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arXiv:math/0305103 · Journal of Mathematical Physics

Canonical factorization and diagonalization of Baxterized braid matrices: Explicit constructions and applications

Amitabha Chakrabarti

Published Oct 24, 2003• 10 citations
arXivHTMLPDF

Abstract

Braid matrices R̂(θ), corresponding to vector representations, are spectrally decomposed obtaining a ratio fi(θ)/fi(−θ) for the coefficient of each projector Pi appearing in the decomposition. This directly yields a factorization (F(−θ))−1F(θ) for the braid matrix, implying also the relation R̂(−θ)R̂(θ)=I. This is achieved for GLq(n), SOq(2n+1), SOq(2n), Spq(2n) for all n and also for various other interesting cases including the eight-vertex matrix. We explain how the limits θ→±∞ can be interpreted to provide factorizations of the standard (non-Baxterized) braid matrices. A systematic approach to diagonalization of projectors and hence of braid matrices is presented with explicit constructions for GLq(2), GLq(3), SOq(3), SOq(4), Spq(4) and various other cases such as the eight-vertex one. For a specific nested sequence of projectors diagonalization is obtained for all dimensions. The canonical property implemented in the diagonalizers is mutual orthogonality of the rows. In each factor F(θ) our diagonalizer again factors out all dependence on the spectral parameter θ as a diagonal matrix. Applications of our formalism to the construction of L-operators and transfer matrices are indicated. In an Appendix our type of factorization is compared to another one proposed by other authors.

DOI
10.1063/1.1613378

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Braid matrices R̂(θ), corresponding to vector representations, are spectrally decomposed obtaining a ratio fi(θ)/fi(−θ) for the coefficient of each projector Pi appearing in the decomposition. This directly yields a factorization (F(−θ))−1F(θ) for the braid matrix, implying also the relation R̂(−θ)R̂(θ)=I. This is achieved for GLq(n), SOq(2n+1), SOq(2n), Spq(2n) for all n and also for various other interesting cases including the eight-vertex matrix. We explain how the limits θ→±∞ can be interpreted to provide factorizations of the standard (non-Baxterized) braid matrices. A systematic approach to diagonalization of projectors and hence of braid matrices is presented with explicit constructions for GLq(2), GLq(3), SOq(3), SOq(4), Spq(4) and various other cases such as the eight-vertex one. For a specific nested sequence of projectors diagonalization is obtained for all dimensions. The canonical property implemented in the diagonalizers is mutual orthogonality of the rows. In each factor F(θ) our diagonalizer again factors out all dependence on the spectral parameter θ as a diagonal matrix. Applications of our formalism to the construction of L-operators and transfer matrices are indicated. In an Appendix our type of factorization is compared to another one proposed by other authors.
A basis of N2 projectors, each an N2×N2 matrix with constant elements, is implemented to construct a class of braid matrices R̂(θ), θ being the spectral parameter. Only odd values of N are considered here. Our ansatz for the projectors Pα appearing in the spectral decomposition of R̂(θ) leads to exponentials exp(mαθ) as the coefficient of Pα. The sums and differences of such exponentials on the diagonal and the antidiagonal, respectively, provide the (2N2−1) nonzero elements of R̂(θ). One element at the center is normalized to unity. A class of supplementary constraints imposed by the braid equation leaves 12(N+3)(N−1) free parameters mα. The diagonalizer of R̂(θ) is presented for all N. Transfer matrices t(θ) and L(θ) operators corresponding to our R̂(θ) are studied. Our diagonalizer signals specific combinations of the components of the operators that lead to a quadratic algebra of N2 constant N×N matrices. The θ dependence factors out for such combinations. R̂(θ) is developed in a power series in θ. The basic difference arising for even dimensions is made explicit. Some special features of our R̂(θ) are discussed in a concluding section.
We construct (2n)2×(2n)2 unitary braid matrices R̂ for n⩾2 generalizing the class known for n=1. A set of (2n)×(2n) matrices (I,J,K,L) is defined. R̂ is expressed in terms of their tensor products (such as K⊗J), leading to a canonical formulation for all n. Complex projectors P± provide a basis for our real, unitary R̂. Baxterization is obtained. Diagonalizations and block diagonalizations are presented. The loss of braid property when R̂ (n>1) is block diagonalized in terms of R̂ (n=1) is pointed out and explained. For odd dimension (2n+1)2×(2n+1)2, a previously constructed braid matrix is complexified to obtain unitarity. R̂LL and R̂TT algebras, chain Hamiltonians, potentials for factorizable S matrices, and complex noncommutative 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.
Statistical models corresponding to a new class of braid matrices (ôN;N⩾3) presented in a previous paper are studied. Indices labeling states spanning the Nr dimensional base space of T(r)(θ), the rth order transfer matrix are so chosen that the operators W (the sum of the state labels) and (CP) (the circular permutation of state labels) commute with T(r)(θ). This drastically simplifies the construction of eigenstates, reducing it to solutions of relatively small number of simultaneous linear equations. Roots of unity play a crucial role. Thus for diagonalizing the 81 dimensional space for N=3, r=4, one has to solve a maximal set of five linear equations. A supplementary symmetry relates invariant subspaces pairwise [W=(r,Nr) and so on] so that only one of each pair needs study. The case N=3 is studied fully for r=(1,2,3,4). Basic aspects for all (N,r) are discussed. Full exploitation of such symmetries lead to a formalism quite different from, possibly generalized, algebraic Bethe ansatz. Chain Hamiltonians are studied. The specific types of spin flips they induce and propagate are pointed out. The inverse Cayley transform of the YB matrix giving the potential leading to factorizable S matrix is constructed explicitly for N=3 as also the full set of R̂tt relations. Perspectives are discussed in a final section.
We study statistical models, specifically transfer matrices corresponding to a multiparameter hierarchy of braid matrices of (2n)2×(2n)2 dimensions with 2n2 free parameters (n=1,2,3,…). The simplest, 4×4 case, is treated in detail. Powerful recursion relations are constructed by explicitly giving the dependence on the spectral parameter θ of the eigenvalues of the transfer matrix at each level of coproduct sequence. A brief study of higher dimensional cases (n⩾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 4×4 case is compared to standard six vertex and eight vertex models.
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 N2×N2 matrices for odd N are studied here. Presence of 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 12(N−1) parameters. For order r, N eigenvalues constitute the trace and the remaining (Nr−N) eigenvalues involving the full range of parameters come in zero-sum multiplets formed by the rth 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 eμθ, where μ is a linear combination of the free parameters, θ 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.
We study quantum entanglements induced on product states by the action of 8-vertex braid matrices, rendered unitary with purely imaginary spectral parameters (rapidity).The unitarity is displayed via the 'canonical factorization' of the coefficients of the projectors spanning the basis.This adds one more new facet to the famous and fascinating features of the 8vertex model.The double periodicity and the analytic properties of the elliptic functions involved lead to a rich structure of the 3-tangle quantifying the entanglement.We thus explore the complex relationship between topological and quantum entanglement.