Convergent method for calculating the properties of many interacting electrons

Abstract

A method is presented for calculating binding energies and other properties of extended interacting systems using the projected density of transitions (PDoT) which is the probability distribution for transitions of different energies induced by a given localized operator, the operator on which the transitions are projected. It is shown that the transition contributing to the PDoT at each energy is the one which disturbs the system least, and so, by projecting on appropriate operators, the binding energies of equilibrium electronic states and the energies of their elementary excitations can be calculated. The PDoT may be expanded as a continued fraction by the recursion method, and as in other cases the continued fraction converges exponentially with the number of arithmetic operations, independent of the size of the system, in contrast to other numerical methods for which the number of operations increases with system size to maintain a given accuracy. These properties are illustrated with a calculation of the binding energies and zone-boundary spin-wave energies for an infinite spin-1/2 Heisenberg chain, which is compared with analytic results for this system and extrapolations from finite rings of spins.

Site dilution of quantum spins in the honeycomb lattice

“…34 It is worth mentioning that the recursion method has proved to be a powerful technique even in the presence of interactions. 35 Actually, the continued fraction representation of the Fourier components of the one particle propagator, the basis of the recursion method, is also an essential point in the Padé analytical continuation which usually arises in the many-body problem. 36 We define the following set of zero temperature retarded Green's functions in the standard way,…”

Multiple scattering of electrons in solids and molecules: A cluster-model approach

“…͑15͒: ͑a͒ direct Jacobi iteration; ͑b͒ simultaneous relaxation, 43 previously used in this context; 42 and ͑c͒ the Haydock recursion method. [44][45][46] (a) Direct Jacobi iteration. This method is based upon the iterative evaluation of Eq.…”

“…Haydock's recursion method 44,46 permits one to obtain this matrix element by iterative refinement. Here plays the same role as the energy in calculations of solid ground-state properties.…”

“…Tong 42 reported a lack of convergence in the exact MS expansion, and claimed that this problem can be overcome by using the simultaneous relaxation method, 43 consisting of both mixing the result of each iteration with that of the previous one and using the updated components of the wave function as they are calculated rather than waiting for a given iteration to be completed. That iteration procedure is compared in the present paper with the Haydock recursion method, [44][45][46] which was shown to be more robust and to lead to full convergence even in cases where the former fails to converge. In addition, the recursion method results in faster convergence as compared with either the direct MS expansion or the simultaneous relaxation method.…”

Classical localization of an unbound particle in a two-dimensional periodic potential and surface diffusion

“…14 the projected resolvent determines the diffusion constant, and from previous work, see Ref. [14], it is the eigenvalues and eigenfunctions of the Liouvillian which determine the projected resolvent. In this case it is the eigenfunctions with eigenvalues near zero frequency which are important, and one of these is the constant function on phase space which is always an eigenfunction of the Liouvillian with zero frequency because its gradients are all zero.…”