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406 chapter 15xyFigure 15.6 The numbering scheme used for our 8 × 8 2-D lattice of spins.15.6.1 WLS Ising Model ImplementationWe assume an Ising model with spin–spin interactions between nearest neighborslocated in an L × L lattice (Figure 15.6). To keep the notation simple, we set J =1so thatN∑E = − σ i σ j , (15.31)i↔jwhere ↔ indicates nearest neighbors. Rather than recalculate the energy each timea spin is flipped, only the difference in energy is computed. For example, for eightspins in a 1-D array,−E k = σ 0 σ 1 + σ 1 σ 2 + σ 2 σ 3 + σ 3 σ 4 + σ 4 σ 5 + σ 5 σ 6 + σ 6 σ 7 + σ 7 σ 0 , (15.32)where the 0–7 interaction arises because we assume periodic boundary conditions.If spin 5 is flipped, the new energy is−E k+1 = σ 0 σ 1 + σ 1 σ 2 + σ 2 σ 3 + σ 3 σ 4 − σ 4 σ 5 − σ 5 σ 6 + σ 6 σ 7 + σ 7 σ 0 , (15.33)and the difference in energy is∆E = E k+1 − E k =2(σ 4 + σ 6 )σ 5 . (15.34)This is cheaper to compute than calculating and then subtracting two energies.When we advance to two dimensions with the 8 × 8 lattice in Figure 15.6, thechange in energy when spin σ i,j flips is∆E =2σ i,j (σ i+1,j + σ i−1,j + σ i,j+1 + σ i,j−1 ), (15.35)−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 406