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Physical review. B, Condensed matter

Linked-cluster expansion around the atomic limit of the Hubbard model

Walter Metzner

Published Apr 1, 1991• 239 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.43.8549

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Physical review. B, Condensed matter•Jul 15, 1996•28 citations
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.
We show that mean-field-type symmetry-breaking approximations for correlated fermions can be constructed by performing vertex renormalization (VR) in the linked-cluster expansion, just as for the case of lattice spin models. In the limit of infinite spatial dimensions we sum up exactly the renormalized diagrams for the grand-canonical potential of the Hubbard model and, using the diagrammatic technique for Hubbard operators, we construct explicitly an approximate VR. The procedure leads to a self-consistent symmetry breaking analogous to that of the mean-field theory for spin systems. We present solutions for the case of half-filled energy band. They are valid for arbitrary temperatures and become exact for both large-U limit and for U=0. The antiferromagnetic order parameter is found to have the conventional mean-field form and, in the regime of strong correlations, the obtained critical temperature coincides with the mean-field-theory result for the N\'eel temperature of the effective Heisenberg Hamiltonian.
Using the diagrammatic technique for Hubbard operators we construct a linked-cluster perturbation expansion around the atomic limit of the Hubbard model. We first perform a computer-aided fourth-order calculation. The results clarify discrepances in earlier expansions around the large- U limit. Next, we construct a partial vertex renormalization in the second order of the linked-cluster expansion. The vertex renormalization leads to a selfconsistent magnetic symmetry breaking and allows to study explicitly magnetic oderings of the system in the whole range of temperatures and for arbitrary electron density. The presented results for the case of a half-filled energy band indicate that our approach for the Hubbard model is analogous to the mean-field theory for spin systems, and can serve as a starting point for developing advanced selfconsistent schemes which become exact in the large- U limit. We present results of a systematic numerical study of the non-half-filled-band case. In particular, the vertex-renormalized second-order calculation leads to a qualitatively correct sharp decrease of the Néel temperature as a function of concentration of holes.
We develop a diagrammatic approach for calculating the high temperature expansion of dynamic correlation functions, such as the electron Green's function and the time-dependent density-density and spin-spin correlation functions, for the infinite-U Hubbard Model with any number of spin species. The formalism relies on the use of restricted lattice sums, in which distinct vertices of the diagram represent distinct sites on the lattice. We derive a new formula for the restricted lattice sum of a disconnected diagram consisting of several connected components, and use it to prove the linked cluster theorem with respect to "generalized connected diagrams", formed by overlapping the original connected components on the lattice. This enables us to express all quantities as a sum over these generalized connected diagrams. We compute the Green's function to 4th order in βt for the case of m spin species on a d-dimensional hypercube by hand. We take the m\to\infty limit, enabling us to obtain expressions for the Dyson-Mori self-energy to 4th order in βt for the case of an infinite number of spin species. This may have connections to slave boson techniques used for the study of this model. Our approach is computationally more efficient than any used previously for the calculation of the high temperature expansion of dynamic correlation functions, and high order results for both the Green's function and the time-dependent density-density and spin-spin correlation functions shall be presented in a separate paper [25].
A perturbation theory scheme in terms of electron hopping, which is based on the Wick theorem for Hubbard operators, is developed. Diagrammatic series contain single-site vertices connected by hopping lines and it is shown that for each vertex the problem splits into the subspaces with ``vacuum states'' determined by the diagonal Hubbard operators and only excitations around these vacuum states are allowed. The rules to construct diagrams are proposed. In the limit of infinite spatial dimensions the total auxiliary single-site problem exactly splits into subspaces that allows to build an analytical thermodynamically consistent approach for a Hubbard model. Some analytical results are given for the simple approximations when the two-pole (alloy-analogy solution) and four-pole (Hartree-Fock approximation) structure for Green's function is obtained. Two poles describe contribution from the Fermi-liquid component, which is dominant for small electron and hole concentrations (``overdoped case'' of high-Tc′s),{T}_{c}'\mathrm{s}),Tc′​s), whereas other two describe contribution from the non-Fermi liquid and are dominant close to half-filling (``underdoped case'').
We implement a highly efficient strong-coupling expansion for the Green's function of the Hubbard model. In the limit of extreme correlations, where the onsite interaction is infinite, the evaluation of diagrams simplifies dramatically enabling us to carry out the expansion to the eighth order in powers of the hopping amplitude. We compute the finite-temperature Green's function analytically in the momentum and Matsubara frequency space as a function of the electron density. Employing Padé approximations, we study the equation of state, Kelvin thermopower, momentum distribution function, quasiparticle fraction, and quasiparticle lifetime of the system at temperatures lower than, or of the order of, the hopping amplitude. We also discuss several different approaches for obtaining the spectral functions through analytic continuation of the imaginary frequency Green's function, and show results for the system near half filling. We benchmark our results for the equation of state against those obtained from a numerical linked-cluster expansion carried out to the eleventh order.
We discuss the infinite dimension limit of the Hubbard model by means of a perturbative expansion of the one-particle Green's function around the atomic limit. The diagrammatic structure is simplified in this limit, allowing a formal resummation that reproduces a previously proposed mapping to a single-site mean-field problem. The method provides a natural way of addressing this effective problem by means of a perturbative expansion in the local mean field. This gives the correct exact result for the Falicov-Kimball model, which is used as a starting point to study the Hubbard case. \textcopyright{} 1996 The American Physical Society.