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arXiv:cond-mat/0703490 · Physical Review Letters

Thermodynamical Limit of the Lipkin-Meshkov-Glick Model

Pedro Ribeiro, Julien Vidal, Rémy Mosseri

Published Aug 2, 2007• 223 citations
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Abstract

A method based on the analysis of the Majorana polynomial roots is introduced to compute the spectrum of the Lipkin-Meshkov-Glick model in the thermodynamical limit. A rich structure made of four qualitatively different regions is revealed in the parameter space, whereas the ground state study distinguishes between only two phases.
DOI
10.1103/physrevlett.99.050402

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The spectrum of the Lipkin-Meshkov-Glick model is exactly derived in the thermodynamic limit by means of a spin-coherent-state formalism. In the first step, a classical analysis allows one to distinguish between four distinct regions in the parameter space according to the nature of the singularities arising in the classical energy surface; these correspond to spectral critical points. The eigenfunctions are then analyzed more precisely in terms of the associated roots of the Majorana polynomial, leading to exact expressions for the density of states in the thermodynamic limit. Finite-size effects are also analyzed, leading in particular to logarithmic corrections near the singularities occurring in the spectrum. Finally, we also compute expectation values of the spin operators in a semiclassical analysis in order to illustrate some subtle effects occurring in one region of the parameter space.
An analysis of the classical and quantum phase transitions of the Lipkin-Meshkov-Glick model is presented. It is shown that the classical dynamics is ruled by the energy surface of the system. Applying the catastrophe formalism to this energy surface the separatrix is obtained. It determines the regions in the control parameter space where there are phase transitions. Special attention is given to the compositions of ground and first-excited energy states, which are well described by the even and odd SU(2) coherent states. Phase transitions are shown to be associated with a change in the wave functions from collective to single-particle behavior. Evaluating the distribution of nearest-neighbor spacings it is shown that the separatrix of the system emerges as a useful tool to describe the global behavior of the quantum-level structure and their corresponding wave functions.
Abstract The Agassi model (Agassi 1968 Nucl. Phys. A 116 49) is a schematic two-level model that involves pairing and monopole–monopole interactions. It is, therefore, an extension of the well known Lipkin–Meshkov–Glick model (Lipkin et al 1965 Nucl. Phys. 62 188). In this paper we review the algebraic formulation of an extension of the Agassi model as well as its bosonic realization through the Schwinger representation. Moreover, a mean-field approximation for the model is presented and its phase diagram discussed. Finally, a 1/ j analysis, with j proportional to the degeneracy of each level, is worked out to obtain the thermodynamic limit of the ground state energy and some order parameters from the exact Hamiltonian diagonalization for finite— j .
Abstract We introduce a class of generalized Lipkin–Meshkov–Glick (gLMG) models with s u ( m ) interactions of Haldane–Shastry type. We compute the partition function of these models in closed form by exactly evaluating the partition function of the restriction of a spin chain Hamiltonian of Haldane–Shastry type to subspaces with well-defined magnon numbers. As a byproduct of our analysis, we obtain strong numerical evidence of the Gaussian character of the level density of the latter restricted Hamiltonians, and study the distribution of the spacings of consecutive unfolded levels. We also discuss the thermodynamic behavior of a large family of s u ( 2 ) and s u ( 3 ) gLMG models, showing that it is qualitatively similar to that of a two-level system.
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.