%0 Book Section %1 statphys23_0057 %A Ganikhodjaev, N.N. %A Mukhamedov, F.M. %A Pah, C.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 antiphase diagram model modulated phase potts statphys23 topic-2 %T Modulated Phase of a Potts Model with Competing Interactions on a Cayley Tree %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=57 %X Considering an Ising system with ferromagnetic nearest-neighbor($nn$)interactions and competing (antiferromagnetic)next-nearest-neighbor($nnn$) interactions on a Cayley tree, Vannimenus V was able to find new modulated phases, in addition to the expected paramagnetic($P$) and ferromagnetic($F$) ones.These new phases consist in a period-four one (so-called antiphase and denoted $<2>$), and in a complex set of higher-order commensurate or incommensurate modulated phases (denoted $M$). We consider a Potts model with competing interactions and describe it phase diagram.The model considered consists of Potts spins ($ S_i=1,2,3$) on a Cayley tree of branching ratio 2, so that every spin has three nearest neighbors. Two kinds of bonds are present:$nn$ interactions of strength $J_1$ and $nnn$ interactions $J$, these being restricted to spins belonging to the same branch of the tree. In order to set up our basic equations in a recurrence scheme relating the partition function of an $N$-generation tree to the partition functions of its subsystems, we should take into account the partial functions for all possible configurations of the spins in two successive generations. We identify $Z_N(S_1,S_2,S_3)$ as the partition function of a branch of an $N$-generation tree where the spin in the last generation is $S_2$ and the two spins in the preceding one are $S_1$ and $S_3$. We define the following variables: eqnarray* u_1&=&Z_N(1,1,1); \\ u_2&=&Z_N(2,1,1);\\ u_3&=&Z_N(3,1,1); \\ u_4&=&Z_N(1,2,1); \\ u_5&=&Z_N(1,2,3). eqnarray* It is straightforward to establish the following recursive equations: equation arraylcr u_1^&=&þeta_1 u_1+2 u_2^2, \\ u_2^\prime&=& u_3+ u_4+u_5^2, \\ u_3^\prime&=& u_1+(þeta+1) u_2^2, \\ u_4^\prime&=& þeta_1 u_3+ u_4+ u_5^2, \\ u_5^\prime&=& u_3+ u_4+ u_5^2, array equation where the prime denotes recurrence image with \þeta=\exp(J/T); þeta_1=\exp(J_1/T).\ For discussing the phase diagram, the following choice of reduced variables is convenient: eqnarray* x&=&2u_2+u_3+u_5u_1+u_4;\\ y_1&=&u_1-u_4u_1+u_4; \\ y_2&=&u_2-u_3u_1+u_4; \\ y_3&=&u_2-u_5u_1+u_4. \\ eqnarray* Equations (1) yield $$x^\prime=12þeta_1D((þeta+1)x+2-y_1-y_2-y_3)^2+$$ $$+P(y_1,y_2,y_3);$$ $$y^\prime_1=2D(þeta+x)(y_1+y_2+y_3);$$ equation rc equation $$y^\prime_2=-1þeta_1Dy_1+y_2-y_32+(þeta+1)x-$$ $$-(þeta-1)(y_2-y_3);$$ $$y^\prime_3=1þeta_1D(þeta-1)(y_3-y_2)2+(þeta+1)x-$$ $$-2y_1-(þeta+1)(y_2+y_3);$$ where \D=(þeta+x)^2+(y_1+y_2+y_3)^2\ and $$P(y_1,y_2,y_3)=3y^2_1+(4þeta^2-4þeta+3)y_2^2+$$ $$+(3þeta^2-4þeta+4)y^2_3+2(2þeta+1)y_1y_2+$$ $$+2(þeta+2)y_1 y_3-2(2þeta^2-7þeta+2)y_2y_3;$$ The average magnetization $ m $ for the $N^th$ generation is given by $$ m = 2(1+x)(y_1+y_2 +y_3 ) (1+x)^2+(y_1+y_2+y_3)^2. $$ The resultant phase diagram is shown in Fig. 1. As one can see the domain of $M$ phase is non-convex (compare V) and it contains the peninsula of new phase,which we call a quasi-paramagnetic ($ QP $) phase,since in this case the average magnetization $ m $ is also equal to $ 0$. These results differ from those obtained V and MTA for Ising model.\\ Acknowledgments\par This research is funded by the SAGA Fund P77c by the Ministry of Science, Technology and Innovation (MOSTI) through the Academy of Sciences Malaysia(ASM).\\ V J.Vannimenus, Z.Phys.B 43,141 (1981). \\ MTA A.M.Mariz,C.Tsallis,E.L.Albuquerque,J.Stat.Phys.40,577 (1985).

@incollection{statphys23_0057, abstract = {Considering an Ising system with ferromagnetic nearest-neighbor($nn$)interactions and competing (antiferromagnetic)next-nearest-neighbor($nnn$) interactions on a Cayley tree, Vannimenus [V] was able to find new modulated phases, in addition to the expected paramagnetic($P$) and ferromagnetic($F$) ones.These new phases consist in a period-four one (so-called antiphase and denoted $<2>$), and in a complex set of higher-order commensurate or incommensurate modulated phases (denoted $M$). We consider a Potts model with competing interactions and describe it phase diagram.The model considered consists of Potts spins ($ S_i=1,2,3$) on a Cayley tree of branching ratio 2, so that every spin has three nearest neighbors. Two kinds of bonds are present:$nn$ interactions of strength $J_1$ and $nnn$ interactions $J$, these being restricted to spins belonging to the same branch of the tree. In order to set up our basic equations in a recurrence scheme relating the partition function of an $N$-generation tree to the partition functions of its subsystems, we should take into account the partial functions for all possible configurations of the spins in two successive generations. We identify $Z_N(S_1,S_2,S_3)$ as the partition function of a branch of an $N$-generation tree where the spin in the last generation is $S_2$ and the two spins in the preceding one are $S_1$ and $S_3$. We define the following variables: \begin{eqnarray*} u_1&=&\sqrt{Z_N(1,1,1)}; \\ u_2&=&\sqrt{Z_N(2,1,1)};\\ u_3&=&\sqrt{Z_N(3,1,1)}; \\ u_4&=&\sqrt{Z_N(1,2,1)}; \\ u_5&=&\sqrt{Z_N(1,2,3)}. \end{eqnarray*} It is straightforward to establish the following recursive equations: \begin{equation} \begin{array}{lcr} u_1^\prime &=&\theta_1 [\theta u_1+2 u_2]^2, \\ u_2^\prime&=& [\theta u_3+ u_4+u_5]^2, \\ u_3^\prime&=& [ u_1+(\theta+1) u_2]^2, \\ u_4^\prime&=& \theta_1[ u_3+ \theta u_4+ u_5]^2, \\ u_5^\prime&=& [ u_3+ u_4+ \theta u_5]^2, \end{array} \end{equation} where the prime denotes recurrence image with \[\theta=\exp(J/T); \theta_1=\exp(J_1/T).\] For discussing the phase diagram, the following choice of reduced variables is convenient: \begin{eqnarray*} x&=&\frac{2u_2+u_3+u_5}{u_1+u_4};\\ y_1&=&\frac{u_1-u_4}{u_1+u_4}; \\ y_2&=&\frac{u_2-u_3}{u_1+u_4}; \\ y_3&=&\frac{u_2-u_5}{u_1+u_4}. \\ \end{eqnarray*} Equations (1) yield $$x^\prime=\frac{1}{2\theta_1D}[((\theta+1)x+2-y_1-\theta y_2-y_3)^2+$$ $$+P(y_1,y_2,y_3)];$$ $$y^\prime_1=\frac{2}{D}(\theta+x)(\theta y_1+y_2+y_3);$$ \begin{equation} \label{rc} \end{equation} $$y^\prime_2=-\frac{1}{\theta_1D}[y_1+\theta y_2-y_3][2+(\theta+1)x-$$ $$-(\theta-1)(y_2-y_3)];$$ $$y^\prime_3=\frac{1}{\theta_1D}(\theta-1)(y_3-y_2)[2+(\theta+1)x-$$ $$-2y_1-(\theta+1)(y_2+y_3)];$$ where \[D=(\theta+x)^2+(\theta y_1+y_2+y_3)^2\] and $$P(y_1,y_2,y_3)=3y^2_1+(4\theta^2-4\theta+3)y_2^2+$$ $$+(3\theta^2-4\theta+4)y^2_3+2(2\theta+1)y_1y_2+$$ $$+2(\theta+2)y_1 y_3-2(2\theta^2-7\theta+2)y_2y_3;$$ The average magnetization $ m $ for the $N^{th}$ generation is given by $$ m = \frac{2(1+x)(y_1+y_2 +y_3 )} {(1+x)^2+(y_1+y_2+y_3)^2}. $$ The resultant phase diagram is shown in Fig. 1. As one can see the domain of $M$ phase is non-convex (compare [V]) and it contains the peninsula of new phase,which we call a quasi-paramagnetic ($ QP $) phase,since in this case the average magnetization $ m $ is also equal to $ 0$. These results differ from those obtained [V] and [MTA] for Ising model.\\ Acknowledgments\par \noindent This research is funded by the SAGA Fund P77c by the Ministry of Science, Technology and Innovation (MOSTI) through the Academy of Sciences Malaysia(ASM).\\ {[V]} J.Vannimenus, Z.Phys.B 43,141 (1981). \\ {[MTA]} A.M.Mariz,C.Tsallis,E.L.Albuquerque,J.Stat.Phys.40,577 (1985).}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Ganikhodjaev, N.N. and Mukhamedov, F.M. and Pah, C.H.}, biburl = {https://www.bibsonomy.org/bibtex/2bf334be5af195043c71fc8e778e39e32/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {914fcb7e1284df5b418ad49e1e9f7839}, intrahash = {bf334be5af195043c71fc8e778e39e32}, keywords = {antiphase diagram model modulated phase potts statphys23 topic-2}, month = {9-13 July}, timestamp = {2007-06-20T10:16:10.000+0200}, title = {Modulated Phase of a Potts Model with Competing Interactions on a Cayley Tree}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=57}, year = 2007 }