Renormalization Group for the Two-Dimensional Ising Model

Chapter 8 Renormalization Group for the Two-Dimensional Ising Model The two-dimensional (2D) Ising model is arguably the most important in statistical physics. This special status is due to Lars Onsager s exact solution of the zero-field case 1 and the extension by Chen Ning Yang to an exact solution in finite fields. 2 The impact of Onsager s work cannot be overestimated. Quite apart from explicitly demonstrating the existence of phase transitions within equilibrium statistical mechanics, Onsager s solution established a benchmark for simulations, motivated the search for exact solutions of other models, including alternative solutions of the 2D Ising model, and has served as a testing ground for the efficacy of new theoretical ideas. In this chapter, our interest in this model falls into the last category. The application of the real-space RG to the 1D Ising model in the preceding chapter demonstrated the mathematical machinery of the RG method, introduced key concepts, and reproduced the results of the exact solution we have previously obtained. The corresponding calculation for the 2D Ising model is the subject of this chapter. This calculation proves to be more typical of the real-space RG method than the 1D counterpart in that approximations to the recursion relations are necessary to obtain a tractable computational scheme. Nevertheless, the results obtained go beyond mean-field theory in several respects and can be systematically improved. 3 1 L.Onsager, Crystal statistics. I. A two-dimensional model with an order-disorder transition, Phys. Rev. 65, 117 149 (1944) 2 C. N. Yang, The spontaneous magnetization of a two-dimensional Ising model, Phys. Rev. 85, 808 816 (1952). 3 H. J. Hilhorst, M. Schick, and J. M. J. van Leeuwen, Exact renormalization group equations for the two-dimensional Ising model, Phys. Rev. B 19, 2749 2763 (1979). 111

112 The Renormalization Group in Real Space 8.1 Decimation of the Partition Function The decimation transformation of the 1D Ising model left the form of the energy invariant in that no new interactions were produced by the decimation: the nearest-neighbor interactions were retained, but with a renormalized interaction as the sole effect of the transformation. In general, however, RG transformations produce a hierarchy of interactions that are consistent with the symmetry of the original model. For the RG to be practical, the lower order terms must dominate the energy, so that the additional terms can be treated perturbatively. In this section, we carry out the decimation of the 2D Ising model to illustrate these additional features. We consider a square lattice Fig. 8.1 on each site (i, j) of which is a spin s i,j that points either up (s i,j = 1) or down (s i,j = 1). The spins interact through a ferromagnetic coupling J > 0 such that the energy of nearestneighbor spins is lower if they aligned than if they are opposed. The energy for this model is H = J i,j (s i,j s i+1,j + s i,j s i,j+1 ). (8.1) (i,j 1) (i 1,j) (i,j) (i 1,j) (i,j 1) Figure 8.1: The lattice sites of a 2D Ising model, showing a typical site (i, j) that is coupled by ferromagnetic interactions to its four nearest neighbors at (i±1,j) and (i, j ±1). The dark circles indicate the spins that are eliminated by decimation.

The Renormalization Group in Real Space 113 The partition function is Z = s i,j =±1 e K(s i,js i+1,j +s i,j s i,j+1 ), (8.2) i,j in which K = J/(k B T ). The decimation transformation that we will use for the 2D Ising model 4,5 is shown in Fig. 8.1. The spins that are eliminated by the decimation occupy alternate diagonal rows. After the decimation, the remaining spins form a 2D lattice that is rotated by 45 with respect to the original lattice, with a nearest-neighbor spacing that has been increased by a factor of 2 (Fig. 8.2). To determine the form of the transformed energy, we use the fact none of the spins that is being summed over couple to any other spins that are being summed. Thus, as in Sec. 7.2, the sum over spins on each such site 4 K. G. Wilson, The Renormalization Group: Critical Phenomena and the Kondo Problem, Rev. Mod. Phys. 47, 773 840 (1975) 5 H. J. Maris and L. P. Kadanoff, Teaching the Renormalization Group, Am J. Phys. 46, 652 657 (1978). Figure 8.2: The remaining sites after the decimation, which are indicated by open circles, lie on a (rotated) square lattice with the nearest-neighbor distance increased by a factor of 2. In addition to a renormalization of the nearest-neighbor coupling, indicated by bold solid lines, the decimation produces a new diagonal coupling with next-nearest neighbors, indicated by the bold broken lines, and four-spin coupling between all the spins at the vertices of the square.

114 The Renormalization Group in Real Space can be performed independently. For the site (i, j) shown in Fig.??(a), this summation yields s i,j =±1 e K(s i,js i 1,j +s i,j s i+1,j +s i,j s i,j 1 +s i,j s i,j+1 ) = 2 cosh[k(s i 1,j + s i+1,j + s i,j 1 + s i,j+1 )]. (8.3) We can now see that decimation for the 2D Ising model produces qualitatively different results from that in 1D. The corresponding relation (7.8) produced two conditions from the two independent spin configurations: both spins aligned or opposed. The right-hand side of Eq. (8.3) has three distinct values obtained from the 2 4 = 16 configurations that correspond to (i) all spins aligned, (ii) three aligned, one opposed, and (iii) two aligned and two opposed. This leads to at least three conditions and, therefore, to at least three quantities in the renormalized energy. Since the original energy in Eq. (8.1) has only two parameters (K and the constant resulting from the partial evaluation of the partition function), the renormalized energy obtained after decimation cannot retain its original functional form, i.e. there must be additional interactions generated by the decimation. 8.2 Recursion Relations In view of the foregoing, the partially summed partition function produces a renormalized energy that must contain terms with higher spin products. Accordingly, we take the trial form [cf. Eq. (7.9)] 2 cosh[k(s i 1,j + s i+1,j + s i,j 1 + s i,j+1 )] = z(k) exp [ K 1(s i 1,j s i,j 1 + s i 1,j s i,j+1 + s i+1,j s i,j 1 + s i+1,j s i,j+1 ) +K 2(s i 1,j s i+1,j + s i,j 1 s i,j+1 )+K 3s i 1,j s i+1,j s i,j 1 s i,j+1 ]. (8.4) The coupling constant K 1 represents the renormalized coupling between nearest neighbors, K 2 is a new interaction between next-nearest neighbors, and K 3 is another new interaction between four spins at the vertices of any square of the lattice (Fig. 8.2). For this equation to be an identity, we must be able to solve this equation for f, K 1 K 2, and K 3. The 16 spin configurations yield

The Renormalization Group in Real Space 115 four equations: 2 cosh 4K = z(k)e 2K 1 +2K 2 +K 3, the solutions of which are 2 cosh 2K = z(k)e K 3, (8.5) 2=z(K)e 2K 2 +K 3, 2=z(K)e 2K 1 +2K 2 +K 3, f(k )=2f(K) ln [ 2(cosh 4K) 1/8 (cosh 2K) 1/2], (8.6) K 1 = 1 ln(cosh 4K), (8.7) 4 K 2 = 1 ln(cosh 4K), (8.8) 8 K 3 = 1 ln(cosh 4K) 1 ln(cosh 2K), (8.9) 8 2 where we written ln Z = Nf(K), as for the 1D Ising model. These recursion relations establish the validity of Eq. (8.4). Subsequent decimations of the 2D Ising model generate interactions of increasing complexity. But, as long as the nearest-neighbor interaction dominates, a perturbation expansion can be used to account for remaining terms. We will neglect the presence of the four-spin term K 3 (but will check the consistency of this after we obtain a solution) and account for K 2 approximately. Both K 1 and K 2 are positive and thus favor the alignment of spins. Hence, we will omit the explicit presence of K 2 in the renormalized energy, but increase K 1 accordingly to a new value K = K 1 + K 2 so that the tendency toward alignment is approximately the same. From Eqs. (8.7) and (8.8), we obtain the recursion relation for K as K = 3 ln(cosh 4K). (8.10) 8 The recursion relation (8.6) for f is unaffected by this modification. 8.3 Fixed Points The recursion relation in Eq. (8.10) produces quite different behavior of the coupling constant than that for the 1D Ising model in Eq. (7.12). There are two regions to consider, separated by the critical value K c, which is a solution

116 The Renormalization Group in Real Space 1.2 R(K 2 ) 1 0.8 R(K 0 ) R(K 1 ) K 2 K 0.6 K 0 K 1 0.4 0.2 0 K 1 K 0 K 2 K 3 R(K 0 ) R(K 1 ) R(K 2 ) 0 0.2 0.4 0.6 0.8 1 K Figure 8.3: Graphical representation of successive decimations of the 2D Ising model, as represented by the recursion relation (8.10). The straight line is K = K and the curve is the function K = 1 ln(cosh 4K). The curves cross 8 at K c = 0.50698. For any initial value K > K c, the sequence of values K 1,K 2,... determined by iterating the recursion relation, K n = R(K n 1 ), approaches K n. But, for an initial value K<K c, the sequence of values K 1,K 2,... approaches zero. Hence, K = 0 and K are stable fixed points of the RG transformation, while K = K c is an unstable fixed point. of K c = 3 ln(cosh 4K 8 c) and has the value K c =0.50698. For an initial value K 0 >K c, the sequence of values K n = R(K n 1 ) increases without bound and approaches infinity. Alternatively, for an initial value K<K c, the sequence K n = R(K n 1 ) decreases to zero. Thus, both K = 0 and K are now stable fixed points, while the fixed point K = K c is unstable. This is depicted in Fig. 8.3. The fixed points K = 0 and K are the same fixed points that were obtained for the 1D Ising model in Sec. 7.4. The difference is that, for the 2D Ising model, both of these fixed points are now stable, with the fixed point at K = K c being unstable. The fixed point K = 0 is a trivial fixed point corresponding to the high-temperature limit where the correlations between neighboring spins vanish. The fixed point K is a critical point in the RG picture, but only in the trivial sense because, although the correlation length diverges at T = 0, there are no fluctuations because the

The Renormalization Group in Real Space 117 spins are completely aligned. Thus, the fixed point K = K c represents the true critical point of the 2D Ising model. Figures 8.4 and 8.5 show the effect of successive block-spin transformations for a system above T c (Fig. 8.4) and below T c (Fig. 8.5). The transformations above T c produce a sequence of systems in which the correlations between neighboring remaining spins are decreased. Below T c, the block-spin transformation s produce a sequence of systems in which the correlations between the remaining spins are increased. This is the behavior embodied in Fig. 8.3. The numerical value K c =0.50698 compares favorably with the exact value 0.44069 obtained from the Onsager solution, especially in view of the ad hoc nature of the approximations made to arrive at our recursion relations. When substituted into Eq. (8.8), we obtain K 3(K c )= 0.05323, which suggests that the four-spin couplings can be accounted for within perturbation theory. 8.4 Scaling and Critical Exponents Apart from identifying the existence of a critical point and estimating the critical coupling, the RG recursion relations provide the scaling form of the free energy and the critical exponents associated with the singular behavior of thermodynamic quantities at the critical point. Consider the recursion relation (8.6) for the free energy. The second term on the right-hand side of this equation accounts for the removal of short-range degrees of freedom, which results in an expression that is analytic in K. Thus, this term is not expected to play a role in the phase transition and we neglect it in the following discussion. The resulting recursion relation for the singular part of the free energy f s is written as f s (K) =b d f s (K ), (8.11) where b = 2 and d = 2. The factor 2 represents the change in the nearestneighbor distance, as shown in Fig.??, so b d represents the change in the area of the system. We now suppose that K close enough to the critical value K c that we may express K in terms of a linearized recursion relation (8.10): K = R(K) =K c + dr dk (K K c )+. (8.12) K=Kc

118 The Renormalization Group in Real Space Figure 8.4: Successive block-spin renormalization group transformation for a 2D Ising model at T =1.22 T c. The effect of the transformations is produce a sequence of systems that approach the infinite-temperature limit of disordered uncorrelated spins. K. G. Wilson, Problems in physics with many scales of length, Scientific American, 241(2), 140 157 (1979).

The Renormalization Group in Real Space 119 Figure 8.5: Successive block-spin renormalization group transformation for a 2D Ising model at T =0.99 T c. The effect of the transformations is produce a sequence of systems that approach the zero-temperature limit of aligned spins. K. G. Wilson, Problems in physics with many scales of length, Scientific American, 241(2), 140 157 (1979).

120 The Renormalization Group in Real Space In terms of the quantities δk = K K c and δk = K K c, this equation has the form δk = λδk = b y δk, (8.13) where λ = R Kc and we have written λ = b y because of the composition law of RG transformations, λ(b)λ(b) =λ(b 2 ). Equation (8.11) can now be written as or, since K c is a constant, as Now, f s (K + δk) =b d f s (K c + b y δk), (8.14) f s (δk) =b d f s (b y δk), (8.15) δk = K K c = J k B T J k B T c = J ( ) Tc T k B T T c Kt, (8.16) where t =(T c T )/T c is called the reduced temperature. Because K remains finite at the critical point, we write Eq. (8.15) as f s (t) =b d f s (b y t). (8.17) Finally, since this equation is valid for any b, we can set b = t 1/y, the singular part of the free energy becomes f s (t) = t d/y f s (t/ t ). (8.18) The absolute value sign means that the resulting expression is valid for temperatures above and below the critical temperature. The specific heat is proportional to the second derivative of the free energy with respect to the temperature, ( ) 2 F C V = T. (8.19) T 2 V

The Renormalization Group in Real Space 121 Then, with the singularity of C V characterized by the exponent α in Eq. (4.20), C V T T c α t α, we must have that f s t 2 α, so d/y =2 α and According to Eq. (8.13) we have that from which we obtain f s (t) = t 2 α f s (t/ t ). (8.20) y = ln λ ln b = 1 ( dr ln b ln dk = ln( 3 tanh 4K 2 c ln b ) ) K=Kc = 1.070, (8.21) α =2 d y =0.131, (8.22) which should be compared with the exact result from Onsager s solution that C V has a logarithmic singularity at T c, i.e. α = 0. Systematic corrections to the basic procedure outlined here involve substantial computation, but are capable of yielding critical quantities that are accurate to with a per cent of the Onsager solution 6. 8.5 Summary The procedure we have described for the 1D and 2D Ising models is completely general and demonstrates several important aspects of the RG: 1. Critical exponents associated with a fixed point are calculated by linearizing the RG recursion relations about that fixed point. Where there are recursion relations for n quantities, K i = R i (K 1,K 2,...,K n ), (8.23) 6 T. Niemeijer and J. M. J. van Leeuwen, Renormalizaton Theory for Ising-Like Spin Systems, in Phase Transitions and Critical Phenomena vol. 6, edited by C. Domb and M. S. Green (Academic, New York, 1976) pp. 425 505

122 The Renormalization Group in Real Space for i = 1, 2,..., n, linearization yields n δk i R i = K j j=1 {Ki }={K i } δk j n M ij δk j. (8.24) j=1 The eigenvectors and eigenvectors of the matrix M with entries M ij, MU i = λ i U i = b y i U i, yield the linear scaling elds U i, for i =1, 2,...,n in terms of which the singular part of the free energy is expressed as f s (U 1,U 2,...,U n )=b d f s (b y 1 U 1,b y 2 U 2,...,b yn U n ). (8.25) 2. The form of the singular part of the free energy in Eq. (8.25) is a generalized homogeneous function, whose properties and consequences were discussed in Sec. 6.3. 3. The exponents y i in Eq. (8.25) determine the behavior of the U i under the repeated action of the linear RG recursion relations. If y i > 0, U i is called relevant. If y i < 0, U i is called irrelevant and, if y i = 0, U i is called marginal. We see that if a relevant scaling field is non-zero initially, then the linear recursion relations will transform this quantity away from the critical point. Alternatively, an irrelevant scaling field will transform this quantity toward the critical point, while a marginal variable will be left invariant. Thus, relevant quantities must vanish at a critical point. For the 2D Ising model, the reduced temperature t is clearly a relevant scaling field, as the calculation in Eq. (8.21) demonstrates explicitly. The magnetic field must also vanish at the critical point, so this is another relevant variable. 4. The existence of the relevant, irrelevant, and marginal variables explains the observation of universality, namely, that ostensibly disparate systems (e.g. fluids and magnets) show the same critical behavior near a second-order phase transition,, including the same exponents for analogous physical quantities. In the RG picture, critical behavior is described entirely in terms of the relevant variables, while the microscopic differences between systems is due to irrelevant variables.