We consider a system of N = L² Ising spins placed on a square lattice. Thus the spin placed at the (k,ℓ) lattice point is denoted by σ_kℓ, and one has σ_kℓ = ±1, k,ℓ {1,…,L}. The hamiltonian H({σ}) is given by

(1)

where we have assumed periodic boundary conditions:

(2)

We set h = 0 from now on. Then we have seen in section 5.11 that the partition function can we written

(3)

where

(4)

and the sum runs over all diagrams that can be drawn on the lattice, such that (i) each bond appears at most once, and (ii) at each vertex meet zero, two or four bonds. In this expression || is the number of bonds that appear on the diagram . Then this expression can be written in the form

(5)

where g_r is the total number of diagrams satisfying the two rules above and containing exactly r bonds.

We shall now evaluate this expression by transforming it into a sum over loops. The resulting expression will then be evaluated by reducing it to a random-walk problem.

A generic diagram can be considered as a collection of loops. A loop is the trajectory of a walk that starts and ends on the same site. However, the decomposition of a diagram into loops is ambiguous if there are self-intersections, i.e., if there are vertices where four bonds meet. Let us consider, e.g., the diagram in figure 1. It can be considered as the collection of two loops (which meet at one vertex) (case (a)), or as a single loop whose path does intersect itself (case (c)) or does not (case (b)).

In order to obtain a nonambiguous sum, we assign to each diagram a factor (-1)ⁿ, where n is the number of intersections. In this situation, the contribution of case (c) will be opposite to that of case (b), and they cancel out, leaving only the contribution of case (a). One can easily realize, then, that with this convention, the contribution of diagrams in which three bonds meet at a vertex identically vanishes, as can be seen in figure 2.

In this way the sum over all diagrams is reduced to a sum over all loops, in which each loop appears with a weight proportional to (-1)ⁿ, where n is the number of self-intersections. Notice that we do not allow vertices connected to only one bond, and, therefore, the possibility that a walker gets back in its steps.

Now we can express the number of self-intersections of a loop by the following trick. It is well known that the total angle through which the tangent angle to the trajectory of a walker performing a loop turns around is given by 2π(ℓ + 1), where the parity of ℓ is equal to the parity of the number of intersections n. Thus, if we assign a factor e^iφ∕2 to each lattice point with turning angle φ, then we shall have at the end of the loop a factor (-1)^ℓ+1 = (-1)ⁿ⁺¹, where n is the number of intersections. With this counting, each diagram made up of s loops will give a contribution proportional to (-1)^s+n. Thus we have to multiply this contribution by (-1)^s in order to have the required sign in equation (5).

In order to count up the angle it is convenient to deal with directed loops. Let us denote by f_r the sum over all undirected loops consisting of r bonds (taking into account the factors t^r and e^iφn∕2). Then the sum over all double loops of ℓ bonds will be given by

taking into account the possible permutations of the loops. Thus we have in general

(6)

In going from undirected to directed loops each loop is encountered twice, thus if we denote by v_r the sum of the contributions of directed loops with r bonds, we have

(7)

We shall now evaluate v_r. Given a lattice point (k,ℓ), let us denote the possible directions as follows:

{

Then it is possible to write a linear recursion relation for W_r:

(8)

The transition matrix T = (T_{kℓν,k′ℓ′ν′}) has the expression

(9)

where

(10)

where

(11)

and ω^* is the complex conjugate of ω.

The connection between the weights W and the loop contributions v_r is given by

(12)

where

(13)

The factor 1∕r comes from the fact that a single diagram with r bonds can be obtained from r different walks, depending on the starting point. Now, from equation (9), we have

(14)

where λ_i are the eigenvalues of the matrix T. From this equation, taking into account equations (7) and (12), we obtain

(15)

Thus the problem boils down to the diagonalization of the matrix T. One can see from equation (9) that T depends only on the differences in the indices k,ℓ. It can thus be diagonalized by a Fourier transformation. We set

(16)

We then find that

(17)

where

(18)

Thus, for given values of (m,n), we have

(20)

where the product runs over L consecutive values of m and of n.

Setting p = 2πm∕L and q = 2πn∕L, the Helmholtz free energy is given by

Let us consider the contribution of the integral. The minimum value of the integrand is reached for p = q = 0, and is given by

The argument of the logarithm vanishes for

which corresponds to the transition temperature T_c given by equation (5.72):

In order to understand the behavior of F in the neighborhood of this temperature, let us introduce τ = t - t_c and expand the integrand for small values of τ and of p,q. One has

where c₁ and c₂ are constants. Integrating, one obtains

where a > 0 is a constant. The specific heat C is proportional to -d²F∕dτ². Thus we have

(22)

indicating that the specific heat exhibits a logarithmic divergence at the critical temperature.

The evaluation of the spontaneous magnetization m₀ = proceeds in a similar way [6], starting, e.g., from the relation

One obtains m₀ = 0 for t < t_c given above, and

(23)

for t > t_c, i.e., below the transition temperature. Thus, for small positive values of τ = t - t_c one has

(24)

where the exponent β is given by

(25)

The connection between the Ising model and the statistics of loops can be interpreted more deeply as the equivalence between the model and a system of noninteracting fermions. This correspondence is exploited by Schulz, Mattis and Lieb [4] in their solution of the Ising model.

References

References