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arXiv:1805.12442 · International Journal of Modern Physics E

Analytical and numerical analysis of the complete Lipkin–Meshkov–Glick Hamiltonian

Giampaolo Co’, Stefano De Leo

Published Apr 24, 2018• 12 citations
arXivHTMLPDF

Abstract

The Lipkin–Meshkov–Glick is a simple, but not trivial, model of a quantum many-body system which allows us to solve the many-body Schrödinger equation without making any approximation. The model, which in its unperturbed case is composed only by two energy levels, includes two interacting terms. A first one, the [Formula: see text] interaction, which promotes or degrades pairs of particles, and a second one, the [Formula: see text] interaction, which scatters one particle in the upper and another in the lower energy level. In comparing this model with other approximation methods, the [Formula: see text] term interaction is often set to zero. In this paper, we show how the presence of this interaction changes the global structure of the system, generates degeneracies between the various eigenstates and modifies the energy eigenvalues structure. We present analytical solutions for systems of two and three particles and, for some specific cases, also for four, six and eight particles. The solutions for systems with more than eight particles are only numerical but their behavior can be well understood by considering the extrapolations of the analytical results. Of particular interest is the study of how the [Formula: see text] interaction affects the energy gap between the ground state and the first-excited state.

DOI
10.1142/s0218301318500398

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The time-dependent Hartree-Fock solutions of the two-level Lipkin-Meshkov-Glick model are studied by transforming the time-dependent Hartree-Fock equations into Hamilton's canonical form and analyzing the qualitative structure of the Hartree-Fock energy surface in the phase space. It is shown that as the interaction strength increases these time-dependent Hartree-Fock solutions undergo a qualitative change associated with the ground state phase transition previously studied in terms of coherent states. For twobody interactions stronger than the critical value, two types of time-dependent Hartree-Fock solutions (the "librations" and "rotations" in Hamilton's mechanics) exist simultaneously, while for weaker interactions only the rotations persist. It is also shown that the coherent states with the maximum total pseudospin value are determinants, so that time-dependent Hartree-Fock analysis is equivalent to the coherent state method.NUCLEAR STRUCTURE Lipkin-Meshkov-Glick model, TDHF, transformation of TDHF equations to Hamilton's canonical form, coherent states, ground state phase transition.

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The Lipkin–Meshkov–Glick is a simple, but not trivial, model of a quantum many-body system which allows us to solve the many-body Schrödinger equation without making any approximation. The model, which in its unperturbed case is composed only by two energy levels, includes two interacting terms. A first one, the [Formula: see text] interaction, which promotes or degrades pairs of particles, and a second one, the [Formula: see text] interaction, which scatters one particle in the upper and another in the lower energy level. In comparing this model with other approximation methods, the [Formula: see text] term interaction is often set to zero. In this paper, we show how the presence of this interaction changes the global structure of the system, generates degeneracies between the various eigenstates and modifies the energy eigenvalues structure. We present analytical solutions for systems of two and three particles and, for some specific cases, also for four, six and eight particles. The solutions for systems with more than eight particles are only numerical but their behavior can be well understood by considering the extrapolations of the analytical results. Of particular interest is the study of how the [Formula: see text] interaction affects the energy gap between the ground state and the first-excited state.
The well known and analytically solvable Lipkin–Meshkov–Glick model was devised to test several many-body approximation approaches used in nuclear physics. Here, we use this model to illustrate a method that sets lower bounds for the ground state energy of a quantum Hamiltonian. The idea of the method consists of dividing the Hamiltonian into identical groups of terms that represent subdivisions in the particle population of the system, and then to make an appropriate use of the ground energy of one of these groups. The bounds clearly improve as the size of these groups increases.
Abstract The Lipkin–Meshkov–Glick (LMG) model was devised to test the validity of different approximate formalisms to treat many-particle systems. The model was constructed to be exactly solvable and yet non-trivial, in order to capture some of the main features of real physical systems. In the present contribution, we explicitly review the fact that different many-body approximations commonly used in different fields in physics clearly fail to describe the exact LMG solution. With similar assumptions as those adopted for the LMG model, we propose a new Hamiltonian based on a general two-body interaction. The new model (extended LMG) is not only more general than the original LMG model and, therefore, has a potentially larger spectrum of applicability, but also the physics behind its exact solution can be much better captured by common many-body approximations. At the basis of this improvement lies a new term in the Hamiltonian that depends on the number of constituents and polarizes the system; the associated symmetry breaking is discussed, together with some implications for the study of more realistic systems.
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.
We present a family of many-body models which are exactly solvable analytically. The models are an extended n-body interaction Lipkin-Meshkov-Glick model which considers spin-flip terms which are associated with the interaction of an external classical field which coherently manipulates the state of the system in order to, for example, process quantum information. The models also describe a two-mode Bose-Einstein condensate with a Josephson-type interaction which includes n-particle elastic and inelastic collisions. One of the models corresponds to the canonical two-mode Bose-Einstein Hamitonian plus a term which we argue must be considered in the description of the two-mode condensate. Intriguingly, this extra term allows for an exact and analytical solution of the two-particle collision two-mode BEC problem. Our results open up an arena to study many-body system properties analytically.
The Lipkin-Meshkov-Glick (LMG) model originally describes a Fermionic many-body system in nuclear physics. However, in recent years, it has been widely found in condensed matter physics, quantum information systems, and quantum optics, and it is of wider and wider interest. Previous studies on this model mainly focused on the physics under the thermal dynamical limit, such as quantum phase transitions and quantum entanglement. There are also some researches about LMG model with finite size in some special limits, but the finite-size effect on energy spectrum is not very clear yet. This is the main motivation of this work. In this paper, the exact diagonalization method and the quantum perturbation theory are used to calculate and analyze the energy-level structure of the LMG model at a finite N. To solve it, we first map this model into the angular-momentum space to obtain a reduced LMG model. By this mapping, the dimension of Hilbert space is reduced to N+1 from 2N. The exact solution of its energy levels can be obtained easily in the U(1) limit where the total spin is conserved. We find that the levels are woven into a fishing-net structure in the U(1) limit. While away from the U(1) limit, the crossings between even and odd levels will open a gap, and the system's energy levels will be grouped into pairs with an odd and an even level, forming some bound states, called doublet states, and the parity of each doublet state will oscillate as the Zeeman field increases. This work gives the values of the critical Zeeman field for the parity crossings. These critical values shift as the interacting parameters and disappear at zero in the Z2 limit. In the Z2 limit, the system energy levels form splittings near the zero Zeeman field. In this article, we analytically calculate the relationship between these energy gaps and the Zeeman field. For odd and even number N, the parity of each state has a different behavior. Specifically, the ground state and the doublet excited states of the system with odd N will suffer a parity reversion at zero Zeeman field, while the states with even N will not. By tuning the interacting parameters, we also study the crossover from the U(1) limit to the Z2 limit. The parity oscillation we find in this system is a very important physical phenomenon, which also exists in some other systems like optical cavity quantum electrodynamics and magnetic molecule system.
The time-dependent Hartree-Fock solutions of the two-level Lipkin-Meshkov-Glick model are studied by transforming the time-dependent Hartree-Fock equations into Hamilton's canonical form and analyzing the qualitative structure of the Hartree-Fock energy surface in the phase space. It is shown that as the interaction strength increases these time-dependent Hartree-Fock solutions undergo a qualitative change associated with the ground state phase transition previously studied in terms of coherent states. For twobody interactions stronger than the critical value, two types of time-dependent Hartree-Fock solutions (the "librations" and "rotations" in Hamilton's mechanics) exist simultaneously, while for weaker interactions only the rotations persist. It is also shown that the coherent states with the maximum total pseudospin value are determinants, so that time-dependent Hartree-Fock analysis is equivalent to the coherent state method.NUCLEAR STRUCTURE Lipkin-Meshkov-Glick model, TDHF, transformation of TDHF equations to Hamilton's canonical form, coherent states, ground state phase transition.