Abstract
For a class of multiparameter statistical models based on braid matrices, the eigenvalues of the transfer matrix are obtained explicitly for all . 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 matrices. The role of free parameters, increasing as withN, is emphasized throughout. Spin chain Hamiltonians are constructed and studied for allN. Inverse Cayley transforms of the Yang-Baxter matrices corresponding to our braid matrices are obtained for allN. They provide potentials for factorizableS-matrices. Main results are summarized, and perspectives are indicated in the concluding remarks.
Highlights
Statistical models “exact” in the sense of Baxter 1 satisfy “star-triangle” relations leading to transfer matrices commuting for different values of the spectral parameter
Even for the extensively studied 6-vertex and 8-vertex models based on 4 × 4 braid matrices the braid property guaranteeing star-triangle relations, after the first few simple steps, one has to resort to numerical computations
The basic reason for such situation is that, starting with 2 × 2 blocks Tab of the transfer matrix and constructing 2r × 2r blocks via coproduct rules for order r r 1, 2, 3, . . . , one faces increasingly complicated nonlinear systems of equations to be solved in constructing eigenstates and eigenvalues
Summary
Statistical models “exact” in the sense of Baxter 1 satisfy “star-triangle” relations leading to transfer matrices commuting for different values of the spectral parameter. Even for the extensively studied 6-vertex and 8-vertex models based on 4 × 4 braid matrices the braid property guaranteeing star-triangle relations , after the first few simple steps, one has to resort to numerical computations. For the 8-vertex case one explores analytical properties to extract informations See 4 , e.g. the number of free parameters remains strictly limited for such models, including the multistate generalization of the 6-vertex one 2. This is a consequence of our starting point: braid matrices constructed on a basis of “nested sequence” of projectors 5. It is possible to implement a general and efficient approach. This will be illustrated for all N and r 1, 2, 3, 4, 5. Certain other features will be explored with comments in conclusion
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