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Physica Scripta

Phase space analysis of first-, second- and third-order quantum phase transitions in the Lipkin–Meshkov–Glick model

E. Romera, Manuel Calixto, Octavio Castaños

Published Aug 20, 2014• 40 citations
Publisher

Abstract

We present a phase-space study of first-, second- and third-order quantum phase transitions in the Lipkin–Meshkov–Glick model by means of the Husimi function. By analyzing the distribution of zeros of the ground state Husimi function we have characterized each phase and each type of quantum phase transition in this model. We show that Renyi–Wehrl entropies of the ground state Husimi function give a good description of quantum phase transitions. The study has been done using a numerical treatment and a variational approximation in terms of coherent states. Additionally, we have analyzed quantum phase transitions using the fidelity and fidelity susceptibility concepts.

DOI
10.1088/0031-8949/89/9/095103

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We present a phase-space study of first-, second- and third-order quantum phase transitions in the Lipkin–Meshkov–Glick model by means of the Husimi function. By analyzing the distribution of zeros of the ground state Husimi function we have characterized each phase and each type of quantum phase transition in this model. We show that Renyi–Wehrl entropies of the ground state Husimi function give a good description of quantum phase transitions. The study has been done using a numerical treatment and a variational approximation in terms of coherent states. Additionally, we have analyzed quantum phase transitions using the fidelity and fidelity susceptibility concepts.
This research demonstrates that entropy-based measures, calculated from marginals of the Husimi functions, effectively identify and characterize quantum phase transitions in the Lipkin-Meshkov-Glick model. The study reveals that changes in entropy, particularly Wehrl entropy, marginal entropies, and mutual information, act as sensitive indicators of criticality, distinguishing between different transition orders and highlighting the breakdown of separability near critical points. These findings establish entropy analysis as a powerful, complementary tool to traditional methods for understanding quantum criticality in many-body 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.
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.
The Husimi distribution is proposed for a phase-space analysis of quantum phase transitions in the Dicke model of spin-boson interactions. We show that the inverse participation ratio and Wehrl entropy of the Husimi distribution give sharp signatures of the quantum phase transition. The analysis is done using two frameworks: a numerical treatment and an analytical variational approximation. Additionally, we propose a characterization of the Dicke model quantum phase transition by means of the zeros of the Husimi distribution in the variational approach.
Using the Husimi quasiprobability distribution, we investigate the phase space signatures of excited-state quantum phase transitions (ESQPTs) in the Lipkin-Meshkov-Glick and coupled top models. We show that the ESQPT is evinced by the dynamics of the Husimi function, that exhibits a distinct time dependence in the different ESQPT phases. We also discuss how to identify the ESQPT signatures from the long-time averaged Husimi function and its associated marginal distributions. Moreover, from the calculated second moment and Wherl entropy of the long-time averaged Husimi function, we estimate the critical points of the ESQPT in both models, obtaining a good agreement with analytical (mean field) results. We provide a firm evidence that phase space methods are both a new probe for the detection and a valuable tool for the study of ESQPTs.