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Physical review. B, Condensed matter

Renormalized linked-cluster expansion for strongly correlated lattice fermions ind=∞ dimensions

M. Bartkowiak, K. A. Chao

Published Feb 15, 1993• 8 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.47.4193

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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.
We have shown that mean-field-type symmetry-breaking approximations for correlated fermions can be constructed by performing vertex renormalization (VR) in the linked-cluster expansion (LCE), just as for the case of lattice spin models. Using the diagrammatic technique for Hubbard operators, we have constructed explicitly a partial VR in the second order of the LCE for the half-filled Hubbard model around the atomic limit. We obtain a crossover from large-U Heisenberg mean-field behavior to zero Ne\ifmmode\acute\else\textasciiacute\fi{}el temperature at U=0.
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 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 consider 1/d1/d1/d expansions for classical spin systems based on the vertex renormalized linked cluster expansion (LCE). The free multiplicities of the LCE graphs on a hypercubic lattice in an arbitrary dimension d are calculated using generating functions. The technique is applied to the Ising model and to a two-component classical lattice gas corresponding to an extended Hubbard model at half filling in the zero-bandwidth limit. We use a resummation of the LCE to generate 1/d1/d1/d expansions for the equation of state and for the critical temperature. The method, which is rather general and applicable to a wide range of models, proves convenient for calculating asymptotic power series expansions in 1/d.1/d.1/d. The vertex renormalized expansion is shown to break down at low temperature in higher order approximations, barring attempts to construct simple approximations that are both self-consistent and exact to some finite order in 1/d.1/d.1/d.
We show that even for fermionic lattice models with on-site interaction a mean-field Hamiltonian can be constructed, which--in analogy with spin-lattice models--becomes exact in the limit of high spatial dimensions d. Here the mean fields are fermion operators, rather than numbers. We use this method to obtain the exact solution of a simplified Hubbard model on a Bethe lattice for all temperatures. The order parameter is found to have the conventional mean-field form, but exhibits unusually rich behavior as a function of the interaction.
A general expansion scheme based on the concept of linked-cluster expansion from the theory of classical spin systems is constructed for models of interacting electrons. It is shown that with a suitable variational formulation of mean-field theories at weak (Hartree-Fock) and strong (Hubbard-III) coupling the expansion represents a universal and comprehensive tool for systematic improvements of static mean-field theories. As an example of the general formalism we investigate in detail an analytically tractable series of ring diagrams that correctly capture dynamical fluctuations at weak coupling. We introduce renormalizations of the diagrammatic expansion at various levels and show how the resultant theories are related to other approximations of similar origin. We demonstrate that only fully self-consistent approximations produce global and thermodynamically consistent extensions of static mean-field theories. A fully self-consistent theory for the ring diagrams is reached by summing the so-called noncrossing diagrams.