Volume 135, number 6,7 PHYSICSLETTERS A 6 March 1989 DYNAMICS OF THE ISING MODEL IN A CAYLEY TREE N.R. DA SILVA and J.M. SILVA Departamento de Fisica, UniversidadeFederal da Paraiba, (58.059) Jo~o Pessoa, PB, Brazil Received29 November 1988;acceptedfor publication 19 December 1988 Communicatedby V.M. Agranovich The relaxationof the central spin in a Cayleytree is calculatedby the Glaubermethod. The effective time relaxationgrowsfor temperaturesbelowthe critical temperaturebut not divergence is observed. The partition function in zero field for a system of Ising variables in a Cayley tree can be calculated ex- actly [ 1 ]. The fractal geometry of its structure allows a phase transition at finite temperature in spite of the partition function being analytical at any tem- perature. More precisely, for temperatures below T¢= 2J{ln [ ~/( ~,- 2 ) ] }- ', where J is the interaction coupling and ;~ the coordination number, there is a discontinuity in the specific heat and the central spin shows a spontaneous magnetization (1 +ytgh flJ)~- (1 -ytgh flJ)Y (ao)=. (l+ytghflj)~+(l_ytghflj) ~ , (1) where y is the largest positive root of the equation l+y (l+ytghflJ~ ~-I l-y- l-ytghflJ] ' (2) which is zero for T~> T¢ and non-zero for T< T~. The ratio between the probabilities of finding the central spin in the states ao = 1 and ao = - 1 is given by Po(1) _ (1 +ytgh flflJj)'. (3) Po(-l) \l-ytgh Recent works [ 2,3 ] have shown an anomalous be- haviour for the dynamical transition in hierarchical Ising systems. Thus it is interesting to study the dy- namics of the Cayley tree, specially near the transition. The method used here to write the dynamical equations is essentially the same as that of Glauber [ 4 ], with the difference that the dynamical variables will no longer be the spins in the sites, but the prod- uct 0a=~,~s of the spins at the ends of each bond, and the central spin ao. To each configuration of spins {at}, corresponds a configuration {ao, {0a}}. The master equation for the probability P({ao, {0,~}}; t) of a configuration {t~o, {0~)} at the instant t will be d P({ao, {0.)); t) = -Y, W,~(O,)P({ao, {0,}}; t) Ot +Z w.(-ao)P({ao, 0, ..... -0 . .... };t) Ot - Wo(ao)P({ao, {0,}}; t) + Wo( - ao)P({- ao, {0.}}; t). (4) W~ (0,~) is the transition rate of 0~ to - 0~, keeping all other bonds and ~o fixed. Wo(ao) is the analogous rate for the central spin. The advantage of taking the bonds as basis to de- scribe the states is that the Hamiltonian becomes de- coupled and, as there are no closed loops in a Cayley tree, the sum over the states is not restricted. These facts will make it easy to write an expression for w.(o.). The question which can be asked is whether the flipping of a sole bond requires the flipping of all the spins in the next ramifications. If these flippings oc- cur through the motion of a domain wall, there would 0375-9601/89/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division ) 373