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Bethe equations for generalized Hubbard models

Victor Fomin, L. Frappat, E. Ragoucy

Published Jan 1, 2009• 5 citations
OpenAlex

Abstract

We compute the eigenfunctions, energies and Bethe equations for a class of generalized integrable Hubbard models based on gl(n|m)\\oplus gl(2) superalgebras. The Bethe equations appear to be similar to the Hubbard model ones, up to a phase due to the integration of a subset of `simple' Bethe equations. We discuss relations with AdS/CFT correspondence, and with condensed matter physics.

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We compute the eigenfunctions, energies and Bethe equations for a class of generalized integrable Hubbard models based on gl(n|m)\oplus gl(2) superalgebras. The Bethe equations appear to be similar to the Hubbard model ones, up to a phase due to the integration of a subset of `simple' Bethe equations. We discuss relations with AdS/CFT correspondence, and with condensed matter physics.
This thesis is devoted to the one-dimensional integrable Hubbard model and its generalizations. The Hubbard model is one of the fundamental models in condensed matter physics which describes interacting electrons on the lattice. It has very rich physical structure. Inspite of its construction simplicity it has been applied to diverse problems as high-Tc superconductivity, band magnetism and the metal-insulator transition. In one dimension the Hubbard model is an integrable model which has been intensively studied and served as a theoretical laboratory for the condensed matter physics. Recently, the integrable systems and in particular the Hubbard model, have surprisingly appeared in the AdS/CFT correspondence context. The intersection point between the fields is the Bethe equations: the ones of new integrable models and the generalizations of existing ones are relevant in the application in AdS/CFT duality. In the first part of the thesis, we present basis notions of the quantum integrability: the R-matrix formalism, the Yang-Baxter equation, integrable spin chains etc. In the second part we review several fundamental results of the one-dimensional Hubbard model: the Coordinate Bethe Ansatz solution, real solutions of the Lieb-Wu equations etc. Moreover, applications in the AdS/CFT duality are considered. However, it turns out that certain modifications of the integrable Hubbard model are necessary to reproduce the correct results of the AdS/CFT context. This is one of the main motivations of the studies of generalized Hubbard models. The fourth chapter is devoted to generalizations of the Hubbard model and we focus our attention on supersymmetric ones. The fifth chapter con- tains the results obtained in the framework of this thesis on the supersymmetric generalizations of the Hubbard models. Namely, the Coordinate Bethe Ansatz solution and real solutions of the Bethe equations in the thermodynamic limit are exposed. We point out that the obtained Bethe equations differs from the Lieb-Wu ones by phases which appearance is encouraging sign for the application in the AdS/CFT context. We also discuss possible applications in the AdS/CFT duality and in con- densed matter physics.
We construct the XX and Hubbard-like models based on unitary superalgebras gl(N|M) generalizing Shastry's and Maassarani's approach. We introduce the R-matrix of the gl(N|M) XX-type model; the one of the Hubbard-like model is defined by "coupling" two independent XX models. In both cases, we show that the R-matrices satisfy the Yang-Baxter equation. We derive the corresponding local Hamiltonian in the transfer matrix formalism and we determine its symmetries. A perturbative calculation "à la Klein and Seitz" is performed. Some explicit examples are worked out. We give a description of the two-particle scattering.
We solve the gl(1|2) generalized model by means of the algebraic Bethe ansatz. The resulting eigenvalue of the transfer matrix and the Bethe ansatz equations depend on three complex functions, called the parameters of the generalized model. Specifying the parameters appropriately, we obtain the Bethe ansatz equations of the supersymmetric t-J model, the Hubbard model, or of Yang’s model of electrons with delta interaction. This means that the Bethe ansatz equations of these (and many other) models can be obtained from a common algebraic source, namely from the Yang-Baxter algebra generated by the gl(1|2) invariant R-matrix.
We present in a unified and detailed way the nested Bethe ansatz for closed spin chains based on or (super)algebras, with arbitrary representations (i.e. 'spins') on each site of the chain. In particular, the case of indecomposable representations of superalgebras is studied. The construction extends and unifies the results already obtained for spin chains based on or and for some particular super-spin chains. We give the Bethe equations and the form of the Bethe vectors. The case of gl (2|1), gl (2|2) and gl (4|4) superalgebras (that are related to AdS/CFT correspondence) is also detailed.