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arXiv:cond-mat/9702128 · Modern Physics Letters A

A NEW FAMILY OF INTEGRABLE EXTENDED MULTI-BAND HUBBARD HAMILTONIANS

Jan Ambjørn A, A. Sedrakyan, A. Avakyan, Tigran Hakobyan

Published Mar 7, 1998• 4 citations
arXivHTMLPDF

Abstract

We consider exactly solvable 1D multi-band fermionic Hamiltonians, which have affine quantum group symmetry for all values of the deformation. The simplest Hamiltonian is a multi-band t-J model with vanishing spin–spin interaction, which is the affinization of an underlying XXZ model. We also find a multi-band generalization of standard t-J model Hamiltonian.

DOI
10.1142/s0217732398000553

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We consider exactly solvable 1D multi-band fermionic Hamiltonians, which have affine quantum group symmetry for all values of the deformation. The simplest Hamiltonian is a multi-band t-J model with vanishing spin–spin interaction, which is the affinization of an underlying XXZ model. We also find a multi-band generalization of standard t-J model Hamiltonian.
We construct the family of spin-chain Hamiltonians, which have affine quantum group symmetry Uqĝ. Their eigenvalues coincide with the eigenvalues of the usual spin-chain Hamiltonians, but have the degeneracy of levels, corresponding to affine Uqĝ. The space of states of these spin-chains is formed by the tensor product of fully reducible representations. The fermionic representations of spin-chain Hamiltonians, which have affine quantum group symmerty, was consructed. They correspond to new extensions of Hubbard Hamiltonians. The exact ground state of some examples is presented, exhibiting superconducting behavior via η-pairing mechanism.
We consider the most general form of extended Hubbard Hamiltonian conserving the total spin and number of electrons, and find all the 1-dimensional completely integrable models which can be derived from first degree polynomial solution of the Yang–Baxter equation. It is shown that such models are 96. They are identified with the 16-dimensional representations of the class of solutions of symmetric group relations acting as generalized permutators. As particular examples, the EKS and some other known models are obtained. A method for determining the physical features of the above models is outlined.
Abstract In this paper we find new integrable one-dimensional lattice models of electrons. We describe all such nearest-neighbour integrable models with s u ( 2 ) × s u ( 2 ) symmetry by classifying solutions of the Yang–Baxter equation following the procedure first introduced in [1]. We find 12 R-matrices of difference form, some of which can be related to known models such as the XXX spin chain and the free Hubbard model, and some are new models. In addition, integrable generalizations of the Hubbard model are found by keeping the kinetic term of the Hamiltonian and adding all terms which preserve fermion number. We find that most of the new models cannot be diagonalized using the standard nested Bethe Ansatz.
In this article we review recent developments in the one-dimensional Bethe ansatz solvable multi-chain quantum models. The algebraic version of the Bethe ansatz (the quantum inverse scattering method) permits us to construct new families of integrable Hamiltonians using simple generalizations of the well known constructions of the single-chain model. First we consider the easiest example (`basic' model) of this class of models: the antiferromagnetic two-chain spin-1/2 model with the nearest-neighbour and next-nearest-neighbour spin-frustrating interactions (zigzag chain). We show how the algebra of the quantum inverse scattering method works for this model, and what are the important features of the Hamiltonian (which reveal the topological properties of two dimensions together with the one-dimensional properties). We consider the solution of the Bethe ansatz for the ground state (in particular, commensurate-incommensurate quantum phase transitions present due to competing spin-frustrating interactions are discussed) and construct the thermal Bethe ansatz (in the form of the `quantum transfer matrix') for this model. Then possible generalizations of the basic model are considered: an inclusion of a magnetic anisotropy, higher-spin representations (including the important case of a quantum ferrimagnet), the multi-chain case, internal degrees of freedom of particles at each site, etc. We observe the similarities and differences between this class of models and related exactly solvable models: other groups of multi-chain lattice models, quantum field theory models and magnetic impurity (Kondo-like) models. Finally, the behaviour of non-integrable (less constrained) multi-chain quantum models is discussed.
I discuss many-body models for correlated fermions in two space dimensions which can be solved exactly using group theory. The simplest example is a model of a quantum Hall system: two-dimensional (2D) fermions in a constant magnetic field and a particular non-local four-point interaction. It is exactly solvable due to a dynamical symmetry corresponding to the Lie algebra gl ∞ ⊕ gl ∞ . There is an algorithm to construct all energy eigenvalues and eigenfunctions of this model. The latter are, in general, many-body states with spatial correlations. The model also has a non-trivial zero temperature phase diagram. I point out that this QH model can be obtained from a more realistic one using a truncation procedure generalizing a similar one leading to mean field theory. Applying this truncation procedure to other 2D fermion models I obtain various simplified models of increasing complexity which generalize mean field theory by taking into account non-trivial correlations but nevertheless are treatable by exact methods.
We propose a new, alternative method for ab-initio calculations of the electronic structure of solids, which has been specifically adapted to treat many-body effects in a more rigorous way than many existing ab-initio methods. We start from a standard band-structure calculation for an effective one-particle Hamiltonian approximately describing the material of interest. This yields a suitable set of one-particle basis functions, from which well localized Wannier functions can be constructed using a method proposed by Marzari and Vanderbilt. Within this (minimal) basis of localized Wannier functions the matrix elements of the non-interacting (one-particle) Hamiltonian as well as the Coulomb matrix elements can be calculated. The result is a many-body Hamiltonian in second quantization with parameters determined from first principles calculations for the material of interest. The Hamiltonian is in the form of a multi-band Hamiltonian in second quantization (a kind of extended, multi-band Hubbard model) such that all the standard many-body methods can be applied. We explicitly show how this approach can be solved in the simplest many-body approximation, the mean-field Hartree-Fock approximation (HFA), which takes into account exact exchange and corrects for self-interaction effects.