We discuss the one-dimensional Hubbard model for a finite chain of atoms, that is within the single band approximation, in the context of the transfer matrix. We show two ways of constructing the finite Hilbert space, one with the basis of electron configurations, and the second created as the antisymmetric part of the orbital and spin function tensor product. We classify the set of all electron configurations according to the action of the symmetric group. We present the Bethe Ansatz (BA) solution for the example of three electrons moving on a ring with N = 6 nodes. After that, we concentrate on giving the motivation for exploring the properties of the transfer matrix. We discuss the scattering and the winding operators in the tensor product space, resulting from the physical and the auxiliary spaces. We consider the transfer matrix as the partial trace over the auxiliary space from the winding operator for the auxiliary particle. After that, we derive the explicit expressions for some elements of the transfer matrix in the basis of the spin electron configurations.
We discuss the one-dimensional Hubbard model for a finite chain of atoms, that is within the single band approximation, in the context of the transfer matrix. We show two ways of constructing the finite Hilbert space, one with the basis of electron configurations, and the second created as the antisymmetric part of the orbital and spin function tensor product. We classify the set of all electron configurations according to the action of the symmetric group. We present the Bethe Ansatz (BA) solution for the example of three electrons moving on a ring with N = 6 nodes. After that, we concentrate on giving the motivation for exploring the properties of the transfer matrix. We discuss the scattering and the winding operators in the tensor product space, resulting from the physical and the auxiliary spaces. We consider the transfer matrix as the partial trace over the auxiliary space from the winding operator for the auxiliary particle. After that, we derive the explicit expressions for some elements of the transfer matrix in the basis of the spin electron configurations.
The algebraic structure of the 1D Hubbard model is studied by means of the fermionic R-operator approach. This approach treats the fermion models directly in the framework of the quantum inverse scattering method. Compared with the graded approach, this approach has several advantages. First, the global properties of the Hamiltonian are naturally reflected in the algebraic properties of the fermionic R-operator. We want to note that this operator is a local operator acting on fermion Fock spaces. In particular, SO(4) symmetry and the invariance under the partial particle hole transformation are discussed. Second, we can construct a genuinely fermionic quantum transfer transfer matrix (QTM) in terms of the fermionic R-operator. Using the algebraic Bethe Ansatz for the Hubbard model, we diagonalize the fermionic QTM and discuss its properties.
The ground state of the one-dimensional Hubbard Hamiltonian is discussed on the basis of the Bethe ansatz solution of Lieb and Wu. A simple analytical representation of the distribution functions for charge and spin degrees of freedom is derived. The elementary excitations are associated with electron-hole pairs of two pseudo-fermion systems, one for the charge, the other for the magnetic excitations. Spinons and holons correspond to ground state configurations of odd-numbered rings. It is argued that holons will form bound pairs in weakly coupled chains.
We derive a matrix product representation of the Bethe ansatz state for the XXX and XXZ spin-1 2 Heisenberg chains using the algebraic Bethe ansatz.In this representation, the components of the Bethe eigenstates are expressed as traces of products of matrices which act on H, the tensor product of auxiliary spaces.By changing the basis in H, we derive explicit finite-dimensional representations for the matrices.These matrices are the same as those appearing in the recently proposed matrix product ansatz by Alcaraz and Lazo (2006 J. Phys.A: Math.Gen. 39 11335) apart from normalization factors.We also discuss the close relation between the matrix product representation of the Bethe eigenstates and the six-vertex model with domain wall boundary conditions (Korepin 1982 Commun.Math.Phys.86 391) and show that the change of basis corresponds to a mapping from the six-vertex model to the five-vertex model.
We examine the nature, number, and interrelation of conservation laws in the one-dimensional Hubbard model. In previous work by Shastry [Phys. Rev. Lett. 56, 1529 (1986); 56, 2334 (1986); 56, 2453 (1986); J. Stat. Phys. 50, 57 (1988)], who studied the model on a large but finite number of lattice sites (Na), only Na+1 conservation laws, corresponding to Na+1 operators that commute with themselves and the Hamiltonian, were explicitly identified, rather than the $\ensuremath{\sim}{2N}_{a}$ conservation laws expected from the solvability and integrability of the model. Using a pseudoparticle approach related to the thermodynamic Bethe ansatz, we discover an additional Na+1 independent conservation laws corresponding to nonlocal, mututally commuting operators, which we call transfer-matrix currents. Further, for the model defined in the whole Hilbert space, we find there are two other independent commuting operators (the squares of the \ensuremath{\eta}-spin and spin operators) so that the total number of local plus nonlocal commuting conservation laws for the one-dimensional Hubbard model is 2Na+4. Finally, we introduce an alternative set of 2Na+4 conservation laws which assume particularly simple forms in terms of the pseudoparticle and Yang-particle operators. This set of mutually commuting operators lends itself more readily to calculations of physically relevant correlation functions at finite energy or frequency than the previous set.
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.
One-dimensional conductors are a long-standing topic of research with direct applications to organic conductors and mesoscopic rings. The discovery of the ceramic high-temperature superconductors has revitalized the interest in low-dimensional charge and spin fluctuations of highly correlated electron systems. Several mechanisms proposed to explain the high-T c superconductors invoke properties of the two-dimensional Hubbard model, but probably also some one-dimensional aspects are relevant. Numerous one-dimensional models for correlated electrons have been studied with various approximate, asymptotically exact and exact methods. These results lead to the concept of Luttinger liquid for interacting electron gases without excitation gaps (metallic systems). Characteristic of Luttinger liquids are the charge and spin separation, marginal Fermi liquid properties, e.g. the absence of quasiparticles in the vicinity of the Fermi surface, nonuniversal power-law singularities in the one-particle spectral function and the related absence of a discontinuity in the momentum distribution at the Fermi level, the power-law decay of correlation functions for long times and large distances, persistent currents in finite rings, etc. Due to the peculiarities of the phase space in one dimension some of the models have sufficient conserved currents to be completely integrable. We review exact results derived within the framework of Bethe's ansatz for integrable one-dimensional models of correlated electrons. The Bethe-ansatz method is presented by explicitly showing the steps leading to the solution of the N-component electron gas interacting via a δ-function potential (repulsive and attractive interaction), which is probably the simplest model of correlated electrons. Emphasis is given to the procedure to extract the groundstate properties, the classification of states, the excitation spectrum, the thermodynamics and finite size effects, such as critical exponents of correlation functions and persistent currents. The method is then applied to numerous other models, e.g. (i) a two-band model involving attractive and repulsive potentials and crystalline fields splitting the bands, (ii) the traditional Hubbard chain with attractive and repulsive U, (iii) the degenerate Hubbard model with repulsive U, which displays a metal–insulator transition at a finite U, (iv) a two-band Hubbard model with repulsive U, (v) the traditional supersymmetric t–J model (vi) a two-band supersymmetric t–J model with band-splitting and (vii) the N-component supersymmetric t–J model. Finally, results for models with long-range interactions, in particular r-2 and sinh -2(r) potentials, are briefly reviewed.