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 P VI.F Summing over Phantom Loops The high temperature series can be approximately summed so as to reproduce the Gaussian model. This correspondence provides a better understanding of why Gaussian behavior is applicable in high dimensions, and also prepares the way for the exact summa-tion of the series in two dimensions i n the next section. The high temperature series for the partition function of the Ising model on a d–dimensional hypercubic lattice is obtai ned from Z = e K hiji σiσj = 2N coshdN K × S, (VI.35) {σi} where S is the sum over all allowed graphs on the lattice, each weighted by t ≡ tanh K raised to the power of the number of bonds in the graph. The allowed g raphs have an even number of bonds per site. The simplest graphs have the t opol o gy of a single closed loop. There are also graphs composed of disconnected closed loops. Keeping in mind the cumulant expansion, we can set Ξ = sum over contribution of all graphs wi th one loo p, (VI.36) and introduce another sum, S ′ = exp (Ξ) =1 + Ξ + 1(Ξ)2 +1(Ξ)3 + 2 6 ··· (VI.37) =1 + (1 loop graphs) + (2 loop graphs) + (3 loop graphs) + .··· Despite their similarities, the sums S and S ′ are not identical: T here are ambiguities associated with loops that intersect at a single site, which will be discussed more fully in the next section. More impo rtantly, S ′ includes additional graphs where a particular bond contributes more than once, while in the original sum S, each lattice bond contributes a factor of 1 or t. This arises because after raising Ξ t o a power ℓ, a particular bond may contribute up to ℓ times for a factor of tℓ . In the spirit of the approximation that includes multiple appearances of a bond, we shall allow additional closed paths in Ξ, in w hich a particular bond is traversed more than once in completing the loop. Qualitatively, S is the partition function of a gas of self–avoiding polymer loops with a monomer fugacity of t. The self–avoiding constraint is left o ut in the partiti on function S ′ , which thus corresponds to a gas of phantom polymer loops, which may pass through each other wi th impunity. 99X X Y X Loops o f various shapes can be constructed from closed random walks on the lattice, and the corresponding free energy of phantom loops is ln S ′ = all closed random walks on the latt ice × tlength of walk X tℓ (VI.38) =N (number of closed walks of ℓ steps starting and ending at 0). ℓ ℓ Note that extensivity is guaranteed since (up to boundary effects) the same loop can be started from any point on the lat tice. The overall factor of 1/ℓ accounts for the ℓ possible starting poi nts for a loop of length ℓ. A transfer matrix method can be used to count all possible (phantom) random walks on the lattice. Let us introduce a set of N ×N matrices, hi|W (ℓ)|ji ≡ number of walks from j to i in ℓ steps, (VI.39) in terms of which eq.(VI.38) b ecomes ln S ′ 1 X tℓ N =2 ℓ h0|W (ℓ)|0i. (VI.40) ℓ The additional factor of 2 arises since the same loop can be traversed by two random walks moving in opposite directions. Similarly, the spin–spin correlation function 1 hσ(0)σ(r)i = σ(0)σ(r) (1 + tσiσj), (VI.41) Z {σi} hiji is related to t he sum over all paths connecting the points 0 and r on the lattice. In addition to the simple paths that directly connect t he two points, t here are disconnected graphs that contain additional closed loops. In the same approximation of allowing all intersections between paths, the partit ion function S ′ can be factored out of t he numerator and denominator of eq.(VI.41), and hσ(0)σ(ri ≈ tℓ hr|W (ℓ)|0i. (VI.42) ℓ The counting of phantom paths on a lattice is easily accomplished by taking advantage of their Markovian property. This is the property that each step of a random walk proceeds from its last locat ion and is independent of its previous steps. Hence, the number of walks can be calculated recursively. First, note that the any walk from 0 to r in ℓ steps can be 100X X ′ accomplished as a walk from 0 to some other point r in ℓ − 1 steps, followed by a single ′ step from r to r. Summing over all possible locations of the intermediate p oint leads to ′ ′ hr|W (ℓ)|0i = r hr|W (1)|r i × h r |W (ℓ −1)|0i (VI.43) ′ = hr|T W (ℓ − 1)|0i, where the sum corresponds to the product of two matrices, and we have defined T ≡ W (1). The recursion process can be continued and W (ℓ) = T W (ℓ −1) = T2W (ℓ − 2 )2 = = Tℓ . (VI.44) ··· Thus al l latti ce random walks are generated by the transfer matrix T , whose elements are n ′ ′ 1 if r and r are nearest neighbors hr|T |r i = 0 otherwise . (VI.45) (It is also called the adjacency, or connectivity matrix.) For example in d = 2, ′ ′ x ′ ′ ′ ′ ′hx, y|T | , y i = δy,y (δx,x +1 + δx,x −1) + δx,x (δy,y +1 + δy,y ′−1) , (VI.46) and successive actions of T on a walker starting at the origin x, y >= δx,0δy,0, generate |the pat terns 0 0 1 0 0 0 0 0 0 1 0 0 2 0 2 0 0 0 1 0 0 0 T −→ 1 0 0 1 1 0 T −→ 1 0 0 2 4 0 0 2 1 0 T −→ · · · . 0 0 1 0 0 The value at each site is the number of walks ending at that point after ℓ steps. Various properties of random walks can be deduced from dia gonalizing the matrix T . Due to t he translational symmetry of the lattice, this is achieved in the Fourier basis hr|qi = eiq·r/√N. For example in d = 2, starting from eq (VI.46), it can be checked that ′ ′ ′ ′ hx, y|T |qx, qyi = hx, y|T |x , y ihx , y |qx, qyi′ ′ x ,y h    i =1 eiqy y eiqx(x+1) + eiqx (x−1) + eiqx x eiqy