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physica status solidi (b)

On the dynamics of electrons in the strong‐coupling Hubbard chain

Rainer J. Jelitto, M. Heise

Published Feb 1, 1978• 0 citations
Publisher

Abstract

Abstract The dynamics of electrons is investigated in a Hubbard chain in the strong‐coupling limit. This problem may be figured as the motion of a double‐occupancy (consisting of one spin‐up and one spin‐down electron fixed at the same lattice site) on a linear chain, all other sites of which are occupied by single electrons of arbitrary spin orientation. The spectrum of this problem is derived exactly and from it the partition function is calculated. It agrees with the result derived by Beni and coworkers formerly with the aid of thermodynamic perturbation theory.

DOI
10.1002/pssb.2220850216

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Abstract The dynamics of electrons is investigated in a Hubbard chain in the strong‐coupling limit. This problem may be figured as the motion of a double‐occupancy (consisting of one spin‐up and one spin‐down electron fixed at the same lattice site) on a linear chain, all other sites of which are occupied by single electrons of arbitrary spin orientation. The spectrum of this problem is derived exactly and from it the partition function is calculated. It agrees with the result derived by Beni and coworkers formerly with the aid of thermodynamic perturbation theory.
Abstract We expose the relevance of double occupancy conservation symmetry in application of the Hubbard‐I approach to strongly correlated electron systems. We propose the utility of a composite method, viz. the Hubbard‐I method in conjunction with strong coupling perturbation expansion, for studying systems violating the afore‐mentioned symmetry. We support this novel approach by presenting a first successful Hubbard‐I type calculation for the description of the metal‐insulator paramagnetic Mott transition in a strongly correlated electron system with conserved double occupancies, which is a constrained Hubbard Hamiltonian equivalent to the Hubbard bond charge Hamiltonian at a symmetry point X = t. In particular, we obtain the phase diagram of this system for arbitrary fillings, as well as new, universal critical indices of the Mott transition at half‐filling in 1,2, and 3 dimensions. We also compare the Hubbard‐I band–splitting Mott transition description with results obtained using the standard Gutzwiller Approximation (GA), and show that the two approximate approaches lead to qualitatively different results. In contrast to the GA applied to the system studied here, the Hubbard‐I approach compares favorably with known exact results for the d = 1 dimensional chain.
Abstract By use of the conservation laws a four‐site Hubbard model coupled to a particle bath within an external magnetic field in z ‐direction was diagonalized. The analytical dependence of both the eigenvalues and the eigenstates on the interaction strength, the chemical potential and magnetic field was calculated. It is demonstrated that the low temperature behaviour is determined by a delicate interplay between many‐particle states differing in electron number and spin if the electron density is away from half‐filling. The grand partition sum is calculated and the specific heat, the susceptibility as well as various correlation functions and spectral functions are given in dependence of the interaction strength, the electron occupation and the applied magnetic field. For both the grand canonical and the canonical ensemble the high‐temperature crossing points of the specific heat are calculated. Whereas in the weak correlation regime the universal value calculated by second order perturbation theory for several Hubbard systems being in the thermodynamic limit is confirmed, these crossing points vanish for intermediate to strong correlation.
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 reconsider the strong-coupling expansion for the Hubbard model recently introduced by Sarker and Pairault et al. By introducing slave particles that act as projection operators onto the empty, singly occupied and doubly occupied atomic states, the perturbation theory around the atomic limit distinguishes between processes that do conserve or do not conserve the total number of doubly occupied sites. This allows for a systematic t/U expansion that does not break down at low temperature (t being the intersite hopping amplitude and U the local Coulomb repulsion). The fermionic field becomes a two-component field, which reflects the presence of the two Hubbard bands. The single-particle propagator is naturally expressed as a function of a 2×2 matrix self-energy. Furthermore, by introducing a time- and space-fluctuating spin-quantization axis in the functional integral, we can expand around a "non-degenerate" ground-state where each singly occupied site has a well defined spin direction (which may fluctuate in time). This formalism is used to derive the effective action of charge carriers in the lower Hubbard band to first order in t/U. We recover the action of the t–J model in the spin-hole coherent-state path integral. We also compare our results with those previously obtained by studying fluctuations around the large-U Hartree–Fock saddle point.
Exact relations are derived for the Fermi Hubbard spectral weight function for infinite U at zero temperature in the thermodynamic limit for any dimension,any lattice structure and general hopping matrix. These relations involve moments of the spectral weight function but differ from similar work by (a) restricting the moments over the interesting low energy (lower Hubbard band) spectrum and (b) without any of the usual approximations (e.g. decoupling) for the requisite higher order correlation functions. The latter are determined exactly as explicit functions of the single particle correlation function in the infinite U limit. The methods behind these results have the additional virtue of simplicity - cumbersome, but entirely straightforward. In a companion paper, the relations offer rich insights into the model behavior and offer severe tests - a computer lab - of any proposed solution to the Hubbard model in the strong correlation (large finite U) region. As the electron density approaches one electron per site, then the correlation functions become local, so are trivial to compute exactly. In that limit and for the paramagnetic case, the second central moment of the spectral weight function is not zero, but quite large.