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Physical review. B, Solid state

Hubbard Model. I. Degeneracy in the Atomic Light

D. M. Esterling, Robert Lange

Published Mar 1, 1970• 12 citations
Publisher

Abstract

The atomic limit of the Hubbard Hamiltonian is extremely degenerate. The ground-state degeneracy, in particular, makes the zero-temperature Green's function for this limit ambiguous. The manifestations of the degeneracy in the one- and two-particle Green's functions are found. The degeneracy gives rise to nonlocal spatial correlation of physical significance which must be taken into account in narrow-band-regime perturbation schemes for the model.

DOI
10.1103/physrevb.1.2231

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The atomic limit of the Hubbard Hamiltonian is extremely degenerate. The ground-state degeneracy, in particular, makes the zero-temperature Green's function for this limit ambiguous. The manifestations of the degeneracy in the one- and two-particle Green's functions are found. The degeneracy gives rise to nonlocal spatial correlation of physical significance which must be taken into account in narrow-band-regime perturbation schemes for the model.
The "atomic limit" (kinetic energy zero) of the Hubbard Hamiltonian has extreme degeneracy under very general circumstances. The kinetic energy or hopping term removes this degeneracy. It has been shown by Harris and Lange that there are important first-order manifestations of this fact. For instance, nonlocal correlations exist which invalidate usual Green's function truncation schemes. We present a mass operator perturbation scheme devised on the principle that, first of all, the role of kinetic energy as a breaker of degeneracy must be accounted for. A sort of Taylor's expansion of the mass operator in terms of the kinetic energy is found, the coefficients of which are functionals of the exact Green's functions at zero kinetic energy. The ambiguity due to the degeneracy is removed from these functions self consistently and thereby the degeneracy properly treated. We present a Green's function which accounts for all first-order effects of the kinetic energy.
A solution for the Hubbard Hamiltonian in the half-filled-band case is presented, which takes into account the degeneracy of the ground state in the atomic limit and also correctly interpolates to the limit Coulomb energy U to 0. It is shown explicitly which approximations lead to Hubbard's improved decoupling scheme and other existing approximate solutions. Taking into account the antiferromagnetic character of the ground state the splitting of the two sub-bands in the case of large U is enhanced. The band gap is shown to decrease with increasing temperatures.
The asymptotic expression for the one-particle Green's function in the strong coupling limit of the double degenerate Hubbard model containing multiplet terms is obtained for arbitrary band filling within the irreducible Green's function approach. An explicit analytic expression is obtained for the mean-field Greens's function, conserving the sum rules for the first few spectral moments. Stability of homogeneous phases is analised in terms of a single flip spin and orbital excitations. It is shown that degeneracy has an additional enhancing effect on correlation driven interband transfer of spectral weight. Discovery of superconductivity in fullerides and of materials with colossal magnetoresistance during recent years have increased the interest for electron systems with physical properties being essentially dependent on orbital degeneracy and strong correlation effects. In many cases the orbitally degenerate Hubbard model (DHM) incorporates the key features of such materials [1], although there are even fewer exact results established for DHM than for the nondegenerate Hubbard model (NHM). In view of complexity of correlated systems, corrections to the atomic limit of one-particle excitation spectrum and spectral weights of correlated bands found by Harris and Lange for NHM [2] by means of perturbation theory based on canonical transformations provide an important exact result which serves as a test for different approximations. A similar derivation for DHM has not been carried out yet. In the present paper the Green's function for the strong coupling limit of the DHM is obtained by means of irreducible Green's function approach [3] - [6] which is closely related to canonical perturbation theory because of its spectral moment conserving construction. As was explicitly proved in [7] for the NHM, the first couples of spectral moments of correlated bands obtained within mean-field decoupling are identical to those of canonical perturbation theory to first order in / tU ( t is the n.n. hopping amplitude and U the Coulomb repulsion).
Unusual properties of the Hubbard chain of arbitrary degeneracy N and excluded-site occupations of more than two electrons are presented: (i) In zero field and in the limit of large N the ground state resembles an interacting Bose gas. The 1/N contributions vanish identically, so that corrections to leading order are 1/N2{\mathrm{N}}^{2}N2. (ii) The susceptibility at T=0 shows logarithmic singularities as H\ensuremath{\rightarrow}0. (iii) The Hubbard model and the Heisenberg chain have the same low-energy spin excitations for all U. (iv) The specific-heat coefficient \ensuremath{\gamma} is singular. These properties hold for arbitrary band filling and (ii)--(iv) hold for all N\ensuremath{\ge}2.