%0 Book Section %1 statphys23_0476 %A Santos, F.A.N. %A Coutinho-Filho, M.D. %A Montenegro-filho, R.R. %A Leandro, E.S.G. %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 frustration mean-field model phase statphys23 topic-2 topology transitions xy %T Topology, symmetry and phase transitions on the mean-field frustrated XY model at zero temperature %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=476 %X We use a topological approach 1 to describe the phase transition exhibited by the frustrated mean-field XY model on the AB$_2$ chain (three spins per unit cell) at zero-temperature ($T$). This $T=0$ classical description has physical interest since it can be shown 2 that, at the mean-field level, a similar phase transition occurs in a quantum rotor model on the same chain. In fact, the system undergoes a continuous transition from a ferrimagnetic phase to a canted one at $JJ_2/J_1=1$, where $J_1$ and $J_2$ are the couplings of the antiferromagnetic interactions between spins at sites A and B and between B sites, respectively. Paving the way to apply Morse theory, we discover a continuum of critical points after the transition, even for a finite number of unit cells in the canted phase. Moreover, we analyze the behavior of the equipotential submanifolds of the Hamiltonian $H$ around the transition at $J=1$ ($H=-3J_1$), and deduce the necessity of a topological change 3, although a proof exists only for smooth, finite-range, confining microscopic interaction potentials. However, the suffiency is not necessarily true 4, as also demonstrated for mean-field models 5,6. In fact, we find two classes presenting a continuum of topological changes for $0J1$ and $J>1$, respectively, with $J=1$ being a saddle point associated with the phase transition mentioned above. If, instead, we restrict the analysis to the minima of $H(J)$, a topological change is shown to occur only at $J=1$. Moreover, at $J=2$ (with $H$ also equal to $-3J_1$) the system displays a spin structure of high symmetry, in which case the angle $þeta$ between the three spins of the unit cell matches $2\pi/3$ (zero magnetization). In addition, we identify that a topological change takes place at $H=0$, but the associated configurations do not satisfy the energy minimization condition. Finally, we also show that the presence of a magnetic field $h$ does not destroy neither the transition nor the topological change, but shifts the transition point to $J=1+(h/2)$.\\ 1) L. Caiani, L. Casetti, C. Clementi, and M. Pettini, Phys. Rev. Lett. 79, 4361 (1997); L. Casetti, M. Pettini, and E.G.D. Cohen, Phys. Rep. 337, 237 (2000).\\ 2) A. S. F. Tenório, Master's thesis, UFPE, 2005.\\ 3) R. Franzosi and M. Pettini, Phys. Rev. Lett. 92, 060601 (2004).\\ 4) S. Risau-Gusman $et$ $al$., Phys. Rev. Lett. 95, 145702 (2005).\\ 5) L. Casetti, M. Pettini, and E. G. D. Cohen, J. Stat. Phys. 111, 1091 (2003).\\ 6) A. C. Ribeiro Teixeira and D. A. Stariolo, Phys. Rev. E 70, 016113; I. Hahn and M. Kastner, Phys. Rev. E 72, 056134 (2005).

@incollection{statphys23_0476, abstract = {We use a topological approach [1] to describe the phase transition exhibited by the frustrated mean-field XY model on the AB$_2$ chain (three spins per unit cell) at zero-temperature ($T$). This $T=0$ classical description has physical interest since it can be shown [2] that, at the mean-field level, a similar phase transition occurs in a quantum rotor model on the same chain. In fact, the system undergoes a continuous transition from a ferrimagnetic phase to a canted one at $J\equiv J_2/J_1=1$, where $J_1$ and $J_2$ are the couplings of the antiferromagnetic interactions between spins at sites A and B and between B sites, respectively. Paving the way to apply Morse theory, we discover a continuum of critical points after the transition, even for a finite number of unit cells in the canted phase. Moreover, we analyze the behavior of the equipotential submanifolds of the Hamiltonian $H$ around the transition at $J=1$ ($H=-3J_1$), and deduce the necessity of a topological change [3], although a proof exists only for smooth, finite-range, confining microscopic interaction potentials. However, the suffiency is not necessarily true [4], as also demonstrated for mean-field models [5,6]. In fact, we find two classes presenting a continuum of topological changes for $0\leq J\leq 1$ and $J>1$, respectively, with $J=1$ being a saddle point associated with the phase transition mentioned above. If, instead, we restrict the analysis to the minima of $H(J)$, a topological change is shown to occur only at $J=1$. Moreover, at $J=2$ (with $H$ also equal to $-3J_1$) the system displays a spin structure of high symmetry, in which case the angle $\theta$ between the three spins of the unit cell matches $2\pi/3$ (zero magnetization). In addition, we identify that a topological change takes place at $H=0$, but the associated configurations do not satisfy the energy minimization condition. Finally, we also show that the presence of a magnetic field $h$ does not destroy neither the transition nor the topological change, but shifts the transition point to $J=1+(h/2)$.\\ 1) L. Caiani, L. Casetti, C. Clementi, and M. Pettini, Phys. Rev. Lett. {\bf 79}, 4361 (1997); L. Casetti, M. Pettini, and E.G.D. Cohen, Phys. Rep. {\bf 337}, 237 (2000).\\ 2) A. S. F. Ten\'orio, Master's thesis, UFPE, 2005.\\ 3) R. Franzosi and M. Pettini, Phys. Rev. Lett. {\bf 92}, 060601 (2004).\\ 4) S. Risau-Gusman $et$ $al$., Phys. Rev. Lett. {\bf 95}, 145702 (2005).\\ 5) L. Casetti, M. Pettini, and E. G. D. Cohen, J. Stat. Phys. {\bf 111}, 1091 (2003).\\ 6) A. C. Ribeiro Teixeira and D. A. Stariolo, Phys. Rev. E {\bf 70}, 016113; I. Hahn and M. Kastner, Phys. Rev. E {\bf 72}, 056134 (2005).}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Santos, F.A.N. and Coutinho-Filho, M.D. and Montenegro-filho, R.R. and Leandro, E.S.G.}, biburl = {https://www.bibsonomy.org/bibtex/253a6dd7e6ceb6972fb506f49673a8a3b/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {4886fb43755743f7c0609fc1683ccbad}, intrahash = {53a6dd7e6ceb6972fb506f49673a8a3b}, keywords = {frustration mean-field model phase statphys23 topic-2 topology transitions xy}, month = {9-13 July}, timestamp = {2007-06-20T10:16:21.000+0200}, title = {Topology, symmetry and phase transitions on the mean-field frustrated XY model at zero temperature}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=476}, year = 2007 }