,
Modulated Phase of a Potts Model with Competing Interactions on a Cayley Tree
Abstract Book of the XXIII IUPAP International Conference on Statistical Physics, Genova, Italy, (9-13 July 2007)
Аннотация
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).
