beautiful and powerful method for analyzing physical phenomena (especially so called critical phenomena). RG is based on the invariance of Hamiltonian under some scale transformations. e.g.) 1 dim Ising model The Renormalization Group in Real Space his chapter, we develop the real-space RG for the one-dimensional ing model. The Ising model is especially useful for illustrating the ion of Kadanoﬀ averaging and its eﬀect on the partition function. This was solved exactly in Chapter 3, so comparisons with those results will the eﬃcacy of the RG method to be assessed. In the next chapter, we ply the RG method to the two-dimensional (2D) Ising model. The One-Dimensional Ising Model ergy E of the 1D Ising model is H = −J i si si+1 , (7.1) > 0 is the coupling constant. The partition function, Z = {si=±1} e−H/kBT = {si=±1} i eKsisi+1 , (7.2) K = J/kB T, kB is Boltzmann’s constant, and T is the absolute tem- e, determines the thermodynamic properties of the Ising model in f the Helmholtz free energy F = −kB T ln Z . (7.3) ution of this model was derived in Sec. 3.2. Renormalization by Decimation are several ways of implementing the real-space RG because of the ty in choosing the block spins. This can rarely be performed exactly, tematic approximations are available to make the calculation man- . The 1D Ising model is an exception in that the RG calculations can ed out exactly. In this chapter, we develop the real-space RG for the one-dimensional (1D) Ising model. The Ising model is especially useful for illustrating the realization of Kadanoﬀ averaging and its eﬀect on the partition function. This model was solved exactly in Chapter 3, so comparisons with those results will enable the eﬃcacy of the RG method to be assessed. In the next chapter, we will apply the RG method to the two-dimensional (2D) Ising model. 7.1 The One-Dimensional Ising Model The energy E of the 1D Ising model is H = −J i si si+1 , (7.1) where J > 0 is the coupling constant. The partition function, Z = {si=±1} e−H/kBT = {si=±1} i eKsisi+1 , (7.2) where K = J/kB T, kB is Boltzmann’s constant, and T is the absolute tem- perature, determines the thermodynamic properties of the Ising model in terms of the Helmholtz free energy F = −kB T ln Z . (7.3) The solution of this model was derived in Sec. 3.2. 7.2 Renormalization by Decimation There are several ways of implementing the real-space RG because of the ﬂexibility in choosing the block spins. This can rarely be performed exactly, but systematic approximations are available to make the calculation man- ageable. The 1D Ising model is an exception in that the RG calculations can be carried out exactly. (1D) Ising model. The Ising model is especially useful for illustr realization of Kadanoﬀ averaging and its eﬀect on the partition func model was solved exactly in Chapter 3, so comparisons with those re enable the eﬃcacy of the RG method to be assessed. In the next ch will apply the RG method to the two-dimensional (2D) Ising mode 7.1 The One-Dimensional Ising Model The energy E of the 1D Ising model is H = −J i si si+1 , where J > 0 is the coupling constant. The partition function, Z = {si=±1} e−H/kBT = {si=±1} i eKsisi+1 , where K = J/kB T, kB is Boltzmann’s constant, and T is the abso perature, determines the thermodynamic properties of the Ising terms of the Helmholtz free energy F = −kB T ln Z . The solution of this model was derived in Sec. 3.2. 7.2 Renormalization by Decimation There are several ways of implementing the real-space RG becau ﬂexibility in choosing the block spins. This can rarely be performe but systematic approximations are available to make the calculat ageable. The 1D Ising model is an exception in that the RG calcula be carried out exactly. We will perform a decimation, whereby a partial evaluation of the function (7.2) is carried out by to summing over alternate spins on th ﬃcacy of the RG method to be assessed. In the next chapter, we e RG method to the two-dimensional (2D) Ising model. he One-Dimensional Ising Model E of the 1D Ising model is H = −J i si si+1 , (7.1) is the coupling constant. The partition function, Z = {si=±1} e−H/kBT = {si=±1} i eKsisi+1 , (7.2) J/kB T, kB is Boltzmann’s constant, and T is the absolute tem- termines the thermodynamic properties of the Ising model in Helmholtz free energy F = −kB T ln Z . (7.3) of this model was derived in Sec. 3.2. enormalization by Decimation veral ways of implementing the real-space RG because of the choosing the block spins. This can rarely be performed exactly, tic approximations are available to make the calculation man- 1D Ising model is an exception in that the RG calculations can ut exactly. erform a decimation, whereby a partial evaluation of the partition ) is carried out by to summing over alternate spins on the lattice. where J > 0 is the coupling constant. The partition function, Z = {si=±1} e−H/kBT = {si=±1} i eKsisi+1 , (7.2) where K = J/kB T, kB is Boltzmann’s constant, and T is the absolute tem- perature, determines the thermodynamic properties of the Ising model in terms of the Helmholtz free energy F = −kB T ln Z . (7.3) The solution of this model was derived in Sec. 3.2. 7.2 Renormalization by Decimation There are several ways of implementing the real-space RG because of the ﬂexibility in choosing the block spins. This can rarely be performed exactly, but systematic approximations are available to make the calculation man- ageable. The 1D Ising model is an exception in that the RG calculations can be carried out exactly. We will perform a decimation, whereby a partial evaluation of the partition function (7.2) is carried out by to summing over alternate spins on the lattice. The decimation proceeds by summing over the spins on, say, the odd sites. This is shown schematically in Fig. 7.1. The ﬁrst step is to write the partition function in a form where the spins on the even and odd sites are separated: Z = {s2i=±1} 2i {s2i+1=±1} 2i+1 eK(s2is2i+1+s2i+1s2i+2 . (7.4) K K K K K K K K Figure 7.1: Decimation of the 1D Ising model. The spins on the sites marke by ﬁlled circles can be summed over independently, leaving a partition fun tion expressed only in terms of spins on sites marked by open circles. As indicated in Fig. 7.1, in a model that has interactions only between ad jacent spins,the spins on odd sites do not interact with one another. Thu the sums in the partition function over alternate sites may be evaluated in dependently of one another. The contributions from the odd sites can b determined by focussing on the contribution from a typical odd spin s2n+ Performing the partition sum over odd spins leaves only even spins: s2n+1=±1 eK(s2ns2n+1+s2n+1s2n+2) = eK(s2n+s2n+2) + e−K(s2n+s2n+2) = 2 cosh K(s2n + s2n+2 ) . (7.5 in terms of which the partition function becomes Z = {s2i=±1} 2i 2 cosh K(s2i + s2i+2 ) . (7.6 As noted above, this is an exact expression. 7.3 Recursion Relations The expression in (7.6) represents the partial evaluation of the partitio function obtained by summing over the spins on odd sites. This term ha two types of contributions: a spin-independent term from the removed (odd spins and a spin-dependent term from the remaining (even) spins. Notice tha the resulting interaction between spins does not appear to have the form the original Ising energy in (7.1). However, we now assert that the eﬀect 2i 2i+2 for every conﬁguration of neighboring spins s2i and s2i+1 . Ther conﬁgurations obtained from s2i = ±1 and s2i+2 = ±1, so we m s2i = +1 , s2i+2 = +1 , z(K)eK′ = 2 cosh 2K s2i = −1 , s2i+2 = −1 , z(K)eK′ = 2 cosh 2K s2i = +1 , s2i+2 = −1 , z(K)e−K′ = 2 , s2i = −1 , s2i+2 = +1 , z(K)e−K′ = 2 . Thus, there are only two distinct conditions to satisfy, one to aligned spins and one corresponding to opposed spins. T of pairs of conﬁgurations results from the up-down symmet model in the absence of a magnetic ﬁeld. We can express the above in terms of K′ and z(K) separately by dividing and m two distinct equations, which yields, respectively, e2K′ = cosh 2K , z2(K) = 4 cosh 2K . The ﬁrst of these equations can be solved for K′ in terms of K K′ = 1 2 ln(cosh 2K) . rmalization Group in Real Space 105 recursion relation for the coupling constant. This recursion relation ed entirely in terms of the original coupling constant, i.e. no new ns are generated by the decimation. This is a special property of ng model; in higher dimensions the RG generates additional inter- tween spins that are consistent with the symmetry of the original on (7.11) can be written in a more useful form by using the fact Helmholtz free energy F = −kB T ln Z is an extensive quantity, so is also extensive: ln Z = Nf(K), where N is the number of sites tem. Hence, Eq. (7.8) may be written as Z(K, N) = [z(K)]N/2Z(K′, N/2) , (7.13) This is the recursion relation for the coupling constant. This recursion relation is expressed entirely in terms of the original coupling constant, i.e. no new interactions are generated by the decimation. This is a special property of the 1D Ising model; in higher dimensions the RG generates additional inter- actions between spins that are consistent with the symmetry of the original model. Equation (7.11) can be written in a more useful form by using the fact that the Helmholtz free energy F = −kB T ln Z is an extensive quantity, so that ln Z is also extensive: ln Z = Nf(K), where N is the number of sites in the system. Hence, Eq. (7.8) may be written as Z(K, N) = [z(K)]N/2Z(K′, N/2) , (7.13) where, on account of Eq. (7.10), Z is the same function of K′ and 1 2 N on the right-hand side of this equation as it is of K and N on the left-hand side. Additionally, since the form of the energy is preserved after each decimation, the functional form of z(K) is preserved as well. Taking logarithms, we have ln Z(K, N) = Nf(K) = 1 2 N ln z(K) + 1 2 Nf(K′) , (7.14) which can be rearranged as f(K′) = 2f(K) − ln z(K) . (7.15) By invoking Eq. (7.11), we obtain f(K′) = 2f(K) − ln 2(cosh 2K)1/2 . (7.16) This is the second recursion relation of the 1D Ising model. Figure 7.2 summarizes the repeated application of the decimation proce- 106 The Renormalization Group in Real K K K K K K K K K K K K K K K K
K K K K K K K K K K K K
K K
Figure 7.2: Schematic illustration of the recursive decimation of the 1 model. At each step, every second spin spin, indicated by the ﬁlled are incorporated into the partition function, which results in an Ising with the same energy, but with a renormalized coupling constant. 7.4 Fixed Points The behavior of the coupling constant under repeated decimation examined by ﬁrst writing the recursion relation in Eq. (7.10) as K′ = Thus, beginning with an initial value K0 of the coupling constant, deci yields K1 = R(K0 ), a second decimation produces K2 = R(K1 ), and This can be represented diagrammatically as in Fig. 7.3. Suppose we have some initial value of the coupling constant K0 , Hamiltonian Partition func Free energy ( ) Sum over odd spins, → We can get the parameter relation between the original and coarse grained system : 104 The Renormalization Group in Real Spa this assertion. For the moment, we write Z = {s2i=±1} 2i 2 cosh K(s2i + s2i+2 ) = {s2i=±1} 2i z(K)eK′s2is2i+2 (7 = [z(K)]N/2 {s2i=±1} 2i eK′s2is2i+2 , (7 in which z(K) is the spin-independent part of the partition function a K′ is the new (i.e. renormalized) coupling constant. Consistency betw Eqs. (7.2) and (7.7) requires that 2 cosh K(s2i + s2i+2 ) = z(K)eK′s2is2i+2 (7 for every conﬁguration of neighboring spins s2i and s2i+1 . There are four s conﬁgurations obtained from s2i = ±1 and s2i+2 = ±1, so we must have t K′ If we denote , we can derive For fixed point ( ), we get 108 The Renormalization Group in Real Space and the recursion relation reduces to f(K∗) ≈ ln 2(cosh 2K∗)1/2 = ln 2 + 1 2 ln(cosh 2K∗) . (7.18) As K → 0, cosh K → 1, so f(K∗) → ln 2 , (7.19) and the free energy F = −NkB Tf(K∗) = −NkB T ln 2 , (7.20) (For 2 dim, we get non-trivial result : ) 33/34