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arXiv:1905.07745 · Condensed Matter Physics

REFERENCE APPROACH IN THEORY OF PSEUDOSPIN SYSTEMS

Levitskii, Sorokov, O. R. Baran

Published Jan 1, 2000• 4 citations
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Abstract

For theoretical description of pseudospin systems with essential shortrange and long-range interactions we use the method based on calculations of the free energy functional taking into account the short-range interactions within the reference approach in cluster approximation.We propose a consistent formulation of the cluster expansion method for quantum pseudospin systems.We develop a method allowing one to obtain within the cluster approximation an Ornstein-Zernike type equation for reference cumulant Green function of an arbitrary order.In the two-particle cluster approximation we derived an explicit expression for pair temperature cumulant Green function of the reference system.In the cluster random phase approximation we calculated and studied thermodynamic characteristics, elementary excitation spectrum, and integral intensities of the Ising model in transverse field.

DOI
10.5488/cmp.3.3.515

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For theoretical description of pseudospin systems with essential shortrange and long-range interactions we use the method based on calculations of the free energy functional taking into account the short-range interactions within the reference approach in cluster approximation.We propose a consistent formulation of the cluster expansion method for quantum pseudospin systems.We develop a method allowing one to obtain within the cluster approximation an Ornstein-Zernike type equation for reference cumulant Green function of an arbitrary order.In the two-particle cluster approximation we derived an explicit expression for pair temperature cumulant Green function of the reference system.In the cluster random phase approximation we calculated and studied thermodynamic characteristics, elementary excitation spectrum, and integral intensities of the Ising model in transverse field.
The method for calculation of the correlation functions of the Ising-type systems with short-range interaction and with arbitrary value of spin is developed within cluster approximation. For the Ising model (spin Sz=±1S^z=\pm1Sz=±1) the expressions for pair and ternary correlation functions within two-particle approximation in { q }\set{q}{q}-space are obtained for the hypercubic Bravais lattices. In the 1D case the exact expressions for them in the site space is obtained as well. On the basis of the Glauber equation within two-particle cluster approximation the longitudinal dynamical susceptibility χ({ q },E)χ(\set{q},E)χ({q},E) is found. In the 1D case and in the absence of external field the expression for χ({ q },E)χ(\set{q},E)χ({q},E) is exact. For the Emery-Blume-Griffiths model (Sz=−2,0,2S^z=-2,0,2Sz=−2,0,2) within two-particle approximation the pair correlation functions are calculated. The four-particle cluster approximation is used for calculation of static susceptibility χ({ q })χ(\set{q})χ({q}) of KD2PO4KD_2PO_4KD2​PO4​ ferroelectrics.
This is the first of a series of papers dealing with many-particle systems from a unified, nonperturbative point of view. It contains derivations and discussions of various field-theoretical techniques which will be applied in subsequent papers. In a short introduction the general method of approach is summarized, and its relationship to other field-theoretic problems indicated. In the second section the macroscopic properties of the spectra of many-particle systems are described. Asymptotic evaluations are performed which characterize these macroscopic features in terms of intensive parameters, and the relationship of these parameters to thermodynamics is discussed. The special characteristics of the ground state are shown to follow as a limiting case of the asymptotic evaluations. The third section is devoted to the time-dependent field correlation functions, or Green's functions, which describe the microscopic behavior of a multiparticle system. These functions are defined, and related to intensive macroscopic variables when the energy and number of particles are large. Spectral representations and other properties of various one-particle Green's functions are derived. In the fourth section the treatment of non-equilibrium processes is considered. As a particular example, the electromagnetic properties of a system are expressed in terms of the special two-particle Green's function which describes current correlation. The discussion yields specifically a fluctuation-dissipation theorem, a sum rule for conductivity, and certain dispersion relations. The fifth section deals with the differential equations which determine the Green's functions. The boundary conditions that characterize the Green's function equations are exhibited without reference to adiabatic decoupling. A method for solving the equations approximately, by treating the correlations among successively larger numbers of particles, is considered. The first approximation in this sequence is shown to yield a generalized Hartree-like equation. A related, but rigorous, identity for the single-particle Green's function is then derived. A second approximation, which takes certain two-particle correlations into account, is shown to produce various additional effects: The interaction between particles is altered in a manner characterized by the intensive macroscopic parameters, and the modification and spread of the energy-momentum relation come into play. In the final section compact formal expressions for the Green's functions and other physical quantities are derived. Alternative equations and systematic approximations for the Green's functions are obtained.
The previous configuration-space cluster integral treatment of the properties of the ground state of a many-boson system is modified by including those higher order diagrams consistent with the pair excitation approximation, which were previously omitted. The cluster expansions involve a parameter, analogous to the fugacity in classical expansions, whose definition automatically accounts for the depletion of the free particle ground state. The resulting expectation value for the Hamiltonian is in agreement with that obtained by field-theoretic methods for states of pair-excitation type.
The infinite-range quantum Ising spin-glass model in a transverse field \ensuremath{\Gamma} is studied within the cluster-expansion method formulated originally for spin systems by Morita and Tanaka. With use of the pair approximation, the mean-field equation and condition for critical values of ${\ensuremath{\Gamma}}_{c}$ and Tc{T}_{c}Tc​ are obtained including the case Tc{T}_{c}Tc​\ensuremath{\rightarrow}0.
We consider the cumulant expansion of the PAM employing the hybridization as perturbation (Phys. Rev. B 50, 17933 (1994)), and we obtain formally exact one-electron Green's functions (GF). These GF contain effective cumulants that are as difficult to calculate as the original GF, and the Atomic Approximation consists in substituting the effective cumulants by the ones that correspond to the atomic case, namely by taking a conduction band of zeroth width and local hybridization. This approximation has already been used for the case of infinite electronic repulsion U (Phys. Rev. B 62, 7882 (2000)), and here we extend the treatment to the case of finite U. The method can also be applied to the single impurity Anderson model (SIAM), and we give explicit expressions of the approximate GF both for the PAM and the SIAM.
Introduction Among pseudospin systems studied theoretically, a great attention is paid to Ising-type models, that is, to systems with the Hamiltonian containing only z-component of pseudospin. That is so due to relative simplicity with which approximate calculations for these models can be carried out and tested and possibility of their application to a wide class of real objects. Thus, the Blume-Emery-Griffiths model (BEG) corresponding to the Isingtype model with j = 2 (hereafter S z = \\Gammaj; \\Gammaj +2; : : : ; j \\Gamma 2; j) was proposed for the investigation of phase transitions (PT) in He 3 \\Gamma He 4 mixture [1]. A general form of the Ising-type model with j = 2 contains extra terms, as compared to the BEG model, like S z i (S z j<