The Rushbrooke and Griffiths inequalities, (7.140) and (7.141), involve the critical exponents derived from the free energy, α,β,γ,δ. Can the model-independent equalities that we see in Table 8.1, α+2β+γ=2 and α+β(1+δ)=2, be explained? In 1965, B. Widom proposed a mechanism to account for these relations.[121] Widom’s hypothesis assumes the free energy function can be decomposed into a singular part, which we’ll denote¹ G_s, and a regular part, G_r, G=Gr+Gs. The regular part G_r varies slowly in the critical region; critical phenomena are associated with G_s. The Widom scaling hypothesis is that G_s is a generalized homogeneous function, such that

λ−1Gs(λat,λbB)=Gs(t,B) ,

(8.1)

where λ>0 is the scaling parameter, t denotes the reduced temperature, Eq. (7.33), and a,b are constants, the scaling exponents.²,³ Equation (8.1) asserts a geometric property of Gs(t,B) that, when t is “stretched” by λ^a, t→λat, and at the same time B is stretched by λ^b, B→λbB, Gs(λat,λbB) has the value Gs(t,B) when scaled by λ−1. Let’s see how this idea helps us.

Differentiate Eq. (8.1) with respect to B, which we can write

λb∂∂(λbB)Gs(λat,λbB)=λ∂∂BGs(t,B) .

(8.2)

Equation (8.2) implies (using Eq. (7.61)) that the critical magnetization satisfies a scaling relation,

M(t,B)=λb−1M(λat,λbB) .

(8.3)

Equation (8.3) indicates, as a consequence of Eq. (8.1), that in the critical region, under t→λat and B→λbB, M(λat,λbB) has the value M(t,B) when scaled by λb−1. The scaling hypothesis asserts the existence of the exponents a,b, but does not specify their values. They are chosen to be consistent with critical exponents, as we now show.

Two exponents are associated with the equation of state in the critical region: β as t→0 for B=0, and δ as B→0 for t=0. The strategy in working with scaling relations such as Eq. (8.3) is to recognize that if it holds for all values of λ, it holds for particular values as well. Set B=0 in Eq. (8.3): λbM(λat,0)=λM(t,0). Now let λ=t−1/a, in which case (show this)

M(t,0)=t(1−b)/aM(1,0) .

(8.4)

Comparing with the definition of β ( M(t,0)~t→0tβ), we identify

1−ba=β .

(8.5)

Now play the game again. Set t=0 in Eq. (8.3) and let λ=B−1/b: M(0,B)=B(1/b)−1M(0,1). Comparing with the definition of δ ( M(0,B)~B→0B1/δ), we identify

1b−1=1δ .

(8.6)

We therefore have two equations in two unknowns (Eqs. (8.5) and (8.6)), which are readily solved:

a=1β(δ+1)    b=δδ+1 .

(8.7)

If we know β,δ, we know a,b.

What about α,γ ? Equations (8.5), (8.6) follow from the equation of state, obtained from the first derivative of G, M=−∂G/∂BT, Eq. (7.61).⁴ The critical exponents α,γ are associated with second derivatives of G, ∂2G/∂T2B=−CB/T and ∂2G/∂B2T=−χ, Eqs. (7.62) and (P7.14). Differentiating Eq. (8.1) twice with respect to B,

λ2b∂2∂(λbB)2Gs(λat,λbB)=λ∂2∂B2Gs(t,B)⇒χ(t,B)=λ2b−1χ(λat,λbB) .

(8.8)

Equation (8.8) is a scaling relation for χ in the critical region. Set B=0 in Eq. (8.8) and let λ=t−1/a, χ(t,0)=t(1−2b)/aχ(1,0). Comparing with the definition of γ, χ~t−γ, we identify

2b−1a=γ .

(8.9)

Differentiating Eq. (8.1) twice with respect to t, we find CB(t,B)=λ2a−1CB(λat,λbB). Set B=0 and λ=t−1/a, implying CB(t,0)=t(1/a)−2CB(1,0), and thus for CB(t,0)~t−α,

2−1a=α .

(8.10)

Equations (8.9) and (8.10) are two equations in two unknowns, implying

a=12−α    b=121+γ2−α .

(8.11)

If we know α,γ, we know a,b. Note that we also know a,b if we know β,δ; Eq. (8.7).

Thus, through application of thermodynamics to the proposed scaling form of the free energy, Eq. (8.1), we’ve found four equations in the two unknowns a,b (Eqs. (8.5), (8.6), (8.9), (8.10)), implying the existence of relationships among α,β,γ,δ. We’ve just shown that if we know β,δ, we know α,γ, and conversely (see Exercise 8.7). Using Eq. (8.7) for a,b in Eq. (8.9), we find

γ=β(δ−1) .

(8.12)

We noted in (7.146) the Griffiths inequality γ≥β(δ−1), which is satisfied as an equality among the classical exponents and those for the d=2 and d=3 Ising models, within numerical uncertainties. The scaling hypothesis therefore accounts for the equality in (8.12) that otherwise we would have known only empirically. Equating the result for a from Eq. (8.7) with that in Eq. (8.11), we find

α+β(δ+1)=2 ,

(8.13)

a relation that satisfies Griffiths inequality (7.141) as an equality. By eliminating δ between Eqs. (8.12) and (8.13), we find α+2β+γ=2, Rushbrooke’s inequality (7.140), satisfied as an equality.

The scaling hypothesis is phenomenological: It’s designed to account for the relations among critical exponents that we’ve found to be true empirically. We have (at this point) no microscopic justification for scaling behavior of singular thermodynamic functions. Because of its simplicity, however, and because of its successful predictions, it stimulated research into a fundamental understanding of scaling (the subject of coming sections). If the scaling hypothesis is lacking theoretical support (at this point), what about experimental? In Eq. (8.3) let λ=|t|−1/a,

M(t,B)=|t|−(b−1)/aMt|t|,B|t|b/a=|t|βM±1,B|t|β+γ ,

(8.14)

where we’ve used Eq. (8.5) and b/a=β+γ (Exercise 8.8). Equation (8.14) can be inverted,

B|t|β+γ=f±(|t|−βM(t,B)) ,

(8.15)

i.e., B/(|t|β+γ) is predicted to be a function f (with two branches f_±, corresponding to t>0 and t<0) of the single variable |t|−βM(t,B). Equation (8.15) is a definite prediction of the scaling hypothesis, subject to validation. Figure 8.1 is a plot of the scaled magnetic field versus the scaled magnetization for the ferromagnet CrBr₃[122]. The critical exponents β,γ were independently measured in zero field, the values of which were used to calculate the scaled field and magnetization. Measurements of M,B were made along isotherms for T<Tc and T>Tc, and the data beautifully fall on two curves. Figure 8.1 offers compelling evidence for the scaling hypothesis.

The scaling hypothesis accounts for the relations we find among critical exponents and it has experimental support. It’s incumbent upon us therefore to find a microscopic understanding of scaling. We present the physical picture developed by L.P. Kadanoff[123] that provides a conceptual basis for scaling and which has become part of the standard language of critical phenomena.

Consider a square lattice of lattice constant a, such as in the left part of Fig. 8.2, on which we have Ising spins (∙) coupled through nearest-neighbor interactions, with Hamiltonian

H=−J∑〈ij〉σiσj−b∑iσi ,

(8.16)

where 〈ij〉 indicates a sum over nearest neighbors, on any lattice.⁵ In the critical region the correlation length ξ becomes macroscopic (critical opalescence, Section 6.7), implying the obvious inequality ξ≫a, yet this simple observation lies at the core of Kadanoff’s argument. So far, the lattice constant has been irrelevant in our treatment of phase transitions (often set to unity for convenience and forgotten thereafter). In many areas of physics, the length scale determining the relevant physics is fixed by fundamental constants.⁶ The correlation length ξ is an emergent length that develops macroscopic proportions as T→Tc, and is not fixed once and for all. A divergent correlation length in essence describes a new state of matter of highly correlated fluctuations.

Following Kadanoff, redefine the lattice constant, a→a′≡La, L≪ξ/a, an operation specifying a new lattice with the same symmetries as the original; the case for L=3 on the square lattice is shown in the right part of Fig. 8.2. On the original lattice there is one spin per unit cell. The unit cell of the scaled lattice contains L^d Ising spins, where we allow for an arbitrary dimension d, not just d=2 as in Fig. 8.2. That is, spins are still in their places; we’ve just redefined the unit of length (a scale transformation, or a dilatation). Label unit cells on the scaled lattice with capital Roman letters. Define a variable S˜I representing the degrees of freedom in the Ith cell,

S˜I≡∑i∈Iσi≈ξ≫La±Ld≡SILd ,

(8.17)

where SI=±1 is a new Ising spin, the block spin (denoted ⊗ in Fig 8.2). Technically S˜I is a function of cell-spin configurations, S˜I=S˜I(σ1,⋯,σLd), having (Ld+1) possible values (show this). The two values assigned to S˜I in Eq. (8.17) represent the two cases of all cell spins correlated—all spins aligned—either up or down. Any other cell-spin configurations are unlikely to occur in the regime ξ≫La. The validity of the mapping in Eq. (8.17) that selects from the (Ld+1) possible values of S˜I, the two values ±Ld is therefore justified only when ξ≫La. Spins correlated over the size of a cell effectively act as a single unit—block spins.

A key issue is the nature of interactions among block spins. Referring to Fig. 8.3, interactions between near-neighbor block spins (solid lines) are induced through the interactions of the spins on the original lattice (shown as open circles). New types of interactions, however, not in the original Hamiltonian are possible among block spins: next-near neighbor (dashed lines) or four-spin interactions. That is a complication we’ll have to address. For now, assume that block spins interact only with nearest neighbors and the magnetic field. Let’s write, therefore, a Hamiltonian for block spins:

HL≡−JL∑〈IJ〉SISJ−bL∑ISI ,

(8.18)

where J_L is the near-neighbor interaction strength between block spins associated with lattice constant La; likewise b_L is the coupling of S_I to the magnetic field. Clearly J1≡J and b1≡b. How we might calculate JL,bL will concern us in upcoming sections; for now we take them as given.

We’re considering, from a theoretical perspective, a change in the unit of length for systems in their critical regions, a→La, where ξ≫La>a. The correlation length is a function of K≡βJ, B≡βb, ξ=ξ(K,B). On the scaled lattice, ξ=ξ(KL,BL). Because it’s the same length, however, measured in two ways,⁷

ξ(KL,BL)=L−1ξ(K,B) .

(8.19)

For N spins on the original lattice, there are N′≡N/Ld block spins on the scaled lattice. Because it’s the same system described in two ways, the free energy is invariant. Let g(t,B)≡G(t,B)/N denote the Gibbs energy per spin. Thus,

Ng(t,B)=N′g(tL,BL)⇒g(t,B)=L−dg(tL,BL) ,

(8.20)

where t=(Kc/K)−1. Equations (8.19) and (8.20) constrain the form of K_L, B_L; such relations hold only in the critical region, ξ≫La, and thus they apply for T near T_c and for small B.

Because ξ(KL,BL)<ξ(K,B) (Eq. (8.19)), the transformed system is further from the critical point (where ξ→∞). Let’s assume, in order to satisfy Eqs. (8.19), (8.20), that t_L and B_L are in the form

tL=Lxt    BL=LyB ,

(8.21)

where x,y are new scaling exponents (independent of L). We require x,y>0 so that we move away from the critical point under a→La for L>1. Combining Eq. (8.21) with Eq. (8.20),

g(t,B)=L−dg(Lxt,LyB) .

(8.22)

Under the assumptions of Kadanoff’s analysis, the free energy per spin is a generalized homogeneous function. Whereas Eq. (8.1) is posited to hold for an arbitrary mathematical parameter λ, the parameter L in Eq. (8.22) has physical meaning (and is not entirely arbitrary: L≪ξ/a). If we let L→L1/d, Eq. (8.22) has the form of Eq. (8.1):

g(t,B)=L−1g(Lx/dt,Ly/dB) .

(8.23)

The exponents a,b in Eq. (8.1) correspond to Kadanoff’s exponents, a↔x/d and b↔y/d.

Kadanoff has shown therefore, for systems in their critical regions, how, if under a change in length scale a→La the couplings tL,BL transform as in Eq. (8.21), then the free energy exhibits Widom scaling. Of the two assumptions on which the theory rests, that block spins interact with nearest neighbors and t_L, B_L scale as in Eq. (8.21), the latter is most in need of justification, and we’ll come back to it (Sections 8.4, 8.6). The larger point, however, is that the block-spin picture implies a new paradigm, that changes in length scale (a spatial quantity) induce changes in couplings, thereby providing a physical mechanism for scaling.

We’ve noted previously (Section 6.5.3) the distinction between thermodynamic and structural quantities (correlation functions); the former can be derived from the free energy, the latter cannot. Kadanoff’s scaling picture, which links thermodynamic quantities with spatial considerations, can be used to develop a scaling theory of correlation functions, thereby relating the critical exponents ν,η to the others, α,β,γ,δ. Define the two-point correlation function for block spins,

C(rL,tL,BL)≡〈SISJ〉−〈SI〉〈SJ〉 ,

(8.24)

where r_L is the distance between the cells associated with block spins S_I, S_J (measured in units of La). Let r denote the same distance measured in units of a; thus, rL=L−1r (see Eq. 8.19). In order for block-spin correlations to be well defined, we require the inter-block separation be much greater than the size of cells, rL≫La⇒r≫L2a>a. A scaling theory of correlation functions is possible only for long-range correlations, r≫a.

In formulating the block spin idea, we noted in Eq. (8.17) the approximate correspondence SI≈L−d ∑i∈Iσi. How accurate is that association, and does it hold the same for all dimensions d? Consider the interaction of the spin system with a magnetic field, supposing the inter-spin couplings have been turned off. Because we’re looking at the same system in two ways,

bL∑ISI=↓what we wantb∑Iσi=↓by definitionb∑I∑i∈Iσi≈↓Eq. (8.17)bLd∑ISI .

(8.25)

Equation (8.25) invites us to infer bL=bLd, which, comparing with Eq. (8.21), implies the scaling exponent y=d. If that were the case, it would imply the Widom scaling exponent b=y/d=1, and thus δ→∞ (Eq. (8.6)). To achieve consistency, we take, instead of Eq. (8.17),

SI=L−y∑i∈Iσi ,

(8.26)

where we expect y≲d. Nowhere in our analysis did we make explicit use of Eq. (8.17).

Substituting Eq. (8.26) in Eq. (8.24),

C(rL,tL,BL)=〈SISJ〉−〈SI〉〈SJ〉=L−2y∑i∈I∑j∈J〈σiσj〉−〈σi〉〈σj〉=L−2yL2d〈σiσj〉−〈σi〉〈σj〉≡L2(d−y)C(r,t,B) .

(8.27)

Equation (8.27) implies, using Eq. (8.21) and rL=r/L, the scaling form for correlation functions:

C(r,t,B)=L2(y−d)C(r/L,Lxt,LyB) .

(8.28)

Let L=|t|−1/x, and thus, from Eq. (8.28),

C(r,t,B)=|t|2(d−y)/xf±|t|1/xr,B|t|y/x=1r2(d−y)|t|1/xr2(d−y)f±|t|1/xr,B|t|y/x≡1r2(d−y)g±|t|1/xr,B|t|y/x ,

(8.29)

where f±,g± are scaling functions.

Set B=0 in Eq. (8.29) and then let |t|=0. Correlation functions decay algebraically with distance at T=Tc (Section 7.6), and thus we identify, using Eq. (7.75),

2(d−y)=d−2+η⇒y=12(d+2−η) .

(8.30)

Again set B=0; we know as |t|→0, long-range correlation functions are associated with the correlation length ξ. We require that r/ξ~r|t|ν=r|t|1/x, implying

x=1ν .

(8.31)

With these identifications of x,y, combined with the Widom scaling exponents x=ad, y=bd, and using the expressions for a,b in Eqs. (8.7), (8.11), there are numerous interrelations involving ν,η with the other critical exponents. For example, one can show

2−η=dδ−1δ+1    dν=2−α    γ=ν(2−η) .

(8.32)

The Widom scaling hypothesis Eq. (8.1) accounts for relations among the exponents α,β,γ,δ. The block-spin picture (Section 8.2) motivates Eq. (8.1) and shows the exponents η,ν are related to α,β,γ,δ and d. The upshot is, that of the six critical exponents α,β,γ,δ,η,ν, only two are independent. That statement hinges on the validity of Eq. (8.21), Kadanoff’s scaling form for block-spin couplings. It’s time we got down to the business of calculating block-spin couplings, a process known as renormalization. We’ll see (Sections 8.4, 8.6) how understanding the physics behind Eq. (8.21) leads to a more general theory known as the renormalization group.⁸

8.3.1 Decimation method for one-dimensional systems

We start with the d=1 Ising model with near-neighbor couplings where we double the lattice constant, a system simple enough that we can carry out all steps exactly. Figure 8.4 shows a one-dimensional lattice with lattice constant a, with Ising spins σ on lattice sites, except that on every other site we’ve renamed the spins μ, which shall be the block spins. One way to define block spins (but not the only way) is simply to rename a subset of the original Ising spins, μ. One finds the interactions between μ-spins by summing over the degrees of freedom of the σ-spins in a partial evaluation of the partition function, a technique known as decimation. To show that, it’s convenient to work with a dimensionless Hamiltonian, H≡−βH. Thus, for the d=1 Ising model, H=K∑iσiσi+1+B∑iσi. Referring to Fig. 8.4, we can rewrite the Hamiltonian (exactly) assuming that even-numbered spins σ2i are named μ_i,

H=∑i=1N(Kσiσi+1+Bσi)→σ2i→μi∑i=1N/2(σ2i+1[K (μi+μi+1)+B]+12B(μi+μi+1)) ,

(8.33)

where we’ve written the coupling of the μ-spins to the B-field in the form 12B(μi+μi+1) to “share” the μ-spins surrounding each odd-numbered σ-spin.⁹ For the partition function,

ZN(K,B)=∑{σ}eH≡∏i=1N∑σi=−11exp(∑j=1N(Kσjσj+1+Bσj))→σ2i→μi∏i=1N/2∑μi=−11∑σ2i+1=−11exp (∑n=1N/2[Kσ2n+1(μn+μn+1)+Bσ2n+1+12B(μn+μn+1)]) .

(8.34)

The strategy is to evaluate Eq. (8.34) in two steps: Sum first over the σ-degrees of freedom—a partial evaluation of the partition function—and then those associated with the μ-spins. The sum over σ2i+1 in Eq. (8.34) is straightforward:

∑σ2n+1=−11eσ2n+1[K(μn+μn+1)+B]=2 cosh [K (μn+μn+1)+B] .

(8.35)

Our goal is to exponentiate the terms we find after summing over σ2n+1, i.e., fit the right side of Eq. (8.35) to an exponential form. We want the following relation to hold:

2 exp (12B(μn+μn+1)) cosh [K(μn+μn+1)+B]             ≡exp (K0+K′μnμn+1+12B′(μn+μn+1)) .

(8.36)

 

Three parameters, K0,K′,B′, are required to match the three independent configurations of μ_n, μn+1: both up, ↑↑; both down, ↓↓; and anti-aligned, ↑↓ or ↓↑. We require

↑↑:      2eB cosh (2K+B)=eK0+K′+B′       1↑↓:            2 cosh B=eK0−K′           2↓↓:      2e−B cosh (2K−B)=eK0+K′−B′ .        3

(8.37)

We can isolate K0,K′,B′ through combinations of the equations in (8.37),

1×22×3⇒e4K0=16cosh⁡2Bcosh⁡2K+Bcosh⁡2K−B1×322⇒e4K′=cosh⁡2K+Bcosh⁡2K−Bcosh⁡2B13⇒e2B′=e2Bcosh⁡2K+Bcosh⁡2K−B .

(8.38)

Thus, we have in Eq. (8.38) explicit expressions for K0,K′,B′; they are known quantities. Combining Eq. (8.36) with Eq. (8.34), we find

ZN(K,B)=∑{σ}eH=eNK0/2∑{μ}eH′=eNK0/2ZN/2(K′,B′) ,

(8.39)

where ∑{μ}≡∏i=1N/2∑μi=−11 and H′≡∑i=1N/2K′μiμi+1+B′μi. The quantities K′,B′ are known as renormalized couplings, with H′ the renormalized Hamiltonian.¹⁰ In stretching the lattice constant a→2a, we have at the same time “thinned,” or coarse grained, the number of spins N→N/2, which interact through effective couplings K′,B′. Equation (8.39) indicates that while the number of states available¹¹ to the renormalized system is less than the original ( ZN/2<ZN), the equality ZN(K,B)=eNK0/2ZN(K′,B′) is maintained¹² through the factor of eNK0/2, where we note that K0≠0, even if K=0 or B=0 (see (8.38)).

We’ve achieved the first part of the Kadanoff construction. Starting with a near-neighbor model with couplings K,B, we have, upon a→2a, another near-neighbor model with couplings K′,B′. By the scaling hypothesis, K′,B′ should occur further from the critical point at T=0, B=0 with K′<K and B′>B. To examine the recursion relations K′=K′(K,B), B′=B′(K,B), it’s easier in this case (because Tc=0) to work with x≡e−4K and y≡e−2B, in terms of which the critical point occurs at x=0,y=1. With x′≡e−4K′, y′≡e−2B′, we find from (8.38) an equivalent form of the recursion relations

x′=x(1+y)2x(1+y2)+y(1+x2)      y′=yy+x1+xy ,

(8.40)

from which it’s readily shown that y′<y if y<1 and x′>x if x<1. Figure 8.5 shows the flows that occur under iteration of the recursion relations in (8.40). Starting from a given point in the x–y plane, the renormalized parameters do indeed occur further from the critical point.

We see from Fig. 8.5 that systems with coupling constants e−4K≪1 (for any B-field strength) are transformed under successive renormalizations into equivalent systems characterized by K=0. What may have been a difficult problem for K≫0 is, by this technique, transformed into an equivalent problem associated with K=0, which is trivially solved.¹³ What started as a way to provide a physical underpinning to scaling has turned into a method of solving problems in statistical mechanics. But, back to scaling. Let’s check if Eq. (8.21) is satisfied.

 

We start by examining the recursion relations (8.40) for the occurrence of fixed points, values x*,y* invariant under the block-spin transformation:

x*=x*(1+y*)2x*(1+y*2)+y*(1+x*2)      y*=y*y*+x*1+x*y* .

(8.41)

Analysis of (8.41) shows fixed points for x*=0,y*=1 —the critical point—and a line of “high temperature” fixed points x*=1 for all y. From Fig. 8.5, for K,B near the critical point, the renormalization flows are away from the fixed point, an unstable fixed point, and toward the high-temperature fixed points. We therefore have another way of characterizing critical points as unstable fixed points of the recursion relations for couplings.¹⁴ If for a fixed set of couplings (such as at a fixed point) there is no change in couplings under changes in length scale, the correlation length is the same for all length scales, implying fluctuations of all sizes, schematically illustrated in Fig. 8.6. The only way ξ can have the same value for any finite length scale is if ξ→∞. Unstable fixed points imply infinite correlation lengths. At the critical point (and only at the critical point) the system appears the same no matter what length scale you use to look at it. Such systems are said to be self-similar or scale invariant.¹⁵ The critical point is a special state of matter, indeed!

Scale invariance is not typical. In most areas of physics, an understanding of the phenomena requires knowledge of the “reigning physics” at a single scale (length, time, mass, etc.). For example, modeling sound waves in a gas of uranium atoms does not involve subatomic physics. One cannot start with a model of nuclear degrees of freedom, and arrive at the equations of hydrodynamics by continuously varying the length scale. Most theories apply at a definite scale, such as the mean free path between collisions. There are systems besides critical phenomena exhibiting scale invariance—fully developed turbulence in fluids[125][126], in elementary particle physics,¹⁶ and in fractal systems.¹⁷ Scale invariance is also observed in systems where the framework of equilibrium statistical mechanics does not apply, such as the jamming and yielding transitions in granular media [128, 129]

 

The recursion relations in (8.40) are nonlinear; let’s linearize them in the vicinity of a fixed point. Taylor expand¹⁸ x′=x′(x,y), y′=y′(x,y) about x*,y*,

x′=x*+(x−x*)∂x′∂x|x*,y*+(y−y*)∂x′∂y|x*,y*+⋯y′=y*+(x−x*)∂y′∂x|x*,y*+(y−y*)∂y′∂y|x*,y*+⋯ .

(8.42)

Defining δx≡x−x*, δy≡y−y*, and keeping terms to first order,

δx′δy′=∂x′/∂x*∂x′/∂y*∂y′/∂x*∂y′/∂y*δxδy=4002δxδy ,

(8.43)

where we’ve evaluated at x*=0, y*=1 the partial derivatives of x′,y′ obtained from (8.40). The fixed point associated with the critical point is indeed unstable: Small deviations δx, δy from the critical point are, upon a→2a, mapped into larger deviations δx′, δy′. Are the relations in Eq. (8.43) in the scaling form posited by Eq. (8.21)? Because L=2 in this case, we can write (from Eq. (8.43)) δx′=L2δx and δy′=Lδy —nominally the scaling form we seek. The fact, however, that Tc=0 for d=1 complicates the analysis. For small B (near the critical point), δy≡y−1≈−2B; thus δy′=Lδy implies B′=LB≡LysB, where y_s denotes the Kadanoff scaling exponent (to distinguish it from the variable y in use here). Thus, we find ys=1 for the d=1 Ising model. Comparing with Eq. (8.30), we see that ys=1 is precisely what we expect because η=1 exists for d=1 (see Exercise 8.1). For the coupling K, e−4K′=4e−4K (Eq. (8.43)) implies K′=K−12ln⁡2 (for large K). Thus, ∂K′/∂K*=1=L0, implying the scaling exponent xs=0, which in turn implies ν→∞ from Eq. (8.31). The critical exponent ν can’t be defined for d=1 because ξ doesn’t diverge as a power law, but rather exponentially; see Eq. (P6.1). Kadanoff scaling holds in one dimension, but we need to find the “right” scaling variables, here δx,δy.

 

The scaling form tL=Lxt (Eq. (8.21)) is written in terms of t≡(T−Tc)/Tc, where |t|→0 as T→Tc. We can, equivalently, develop a scaling variable involving K=J/(kT), with

KL=Kc−(Kc−K)Lx .

(8.44)

Equation (8.44) indicates that (Kc−K) is the quantity that gets small near the critical point;¹⁹ see Exercise 8.20. We’ve been writing KL,BL in this section as K′,B′. Using Eq. (8.44) and BL=LyB from Eq. (8.21), we can infer the critical exponents η,ν by connecting these scaling forms with Eqs. (8.30) and (8.31):

1x=ν=ln⁡Lln⁡(∂K′/∂K)*    y=12(d+2−η)=ln⁡(∂B′/∂B)*ln⁡L .

(8.45)

The exponents η,ν can be calculated from the recursion relations at unstable fixed points. Once they’re known, the other exponents follow (only two are independent). The renormalization method predicts K_c from the fixed point of the transformation, and, by connecting critical exponents with fixed-point behavior, scaling emerges from the linearized recursion relations at the fixed point. The Kadanoff construction, devised to support the scaling hypothesis, turns out to represent a more comprehensive theory (see Section 8.4).

Let’s see what else we can do with recursion relations besides finding critical exponents. Define a dimensionless free energy per spin, F≡lim⁡N→∞−βF/N=lim⁡N→∞1Nln⁡ZN. Combining F with Eq. (8.39),

F=12K0+12F′ ,

(8.46)

where F′≡F(K′,B′). Make sure you understand the factor of 12 multiplying F′ in Eq. (8.46). Let’s iterate Eq. (8.46) twice:

F=12K0+12F′=12K0+1212K0′+12F′′=12K0+14K0′+18K0′′+18F′′′ ,

where K0′≡K0(K′,B′). Generalize to N iterations:

F=∑n=0N12n+1K0(n)+12N+1F(N+1) ,

(8.47)

where K0(m)≡K0(K(m),B(m)) denotes the mth iterate of K₀, with K0(0)≡K0(K,B). Under successive iterations, K,B are mapped into fixed points at K=0 and some value B=B*. From (8.38), K0(K=0,B)=ln⁡2cosh⁡B. For the Ising model F=ln⁡λ+, where λ₊ is the largest eigenvalue of the transfer matrix. At the high-temperature fixed point, λ+=2cosh⁡B* (Eq. (6.82)). Thus, F is mapped into F=ln⁡(2cosh⁡B*), and hence the sum in Eq. (8.47) converges as N→∞,

F(K,B)=∑n=0∞12n+1K0K(n),B(n) .

(8.48)

 

Renormalization provides a way to calculate the free energy not involving a direct evaluation of the partition function—it’s a new paradigm in statistical mechanics. The function K₀, the “constant” in the renormalized Hamiltonian, is essential for this purpose, as are the recursion relations K′=K′(K,B), B′=B′(K,B). Non-analyticities arise in the limit of an infinite number of iterations of recursion relations, in which all degrees of freedom in the thermodynamic limit have been summed over, as if we had exactly evaluated the partition function.

Recursion relations can be developed for quantities derivable from the free energy by taking derivatives. For the magnetization per spin m≡∂F/∂BK (see Eq. (6.84)), we have by differentiating Eq. (8.46):

m=12∂K0∂B+12∂B′∂Bm′ .

(8.49)

For the zero-field susceptibility per spin χ=∂m/∂BB=0,

χ=12∂2K0∂B2B=0+12∂B′∂BB=02χ′ .

(8.50)

It’s straightforward to write computer programs to iterate recursion relations such as these.

8.3.2 Decimation of the square-lattice Ising model

Let’s try decimation on the square lattice for an Ising model having nearest-neighbor interactions in the absence of a magnetic field. From Fig. 8.7, a square lattice of lattice constant a can be decomposed into interpenetrating sublattices,²⁰ square lattices of lattice constant 2a that have been rotated 45^∘ relative to the original.²¹,²² Let spins on one sublattice be the block spins. Interactions between them can be inferred by summing out the degrees of freedom associated with the other sublattice. Referring to Fig. 8.7, we sum over σ₀:

∑σ0=−11exp (Kσ0(σ1+σ2+σ3+σ4))=2 cosh (K(σ1+σ2+σ3+σ4)) .

(8.51)

 

Just as with Eq. (8.35), our job is to fit the right side of Eq. (8.51) to an exponential form:

2 cosh [K(σ1+σ2+σ3+σ4)]=exp (K0+12K1(σ1σ2+σ2σ3+σ3σ4+σ4σ1)+K2(σ1σ3+σ2σ4)+K4σ1σ2σ3σ4)  ,

(8.52)

where, in addition to K₀, we’ve allowed for nearest-neighbor couplings K₁ (the factor of 12 is because these interactions are “shared” with neighboring cells), next-nearest neighbor couplings K₂, and a four-spin interaction, K₄. Four parameters are required to match the four independent energy configurations of σ1,σ2,σ3,σ4: ↑↑↑↑, ↑↑↑↓, ↑↑↓↓, ↑↓↑↓. Thus, from Eq. (8.52),

↑↑↑↑:      2 cosh⁡4K=eK0+2K1+2K2+K4↑↑↑↓:      2 cosh⁡2K=eK0−K4↑↑↓↓:            2=eK0−2K2+K4↑↓↑↓:            2=eK0−2K1+2K2+K4 .

(8.53)

The latter two relations imply K2=12K1. With K2=12K1, we find

e8K0=256 cosh⁡4K cosh⁡42K    e4K1= cosh⁡4K    e8K4= cosh⁡4K cosh⁡42K .

(8.54)

Thus, starting with a near-neighbor model on the square lattice, we’ve found, using decimation, a model on the scaled lattice having near-neighbor interactions in addition to next-nearest-neighbor and four-spin couplings. To be consistent, we should start over with a model having first, second neighbor and four-spin interactions.²³ If one does that, however, progressively more types of interactions are generated under successive transformations. K.G. Wilson found[130] that more than 200 types of couplings are required to have a consistent set of recursion relations for the square-lattice Ising model.²⁴ For that reason, decimation isn’t viable in two or more dimensions.²⁵ We’d still like a way, however, to illustrate renormalization in two dimensions as there are new features to be learned. We develop in Section 8.3.4 an approximate set of recursion relations for the square lattice. Before doing that, we touch on a traditional approach to critical phenomena, the method of high-temperature series expansions.