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Physical review. B, Condensed matter

Symmetry breaking in the vertex-renormalized linked-cluster expansion for strongly correlated fermions

M. Bartkowiak, K. A. Chao

Published Jan 15, 1993• 4 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.47.1616

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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.
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 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.
A slave-spin representation of fermion operators has recently been proposed for the half-filled single and multiband Hubbard model. We show that with the addition of a gauge variable, the formalism can be extended to finite doping. We solve the resulting spin problem using the cluster mean-field approximation. This approximation takes short-range correlations into account by exact diagonalization on the cluster, whereas long-range correlations beyond the size of clusters are treated at the mean-field level. In the limit where the cluster has only one site and the interaction strength UUU is infinite, this approach reduces to the Gutzwiller approximation. There are some qualitative differences when the size of the cluster is finite. We first compute the critical UUU for the Mott transition as a function of a frustrating nearest-neighbor interaction on lattices relevant for various correlated systems, namely, the cobaltates, the layered organic superconductors and the high-temperature superconductors. For the triangular lattice, we also study the extended Hubbard model with nearest-neighbor repulsion. In addition to a uniform metallic state, we find a $\sqrt{(3)}\ifmmode\times\else\texttimes\fi{}\sqrt{(3)}$ charge density wave in a broad doping regime, including commensurate ones. We find that in the large UUU limit, intersite Coulomb repulsion VVV strongly suppresses the single-particle weight of the metallic state.
We employ the numerical linked-cluster expansion to study finite-temperature properties of the uniform cubic lattice Hubbard model in the thermodynamic limit for a wide range of interaction strengths and densities. We carry out the expansion to the 9th order and find that the convergence of the series extends to lower temperatures as the strength of the interaction increases, giving us access to regions of the parameter space that are difficult to reach by most other numerical methods. We study the precise trends in the specific heat, the double occupancy, and magnetic correlations at temperatures as low as 0.2 of the hopping amplitude in the strong-coupling regime. We show that in this regime, accurate estimates for transition temperatures to the N\'eel ordered phase, in agreement with the predicted asymptotic behavior, can be deduced from the low-temperature magnetic structure factor. We also find evidence for possible instability to the magnetically ordered phase away from, but close to, half filling. Our results have important implications for parametrizing fermionic systems in optical lattice experiments and for benchmarking other numerical methods.