Abstract
We analyze discrete symmetry groups of vertex models in lattice statistical mechanics represented as groups of birational transformations. They can be seen as generated by involutions corresponding respectively to two kinds of transformations onq×q matrices: the inversion of theq×q matrix and an (involutive) permutation of the entries of the matrix. We show that the analysis of the factorizations of the iterations of these transformations is a precious tool in the study of lattice models in statistical mechanics. This approach enables one to analyze two-dimensionalq 4-state vertex models as simply as three-dimensional vertex models, or higher-dimensional vertex models. Various examples of birational symmetries of vertex models are analyzed. A particular emphasis is devoted to a three-dimensional vertex model, the 64-state cubic vertex model, which exhibits a polynomial growth of the complexity of the calculations. A subcase of this general model is seen to yield integrable recursion relations. We also concentrate on a specific two-dimensional vertex model to see how the generic exponential growth of the calculations reduces to a polynomial growth when the model becomes Yang-Baxter integrable. It is also underlined that a polynomial growth of the complexity of these iterations can occur even for transformations yielding algebraic surfaces, or higher-dimensional algebraic varieties.
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References
S. Boukraa, J. M. Maillard, and G. Rollet, Determinantal identities on integrable mappings,Int. J. Mod. Phys. B 8:2157–2201 (1994).
S. Boukraa, J.-M. Maillard, and G. Rollet, Integrable mappings and polynomial growth,Physica A 208:115–175 (1994).
S. Boukraa, J.-M. Maillard, and G. Rollet, Almost integrable mappings,Int. J. Mod. Phys. B 8:137–174 (1994).
J. M. Maillard, Exactly solvable models in statistical mechanics and automorphisms of algebraic varieties,Progress in Statistical Mechanics, Vol. 3, C.-K. Hu, ed. (World Scientific, Singapore, 1988).
M. P. Bellon, J.-M. Maillard, and C.-M. Viallet, Integrable Coxeter groups,Phys. Lett. A 159:221–232 (1991).
M. P. Bellon, J.-M. Maillard, and C.-M. Viallet, Higher dimensional mappings,Phys. Lett. A 159:233–244 (1991).
M. P. Bellon, J.-M. Maillard, and C.-M. Viallet, Infinite discrete symmetry group for the Yang-Baxter equations: Spin models,Phys. Lett. A 157:343–353 (1991).
M. P. Bellon, J.-M. Maillard, and C.-M. Viallet, Infinite discrete symmetry group for the Yang-Baxter equations: Vertex models,Phys. Lett. B 260:87–100 (1991).
M. P. Bellon, J.-M. Maillard, and C.-M. Viallet, Rational mappings, arborescent iterations, and the symmetries of integrability,Phys. Rev. Lett. 67:1373–1376 (1991).
M. P. Bellon, J.-M. Maillard, and C.-M. Viallet, Quasi integrability of the sixteen-vertex model,Phys. Lett. B 281:315–319 (1992).
M. P. Bellon, J.-M. Maillard, and C.-M. Viallet, Dynamical systems from quantum integrability, inYang-Baxter Equations in Paris, J.-M. Maillard, ed. (World Scientific, Singapore, 1993), pp. 95–124.
C. M. Viallet and G. Falqui, Singularity, complexity, and quasi-integrability of rational mappings,Commun. Math. Phys. 154:111–125 (1993).
A. P. Veselov, Growth and integrability in the dynamics of mappings,Commun. Math. Phys. 145:181–193 (1992).
C. M. Viallet, Baxterization, Dynamical systems and the symmetries of integrability, inProceedings of the Third Baltic Rim Student Seminar Helsinki 1993 (Springer, Berlin, 1994).
W. T. Tutte,Studies in Graph Theory (Mathematical Association of America, 175), Part II.
J. M. Maillard and R. Rammal, Some analytical consequences of the inverse relation for the Potts model.J. Phys. A 16:353 (1983).
R. J. Baxter, H. N. V. Temperley, and S. E. Ashley, Triangular Potts model at its transition temperature, and related models,Proc. R. Soc. Lond. A 358:535–559 (1978).
C. L. Schultz, Solvableq-state models in lattice statistics and quantum field theory,Phys. Rev. Lett. 46:629–632 (1981).
Y. G. Stroganov, A new calculation method for partition functions in some lattice models,Phys. Lett. A 74:116 (1979).
A. Gaaf and J. Hijmans, Symmetry relations in the sixteen-vertex model,Physica A 80:149–171 (1975).
M. T. Jaekel and J. M. Maillard, Largeq expansion on the anisotropic Potts model,J. Phys. A 16:175 (183).
R. Rammal and J. M. Maillard,q-State Potts model on the checkerboard lattice,J. Phys. A 16:1073–1081 (1983).
R. J. Baxter, The inversion relation method for some two-dimensional exactly solved models in lattice statistics,J. Stat. Phys. 28:1–41 (1982).
M. T. Jaekel and J. M. Maillard, Inversion relations and disorder solutions on Potts models,J. Phys. A 17:2079–2094 (1984).
M. T. Jaekel and J. M. Maillard, A criterion for disorder solutions of spin models,J. Phys. A 18:1229 (1985).
R. J. Baxter,Exactly Solved Models in Statistical Mechanics (Academic Press, London, 1981).
J. M. Maillard, The inversion relation: Some simple examples (I) and (II),J. Phys. (Paris)46:329 (1984).
J. M. Maillard, Automorphism of algebraic varieties and Yang-Baxter equations,J. Math. Phys. 27:2776 (1986).
A. B. Zamolodchikov and A. B. Zamolodchikov,Ann. Phys. 120:253 (1979).
M. P. Bellon, Addition on curves and complexity of mappings, in preparation.
M. P. Bellon, S. Boukraa, J.-M. Maillard, and C.-M. Viallet, Towards three-dimensional Bethe Ansatz,Phys. Lett. B 314:79–88 (1993).
I. G. Korepanov, Vacuum curves, clssical integrable systems in discrete space-time and statistical physics,Commun. Math. Phys. (1994).
L. D. Faddeev, Integrable models in 1+1 dimensional quantum field theory, inLes Houches Lectures (1982) (Elsevier, Amsterdam, 1984).
M. T. Jaekel and J. M. Maillard, Algebraic invariants in soluble models,J. Phys. A 16:3105 (1983).
I. Krichever, Baxter's equations and algebraic geometry,Funct. Anal. Pril. 15.
M. P. Bellon, Addition on curves and complexity of quasi-integrable rational mappings, in preparation.
M. P. Bellon, J.-M. Maillard, G. Rollet, and C.-M. Viallet, Deformations of dynamics associated to the chiral Potts model,Int. J. Mod. Phys. B 6:3575–3584 (1992).
J. M. Maillard and G. Rollet, Hyperbolic Coxeter groups for triangular Potts models, preprint to be published inJ. Phys. A (1994).
L. P. Kadanoff and F. J. Wegner,Phys. Rev. B 4:3989–3993 (1971).
R. J. Baxter, Eight-vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain. II. Equivalence to a generalized ice-type lattice model,Ann. Phys. 76:25–47 (1973).
S. Boukraa and J.-M. Maillard, Symmetries of lattice models in statistical mechanics and effective algebraic geometry,Phys. (Paris) I3:239–258 (1993).
A. W. Knapp,Elliptic Curves (Princeton University Press, Princeton, New Jersey, 1992).
R. Hartshorne,Algebraic Geometry (Springer-Verlag, Berlin, 1983), Appendix B, p. 438.
P. Griffiths and J. Harris,Principles of Algebraic Geometry (Wiley, New York, 1978).
I. R. Shafarevich,Basic Algebraic Geometry (Springer, Berlin, 1977), p. 216.
J.-M. Maillard, G. Rollet, F. Y. Wu, Inversion relations and symmetry groups for Potts models on the triangular lattice,J. Phys. A 27:3373–3379 (1994).
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Boukraa, S., Maillard, J.M. & Rollet, G. Discrete symmetry groups of vertex models in statistical mechanics. J Stat Phys 78, 1195–1251 (1995). https://doi.org/10.1007/BF02180130
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DOI: https://doi.org/10.1007/BF02180130