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Physical review. B, Condensed matter

Large-Ucluster-Hamiltonian expansion of the Hubbard model

Dimitri Antoniou, Allan H. MacDonald

Published Mar 15, 1996• 2 citations
Publisher

Abstract

We discuss an expansion of the low-energy effective Hamiltonian of the large-U Hubbard model, in which higher-order terms involve lattice clusters with larger numbers of sites rather than larger powers of the hopping amplitude parameter t. Expressions for each term, valid to all orders in t, can be obtained by exactly diagonalizing the Hubbard model for small clusters. Leading terms in this expansion are derived. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.53.6855

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We discuss an expansion of the low-energy effective Hamiltonian of the large-U Hubbard model, in which higher-order terms involve lattice clusters with larger numbers of sites rather than larger powers of the hopping amplitude parameter t. Expressions for each term, valid to all orders in t, can be obtained by exactly diagonalizing the Hubbard model for small clusters. Leading terms in this expansion are derived. \textcopyright{} 1996 The American Physical Society.
We describe a unitary transformation which eliminates terms coupling states with differing numbers of doubly occupied sites from the Hamiltonian of the Hubbard model. The SSS matrix for the transformation, and the transformed Hamiltonian, ${H}^{\ensuremath{'}}$, are generated by an iterative procedure which results in an expansion in powers of the hopping integral ttt divided by the on-site energy UUU. For a half-filled band and in the space with no doubly occupied sites, ${H}^{\ensuremath{'}}$ is equivalent to a spin Hamiltonian. We discuss the implications of our results for ${H}^{\ensuremath{'}}$ on theories of high-temperature superconductivity.
We compare numerical results of O and Cu occupations and optical and magnetic excitations for the three-band Hubbard model in a Cu4{\mathrm{Cu}}_{4}Cu4​O8{\mathrm{O}}_{8}O8​ cluster, with those obtained with an effective Hamiltonian Heff{\mathit{H}}_{\mathit{e}\mathit{f}\mathit{f}}Heff​ in which the Cu-O hopping tpd{\mathit{t}}_{\mathit{p}\mathit{d}}tpd​ is eliminated by means of a canonical transformation. This transformation up to order tpd4{\mathit{t}}_{\mathit{p}\mathit{d}}^{4}tpd4​ is not quantitatively valid when the relevant energy differences are of the order of \ensuremath{\surd}8 tpd{\mathit{t}}_{\mathit{p}\mathit{d}}tpd​ or less. However, an Heff{\mathit{H}}_{\mathit{e}\mathit{f}\mathit{f}}Heff​ with the form of the second-order one, but which retains all higher-order terms contained in a CuO4{\mathrm{CuO}}_{4}CuO4​ (Cu2{\mathrm{Cu}}_{2}Cu2​O7{\mathrm{O}}_{7}O7​) cluster for zero or one added hole (one added electron) to the undoped system, leads to a very good agreement with the results using the full Hilbert space. This Heff{\mathit{H}}_{\mathit{e}\mathit{f}\mathit{f}}Heff​ makes possible the numerical study of the three-band Hubbard model for realistic parameters, using larger clusters.
We develop a perturbation expansion in the intersite hopping around the atomic limit of the Hubbard model. It is valid for arbitrary finite temperatures and interaction strengths. Diagrammatic rules that determine the grand-canonical potential and the Green's functions are derived. They reduce the calculation of any finite-order contribution to simple algebra. This opens the way for series extrapolations from computer-aided high-finite-order evaluations. Discrepancies in earlier expansions around the atomic limit are clarified. The present expansion scheme involves only connected diagrams with unrestricted lattice sums. This allows one to perform a vertex renormalization as for the linked-cluster expansion of the Ising model. The renormalized perturbation expansion can be used to construct self-consistent approximations which are automatically exact in the atomic limit. In the limit of high lattice dimensions, only fully two-particle reducible embeddings of diagrams on the lattice contribute. The single-particle properties of the infinite-dimensional Hubbard model reduce to those of independent tight-binding fermions hopping between dressed sites.
A strong-coupling expansion for the antiferromagnetic phase of the Hubbard model is derived in the framework of the slave-boson mean-field approximation. The expansion can be obtained in terms of moments of the density of states of freely hopping electrons on a lattice, which in turn are obtained for hypercubic lattices in arbitrary dimension. The expansion is given for the case of half-filling and for the energy up to fifth order in the ratio of hopping integral t over on-site interaction U, but can straightforwardly be generalized to the non-half-filled case and be extended to higher orders in t/U. For the energy the expansion is found to have an accuracy of better than 1% for U/t\ensuremath{\ge}8. A comparison is given with an earlier perturbation expansion based on the linear-spin-wave approximation and with a similar expansion based on the Hartree-Fock approximation. The case of an infinite number of spatial dimensions is discussed. \textcopyright{} 1996 The American Physical Society.
Considering Hubbard models with infinite U , N degrees of freedom per site, and hopping terms between nearest and second-nearest neighbours on a square lattice, and using a formulation with X -operators, the leading contributions at large N to the anomalous self-energy and the linearized Eliashberg equation are calculated. Some of these terms have the form of Fock-like terms with an effective two-particle interaction, others have a different structure and have not been discussed previously. Calculating transition temperatures, we find that the new terms found in our 1/ N expansion compensate the other terms nearly perfectly suppressing superconducting instabilities to extremely small temperatures.
Motivated by the recent discovery of a spin-liquid phase for the Hubbard model on the honeycomb lattice at half-filling (Meng et al 2010 Nature 88 487), we apply both perturbative and non-perturbative techniques to derive effective spin Hamiltonians describing the low-energy physics of the Mott-insulating phase of the system. Exact diagonalizations of the so-derived models on small clusters are performed, in order to assess the quality of the effective low-energy theory in the spin-liquid regime. We show that six-spin interactions on the elementary loop of the honeycomb lattice are the dominant sub-leading effective couplings. A minimal spin model is shown to reproduce most of the energetic properties of the Hubbard model on the honeycomb lattice in its spin-liquid phase. Surprisingly, a more elaborate effective low-energy spin model obtained by a systematic graph expansion rather disagrees beyond a certain point with the numerical results for the Hubbard model at intermediate couplings.