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arXiv:0802.0777 · Journal of Physics A Mathematical and Theoretical

Quantum deformations of the one-dimensional Hubbard model

Niklas Beisert, Peter Koroteev

Published May 28, 2008• 137 citations
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Abstract

The centrally extended superalgebra was shown to play an important role for the integrable structures of the one-dimensional Hubbard model and of the planar AdS/CFT correspondence. Here we consider its quantum deformation and derive the fundamental R -matrix. From the latter we deduce an integrable spin-chain Hamiltonian with three independent parameters and the corresponding Bethe equations to describe the spectrum on periodic chains. We relate our Hamiltonian to a two-parametric Hamiltonian proposed by Alcaraz and Bariev which can be considered a quantum deformation of the one-dimensional Hubbard model.

DOI
10.1088/1751-8113/41/25/255204

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The centrally extended superalgebra was shown to play an important role for the integrable structures of the one-dimensional Hubbard model and of the planar AdS/CFT correspondence. Here we consider its quantum deformation and derive the fundamental R -matrix. From the latter we deduce an integrable spin-chain Hamiltonian with three independent parameters and the corresponding Bethe equations to describe the spectrum on periodic chains. We relate our Hamiltonian to a two-parametric Hamiltonian proposed by Alcaraz and Bariev which can be considered a quantum deformation of the one-dimensional Hubbard model.
The integrable structure of the one-dimensional Hubbard model is based on Shastry's R -matrix and the Yangian of a centrally extended superalgebra. Alcaraz and Bariev have shown that the model admits an integrable deformation whose R -matrix has recently been found. This R -matrix is of trigonometric type and here we derive its underlying exceptional quantum affine algebra. We also show how the algebra reduces to the above-mentioned Yangian and to the conventional quantum affine algebra in two special limits.
We investigate the integrable structure of spin chain models with centrally extended and symmetry. These chains have their origin in the planar anti-de Sitter/conformal field theory correspondence, but they also contain the one-dimensional Hubbard model as a special case. We begin with an overview of the representation theory of centrally extended . These results are applied in the construction and investigation of an interesting S -matrix with symmetry. In particular, they enable a remarkably simple proof of the Yang–Baxter relation. We also show the equivalence of the S -matrix to Shastry's R -matrix and thus uncover a hidden supersymmetry in the integrable structure of the Hubbard model. We then construct eigenvalues of the corresponding transfer matrix in order to formulate an analytic Bethe ansatz. Finally, the form of transfer matrix eigenvalues for models with symmetry is sketched.
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.
The spectral properties and phase diagram of the exact integrable spin one quantum chain introduced by Alcaraz and Bariev are presented. The model has a U(1) symmetry and its integrability is associated to an unknown R-matrix whose dependence on the spectral parameters is not of difference form. The associated Bethe ansatz equations, that fix the eigenspectra, are distinct from those associated to other known integrable spin models. The model has a free parameter tpt_ptp​. We show that at the special point tp=1t_p=1tp​=1 the model gets an extra U(1) symmetry and reduces to the deformed SU(3) Perk-Schultz model at a special value of its anisotropy q=exp⁡(i2π/3)q=\exp(i2π/3)q=exp(i2π/3) andin the presence of an external magnetic field. Our analysis is done either by solving the associated Bethe-ansatz equations or by direct diagonalization of thequantum Hamiltonian for small lattice sizes. The phase diagram is calculated by exploring the consequences of conformal invariance on the finite-size corrections of the Hamiltonian eigenspectrum. The model exhibits a critical phase ruled by c=1c=1c=1 conformal field theory separated from a massive phase by first-order phase transitions.
A new integrable model which is a variant of the one-dimensional Hubbard model is proposed. The integrability of the model is verified by presenting the associated quantum R-matrix which satisfies the Yang–Baxter equation. We argue that the new model possesses the SO(4) algebra symmetry, which contains a representation of the η-pairing SU(2) algebra and a spin SU(2) algebra. Additionally, the algebraic Bethe ansatz is studied by means of the quantum inverse scattering method. The spectrum of the Hamiltonian, eigenvectors, as well as the Bethe ansatz equations, are discussed.