Multiparameter statistical models from \(N^2 \times N^2\) braid matrices: explicit eigenvalues of transfer matrices T\(^{(r)}\), spin chains, factorizable scatterings for all \(N\). (English) Zbl 1251.82012
Summary: For a class of multiparameter statistical models based on \(N^2 \times N^2\) braid matrices, the eigenvalues of the transfer matrix 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 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 the Yang-Baxter matrices corresponding to our braid matrices are obtained for all \(N\). They provide potentials for factorizable \(S\)-matrices. The main results are summarized, and perspectives are indicated in the concluding remarks.
MSC:
| 82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |
| 82B23 | Exactly solvable models; Bethe ansatz |
| 81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |
| 15A24 | Matrix equations and identities |
References:
| [1] | R. J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, London, UK, 1982. · Zbl 0538.60093 |
| [2] | H. J. de Vega, “Yang-Baxter algebras, integrable theories and quantum groups,” International Journal of Modern Physics A, vol. 4, no. 10, pp. 2371-2463, 1989. · Zbl 0693.58045 · doi:10.1142/S0217751X89000959 |
| [3] | V. E. Korepin, N. M. Bogoliubov, and A. G. Izergin, Quantum Inverse Scattering Method, Correlation Functions and Algebraic Bethe Ansatz, Cambridge University Press, Cambridge, UK, 1993. · Zbl 0787.47006 · doi:10.1017/CBO9780511628832 |
| [4] | M. Jimbo, T. Miwa, and A. Nakayashiki, “Difference equations for the correlation functions of the eight-vertex model,” Journal of Physics A, vol. 26, no. 9, pp. 2199-2209, 1993. · Zbl 0773.60090 · doi:10.1088/0305-4470/26/9/015 |
| [5] | A. Chakrabarti, “Canonical factorization and diagonalization of Baxterized braid matrices: explicit constructions and applications,” Journal of Mathematical Physics, vol. 44, no. 11, pp. 5320-5349, 2003. · Zbl 1063.17012 · doi:10.1063/1.1613378 |
| [6] | A. Chakrabarti, “A nested sequence of projectors and corresponding braid matrices R^(\theta ):(1). I. Odd dimensions,” Journal of Mathematical Physics, vol. 46, no. 6, p. 18, 2005. · Zbl 1110.82019 · doi:10.1063/1.1900291 |
| [7] | B. Abdesselam and A. Chakrabarti, “Nested sequence of projectors. II. Multiparameter multistate statistical models, Hamiltonians, S-matrices,” Journal of Mathematical Physics, vol. 47, no. 5, p. 25, 2006. · Zbl 1111.82012 · doi:10.1063/1.2197690 |
| [8] | B. Abdesselam, A. Chakrabarti, V. K. Dobrev, and S. G. Mihov, “Higher dimensional multiparameter unitary and nonunitary braid matrices: even dimensions,” Journal of Mathematical Physics, vol. 48, no. 10, p. 6, 2007. · Zbl 1152.81302 · doi:10.1063/1.2793571 |
| [9] | B. Abdesselam and A. Chakrabarti, “A new eight vertex model and higher dimensional, multiparameter generalizations,” Journal of Mathematical Physics, vol. 49, no. 5, p. 21, 2008. · Zbl 1152.81301 · doi:10.1063/1.2918142 |
| [10] | B. Abdesselam, A. Chakrabarti, V. K. Dobrev, and S. G. Mihov, “Exotic bialgebras from 9\times 9 unitary braid matrices,” Physics of Atomic Nuclei, vol. 74, no. 6, pp. 824-831, 2011, (851-857 Russian edition). |
| [11] | J. B. Saleur and H. Zuber, “Integrable lattice models and quantum groups,” in proceedings of the Trieste Spring School on String Theory and Quantum Gravity, 1990. |
| [12] | D. Arnaudon, A. Chakrabarti, V. K. Dobrev, and S. G. Mihov, “Exotic bialgebra S\varphi 3: representations, Baxterisation and applications,” Annales Henri Poincaré, vol. 7, no. 7-8, pp. 1351-1373, 2006. · Zbl 1228.16035 · doi:10.1007/s00023-006-0283-7 |
| [13] | A. B. Zamolodchikov and A. B. Zamolodchikov, “Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models,” Annals of Physics, vol. 120, no. 2, pp. 253-291, 1979. · doi:10.1016/0003-4916(79)90391-9 |
| [14] | B. Abdesselam, A. Chakrabarti, V. K. Dobrev, and S. G. Mihov, “Higher dimensional unitary braid matrices: construction, associated structures, and entanglements,” Journal of Mathematical Physics, vol. 48, no. 5, p. 21, 2007. · Zbl 1144.81301 · doi:10.1063/1.2737266 |
| [15] | S. J. Kauffman and L. H. Lomonaco Jr, “Braiding operators are universal quantum gates,” New Journal of Physics, vol. 6, p. 34, 2004. · doi:10.1088/1367-2630/6/1/134 |
| [16] | Y. Zhang, L. H. Kauffman, and M.-L. Ge, “Yang-Baxterizations, universal quantum gates and Hamiltonians,” Quantum Information Processing, vol. 4, no. 3, pp. 159-197, 2005. · Zbl 1130.81028 · doi:10.1007/s11128-005-7655-7 |
| [17] | J. H. H. Perk and C. L. Schultz, “New families of commuting transfer matrices in Q-state vertex models,” Physics Letters A, vol. 84, no. 8, pp. 407-410, 1981. · doi:10.1016/0375-9601(81)90994-4 |
| [18] | H. Perk and J. H. H. Au-Yang, “Yang-Baxter equations,” Encyclopedia of Mathematical Physics, vol. 5, pp. 465-473, 2006. |
| [19] | C. L. Schultz, “Eigenvectors of the multicomponent generalization of the six-vertex model,” Physica A, vol. 122, no. 1-2, pp. 71-88, 1983. · doi:10.1016/0378-4371(83)90083-3 |
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