
Phase transitions for an open system consisting of an ensemble of interacting quantum subsystems with discrete spectrum are studied in the mean-field approximation. In the considered model, the change of an internal symmetry of a thermodynamic system upon the second-order phase transition is due to changing symmetry of distribution of charge/spin density inside each quantum subsystem. The latter can be caused by either splitting of one of lowest degenerated energy level or closing a gap between the levels and appearance of avoided crossing. The effect of external parameters (pressure, field, composition, etc.) results in direct change of internal control parameters: level spacing and/or the strength of interaction between adjacent quantum subsystems. Considering a simplest case of the two-level quantum subsystems, expressions for the free energy as a function of the internal control parameters were obtained in analytical form. The behavior of the heat capacity and susceptibility for different regions of the low-temperature phase diagram including the area of quantum fluctuations was determined.