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Molecular Physics

Heisenberg-model cluster expansion for half-filled Hubbard and PPP models

R. D. Poshusta, Thomas G. Schmalz, Douglas J. Klein

Published Feb 10, 1989• 26 citations
Publisher

Abstract

Pariser-Parr-Pople, Hubbard, and Heisenberg models for systems with one electron per site, such as the π-networks of neutral conjugated hydrocarbons, are considered. It is argued that for usual systems the Hubbard model describes reasonably only those states that are also capable of being so described by a Heisenberg model. Such Heisenberg models are given by a cluster expansion, surpassing in accuracy the results of correspondingly low-order degenerate perturbation expansions.

DOI
10.1080/00268978900100151

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Pariser-Parr-Pople, Hubbard, and Heisenberg models for systems with one electron per site, such as the π-networks of neutral conjugated hydrocarbons, are considered. It is argued that for usual systems the Hubbard model describes reasonably only those states that are also capable of being so described by a Heisenberg model. Such Heisenberg models are given by a cluster expansion, surpassing in accuracy the results of correspondingly low-order degenerate perturbation expansions.
Static-spin-correlation functions 4〈spz{s}_{\mathrm{pz}}spz​sqz{s}_{\mathrm{qz}}sqz​〉 are evaluated via diagrammatic valence-bond theory for exact states of finite Hubbard, Pariser-Parr-Pople (PPP), and related quantum cell models. Several properties of 〈spz{s}_{\mathrm{pz}}spz​sqz{s}_{\mathrm{qz}}sqz​〉 that are independent of intersite potentials Vpp′{V}_{\mathrm{pp}\mathcal{'}}Vpp′​ in the Hartree-Fock (HF) approximation for half-filled alternant systems are shown to be sensitive to correlations in the ground state of N\ensuremath{\le}12 Hubbard and PPP chains. Accurate N\ensuremath{\rightarrow}\ensuremath{\infty} extrapolations are found for near neighbors. The correct ground-state charge density 〈np{n}_{p}np​〉=1 is a property of half-filled alternant systems and thus not a test of HF or other one-electron approximations. Spin correlations in PPP models with standard molecular parameters and long-range Coulomb interactions are approximately given by on-site interactions of U\ensuremath{\simeq}2\ensuremath{\Vert}t\ensuremath{\Vert} in Hubbard models, although the general semiempirical PPP approach to conjugated molecules is preferable for wide-band systems.
I present a scheme for performing weak-coupling expansions for the Hubbard model on square clusters near half-filling. Using this scheme the 6 × 6 cluster with 2, 3 and 4 holes is considered. I have calculated the ground-state quantum numbers for the 3- and 4-hole cases. In the 3-hole case I find a nonzero momentum near the (±π/2, ±π/2) points, while for the four-hole case the momentum is (0, 0). The calculation of low-lying excited states indicates the opening of a spin gap as one dopes away from half-filling.
A new bond auxiliary particle method for the Hubbard, t-J and Heisenberg Hamiltonians is presented. First, the one-dimensional (1D) Heisenberg Hamiltonian is studied as a test case. The exact ground state is approached rapidly via perturbation theory. At the level of second-order perturbation theory, the ground state energy is accurate within a few per cent, while in the 15th order it is accurate to five figures. Without any renormalization, the spin velocity is given with 10% accuracy, and the spectrum is correctly gapless. The degeneracy is also correct, giving a Wilson ratio R=2. The method is used to calculate analytically the full Green's function for a single hole in a half-filled (one electron per site) t-J model.
The ground-state energy E and momentum distribution n k of the electrons are expanded from the atomic limit for the half-filled Hubbard model. The coefficients of expansion are represented by the spin correlation functions of the spin 1 / 2 Heisenberg model. Using the spin-wave theory, approximate values of coefficients are calculated for the square lattice and the simple cubic lattice. In one dimension, the theory shows a good agreement with the exact solution. An effective spin Hamiltonian for the half-filled Hubbard model with arbitrary hopping integrals is obtained up to the fifth order. It is shown that the fourth term contains four spin interactions.