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Modern Physics Letters B, Vol. 23, No. 4 (2009) 567–573 c World Scientific Publishing Company

EXACT SOLUTION OF ISING MODEL IN 2D SHORTCUT NETWORK

O. SHANKER Hewlett Packard Company, 16399 W Bernardo Dr., San Diego, CA 92130, USA [email protected] http:// Received 17 April 2008 Revised 18 April 2008 We give the exact solution to the Ising model in the shortcut network in the 2D limit. The solution is found by mapping the model to the square lattice model with Brascamp and Kunz boundary conditions. Keywords: Complex networks; Ising model; shortcut model.

1. Introduction For any 2D graph, the exact solution of the Ising model appears to be a more or less challenging problem ever since the celebrated solution for square lattices by Onsager.1 This work solves the Ising model in the shortcut complex network. This is done by mapping the problem to the solution of Brascamp and Kunz2 with particular boundary conditions on a square lattice. The shortcut network was introduced3,4 while studying the dimension5 of complex networks (graphs).6–17 Section 2 reviews the shortcut model. Section 3 shows the mapping to a square lattice, and presents the solution to the Ising model. Finally, the conclusion is presented. 2. Shortcut Model In this section, we give a brief definition of the shortcut model.3,4 The model has fractal dimension as defined by the complex network zeta function,4 and transitions from a one-dimensional system to a two-dimensional system. The starting network is a one-dimensional lattice of N vertices with periodic boundary conditions. Each vertex is joined to its neighbors on either side, which results in a system with N edges. The network is extended by taking each node in turn and, with probability p, adding an edge to a new location m nodes ahead. We require that N  m  1, 567

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5

4 3

6 2

7 1

8

0

9

15

10

14

11

13 12

Fig. 1.

say m =

√

1D shortcut model.

N. The graphs are parametrized by: size = N ,

(1)

shortcut distance = m ,

(2)

shortcut probability = p .

(3)

When the shortcut probability p = 0, the network is a one-dimensional regular lattice of size N . Figure 1 shows the one-dimensional node numbering. The nodes are connected by edges represented by the arcs of the circle. When p = 1, every node is connected by a shortcut edge to a new location. Each node now has edges in two directions, the first along the original direction (on the circle), and the second along the shortcut edges. The graph is essentially a two-dimensional graph with m and N/m nodes in each direction. We will denote N/m by n. Figure 2 shows the two-dimensional node numbering, the x-coordinate represents the original edges, and the y-direction represents the shortcut edges. For p between 0 and 1 (Fig. 3), we have a graph which interpolates between the one and two-dimensional systems. We briefly mention the difference between the shortcut model and the “smallworld model” of Watts and Strogatz.18–20 In the small world model, one also starts with a regular lattice and adds shortcuts with probability p. However, the shortcuts are not constrained to connect to a node a fixed distance ahead. Instead, the other end of the shortcut can connect to any randomly chosen node. As a result, the small world model tends to a random graph rather than a two-dimensional graph as the shortcut probability is increased.

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(0,1)

569

(3,0)

(2,1) (2,0)

(1,0)

(3,1)

(0,0) (0,2)

(3,3) (1,2)

(2,2)

(2,3)

(3,2)

(1,3) (0,3)

Fig. 2.

5

2D shortcut model.

4

3

6 2

1

7

0 8

15 9

10

14

11

13 12

Fig. 3.

Shortcut model in an intermediate case.

3. Mapping to Square Lattice In this section, we show explicitly the mapping of the 2D shortcut network to the two-dimensional square lattice. We discuss the appropriate boundary conditions which allow us to find the exact solution to the Ising model. Figure 4 shows the representation of the 2D complex network in Fig. 2 as a two-dimensional lattice. The links along the circle are represented by the horizontal and the slanting edges. The shortcut edges are the vertical edges in Fig. 4. The system is periodic in the

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Shortcut Direction

2

1

0 0

1

2

3

Original Direction

Fig. 4.

Square lattice representation of a 2D shortcut network.

vertical direction with period n = N/m. This can be seen easily from Fig. 2, where n jumps along the shortcut edges brings one back to the starting point. The chief difference of the shortcut network from the usual square lattices is in the links at the right side of the lattice. Usually they are either free, or connect back to the node at the left end of the same row. The usual lattices can have periodic, anti-periodic or free boundary conditions. For the shortcut network, the unusual aspect is that the right node connects to the left node one level above instead of to the node on the same row. Let us write down the Ising model in terms of the couplings Ja (coupling along the outer edges in Fig. 2, represented as horizontal and inclined edges in Fig. 4) and Jb (coupling along the shortcut edges). The interaction energy is the sum of two terms, E = E a + Eb , Ea = J a

n−1 X

(4)

sm−1,ν s0,ν+1 + Ja

m−1 X n−1 X

sµν sµ+1,ν ,

(5)

µ=0 ν=0

ν=0

Eb = J b

m−2 X n−1 X

sµν sµ,ν+1 ,

(6)

µ=0 ν=0

where the sµν are spins at the nodes and can take the values ±1. We identify rows

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(1,1) (0,1)

(3,0)

(2,1) (2,0)

(1,0)

(3,1)

(0,0) (0,2)

(3,3) (1,2)

(2,3)

(2,2)

(1,3) (3,2)

Fig. 5.

(0,3)

Brascamp–Kunz boundary condition in the shortcut network.

0 and n. The partition function can be written as Z(a, b) =

X s0,0

···

X

exp(−βE) ,

(7)

sm−1,n−1

where a = −βJa , b = −βJb and β is the inverse temperature, β −1 = kT . The couplings along the inclined edges in Fig. 4 (the term with the single sum in Eq. (5)) complicate the application of the usual solution methods.21–31 The key to solving the model above is to choose the proper boundary conditions. Brascamp and Kunz2 gave a solution for a two-dimensional Ising model on a square lattice with periodic boundary conditions in the vertical direction and fixed boundary conditions in the horizontal direction. The spins at the left boundary are fixed at +1, while the spins at the right boundary alternate between −1 and +1, so the number of rows n must be even. Since all the spins along the left boundary are the same, the fact that the inclined edges connect nodes in different rows becomes irrelevant. The inclined edges contribute exactly the same interaction energy that would have been contributed by edges going from the right boundary to the left boundary along the same row, and in fact the single sum term which was creating the difficulty in finding a solution now becomes zero. Thus with Brascamp–Kunz boundary conditions, the solution of the 2D shortcut network becomes identical with the solution of the square lattice. Figure 5 shows the Brascamp–Kunz boundary conditions on the shortcut graph. The nodes represented as blocks are the free spins. The + and − represent nodes whose spin is fixed. Of the fixed spins, the ones with x = 0 represent the spins on the left edge, while the spins with x = 3

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represent the spins on the right edge. The resulting partition function is2 (m,n)

ZBK

(a, b) = 2mn

m n/2 Y Y

p=1 q=1

− cos

cosh(2a) cosh(2b) − cos

 (2q − 1)π sinh(2b) . n

pπ sinh(2a) m+1 (8)

The solution found in Eq. (8) has some desirable properties. The partition function has a multiplicative form. This is very helpful in the analysis of finite-size scaling.32,33 The analytic properties of finite-size scaling provide a good guide when trying to describe critical properties of models whose exact solution is unknown, e.g., the shortcut networks when the shortcut probability p lies between 0 and 1. Reference 33 derived exact finite-size corrections to the free energy and the specific heat. The finite size corrections are integer powers of the size, except for the leading logarithmic term in the specifc heat. Reference 32 derived exact expressions for the finite-size scaling of the Fisher zeros, the critical specific heat, the effective critical points, and the specific heat peak. The Fisher zeros of the partition function have been calculated for finite sizes, and lie on the unit circle in the complex plane. Reference 34 relates the partition function found above to the partition funtions found with other boundary conditions. 4. Conclusion In this work, we gave an exact solution of the Ising model on the 2D shortcut network, for Brascamp–Kunz boundary conditions. The partition function can be expressed in multiplicative form. The boundary conditions make the solution useful for studying the finite size corrections to the estimates of quantities like the specific heat. References 1. 2. 3. 4. 5. 6. 7.

8. 9.

L. Onsager, Phys. Rev. 65 (1944) 117. H. J. Brascamp and H. Kunz, J. Math. Phys. 15 (1974) 65. O. Shanker, Mod. Phys. Lett. B 21 (2007) 321–326. O. Shanker, Mod. Phys. Lett. B 21 (2007) 639–644. K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, 2nd edn. (Wiley, 2003). S. Boccaletti, V. Latora, Y. Moreno, M. Chavez and D.-U. Hwang, Physics Reports 424 (2006) 175–308. O. Shanker and G. Motta, Use of Word Relationship Graphs to Analyze Texts in Different Languages, Technical Report (2007), http://www.geocities.com/oshanker/ TextAnalysis.pdf. O. Shanker, Complex Network Dimension and Path Counts, Technical Report (2007), http://www.geocities.com/oshanker/complexNetworkZetaFunction.pdf. M. E. J. Newman, SIAM Rev. 45 (2003) 167–256.

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