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International Journal of Modern Physics B

INTEGRABLE EXTENDED HUBBARD HAMILTONIANS FROM SYMMETRIC GROUP EQUATIONS

Fabrizio Dolcini, Arianna Montorsi

Published Jul 10, 2000• 6 citations
Publisher

Abstract

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.

DOI
10.1142/s0217979200001540

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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.
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 representation of a new class of solutions of symmetric group relations, acting as generalized permutators. As particular examples, the EKS and some other known models are obtained.
We propose a constructive method to prove the integrability of a given physical Hamiltonian in one dimension, which amounts to looking for appropriate polynomial R-matrices, solutions of GYBE, whose first coefficient in the power expansion with respect to the spectral parameter is the Hamiltonian itself. The method is applied to the extended Hubbard Hamiltonian, in particular to the cases in which it exhibits so(4) or gl(2,1) symmetries. We show that in the latter case the R-matrices are at most polynomial of second degree, whose coefficients are nothing but the Hamiltonian, the identity and the permutation operator. In this way, all known completely integrable cases are recovered. Also, the method allows to recognize that the possible integrability of the most general gl(2,1) invariant Hamiltonian depends on the existence of a non-additive R-matrix.
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.
The generalized Yang-Baxter equations are multi-site versions of the standard Yang-Baxter equation. When spectral parameters are included, such equations are expected to lead to integrable Hamiltonians with local interactions involving multiple degrees of freedom. In this work we characterize both the continuous and discrete symmetries of these equations required to establish an equivalence class of solutions. We find that the set of such symmetries depends on the number of sites on which the equation is supported. In several cases there are more symmetries than the standard Yang-Baxter equation, thus placing heavy constraints on the number of inequivalent solutions and the associated integrable models. As an application we show how to restrict the generic 16×1616\times1616×16 ansatz to the 30-dimensional subspace invariant under a chosen discrete symmetry subgroup. An explicit 16×1616\times 1616×16 solution is also written down. 21 pages + References ; v2 31 pages, Includes several new symmetries, with an exhaustive account of permutation symmetries, and a new section on application of these methods, Published in "A Special Issue in Honour of Jarmo Hietarinta''
Within the framework of quantum groups and additive R -matrices, the fusion procedure allows one to construct higher-dimensional solutions of the Yang-Baxter equation. These solutions lead to integrable one-dimensional spin-chain Hamiltonians. Here fusion is shown to generalize naturally to non-additive R -matrices, which therefore do not have a quantum group symmetry. This method is then applied to the generalized Hubbard models. Although the resulting integrable models are not as simple as the starting ones, the general structure is that of two spin-( s × s ´) sl (2) models coupled at the free-fermion point. An important issue is the probable lack of regular points which give local Hamiltonians. This problem is related to the existence of second-order zeros in the unitarity equation, and arises for the XX models of higher spins, the building blocks of the Hubbard models. A possible connection between some Lax operators L and R -matrices is noted.