A nested sequence of projectors (2): Multiparameter multistate statistical models, Hamiltonians, S-matrices

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 $\frac{1}{2}(N+3)(N-1)$ 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 $\frac{1}{2}(N-1)$ 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.

32 pages, no figure, few mistakes are corrected

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Cited by in corpus (4)

• Higher Dimensional Multiparameter Unitary and Nonunitary Braid Matrices: Even Dimensions
• Higher Dimensional Unitary Braid Matrices: Construction, Associated Structures and Entanglements
• A new eight vertex model and higher dimensional, multiparameter generalizations
• A new class ${\hat{o}}_N$ of statistical models: Transfer matrix eigenstates, chain Hamiltonians, factorizable $S$-matrix