Yao-Zhong Zhang

New Supersymmetric and Exactly Solvable Model of Correlated Electrons.

A new lattice model is presented for correlated electrons on the unrestricted 4L-dimensional electronic Hilbert space n=1LC4 (where L is the lattice length). It is a supersymmetric generalization of the Hubbard model, but differs from the extended Hubbard model proposed by Essler, Korepin, and Schoutens. The supersymmetry algebra of the new model is superalgebra gl(2 |1). The model contains one symmetry-preserving free real parameter which is the Hubbard interaction parameter U, and has its origin here in the one-parameter family of inequivalent typical 4-dimensional irreducible representations of gl(2 |1). On a one-dimensional lattice, the model is exactly solvable by the Bethe ansatz.

Q-operator and T–Q relation from the fusion hierarchy

We propose that the Baxter Q-operator for the spin-1/2 XXZ quantum spin chain is given by the j -> infinity limit of the transfer matrix with spin-j (i.e., (2j + 1)-dimensional) auxiliary space. Applying this observation to the open chain with general (non-diagonal) integrable boundary terms, we obtain from the fusion hierarchy the T-Q relation for generic values (i.e., not roots of unity) of the bulk anisotropy parameter. We use this relation to determine the Bethe ansatz solution of the eigenvalues of the fundamental transfer matrix. This approach is complementary to the one used recently to solve the same model for the roots of unity case. (c) 2005 Elsevier B.V. All rights reserved.

On the second reference state and complete eigenstates of the open XXZ chain

The second reference state of the open XXZ spin chain with non-diagonal boundary terms is studied. The associated Bethe states exactly yield the second set of eigenvalues proposed recently by functional Bethe Ansatz. In the quasi-classical limit, two sets of Bethe states give the complete eigenstates of the associated Gaudin model.

Exact solution of the XXZ Gaudin model with generic open boundaries

The XXZ Gaudin model with generic integrable boundaries specified by generic non-diagonal K-matrices is studied. The commuting families of Gaudin operators are diagonalized by the algebraic Bethe ansatz method. The eigenvalues and the corresponding Bethe ansatz equations are obtained

On the Construction of Trigonometric Solutions of the Yang-Baxter Equation

We describe the construction of trigonometric R-matrices corresponding to the (multiplicity-free) tensor product of two irreducible representations of a quantum algebra Uq(G). Our method is a generalization of the tensor product graph method to the case of two different representations. It yields the decomposition of the R-matrix into projection operators. Many new examples of trigonometric R-matrices (solutions to the spectral parameter dependent Yang-Baxter equation) are constructed using this approach.

Solutions of the Quantum Yang-Baxter Equation with Extra Nonadditive Parameters

We present a systematic technique to construct solutions to the Yang-Baxter equation which depend not only on a spectral parameter but in addition on further continuous parameters. These extra parameters enter the Yang-Baxter equation in a similar way to the spectral parameter but in a non-additive form. We exploit the fact that quantum non-compact algebras such as Uq(su(1,1)) and type-I quantum superalgebras such as Uq(gl(1 mod 1)) and Uq(gl(2 mod 1)) are known to admit non-trivial one-parameter families of infinite-dimensional and finite-dimensional irreps, respectively, even for generic q. We develop a technique for constructing the corresponding spectral-dependent R-matrices. As examples, we work out the the R-matrices for the three quantum algebras mentioned above in certain representations.

Supersymmetric extension of the sine-Gordon theory with integrable boundary interactions

Integrability and supersymmetry of the supersymmetric extension of the sine-Gordon theory on a half-line are examined and the boundary potential which preserves both the integrability and supersymmetry on the bulk is derived. It appears that unlike the boundary bosonic sine-Gordon theory, integrability and supersymmetry strongly restrict the form and parameters of the boundary potential, so that no free parameter in the boundary term is allowed up to a choice of signs.

Exact polynomial solutions of second order differential equations and their applications

We find all polynomials Z(z) such that the differential equation where X(z), Y(z), Z(z) are polynomials of degree at most 4, 3, 2, respectively, has polynomial solutions S(z) = ?ni = 1(z ? zi) of degree n with distinct roots zi. We derive a set of n algebraic equations which determine these roots. We also find all polynomials Z(z) which give polynomial solutions to the differential equation when the coefficients of X(z) and Y(z) are algebraically dependent. As applications to our general results, we obtain the exact (closed-form) solutions of the Schr?dinger-type differential equations describing: (1) two Coulombically repelling electrons on a sphere; (2) Schr?dinger equation from the kink stability analysis of ?6-type field theory; (3) static perturbations for the non-extremal Reissner?Nordstr?m solution; (4) planar Dirac electron in Coulomb and magnetic fields; and (5) O(N) invariant decatic anharmonic oscillator.

Integrable electron model with correlated hopping and quantum supersymmetry

Abstract We give the q-deformed analogue of a recently introduced electron model which generalizes the Hubbard model with additional correlated hopping terms and electron pair hopping. The model contains two independent parameters and is invariant with respect to the quantum superalgebra Uq(gl(2|1)). It is shown to be integrable in one dimension by means of the quantum inverse scattering method.

Integrable open-boundary conditions for the q-deformed supersymmetric U model of strongly correlated electrons

A general graded reflection equation algebra is proposed and the corresponding boundary quantum inverse scattering method is formulated. The formalism is applicable to all boundary lattice systems where an invertible R-matrix exists. As an application, the integrable open-boundary conditions for the q-deformed supersymmetric U model of strongly correlated electrons are investigated. The diagonal boundary K-matrices are found and a class of integrable boundary terms are determined. The boundary system is solved by means of the coordinate space Bethe ansatz technique and the Bethe ansatz equations are derived. As a sideline, it is shown that all R-matrices associated with a quantum affine superalgebra enjoy the crossing-unitarity property