Conservation of the spectral moments in the n-pole approximation

A formulation of the Green’s function method is presented in the n-pole approximation. Without referring to a specific model we give a general scheme of calculations that easily permits the computation of the “single-particle” Green’s function in terms of the energy matrix. A theorem is proved which states that the moments of the spectral density function are conserved up to the order 2(n−l+1), where l is the order of the composite field. A comparison with the spectral density approach is also discussed. 71.10.-w, 71.15.-m, 71.27.+a Typeset using REVTEX Corresponding author. E-mail address: [email protected] 1 The theoretical analysis of highly correlated electron systems contains many unsolved problems and a complete and systematic formulation is still lacking. Many analytical methods have been developed. Among many we recall the slave boson method [1], the non-crossing approximation [2], the d∞ method [3], the projection operator method [4], the method of equation of motion [5], the spectral density approach [6], the coherent potential approximation [7], the composite operator method [8]. It is not easy to judge the reliability of the various approximations. Certainly, one important aspect is the capability to conserve the symmetries inherent to the model and some relations, which can be derived on general basis. A scheme of approximation, common to various methods [4-6, 8], is the so called n-pole approximation, based on a linearization of the equations of motion. In this article we show that in this scheme the exact relation between the spectral moments and the spectral density is conserved: (i) at any order when the moments are calculated with respect to the linearized equations: (ii) up to the order 2(n− l+1), where l is the order of the composite field, when calculated with respect to the full equations of motion. This aspect is attractive and shows that the approximation can reproduce some essential features of the single-particle function. Let us consider a certain Hamiltonian H = H(φ1, . . . , φp) (1) the set {φi} denotes field operators. In order to close the infinite hierarchy of equations of motion some truncation is necessary. One procedure is to choose a basis of operators {ψ1, . . . , ψn} and linearize the equations of motion as i ∂ ∂t ψ(x) = [ψ(x), H ] = ǫ(−i~ ∇)ψ(x) + δj(x) ≈ ǫ(−i~ ∇)ψ(x) (2) where the eigenvalue or energy matrix ǫ [9] is self-consistently calculated by means of the equation ǫ(−i∇x)〈 [