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 V.C The Niemeijer–van Leeuwen Cumulant Approximation Unfortunately, the decimation procedure cannot be performed exactly in higher di-mensions. For example, the square lattice can be divided into two sublattices. For an RG with b = √2, we can start by decimating the spins on one sublattice. The interac-tions between the four spins surrounding each decimated spin are obtained by generalizing eq.(V.13). If initially h = g = 0, we obtain R(σ1′ , σ 2′ , σ 3′ , σ 4′ ) = e Ks(σ1′ +σ2′ +σ3 ′ +σ4′ ) = 2 cosh [K(σ1 ′ + σ2 ′ + σ3 ′ + σ4′ )] . (V.28) s=±1 Clearly the four spins appear symmetrically in the above expression, and hence are subject to the same two body interaction. This implies that new interactions along the diagonals of the renormalized lattice are also generated, and the nearest neighbor form of the original Hamiltonian is not preserved. There is al so a four point interaction, and R = exp [g ′ + K ′ (σ1′ σ2 ′ + σ2′ σ3 ′ + σ3′ σ4 ′ + σ4′ σ1 ′ + σ1′ σ3 ′ + σ2′ σ4′ ) + K4′ σ1′ σ2′ σ3′ σ4′ ] . (V.29) The number (and range) of new interactions increases with each RG step, and some trun-cating approximation is necessary. Two such schemes are described in the following sec-tions. One of the earliest approaches was develop ed by N iemeijer and van Leeuwen (NvL) for treating t he Ising model on a triangular lattice, subject to the usual nearest neighbor Hamiltonian −βH = K σiσj. The original l attice sit es are grouped into cells of three hiji spins (e.g. in alternating up pointing tria ngles). Labelling the three spins in cell α as {σα1 , σα2 , σα3 }, we can use a majority rule to define the renormalized cell spin as σ ′ = sign  σα 1 + σα 2 + σ3 . (V.30) α α (There is no ambiguity in the rule for any odd number of sites, and the renormali zed spin is two–valued.) The renormalized interactions corresponding t o the above map are obtained from t he constrained sum ′ ′7→σαα[σ ′ X ′iα] . (V.31) e −βH ] −βH[σ= e {σiα} 82X X     To truncate the number of interactions in the renormalized Hamiltonia n, NvL intro-duced a perturbative scheme by setting βH = βH0 + U. The unperturbed Hamiltonian −βH0 = K  σα1 σα 2 + σα2 σα 3 + σα3 σα 1  , (V.32) α involves only intra–cell interactions. Since the cells are decoupled, this part of the Hamil-tonian can be treated exactly. The remaining inter–cell interactions are treated as a perturbation −U = K σβ (1) σα (2) + σβ (1) σα (3) . (V.33) <α,β> The sum i s over all neighboring cells, each connected by two bonds. (The actual spins involved depend on the relat ive orientations of the cells.) Eq.(V.31 ) is now evaluated perturbatively as ′ 2 ′7→σα[σ ′ X ′ αiα] 1 − U + U 2 − .··· (V.34) e −βH ] −βH0[σ= e {σiα} The renormalized Hamiltonian is given by the cumulant series βH ′ [σα′ ] = −ln Z0[σα′ ] + �U�0 −21  U 2 0 − �U� 02 + O(U 3), (V.35) where ��0 refers to the ex pectation values with respect to βH0, with the restriction of fixed [σα′ ], and Z0 is the corresponding partition function. To proceed, we construct a table of all possible configurations of spins within a cell, their renormalized value, and contribution to the cell energy: σ ′ σ1 σ2 σ3 exp [−βH0]α α α α + + + + e3K + + + e−K− + + + e−K− + + + e−K− 3K− − − − e− + − − e−K − − + − e−K − − − + e−K 83Y X X X h i D E           The restricted partition function is the product of contributions from the indep endent cells,   ′ 1 ασ2 α+σ2 ασ3 α+σ3 ασ1 α)  =  e 3K + 3e −KN/3 . (V.36) Z0[σα′ ] =  e K(σ7→σ′ αα {σiα} It is independent o f [σα′ ], thus contributing an additive constant to the Hamiltonian. The first cumulant of the interaction is −�U�0 = K σ1 σ2 + σ1 σ3 = 2Kσi σj 0 , (V.37) β 0 α 0 β 0 α 0 α 0 β <α,β> <α,β> where we have taken advantage of the equivalence of the three spins in each cell. Using the table, we can evaluate the restricted average of site spins as    +e3K e−K + 2e−K σ ′   −for = +1  −K   −Kα  e3K + eσi = e3K + 3eσ ′ . (V.38) α 0  3K + e−K −K  ≡e3K + 3e−Kα  −e − 2eσ ′   e3K + 3e−K for α = −1 Substituting in eq.(V.37) leads to N   e3K + e−K 2 X −βH ′ [σα′ ] =3ln e 3K + 3e −K+ 2Ke3K + 3e−K σα′ σβ ′ + O(U 2). (V.39) hαβi At this order, the renormalized Hamiltonian involves only nearest neighbor interactions, with the recursion relation  −K 2 e3K + eK ′ = 2K −K . (V.40) e3K + 3e1. Eq.(V.40) has the following fixed points: (a) The high temperature sink at K∗ = 0. If K ≪ 1 , K ′ ≈ 2K(2/4)2 = K/2 < K, i.e. this fixed point is stable, and has zero correlation length. (b) The low temperature sink at K∗ = ∞. If K ≫ 1, then K ′ ≈ 2K > K, i.e. unlike the one dimensional case, this fixed point is also stable with zero correlati on length. (c) Since both of the above fixed points are unstable, there must be at least one stable fixed point at finite K ′ = K = K∗ . From eq.(V.40), the fixed point position satisfies 1 e3K∗ + e−K∗ 4K∗ 4K∗ √2= e3K∗ + 3e−K∗ , = ⇒√2e + √2 = e + 3. (V.41) 84!  X X   The fixed point value 13 −√2 K ∗ = ln √2 − 1 ≈ 0.3356, (V.42) 4 can be compared to the exactly k nown val ue of 0.2747 for the t ria ngular lattice. 2. Linearizing the recursion relation around the non-trivial fixed point