Abstract:
For a class of multiparameter statistical models based on $N^2\times N^2$ braid matrices the eigenvalues of the transfer matrix ${\bf T}^{(r)}$ are obtained explicitly for all $(r,N)$. Our formalism yields them as solutions of sets of linear equations with simple constant coefficients. The role of zero-sum multiplets constituted in terms of roots of unity is pointed out and their origin is traced to circular permutations of the indices in the tensor products of basis states induced by our class of ${\bf T}^{(r)}$ matrices. The role of free parameters, increasing as $N^2$ with $N$, is emphasized throughout. Spin chain Hamiltonians are constructed and studied for all $N$. Inverse Cayley transforms of Yang-Baxter matrices corresponding to our braid matrices are obtained for all $N$. They provide potentials for factorizable $S$-matrices. Main results are summarized and perspectives are indicated in the concluding remarks.

Abstract:
A class of $(2n)^2\times(2n)^2$ multiparameter braid matrices are presented for all $n$ $(n\geq 1)$. Apart from the spectral parameter $\theta$, they depend on $2n^2$ free parameters $m_{ij}^{(\pm)}$, $i,j=1,...,n$. For real parameters the matrices $R(\theta)$ are nonunitary. For purely imaginary parameters they became unitary. Thus a unification is achieved with odd dimensional multiparameter solutions presented before.

Abstract:
We study chain Hamiltonians derived from a class of multidimensional, multiparameter braid matrices introduced and explored in a series of previous papers. The N2 × N2 braid matrices (for all N) have free parameters for even N and for N odd. We present systematic explicit constructions for eigenstates and eigenvalues of chain Hamiltonians for and all chain lengths r. We derive explicitly the constraints imposed on these states by periodic (circular) boundary conditions. Our results thus cover both open and closed chains. We then indicate how our formalism can be extended for all . The dependence of the eigenvalues on the free parameters is displayed explicitly, showing how the energy levels and their differences vary in a particular simple way with these parameters. Some perspectives are discussed in conclusion. 1. Introduction In a series of previous paper [1–4], we have formulated and studied a class of braid matrices ( ) with free parameters whose numbers increase as . Chain Hamiltonians corresponding to these matrices were also presented. Here, we undertake systematic study of chain Hamiltonians derived from these braid matrices presenting iterative and explicit constructions of eigenstates and eigenvalues for all dimensions and for all orders (chain lengths) . We recapitulate briefly the constructions of the braid matrices satisfying (in standard notations used in [1–4]) Our class has a nested sequence of projectors as a basis which are defined as follows. For even ( , ), where , , , , . Interchanging on the right, one obtains with (We use the notation ( ) for a matrix with only one nonzero element, unity, on row and column .) On such an orthonormal complete basis of projectors, one defines (with domains defined below (2)) The crucial constraints on the free parameters give an braid matrix with free parameters. For odd ( ; ), using our definitions below (2), The braid matrix is now ( , , , , ). The number of free parameters is now (An overall factor and redefinitions of the ’s, , etc. convert our previous convention for odd to the present one, which is more convenient for what follows.) Defining the Hamiltonian acting on a chain of sites ( ) is given by the standard where acts on the sites . For circular boundary conditions (or periodic), The Hamiltonians inherit the free parameters (see (6), (9)) of . This is the most striking features of our construction. The eigenvalues will be seen to depend linearly on these parameters (given by simple sums of the ’s). So, by varying them, one can vary the differences between the energy levels. We will first

Abstract:
Lecture notes on factorizable S-matrices, thermodynamic Bethe Ansatz and integrable perturbations of conformally invariant models; J.A.Swieca Summer School 1991

Abstract:
We construct $(2n)^2\times (2n)^2$ unitary braid matrices $\hat{R}$ for $n\geq 2$ generalizing the class known for $n=1$. A set of $(2n)\times (2n)$ matrices $(I,J,K,L)$ are defined. $\hat{R}$ is expressed in terms of their tensor products (such as $K\otimes J$), leading to a canonical formulation for all $n$. Complex projectors $P_{\pm}$ provide a basis for our real, unitary $\hat{R}$. Baxterization is obtained. Diagonalizations and block-diagonalizations are presented. The loss of braid property when $\hat{R}$ $(n>1)$ is block-diagonalized in terms of $\hat{R}$ $(n=1)$ is pointed out and explained. For odd dimension $(2n+1)^2\times (2n+1)^2$, a previously constructed braid matrix is complexified to obtain unitarity. $\hat{R}\mathrm{LL}$- and $\hat{R}\mathrm{TT}$-algebras, chain Hamiltonians, potentials for factorizable $S$-matrices, complex non-commutative spaces are all studied briefly in the context of our unitary braid matrices. Turaev construction of link invariants is formulated for our case. We conclude with comments concerning entanglements.

Abstract:
Our starting point is a class of braid matrices, presented in a previous paper, constructed on a basis of a nested sequence of projectors. Statistical models associated to such $N^2\times N^2$ matrices for odd $N$ are studied here. Presence of $\frac 12(N+3)(N-1)$ free parameters is the crucial feature of our models, setting them apart from other well-known ones. There are $N$ possible states at each site. The trace of the transfer matrix is shown to depend on $\frac 12(N-1)$ parameters. For order $r$, $N$ eigenvalues consitute the trace and the remaining $(N^r-N)$ eigenvalues involving the full range of parameters come in zero-sum multiplets formed by the $r$-th roots of unity, or lower dimensional multiplets corresponding to factors of the order $r$ when $r$ is not a prime number. The modulus of any eigenvalue is of the form $e^{\mu\theta}$, where $\mu$ is a linear combination of the free parameters, $\theta$ being the spectral parameter. For $r$ a prime number an amusing relation of the number of multiplets with a theorem of Fermat is pointed out. Chain Hamiltonians and potentials corresponding to factorizable $S$-matrices are constructed starting from our braid matrices. Perspectives are discussed.

Abstract:
We study the regularity of the roots of multiparameter families of complex univariate monic polynomials $P(x)(z) = z^n + \sum_{j=1}^n (-1)^j a_j(x) z^{n-j}$ with fixed degree $n$ whose coefficients belong to a certain subring $\mathcal C$ of $C^\infty$-functions. We require that $\mathcal C$ includes polynomial but excludes flat functions (quasianalyticity) and is closed under composition, derivation, division by a coordinate, and taking the inverse. Examples are quasianalytic Denjoy--Carleman classes, in particular, the class of real analytic functions $C^\omega$. We show that there exists a locally finite covering $\{\pi_k\}$ of the parameter space, where each $\pi_k$ is a composite of finitely many $\mathcal C$-mappings each of which is either a local blow-up with smooth center or a local power substitution (in coordinates given by $x \mapsto (\pm x_1^{\gamma_1},...,\pm x_q^{\gamma_q})$, $\gamma_i \in \mathbb N_{>0}$), such that, for each $k$, the family of polynomials $P {\o}\pi_k$ admits a $\mathcal C$-parameterization of its roots. If $P$ is hyperbolic (all roots real), then local blow-ups suffice. Using this desingularization result, we prove that the roots of $P$ can be parameterized by $SBV_{loc}$-functions whose classical gradients exist almost everywhere and belong to $L^1_{loc}$. In general the roots cannot have gradients in $L^p_{loc}$ for any $1 < p \le \infty$. Neither can the roots be in $W_{loc}^{1,1}$ or $VMO$. We obtain the same regularity properties for the eigenvalues and the eigenvectors of $\mathcal C$-families of normal matrices. A further consequence is that every continuous subanalytic function belongs to $SBV_{loc}$.

Abstract:
We present an efficient algorithm for computing the permanent for matrices of size N that can written as a product of L block diagonal matrices with blocks of size at most 2. For fixed L, the time and space resources scale linearly in N, with a prefactor that scales exponentially in L. This class of matrices contains banded matrices with banded inverse. We show that such a factorization into a product of block diagonal matrices gives rise to a circuit acting on a Hilbert space with a tensor product structure and that the permanent is equal to the transition amplitude of this circuit and a product basis state. In this correspondence, a block diagonal matrix gives rise to one layer of the circuit, where each block to a gate acting either on a single tensor component or on two adjacent tensor components. This observation allows us to adopt matrix product states, a computational method from condensed matter physics and quantum information theory used to simulate quantum systems, to evaluate the transition amplitude.

Abstract:
A new method for deriving universal \v{R} matrices from braid group representation is discussed. In this case, universal \v{R} operators can be defined and expressed in terms of products of braid group generators. The advantage of this method is that matrix elements of \v{R} are rank independent, and leaves multiplicity problem concerning coproducts of the corresponding quantum groups untouched. As examples, \v{R} matrix elements of $[1]\times [1]$, $[2]\times [2]$, $[1^{2}]\times [1^{2}]$, and $[21]\times [21]$ with multiplicity two for $A_{n}$, and $[1]\times [1]$ for $B_{n}$, $C_{n}$, and $D_{n}$ type quantum groups, which are related to Hecke algebra and Birman-Wenzl algebra, respectively, are derived by using this method.

Abstract:
We study statistical models, specifically transfer matrices corresponding to a multiparameter hierarchy of braid matrices of $(2n)^2\times(2n)^2$ dimensions with $2n^2$ free parameters $(n=1,2,3,...)$. The simplest, $4\times 4$ case is treated in detail. Powerful recursion relations are constructed giving the dependence on the spectral parameter $\theta$ of the eigenvalues of the transfer matrix explicitly at each level of coproduct sequence. A brief study of higher dimensional cases ($n\geq 2$) is presented pointing out features of particular interest. Spin chain Hamiltonians are also briefly presented for the hierarchy. In a long final section basic results are recapitulated with systematic analysis of their contents. Our eight vertex $4\times 4$ case is compared to standard six vertex and eight vertex models.