%0 Book Section %1 statphys23_0484 %A Sakaniwa, Y. %A Shima, H. %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 critical curved exponent finit-size geometry ising model phase scaling statphys23 topic-2 transition %T Ising phase transition on a negatively curved geometry %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=484 %X \ \ \ \ \ The purpose of the present study is to investigate the critical behavior of Ising lattice models on a curved surface with constant negative curvature. It is widely believed that, at the second-order phase transition, critical behavior of a system is determined only by certain kinds of global symmetry that the system possesses. In fact, for planar Ising models, the values of critical exponents are invariant to the change in microscopic details of the system such as the lattice structures. It is thus significant to make clear which kinds of symmetry are responsible for universal critical behavior of the system. \ \ \ \ \ We point out that, when the Ising model is assigned on a curved surface, critical properties of the system may be affected by intrinsic geometric character of the underlying surface. This is because non-Euclidean property of the underlying surface leads to an alternation in global symmetries of the mounted Ising model, thus possibly affecting scaling behavior of the system. \ \ \ \ \ To make clear the above point, we have numerically investigated the critical exponents of Ising lattice models defined on a pseudosphere. The pseudosphere is a simply connected infinite surface in which the Gaussian curvature at arbitrary points has a constant negative value. It thus serves as a suitable geometry for considering the curvature effect on the critical properties of a system. Furthermore, a wide range of equilateral lattices can be constructed on the pseudosphere: pentagonal, heptagonal, and dodecagonal equilateral lattices are only a few cases in point. The availability of such non-trivial lattices is a direct consequence of non-Euclidean property of a pseudosphere. \ \ \ \ \ In actual calculations, we have employed Monte-Corlo simulations and finite-size scaling analysis to determine the critical exponents, $\beta$ and $\gamma$, of spontaneous magnetization $m(T)$ and magnetic susceptibility $\chi(T)$ for the zero-field, respectively. Numerical results have suggested that two distinct universality classes coexist in the same Ising model on the pseudosphere. The one is associated with the central portion among the whole Ising lattices, which exhibits a mean-field behavior independent of lattice structures. The other class originates in the Ising spins locating near the boundary, of which the critical exponents $\beta$ and $\gamma$ deviate quantitatively from those of planar Ising models and the mean-field system. These findings are evidence that the non-zero curvature of the underlying surface is responsible for the determination of the universality class of the mounted Ising model.\\ H. Shima and Y. Sakaniwa, J. Phys. A $39$ (2006) 4921.\\ H. Shima and Y. Sakaniwa, J. Stat. Mech. P08017 (2006).\\ Y. Sakaniwa, I. Hasegawa and H.Shima, J. Magn. Magn. Mater. $310$ (2007) 1401.\\

@incollection{statphys23_0484, abstract = {\ \ \ \ \ The purpose of the present study is to investigate the critical behavior of Ising lattice models on a curved surface with constant negative curvature. It is widely believed that, at the second-order phase transition, critical behavior of a system is determined only by certain kinds of global symmetry that the system possesses. In fact, for planar Ising models, the values of critical exponents are invariant to the change in microscopic details of the system such as the lattice structures. It is thus significant to make clear which kinds of symmetry are responsible for universal critical behavior of the system. \ \ \ \ \ We point out that, when the Ising model is assigned on a curved surface, critical properties of the system may be affected by intrinsic geometric character of the underlying surface. This is because non-Euclidean property of the underlying surface leads to an alternation in global symmetries of the mounted Ising model, thus possibly affecting scaling behavior of the system. \ \ \ \ \ To make clear the above point, we have numerically investigated the critical exponents of Ising lattice models defined on a pseudosphere. The pseudosphere is a simply connected infinite surface in which the Gaussian curvature at arbitrary points has a constant negative value. It thus serves as a suitable geometry for considering the curvature effect on the critical properties of a system. Furthermore, a wide range of equilateral lattices can be constructed on the pseudosphere: pentagonal, heptagonal, and dodecagonal equilateral lattices are only a few cases in point. The availability of such non-trivial lattices is a direct consequence of non-Euclidean property of a pseudosphere. \ \ \ \ \ In actual calculations, we have employed Monte-Corlo simulations and finite-size scaling analysis to determine the critical exponents, $\beta$ and $\gamma$, of spontaneous magnetization $m(T)$ and magnetic susceptibility $\chi(T)$ for the zero-field, respectively. Numerical results have suggested that two distinct universality classes coexist in the same Ising model on the pseudosphere. The one is associated with the central portion among the whole Ising lattices, which exhibits a mean-field behavior independent of lattice structures. The other class originates in the Ising spins locating near the boundary, of which the critical exponents $\beta$ and $\gamma$ deviate quantitatively from those of planar Ising models and the mean-field system. These findings are evidence that the non-zero curvature of the underlying surface is responsible for the determination of the universality class of the mounted Ising model.\\ H. Shima and Y. Sakaniwa, J. Phys. A $\bf{39}$ (2006) 4921.\\ H. Shima and Y. Sakaniwa, J. Stat. Mech. P08017 (2006).\\ Y. Sakaniwa, I. Hasegawa and H.Shima, J. Magn. Magn. Mater. $\bf{310}$ (2007) 1401.\\}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Sakaniwa, Y. and Shima, H.}, biburl = {https://www.bibsonomy.org/bibtex/21c04d380d65ac0df6931cd818b00c6ff/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {344058d082deb4151f355e4be28dbe2e}, intrahash = {1c04d380d65ac0df6931cd818b00c6ff}, keywords = {critical curved exponent finit-size geometry ising model phase scaling statphys23 topic-2 transition}, month = {9-13 July}, timestamp = {2007-06-20T10:16:21.000+0200}, title = {Ising phase transition on a negatively curved geometry}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=484}, year = 2007 }