%0 Book Section %1 statphys23_0722 %A Fujimoto, M. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K correlation elliptic function group lattice models point renormarization statphys23 topic-2 %T Point Groups, Galois Coverings, and RG Transformations in Lattice Models %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=722 %X Recently, for two-dimensional lattice models, we found that asymptotic behavior of the correlation function is expressed in terms of differential forms on Riemann surfaces of genus 1 1-3. Choosing an elliptic parametrization, we also found a one-to-one correspondence between point groups and Galois groups in covering problems of Riemann surfaces 4. It was proved that the point group of the system essentially determines the differential forms 5. In this presentation we show a close relation between the differential forms and a renormalization group (RG) approach. Firstly, solvable models on the square lattice are considered 2, 6; the point group is $C_4v$ there. Using the fact that $C_4vC_2v$, we construct a Galois extension of an elliptic function field. It follows that Landen's transformation gives a generalization of the RG approach for the one-dimensional Ising model 7; for Landen's transformation, see Chapter 15 of 6. Secondly, it is shown that the same argument can be applied to solvable models possessing sixfold rotational symmetry 3, 6, where $C_4v$ is replaced by $C_6v$, and Landen's transformation by a suitable one. It is expected that the RG approach proposed here is a quite general one which is applicable to a wide class of lattice models including unsolvable ones. Lastly, with the help of series expansions, we discuss the RG approach for unsolvable models. 1) M. Holzer, Phys. Rev. Lett. 64 (1990) 653; Phys. Rev. B42 (1990) 10570; Y. Akutsu and N. Akutsu, Phys. Rev. Lett. 64 (1990) 1189.\\ 2) M. Fujimoto, Physica A 233 (1996) 485.\\ 3) M. Fujimoto, J. Stat. Phys. 90 (1998) 363.\\ 4) M. Namba, Branched Coverings and Algebraic Functions (Longman).\\ 5) M. Fujimoto, J. Phys. A:Math. Gen. 35 (2002) 7553.\\ 6) R. J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic Press).\\ 7) P. Pfeuty and G. Toulouse, Introduction to the Renormalization Group and Critical Phenomena (Wiley).

@incollection{statphys23_0722, abstract = {Recently, for two-dimensional lattice models, we found that asymptotic behavior of the correlation function is expressed in terms of differential forms on Riemann surfaces of genus 1 [1-3]. Choosing an elliptic parametrization, we also found a one-to-one correspondence between point groups and Galois groups in covering problems of Riemann surfaces [4]. It was proved that the point group of the system essentially determines the differential forms [5]. In this presentation we show a close relation between the differential forms and a renormalization group (RG) approach. Firstly, solvable models on the square lattice are considered [2, 6]; the point group is $C_{4v}$ there. Using the fact that $C_{4v}\supset C_{2v}$, we construct a Galois extension of an elliptic function field. It follows that Landen's transformation gives a generalization of the RG approach for the one-dimensional Ising model [7]; for Landen's transformation, see Chapter 15 of [6]. Secondly, it is shown that the same argument can be applied to solvable models possessing sixfold rotational symmetry [3, 6], where $C_{4v}$ is replaced by $C_{6v}$, and Landen's transformation by a suitable one. It is expected that the RG approach proposed here is a quite general one which is applicable to a wide class of lattice models including unsolvable ones. Lastly, with the help of series expansions, we discuss the RG approach for unsolvable models. 1) M. Holzer, Phys. Rev. Lett. 64 (1990) 653; Phys. Rev. B42 (1990) 10570; Y. Akutsu and N. Akutsu, Phys. Rev. Lett. 64 (1990) 1189.\\ 2) M. Fujimoto, Physica A 233 (1996) 485.\\ 3) M. Fujimoto, J. Stat. Phys. 90 (1998) 363.\\ 4) M. Namba, Branched Coverings and Algebraic Functions (Longman).\\ 5) M. Fujimoto, J. Phys. A:Math. Gen. 35 (2002) 7553.\\ 6) R. J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic Press).\\ 7) P. Pfeuty and G. Toulouse, Introduction to the Renormalization Group and Critical Phenomena (Wiley).}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Fujimoto, M.}, biburl = {https://www.bibsonomy.org/bibtex/2ef0fa7efff27ac5ec7105001d5f29bd8/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {cd1412f1b64f2ba51e8e239786d3167d}, intrahash = {ef0fa7efff27ac5ec7105001d5f29bd8}, keywords = {correlation elliptic function group lattice models point renormarization statphys23 topic-2}, month = {9-13 July}, timestamp = {2007-06-20T10:16:28.000+0200}, title = {Point Groups, Galois Coverings, and RG Transformations in Lattice Models}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=722}, year = 2007 }