The Ising and percolation models

Date: 2015-10-07; view: 406.

Random curves and lattice models

In this section, we introduce the lattice models which can be interpreted in terms of random non-intersecting paths on the lattice whose continuum limit will be described by SLE.

The prototype is the Ising model. It is most easily realised on a honeycomb lattice (see Fig. 1). At each site r is an Ising ‘spin' s (r) which takes the values ±1. The partition function is

(1)

where x = tanh βJ, and the sum and product are over all edges joining nearest neighbour pairs of sites. The trace operation is defined as , so that Tr s (r)ⁿ = 1 if n is even, and 0 if it is odd.

(8K)

Fig. 1. Ising model on the honeycomb lattice, with loops corresponding to a term in the expansion of (1). Alternatively, these may be thought of as domain walls of an Ising model on the dual triangular lattice.

At high temperatures (βJ 1) the spins are disordered, and their correlations decay exponentially fast, while at low temperatures (βJ 1) there is long-range order: if the spins on the boundary are fixed say, to the value +1, then s (r) ≠ 0 even in the infinite volume limit. In between, there is a critical point. The conventional approach to the Ising model focuses on the behaviour of the correlation functions of the spins. In the scaling limit, they become local operators in a quantum field theory (QFT). Their correlations are power-law behaved at the critical point, which corresponds to a massless QFT, that is a conformal field theory (CFT). From this point of view (as well as exact lattice calculations) it is found that correlation functions like s (r₁)s (r₂) decay at large separations according to power laws |r₁ − r₂|^−2x: one of the aims of the theory is to obtain the values of the exponents x as well as to compute, for example, correlators depending on more than two points.

However, there is an alternative way of thinking about the partition function (1), as follows: imagine expanding out the product to obtain 2^N terms, where N is the total number of edges. Each term may be represented by a subset of edges, or graph , on the lattice, in which, if the term xs (r)s (r′) is chosen, the corresponding edge (rr′) is included in , otherwise it is not. Each site r has either 0, 1, 2, or 3 edges in . The trace over s (r) gives 1 if this number is even, and 0 if it is odd. Each surviving graph is then the union of non-intersecting closed loops (see Fig. 1). In addition, there can be open paths beginning and ending at a boundary. For the time being, we suppress these by imposing ‘free' boundary conditions, summing over the spins on the boundary. The partition function is then

(2)

where the length is the total of all the loops in . When x is small, the mean length of a single loop is small. The critical point x_c is signalled by a divergence of this quantity. The low-temperature phase corresponds to x > x_c. While in this phase the Ising spins are ordered, and their connected correlation functions decay exponentially, the loop gas is in fact still critical, in that, for example, the probability that two points lie on the same loop has a power-law dependence on their separation. This is the dense phase.

The loops in may be viewed in another way: as domain walls for another Ising model on the dual lattice, which is a triangular lattice whose sites R lie at the centres of the hexagons of the honeycomb lattice (see Fig. 1). If the corresponding interaction strength of this dual Ising model is (βJ) , then the Boltzmann weight for creating a segment of domain wall is e^−2(βJ) . This should be equated to x = tanh (βJ) above. Thus, we see that the high-temperature regime of the dual model corresponds to low temperature in the original model, and vice versa. Infinite temperature in the dual model ((βJ) = 0) means that the dual Ising spins are independent random variables. If we colour each dual site with s (R) = +1 black, and white if s (R) = −1, we have the problem of site percolation on the triangular lattice, critical because for that problem. Thus, the curves with x = 1 correspond to percolation cluster boundaries. (In fact in the scaling limit this is believed to be true throughout the dense phase x > x_c.)

So far we have discussed only closed loops. Consider the spin–spin correlation function

(3)

where the sites r₁ and r₂ lie on the boundary. Expanding out as before, we see that the surviving graphs in the numerator each have a single edge coming into r₁ and r₂. There is therefore a single open path γ connecting these points on the boundary (which does not intersect itself nor any of the closed loops). In terms of the dual variables, such a single open curve may be realised by specifying the spins s (R) on all the dual sites on the boundary to be +1 on the part of the boundary between r₁ and r₂ (going clockwise) and −1 on the remainder. There is then a single domain wall connecting r₁ to r₂. SLE describes the continuum limit of such a curve γ.

Note that we could also choose r₂ to lie in the interior. The continuum limit of such curves is then described by radial SLE (Section 3.6).