VI.G Exact free energy of the Square Lattice Ising model

Transcription

1 VI.G Exact free energy of the Square Lattice Ising mode As indicated in eq.(vi.35), the Ising partition function is reated to a sum S, over coections of paths on the attice. The aowed graphs for a square attice have or bonds per site. Each bond can appear ony once in each graph, contributing a factor of t tanhk. Whie it is tempting to repace S with the exacty cacuabe sum S, of a phantom oops of random waks on the attice, this eads to an overestimation of S. The differences between the two sums arise from intersections of random waks, and can be divided into two categories: (a) There is an over-counting of graphs which intersect at a site, i.e. with bonds through a point. Consider a graph composed of two oops meeting at a site. Since a waker entering the intersection has three choices, this graph can be represented by three distinct random waks. One choice eads to two disconnected oops; the other two are singe oops with or without a sef crossing in the waker s path. (b) The independent random wakers in S may go through a particuar attice bond more than once. Incuding these constraints amounts to introducing interactions between paths. The resuting interacting random wakers are non Markovian, as each step is no onger independent of previous ones and of other wakers. Whie such interacting waks are not in genera amenabe to exact treatment, in two dimensions an interesting topoogica property aows us to make the foowing assertion: S = coections of oops of random waks with no U turns (VI.55) t number of bonds ( 1) number of crossings. The negative signs for same terms reduce the overestimate and render the exact sum. Proof: We sha dea in turn with the two probems mentioned above. (a) Consider a graph with many intersections and focus on a particuar one. A waker must enter and eave such an intersection twice. This can be done in three ways ony one of which invoves the path of the waker crossing itsef (when the waker proceeds straight through the intersection). This configuration carries an additiona 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 probem is removed and the sum over a possibe ways of tracing the graph eads to the correct factor of one. 106

2 (b) Consider a bond that is crossed by two wakers (or twice by the same waker). We can imagine the bond as an avenue with two sides. For each configuration in which the two paths enter and eave on the same side of the avenue, there is another one in which the paths go to the opposite side. The atter invoves a crossing of paths and hence carries a minus sign with respect to the former. The two possibiities thus cance out! The reasoning can be generaized to mutipe passes through any bond. The ony exception is when the doubed bond is created as a resut of a U turn. This is why such backward steps are expicity excuded from eq.(vi.55). Let us abe random wakers with no U turns, and weighted by ( 1) number of crossings, as RW s. Then as in eq.(vi.37) the terms in S can be organized as S = (RW s with 1 oop) + (RW s with oops) + (RW s with 3 oops) + [ ] = exp (RW s with 1 oop). (VI.56) The exponentiation of the sum is justified, since the ony interaction between RW s is the sign reated to their crossings. As two RW oops aways cross an even number of times, this is equivaent to no interaction at a. Using eq.(vi.35), the fu Ising free energy is cacuated as n Z = n + n cosh K + ( RW s with 1 oop t # of bonds). (VI.57) Organizing the sum in terms of the number of bonds, and taking advantage of the transationa symmetry of the attice (up to corrections due to boundaries), n Z ( ) t = n cosh K + 0 W () 0, (VI.58) where 0 W () 0 =number of cosed oops of steps, with no U turns, from 0 to 0 ( 1) # of crossings. (VI.59) The absence of U turns, a oca constraint, does not compicate the counting of oops. On the other hand, the number of crossings is a function of the compete configuration of the oop and is a non Markovian property. Fortunatey, in two dimensions it is possibe to obtain the parity of the number of crossings from oca considerations. The first step is to 107

3 construct the oops from directed random waks, indicated by pacing an arrow aong the direction that the path is traversed. Since any oop can be traversed in two directions, 1 0 W () 0 = directed RW oops of steps, no U turns, from 0 to 0 ( 1) n c, ( ) Θ (n c ) mod = 1 +. (VI.61) mod [ ( )] ( 1) n c = en c Θ = exp 1 + (VI.60) where n c is the number of sef crossings of the oop. We can now take advantage of the foowing topoogica resut: Whitney s Theorem: The number of sef crossings of a panar oop is reated to the tota ange Θ, through which the tangent vector turns in going around the oop by This theorem can be checked by a few exampes. A singe oop corresponds to Θ = ±, whie a singe intersection resuts in Θ = 0. Since the tota ange Θ, is the sum of the anges through which the waker turns at each step, the parity of crossings can be obtained using oca information aone as i = e θ j j=1, (VI.6) where θ j is the ange through which the waker turns on the j th step, eading to 1 0 W () 0 = directed RW oops of steps, with no U turns, from 0 to 0 ( ) 1 exp oca change of ange by the tangent vector. (VI.63) The ange turned can be cacuated at each site, if we keep track of the directions of arriva and departure of the path. To this end, we introduce a abe µ for the directions going out of each site, e.g. µ = 1 for right, µ = for up, µ = 3 for eft, and µ = for down. We next introduce a set of matrices generaizing eq.(vi.39) as x y, µ W () x 1 y 1, µ 1 = directed random waks of steps, with no U turns,. i θ departing (x j j=1 1, y 1 ) aong µ 1, proceeding aong µ after reaching (x, y ) e (VI.6) 108

4 Thus µ specifies a direction taken after the waker reaches its destination. It serves to excude some paths (arriving aong -µ ), and eads to an additiona phase. As in eq.(vi.3), due to their Markovian property, these matrices can be cacuated recursivey as x y, µ W () x 1 y 1, µ 1 = x y, µ T x y, µ x y, µ W ( 1) x 1 y 1, µ 1 x y,µ = x y, µ T W ( 1) x 1 y 1, µ 1 = x y, µ T x 1 y 1, µ 1, (VI.65) where T W (1) describes one step of the wak. The direction of arriva uniquey determines the nearest neighbor from which the waker departed, and the ange between the two directions fixes the phase of the matrix eement. We can thus generaize eq.(vi.6) to a matrix that keeps track of both connectivity and phase between pairs of sites. The steps taken can be represented diagrammaticay as T =, (VI.66) and correspond to the matrix x y T xy = x, y x + 1, y x, y x + 1, y e 0 x, y x + 1, y e x, y x, y + 1 e x, y x, y + 1 x, y x, y + 1 e 0, x 1, y e 0 x, y x, y x 1, y x, y x 1, y e x, y x, y 1 e 0 x, y x, y 1 e x, y x, y 1 (VI.67) where < x, y x, y > δ x,x δ y,y. Because of its transationa symmetry, the matrix takes a bock diagona form in the Fourier basis, xy q x q y = e i(q xx+q y y) /, i.e. x y, µ T xy, µ xy q x q y = µ T (q) µ x y q x q y. xy (VI.68) Each bock is abeed by a wavevector q = (q x, q y ), and takes the form iq + x i(q x ) ) e 0 e i(q x e i(q ) iq y i(q e e y y + ) e 0 T (q) = i(q x ) iq x i(q x + ). 0 e e e i(q y + ) 0 e i(q y ) iq e e y (VI.69) 109

5 To ensure that a path that starts at the origin competes a oop propery, the fina arriva direction at the origin must coincide with the origina one. Summing over a such directions, the tota number of such oops is obtained from 1 1 ( ) 0 W () 0 = 00, µ T 00, µ = xy, µ T xy, µ = tr T. (VI.70) µ=1 xy,µ Using eq.(vi.58), the free energy is cacuated as [ ] nz ( ) 1 t ( ) 1 T t =n cosh K 0 W () 0 = n cosh K tr ( ) =n cosh 1 K + ( ) =n cosh 1 K + tr n(1 tt ) trn (1 tt But for any matrix M with eigenvaues {λ α }, q (q)). tr nm = n λ α = n λ α = n detm. Converting the sum over q in eq.(vi.71) to an integra eads to α n Z ( ) = n cosh K + iqx 1 te te i(q x ) + 0 te i(q ) x te i(q y+ 1 te iq 1 d q ) y te i(q y ) 0 n det iq 1 te x. te i(q x ) te i(q x+ ) () te iq y te i(q y ) te i(q y ) (VI.7) Evauation of the above determinant is straightforward, and the fina resut is nz ( ) [ ] = n cosh 1 d q K + () n ( ) 1 + t ( ) t 1 t (cos q x + cos q y ). (VI.73) Taking advantage of trigonometric identities, the resut can be simpified to nz = n + 1 dqxdq ( ) y α (VI.71) n [ cosh (K) sinh(k) (cos q x + cos q y ) ]. (VI.7) Whie it is possibe to obtain a cosed form expression by performing the integras exacty, the fina expression invoves a hypergeometric function, and is not any more iuminating. 110

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