Mark D. Gould

From Representations of the Braid Group to Solutions of the Yang-Baxter Equation

Abstract A systematic method is developed for constructing solutions of the Yang-Baxter equation from given braid group representations, arising from such finite dimensional irreps of quantum groups that any irrep can be affinized and the tensor product of the irrep with itself is multiplicity-free. The main tool used in the construction is a tensor product graph, whose circuits give rise to consistency conditions. A maximal tree of this graph leads to an explicit formula for the quantum R-matrix when the consistency conditions are satisfied. As examples, new solutions of the Yang-Baxter equation are found, corresponding to braid group generators associated with the symmetric and antisymmetric tensor irreps of Uq[gl(m)], a spinor irrep of Uq[so(2n)]. and the minimal irreps of Uq[E6] and Uq[E7].

Quantum group invariants and link polynomials

A general method is developed for constructing quantum group invariants and determining their eigenvalues. Applied to the universalR-matrix this method leads to the construction of a closed formula for link polynomials. To illustrate the application of this formula, the quantum groupsUq(E8),Uq(so(2m+1) andUq(gl(m)) are considered as examples, and corresponding link polynomials are obtained.

Algebraic Bethe ansatz method for the exact calculation of energy spectra and form factors: applications to models of Bose–Einstein condensates and metallic nanograins

In this review we demonstrate how the algebraic Bethe ansatz is used for the calculation of the-energy spectra and form factors (operator matrix elements in the basis of Hamiltonian eigenstates) in exactly solvable quantum systems. As examples we apply the theory to several models of current interest in the study of Bose-Einstein condensates, which have been successfully created using ultracold dilute atomic gases. The first model we introduce describes Josephson tunnelling between two coupled Bose-Einstein condensates. It can be used not only for the study of tunnelling between condensates of atomic gases, but for solid state Josephson junctions and coupled Cooper pair boxes. The theory is also applicable to models of atomic-molecular Bose-Einstein condensates, with two examples given and analysed. Additionally, these same two models are relevant to studies in quantum optics; Finally, we discuss the model of Bardeen, Cooper and Schrieffer in this framework, which is appropriate for systems of ultracold fermionic atomic gases, as well as being applicable for the description of superconducting correlations in metallic grains with nanoscale dimensions.; In applying all the above models to. physical situations, the need for an exact analysis of small-scale systems is established due to large quantum fluctuations which render mean-field approaches inaccurate.

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.

A q-analogue of Bargmann space and its scalar product

A q-analogue of Bargmann space is defined, using the properties of coherent states associated with a pair of q-deformed bosons. The space consists of a class of entire functions of a complex variable z, and has a reproducing kernel. On this space, the q-boson creation and annihilation operators are represented as multiplication by z and q-differentiation with respect to z, respectively. A q-integral analogue of Bargmanns scalar product is defined, involving the q-exponential as a weight function. Associated with this is a completeness relation for the q-coherent states.

Two variable link polynomials from quantum supergroups

New two variable link polynomials are constructed corresponding to a one-parameter family of representations of the quantum supergroup Uq[gl(2 | 1)]. Their connection with the Kauffman polynomials is also investigated.

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.

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.

Spin‐dependent unitary group approach. I. General formalism

A new spin‐dependent unitary group approach to the many‐electron correlation problem is investigated. It is demonstrated that the matrix elements of the U(2n) generators, in the U(n)×U(2) adapted electronic Gel’fand basis, are determined by the matrix elements of a single U(n) adjoint tensor operator, herein denoted by Δij(1≤i, j≤n), where Δ is a polynomial of degree two in the U(n) matrix E=[Eij]. The method is then applied, in the second paper of the series, to derive a simple segment level formula for the matrix elements of all U(2n) generators. The advantages of this new procedure for computer implementation are discussed and the central role played by the matrix Δ for the determination of molecular spin densities is highlighted.