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arXiv:hep-th/9806019 · Physics Letters B

Integrability of supersymmetric quantum matrix models in the large-N limit

C.-W.H. Lee, S. G. Rajeev

Published Sep 1, 1998• 4 citations
arXivHTMLPDF

Abstract

Many physical systems like supersymmetric Yang-Mills theories are formulated as quantum matrix models. We discuss how to apply the Beth ansatz to exactly solve some supersymmetric quantum matrix models in the large-N limit. Toy models are constructed out of the one-dimensional Hubbard and t-J models as illustrations.
DOI
10.1016/s0370-2693(98)00834-x

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Abstract We give a pedagogical introduction to the Bethe ansatz techniques in integrable QFTs and spin chains. We first discuss and motivate the general framework of asymptotic Bethe ansatz for the spectrum of integrable QFTs in large volume, based on the exact S -matrix. Then we illustrate this method in several concrete theories. The first case we study is the SU (2) chiral Gross–Neveu model. We derive the Bethe equations via algebraic Bethe ansatz, solving in the process the Heisenberg XXX spin chain. We discuss this famous spin chain model in some detail, covering in particular the coordinate Bethe ansatz, some properties of Bethe states, and the classical scaling limit leading to finite-gap equations. Then we proceed to the more involved SU (3) chiral Gross–Neveu model and derive the Bethe equations using nested algebraic Bethe ansatz to solve the arising SU (3) spin chain. Finally we show how a method similar to the Bethe ansatz works in a completely different setting, namely for the 1D oscillator in quantum mechanics.
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.