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Journal of Mathematical Physics

A Soluble Fermi-Gas Model. Validity of Transformations of the Bogoliubov Type

J. des Cloizeaux

Published Dec 1, 1966• 45 citations
Publisher

Abstract

A linear soluble Fermi-gas model is used to test the validity of perturbation theory and the criteria of instability obtained by applying transformations of the Bogoliubov type. The model Hamiltonian describing the interaction of the electrons in a simple band is H(ρ)=−Σ(cosk+ρ)ck+ck+ρΣcos q ck+q+cu−q+cuck. For a half-filled band, the energies of the low-lying states are calculated exactly, by using previous results concerning a linear chain of spins with interaction. It is shown that perturbation theory must be valid for −1 < ρ < 1, and, for values of ρ belonging to this range, the slope of the specific heat for T = 0 is calculated by direct application of Landau's theory. On the other hand, anomalous Hartree-Fock states can be built for all values of ρ, by using transformations of the Bogoliubov type, and these states are more stable than the normal Hartree-Fock state. Thus, it appears that the Bogoliubov method may lead to completely erroneous interpretations, when the coupling constants are small.

DOI
10.1063/1.1704899

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Physical Review•Jan 7, 1966•1 citation
A linear soluble Fermi-gas model is used to test the validity of perturbation theory and the criteria of instability obtained by applying transformations of the Bogoliubov type. The model Hamiltonian describing the interaction of the electrons in a simple band is H(ρ)=−Σ(cosk+ρ)ck+ck+ρΣcos q ck+q+cu−q+cuck. For a half-filled band, the energies of the low-lying states are calculated exactly, by using previous results concerning a linear chain of spins with interaction. It is shown that perturbation theory must be valid for −1 < ρ < 1, and, for values of ρ belonging to this range, the slope of the specific heat for T = 0 is calculated by direct application of Landau's theory. On the other hand, anomalous Hartree-Fock states can be built for all values of ρ, by using transformations of the Bogoliubov type, and these states are more stable than the normal Hartree-Fock state. Thus, it appears that the Bogoliubov method may lead to completely erroneous interpretations, when the coupling constants are small.
A detailed investigation has been made of the application of the Hartree-Fock approximation to the stability and thermodynamic behaviour of spin-density waves in a model with a single electronic energy band and a delta-function-type interaction between electrons. It is shown that for a half-filled band the ground state at zero temperature is always a spin-density wave, provided only that the width of the band is sufficiently small. For other bands the conditions for the relative stability of the spin-density wave, ferromagnetic and paramagnetic states are more complicated, and are studied in detail for a special band form. For a half-filled band the energy per atom and specific heat have been calculated as functions of temperature for the spin-density wave, ferromagnetic and paramagnetic states. The temperature dependence of the amplitude for the spin-density wave and ferromagnetic states, and of the entropy for the spin-density wave and paramagnetic states have also been calculated. The formulation of the problem has been made using a Green function method in order to allow the development in a later paper of a higher-order approximation to the correlation problem.
A simple model of a boson system is considered for which the exact Schr\"odinger equation reduces to a difference equation which can be solved numerically. Application of the Bogoliubov method and comparison of its results with the numerical results then yield some insight into the validity of the Bogoliubov approximations. The simple model considered has, in the absence of interactions, only a zero energy state and two states of unit energy available to each boson. Initially there is assumed to be no interaction between bosons in the degenerate excited states and all existing interactions are assumed to be repulsive. The ground-state energy is calculated in the Bogoliubov approximation, with and without the depletion effect considered, and in another numerical approximation. These results and the results of the numerical solution of the exact Schr\"odinger equation converge to the Bogoliubov result with depletion ignored when the number of particles in the system approaches infinity. It is surprising to note that, for the interaction strengths considered, the Bogoliubov result is within 3% of the numerical result for as few as 32 particles in the system, and within 10% for only 4 particles. A modification of the above system is considered in which there exists an additional two-body interaction between particles in the degenerate excited states which may be attractive or repulsive. It is shown that the ground-state energy, with this additional interaction present, deviates from the Bogoliubov value linearly with the strength of the added interaction, i.e., $W={W}_{\mathrm{Bog}}+\ensuremath{\alpha}F$ (with FFF the strength parameter of the interaction). The factor $\ensuremath{\alpha}$ is found to be small and constant over a wide range of FFF and in this range the deviation of WWW from the Bogoliubov value is small. There is, however, a transition point FT{F}_{T}FT​, beyond which $\ensuremath{\alpha}$ is constant and large and the usual Bogoliubov approximation is invalid for interactions more attractive than that characterized by FT{F}_{T}FT​. A Bogoliubov-like approximation is shown to be quite accurate in this region. It is then shown that $F_{T}^{\ensuremath{-}1}$ is a linear function of the number of particles in the system.
Quantum many-body systems can be studied theoretically from first principles using microscopic models, but the computational cost of numerically exact calculations increases exponentially with the number of particles and the complexity of the interaction, limiting the systems that can be analysed. The Hartree-Fock-Bogoliubov (HFB) and quasiparticle random phase approximation (QRPA) methods, heavily used in nuclear physics, rely on mean-field approximations but scale more favourably with increasing system size and complexity. In this work we apply these methods to the problem of a two-dimensional harmonically trapped Fermi gas with attractive interactions, with the goal of investigating the validity of the approximations in the few-body limit. We implement the approximate methods in code and then compare them with exact calculations. We find that the HFB method can account for pairing correlations that arise due to the attractive interaction, predicting well the ground state energy even at high interaction strengths. Also, the excitation spectra predicted by the QRPA method qualitatively match the true spectra, reproducing the expected subset of excitations. Finally, we see that perturbative corrections can be used to improve the ground state energy estimate of the HFB.
The exactly soluble model, originally considered by Moshinsky, consisting of two spin-1/2 particles interacting through harmonic-oscillator forces and moving in a common harmonic-oscillator potential is extended to include the effects of an external electric field. We solve for the energy exactly, exactly in the Hartree–Fock approximation, and also compare to the results of Hartree–Fock perturbation theory. We find that the Hartree–Fock method yields the same result for the electric polarizability as the exact solution even though the field-independent part of the energy is not the same. However, when second-order Hartree–Fock perturbation theory is employed, the results for the polarizability can be substantially in error. Similar results obtain for the calculation of the matrix element for optical absorption in the dipole approximation.
To describe comprehensively the superconducting phase together with the other phases realized in the copper oxcides, the normal phase, the antiferro-magnetic and the spin-glass phases, an application of the constrained thermal Hartree-Fock-Bogoliubov (CTHFB) approximation is proposed. A set of the electron number- and the magnetization-constrained formalism is exemplified by employing the one-band Hubbard model. The CTHFB equation is applicable to the microscopic interaction whether it includes the phonon-mediated pairing interaction, or it provides a self-sustaining mechanism of the Cooper pair formation. A new type of the order parameter is introduced by extending the generalized Bogoliubov transformation to mix electron spin projections onto the quantization axis (i.e. S^z\hat{S}_{z}S^z​). This yields the lowered minimum of the grand potential, which corresponds to the physical solution violating time-reversal invariance. The gradient method is proposed to solve the CTHFB equation, the non-linear eigenvalue broblem with constraints.
The simple boson model proposed by Foldy, which displayed a discontinuity in its energy as a function of interaction strength, is studied using symmetry arguments, the method of linearized equations of motion, and perturbation theory. The particular symmetry responsible for the discontinuity, or phase transition, is pointed out. The method of linearized equations is shown to predict the transition value of the interaction strength and, like the Bogoliubov approximation, to accurately describe the energy of the first excited state. It is further shown that simple perturbation theory is quite accurate except in the transition region. The validity of that feature of the Bogoliubov method in which excited states are described by independent quasiparticles is examined. This description is found to be excellent in the normal region. All comparisons are made with a numerical solution which provided the complete energy spectrum and the associated eigenvectors. It is hoped that these arguments can be applied in considerations of the validity of the various approximations for realistic systems.