MIT OpenCourseWare http://ocw.mit.edu 8.334 Statistical Mechanics II: Statistical Physics of Fields Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.X VI.G Exact free energy of the Square Lattice Ising model As indicated in eq.(VI.35), the Ising partiti on function is related to a sum S, over collections of paths on the lattice. The allowed graphs for a square lattice have 2 or 4 bonds per site. Each bond can appear only once in each graph, contri buting a factor of t ≡ tanh K. While it is tempting to replace S with the exactly calculable sum S ′ , of all phantom loops of random walks on the lattice, this leads to an overestimation of S. The differences between the two sums arise from intersections of random walks, and can be divided into two categories: (a) There is an over-counting of graphs which intersect at a site, i.e. with 4 bonds through a point. Consider a graph composed of two loops meeting at a site. Since a walker entering the intersection has three choices, this graph can be represented by three distinct random walks. One choice leads to two disconnected l oops; the other two are single loops with or without a self–crossing in the walker’s path. (b) The independent random walkers in S ′ may go through a particular lattice bond more than once. Including these constraints amounts to introducing interactions between paths. The resulting interacting random walkers are non–Markovian, as each step is no longer inde-pendent of previous ones and of other walkers. While such interacting wa lks are not in general amenable to exact treatment, in two dimensions an interesting topological property allows us to make the following assertion: S = collections of loops of random walks with no U turns (VI.55) × tnumber of bonds × (−1)number of crossings. The negative signs for same terms reduce the overestimate and render the exact sum. Proof: We shall deal in turn with the two problems mentioned above. (a) Consider a graph w ith many intersections and focus on a particular one. A walker must enter and leave such an intersection twice. This can be done in three ways only o ne of which involves the path of the walker crossing itself ( w hen the walker proceeds straight through the intersection). This configuration carries an additional factor of (-1) according to eq.(VI.55). Thus, independent of other crossings, these three configurations sum up to contribute a factor of 1. By repeating this reasoning at each intersection, we see that the over-counting problem is removed and the sum over all possible ways of tracing the graph leads to the correct factor of one. 106X X X h i X X X (b) Consider a bond that is crossed by two walkers (or twice by the same walker). We can imagine the bond as an avenue with two sides. For each configuration in which the two paths enter and leave on the same side of the avenue, there is another one in which the paths go to the opposite side. The latter involves a crossing of paths and hence carries a minus sign with respect to the former. The two possibilities thus cancel out! The reasoning can be generalized to multiple passes through any bond. The only exception is when the doubled bond i s created a s a result of a U–turn. This is why such backward steps are explicitly excluded from eq.(VI.55). Let us label random walkers wi th no U–turns, and weighted by (−1)number of crossings, as RW∗ s. T hen as in eq.(VI.37) the terms in S can be organized as S = (RW ∗ s with 1 loop) + (RW ∗ s with 2 loops) + (RW ∗ s with 3 loops) + ··· = exp (RW ∗ s with 1 loop) . (VI.56) The exponentiation of the sum is justified, since the only interaction between RW∗s is the sign related to their crossings. As two RW∗ loops always cross an even number of times, this is equivalent to no interaction at al l. Using eq.(VI.35), the full Ising free energy is calculated as ln Z = N ln 2 + 2N ln cosh K +  RW ∗ s with 1 loop × t# of bonds . (VI.57) Organizing the sum in terms of the number of bonds, and taki ng advantage of the trans-lational symmetry of the lattice (up to corrections due to boundaries), ln Z   ∞ tℓ N = ln 2 cosh2 K + ℓ h0|W ∗ (ℓ)|0i , (VI.58) ℓ where h0|W ∗ (ℓ)|0i =number of closed loops of ℓ steps, with no U turns, from 0 to 0 (VI.59) × (−1)# of crossings. The absence of U–turns, a local constraint, does not complicate the counting of loops. On the other hand, the number of crossings is a function of the complete configuration of the loop and is a non–Markovian property. Fortunately, in two dimensions i t is possible to obtain the parity of the number of crossings from local considerations. The first step is to 107X      X   X X construct the loops from directed random walks, indicated by placing an arrow along the direction that the path is t raversed. Since any l oop can be traversed in two directions, h0|W ∗ (ℓ)|0i =21 directed RW ∗ loops of ℓ steps, no U t urns, from 0 to 0 × (−1)nc , (VI.60) where nc is the number of self–crossings of the loop. We can now take advantage of the follow ing topological result: Whitney’s Theorem: The number of self–crossings of a planar loop is related to the total angle Θ, through which the tangent vector turns in going around the loop by Θ (nc)mod 2 = 1 + . (VI.61) 2π mod 2 This theorem can be checked by a few examples. A single loop corresponds to Θ = ±2π, while a single intersection results in Θ = 0. Since the total angle Θ, is the sum of the angles through which the walker turns at each step, the parity of crossings can be o bta ined using local informatio n a lone as (−1)nc = e iπnc = ex p iπ 1 + Θ = −e2 i Pℓj=1 θj , (VI.62) 2π where