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arXiv:0801.2548 · Journal of Mathematical Physics

A new eight vertex model and higher dimensional, multiparameter generalizations

B. Abdesselam, Amitabha Chakrabarti

Published May 1, 2008• 8 citations
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Abstract

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.

DOI
10.1063/1.2918142

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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.
For a class of multiparameter statistical models based on ð ‘ 2 × ð ‘ 2 braid matrices, the eigenvalues of the transfer matrix ð “ ( ð ‘Ÿ ) are obtained explicitly for all ( ð ‘Ÿ , ð ‘ ) . 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 ð “ ( ð ‘Ÿ ) matrices. The role of free parameters, increasing as ð ‘ 2 with N , is emphasized throughout. Spin chain Hamiltonians are constructed and studied for all N . Inverse Cayley transforms of the 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.
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.
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.
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 1/2N2 free parameters for even N and 1/2N+12-1 for N odd. We present systematic explicit constructions for eigenstates and eigenvalues of chain Hamiltonians for N=2,3,4 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 N,r . 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.
Multiparameter statistical models from N2×N2N^2\times N^2N2×N2 braid matrices: Explicit eigenvalues of transfer matrices T(r){\bf T}^{(r)}T(r), spin chains, factorizable scatterings for all NNN
For a class of multiparameter statistical models based on N2×N2N^2\times N^2N2×N2 braid matrices the eigenvalues of the transfer matrix T(r){\bf T}^{(r)}T(r) are obtained explicitly for all (r,N)(r,N)(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 T(r){\bf T}^{(r)}T(r) matrices. The role of free parameters, increasing as N2N^2N2 with NNN, is emphasized throughout. Spin chain Hamiltonians are constructed and studied for all NNN. Inverse Cayley transforms of Yang-Baxter matrices corresponding to our braid matrices are obtained for all NNN. They provide potentials for factorizable SSS-matrices. Main results are summarized and perspectives are indicated in the concluding remarks.
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.