Series 4 Graph expansion of the Ising model

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Statistical Physics 3 October 27th, 2010

Series 4 Graph expansion of the Ising model

We consider the Hamiltonian fo the Ising model in D dimensions with zero magnetic field:

H = −J ∑

<i, j>

σ _i σ _j

We suppose to have a lattice of N spins with random connectivity, i.e. the lattice is given by a matrix (A _{i j} ) with A _{i j} = 1 if a link exists and A _{i j} = 0 otherwise. The sum is taken over all the pairs of links.

a) Using the identity exp(Lσ _i σ _j ) = cosh(L) + σ _i σ _j sinh(L) with σ _i , σ _j = ±1, show that the parti- tion function can be written as:

Z = (cosh L) ^M ∑

σ

1

=±1 ∑

σ

2

=±1

. . . ∑

σ

N

=±1 ∏

<i j>

(1 + uσ _i σ _j )

with L = β J, u = tanh(L) and M the total number of links.

b) Develope the above product in powers of u and give a graphic interpretation to each term. In particular find the condition for which a term is not zero.

c) Analyse the behaviour of the partition function at high temperature.

d) In the case of a lattice in 1 dimension with links between nearest neighbors and no boundary condition, show that any power of u is zero, except the term u ⁰ . You should find the result of the previous series.

e) Solve the same problem in 1D with periodic boundary conditions.

We remark that in 2 dimensions on a square lattice it is possible to enumerate all the non zero terms of item b). Instead in 3 dimensions, it is possible to prove that it is not possible to enumerate them exactly. See Statistical Physics I by Toda, Kubo et Saitˆo and Statistical Mechanics by K. Huang for more details.

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Series 4 Graph expansion of the Ising model

Statistical Physics 3 October 27th, 2010

Series 4 Graph expansion of the Ising model

We consider the Hamiltonian fo the Ising model in D dimensions with zero magnetic field:

H = −J ∑

<i, j>

σ i σ j

We suppose to have a lattice of N spins with random connectivity, i.e. the lattice is given by a matrix (A i j ) with A i j = 1 if a link exists and A i j = 0 otherwise. The sum is taken over all the pairs of links.

a) Using the identity exp(Lσ i σ j ) = cosh(L) + σ i σ j sinh(L) with σ i , σ j = ±1, show that the parti- tion function can be written as:

Z = (cosh L) M ∑

σ

=±1 ∑

σ

=±1

. . . ∑

σ

=±1 ∏

<i j>

(1 + uσ i σ j )

with L = β J, u = tanh(L) and M the total number of links.

b) Develope the above product in powers of u and give a graphic interpretation to each term. In particular find the condition for which a term is not zero.

c) Analyse the behaviour of the partition function at high temperature.

d) In the case of a lattice in 1 dimension with links between nearest neighbors and no boundary condition, show that any power of u is zero, except the term u 0 . You should find the result of the previous series.

e) Solve the same problem in 1D with periodic boundary conditions.

1

σ _i σ _j

We suppose to have a lattice of N spins with random connectivity, i.e. the lattice is given by a matrix (A _{i j} ) with A _{i j} = 1 if a link exists and A _{i j} = 0 otherwise. The sum is taken over all the pairs of links.

a) Using the identity exp(Lσ _i σ _j ) = cosh(L) + σ _i σ _j sinh(L) with σ _i , σ _j = ±1, show that the parti- tion function can be written as:

Z = (cosh L) ^M ∑

(1 + uσ _i σ _j )

d) In the case of a lattice in 1 dimension with links between nearest neighbors and no boundary condition, show that any power of u is zero, except the term u ⁰ . You should find the result of the previous series.