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arXiv:cond-mat/0407084 · Journal of Physics Condensed Matter

The treatment of zero eigenvalues of the matrix governing the equations of motion in many-body Green function theory

Peter Fröbrich, P. J. Kuntz

Published Feb 5, 2005• 8 citations
arXivHTMLPDF

Abstract

The spectral theorem of many-body Green function theory relates thermodynamic correlations to Green functions. More often than not, the matrix governing the equations of motion has zero eigenvalues. In this case, the standard textbook approach requires both commutator and anti-commutator Green functions to obtain equations for that part of the correlation which does not lie in the null space of the matrix. In this paper, we show that this procedure fails if the projector onto the null space is dependent on the momentum vector, k . We propose an alternative formulation of the theory in terms of the non-null space alone and we show that a solution is possible if one can find a momentum-independent projector onto some subspace of the non-null space. To do this, we enlist the aid of the singular value decomposition of in order to project out the null space, thus reducing the size of the matrix and eliminating the need for the anti-commutator Green function. We extend our previous work (Fröbrich and Kuntz 2003 Phys. Rev. B 68 014410), dealing with a ferromagnetic Heisenberg monolayer and a momentum-independent projector onto the null space, to models where both multilayer films and a momentum-dependent projector are considered. We develop the numerical methods capable of handling these cases and offer a computational algorithm that should be applicable to any similar problem arising in Green function theory.

DOI
10.1088/0953-8984/17/7/010

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The spectral theorem of many-body Green function theory relates thermodynamic correlations to Green functions. More often than not, the matrix governing the equations of motion has zero eigenvalues. In this case, the standard textbook approach requires both commutator and anti-commutator Green functions to obtain equations for that part of the correlation which does not lie in the null space of the matrix. In this paper, we show that this procedure fails if the projector onto the null space is dependent on the momentum vector, k . We propose an alternative formulation of the theory in terms of the non-null space alone and we show that a solution is possible if one can find a momentum-independent projector onto some subspace of the non-null space. To do this, we enlist the aid of the singular value decomposition of in order to project out the null space, thus reducing the size of the matrix and eliminating the need for the anti-commutator Green function. We extend our previous work (Fröbrich and Kuntz 2003 Phys. Rev. B 68 014410), dealing with a ferromagnetic Heisenberg monolayer and a momentum-independent projector onto the null space, to models where both multilayer films and a momentum-dependent projector are considered. We develop the numerical methods capable of handling these cases and offer a computational algorithm that should be applicable to any similar problem arising in Green function theory.
In using the spectral theorem of many-body Green's function theory in order to relate correlations to commutator Green's functions, it is necessary in the standard procedure to consider the anti-commutator Green's functions as well whenever the matrix governing the equations of motion for the commutator Green's functions has zero eigenvalues. We show that a singular-value decomposition of this matrix allows one to reformulate the problem in terms of a smaller set of Green's functions with an associated matrix having no zero eigenvalues, thus eliminating the need for the anti-commutator functions. The procedure is quite general and easy to apply. It is illustrated for the field-induced reorientation of the magnetization of a ferromagnetic Heisenberg monolayer and it is expected to work for more complicated cases as well.
The one-dimensional Heisenberg model with S = 1/2 is treated with the use of the two-time Green's functions. The hierarchy of the equations of motion of the Green'f functions is decoupled at a stage one-step further than Tyablikov's decoupling. The thermal average of the spin component, ≪ Sz >, is set to zero, becaure the long-range order does not exist in one dimension. Instead, our Green's functions are expressed in terms of the correlation functions cn ≡4≪ S0zSnz >. The Green's function is essentially of the form representing undamped spin waves, whose spactrum depends on c1, c2 and one more parameter. They are determined by the requirement that c1 and c2 should be self-consistent and that c0 should be unity. The self-consistency equations have been solved analytically at high- and low-temperature limits, and also solved numerically in the whole range of the temperature. Thermodynamic quantities have been calculated using these solutions. It has turned out that the theory gives the correct high-temperature expansion for the thermodynamic quantities and the correlation functions. The latter is expressed by cn = (J / 4 kBT )n. In the case of ferromagnetic coupling, the correlation function cn at T = 0 is equal to 1/3 for all n's. This is what is expected from the correct ground state of the ferromagnetic Heisenberg system. The spin-wave spectrum at T = 0 also agrees with the correct one. At T ≪J/ kB, we find that the specific heat goes as T1/2 and the susceptibility goes as T-2. The gross feature of the temperature-dependence of the thermodynamic quantities agrees with Bonner and Fisher. In the case of the antiferromagnetic Heisenberg model, we find c1 = -0.55407 and c2 = 0.16100 at T = 0, which are fairly close to the exact values. The thermodynamic quantities are also in gross agreement with Bonner and Fisher.
In the usual theory of the two-time Green's function, decoupling techniques are introduced in order to obtain a self-closed equation for the lowest order Green's function. These decouplings give neither the correct temperature dependence (the absence of a T3{T}^{3}T3 term) of magnetization nor the correct correlation function $〈{{S}_{j}}^{\ensuremath{-}}{{S}_{j}}^{\ensuremath{-}}{{S}_{k}}^{+}{{S}_{l}}^{+}〉$, which should be zero by definition. In this paper the equation of motion for the higher order Green's function is solved in the approximation which will lead to correct results up to terms of order T3{T}^{3}T3 in these quantities. Employing this higher order Green's function, it is found that the correction to the magnetization consists of a term of order T4{T}^{4}T4 and a divergent term and that the correlation function consists of a term of order T3{T}^{3}T3 and a divergent term. A method extracting a significant part from this divergent term is found by analyzing a system of two interacting \textonehalf{} spins. By this method, it is shown that the T3{T}^{3}T3 term in the magnetization is completely eliminated and that the correlation function $〈{{S}_{j}}^{\ensuremath{-}}{{S}_{j}}^{\ensuremath{-}}{{S}_{k}}^{+}{{S}_{l}}^{+}〉$ vanishes identically.
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.
We formulate the spectral-density method in classical statistical mechanics in strict analogy with the known quantum version, and we apply it to the classical Heisenberg ferromagnetic model in an external field. A new formula for the magnetization in the classical formulation of spin-vector Green functions is derived for arbitrary spatial dimensionality. Furthermore, the static properties of the one- and three-dimensional cases are considered in detail. We obtain accurate results over a wide range of temperatures with the external field for the one-dimensional model and without field for the three-dimensional model. In the first case very good agreement with the exact numerical transfer-matrix data is also found in the region of higher temperature where the interacting-boson approach fails. In particular the zero-field susceptibility results are found to be consistent with the exact results obtained by Fisher.
The standard basis operator (SBO) equation of motion is solved exactly for the class of interacting many-body Hamiltonians which can be constructed from a complete set of mutually orthogonal projection operators. This class includes the Ising model Hamiltonians. Two integration procedures are presented: (1) term by term integration over the poles of the SBO double-time temperature-dependent Green’s functions in the spectral representation; (2) direct integration of the time-dependent equation of motion of an SBO. An ensemble average expression relating correlation functions among systems in different energy states is derived. For the Ising model this expression relates correlation functions of spin clusters of different orders, and it is analogous to a relation derived from the master equation under equilibrium conditions.