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Journal of Mathematical Physics

Critical properties of pseudospin Hamiltonians

Robert Gilmore

Published Jul 1, 1984• 11 citations
Publisher

Abstract

The thermodynamic critical properties of a simple class of pseudospin model Hamiltonians are discussed. This class of models includes the spin van der Waals model and the Meshkov–Glick–Lipkin model as particular cases. Second-order thermodynamic phase transitions occur when the spin–spin interaction contributes negatively in a particular direction and the linear interaction term is orthogonal to the direction(s) of greatest energy gain through the spin–spin interaction.

DOI
10.1063/1.526405

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The thermodynamic critical properties of a simple class of pseudospin model Hamiltonians are discussed. This class of models includes the spin van der Waals model and the Meshkov–Glick–Lipkin model as particular cases. Second-order thermodynamic phase transitions occur when the spin–spin interaction contributes negatively in a particular direction and the linear interaction term is orthogonal to the direction(s) of greatest energy gain through the spin–spin interaction.
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.
Phase, as the term is used in thermodynamics, refers to a spatially uniform equilibrium system. Substances undergo phase transitions, changes in phase that occur upon variations of state variables. The chemical potential of substances in coexisting phases has the same value in each of the phases in which coexistence occurs. The specific heat is discontinuous at the van der Waals critical point—a second-order phase transition in the Ehrenfest classification scheme. This chapter shows that one-dimensional systems cannot support phase transitions. Just as with liquid-vapor phase transitions, exponents characterize the critical behavior of magnetic systems. Magnets and fluids are different physical systems, yet they have the same critical behavior. Through the concept of order parameter, Landau theory subsumes the phase-transition behavior of different systems into one theory. A more physical way of demonstrating the existence of a phase transition would be to calculate the order parameter.
To set the stage for our subsequent thorough discussion of dynamic critical phenomena, we first review the theoretical description of second-order equilibrium phase transitions. (Readers already well acquainted with this material may readily move on to Chapter 2.) To this end, we compare the critical exponents following from the van-der-Waals equation of state for weakly interacting gases with the results from the Curie–Weiss mean-field approximation for the ferromagnetic Ising model. We then provide a unifying description in terms of Landau–Ginzburg theory, i.e., a long-wavelength expansion of the effective free energy with respect to the order parameter. The Gaussian model is analyzed, and a quantitative criterion is established that defines the circumstances when non-linear fluctuations need to be taken into account properly. Thereby we identify d c = 4 as the upper critical dimension for generic continuous phase transitions in thermal equilibrium. The most characteristic feature of a critical point turns out to be the divergence of the correlation length that renders microscopic details oblivious. As a consequence, not only the correlation functions, but remarkably the thermodynamics as well of a critical system are governed by an emergent unusual symmetry: scale invariance. A simple scaling ansatz is capable of linking different critical exponents; as an application, we introduce the basic elements of finite-size scaling. Finally, a brief sketch of Wilson's momentum shell renormalization group method is presented, intended as a pedagogical preview of the fundamental RG ideas. Exploiting the scale invariance properties at the critical point, the scaling forms of the free energy and the order parameter correlation function are derived.
The equilibrium and phase properties of the Ising model with three-spin interaction and an external field are studied within the framework of mean-field approximation. The thermodynamic properties of the model reveals two coexistence curves, signifying two distinct second-order phase transitions, dependent on the domain of the interaction parameter. The critical exponents of the magnetic order parameter are calculated in all directions of the phase space and show their agreement with the mean-field universality class.
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.
We study the critical properties of the Lipkin-Meshkov-Glick model in terms of the fidelity susceptibility. By using the Holstein-Primakoff transformation, we obtain explicitly the critical exponent of the fidelity susceptibility around the second-order quantum phase transition point. Our results provide a rare analytic case for the fidelity susceptibility in describing the universality class in quantum critical behavior. The different critical exponents in two phases are nontrivial results, indicating that the fidelity susceptibility is not always extensive.