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Random field Ising model with finite connectivity

F. F. Doria, R. Erichsen Jr., D. Dominguez, M. Gonzlez, S. Magalhes

UFRGS Universidade Federal do Rio grande do Sul

Universidade Estatal del Sur de Manabi

UFF Universidade Federal Fluminense

The model

The random field Ising model Hamiltonian can be written like this

(1)

where

The probability distribution for the connectivity is given as

, (2)

The model

whereas the local random field , follows a

bimodal distribution

(3)

or a gaussian distribution

(4)

The system properties , when in equilibrium with a thermal bath at temperature are derived from the free-energy

The model

whereas the local random field , follows a

bimodal distribution

(5)

where the brackets stand for the disorder

average, is the partition

function and the N-coordinate vector

represents the state of the system.

In order to average over the quenched disorder we follow the replica method.

The replica procedure

The replicated partition function becomes

(6)

is a N-dimensional vector which means the state of the N sites in the replica n, while is a n-dimensional vector which means the state of the n-replicas in the site i.

The replica procedure

After averaging over , we follow the usual procedure, consequently, the free energy for becomes

(7)

The replica procedure

in which is the fraction of sites where the

replica configuration is realized. We search for

solutions satisfying the replica symmetric

ansatz, that is equivalent under permutations of

replicas

, (8)

and we obtain a saddle-point equation for the

local fields distribution

The replica procedure

The average over can be performed using equations (3) and (4). The subsequent result allows a recursive calculation of the local-field distribution . Given , the observables and can be obtained from

(9)

The replica procedure

e

The free-energy can be calculated by introducing the Replica symmetry ansatz, eq. (8). Then we get

(10)

(11)

The replica procedure

where are poissonian weights with average c.

(7)

Results

Results

Results

Results