Chemistry 593: Critical PhenomenaChemistry 593: Critical Phenomena ©David Ronis McGill University 1. Introduction In these lectures some of the general features of the phenomena of

Chemistry 593: Critical Phenomena© David Ronis

McGill University

1. Introduction

In these lectures some of the general features of the phenomena of phase transitions in mat-ter will be examined. We will first review some of the experimental phenomena. We then turn toa discussion of simple thermodynamic and so-called mean-field theoretical approaches to theproblem of phase transitions in general and critical phenomena in particular, showing what theyget right and what they get wrong. Finally, we will examine modern aspects of the problem, thescaling hypothesis and introduce the ideas behind a renormalization group calculation.

Fig. 1. Phase Diagram of Water.1

Fig. 2. Liquid-Vapor P-V phase diagram isothermsnear the critical point.2

Consider the two well known phase diagrams shown in Figs. 1 and 2. Along any of thecoexistence lines, thermodynamics requires that the chemical potentials in the coexisting phasesbe equal, and this in turn gives the well known Clapeyron equation:

dP

dT

coexistence

=∆H

T ∆V, (1.1)

where ∆H and ∆V are molar enthalpy and volume changes, respectively, and T is the tempera-ture. Many of the qualitative features of a phase diagram can be understood simply by using theClapeyron equation, and knowing the relative magnitudes and signs of the enthalpy and volumechanges. Nonetheless, there are points on the phase diagram where the Clapeyron equation can-not be applied naively, namely at the critical point where ∆V vanishes.

1G. W. Castellan, Physical Chemistry, 3rd ed., (Benjamin Pub. Co., 1983), p. 266.2R.J. Silbey and R.A. Alberty, Physical Chemistry, 3rd ed., (John Wiley & Sons, Inc. 2001) p. 16.

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Chemistry 593 -2- Critical Phenomena

The existence of critical points was controversial when it was first considered in the 19thcentury because it means that you can continuously transform a material from one phase (e.g., aliquid) into another (e.g., a gas). We now hav e many experimental examples of systems thathave critical points in their phase diagrams; some of these are shown in Table 1. In each case,the nature of the transition is clearly quite different (from the point of view of the qualitativesymmetries of the phases involved).

TABLE 1. Examples of critical points and their order parameters3

Critical OrderPoint Parameter

Example Tc(oK )

Liquid-gas Density H2O 647.05

Ferromagnetic Magnetization Fe 1044.0

Anti-ferromagnetic Sub-lattice FeF2 78.26magnetization

Super-fluid 4 He-amplitude 4 He 1.8-2.1

Super- Electron pair Pb 7.19conductivity amplitude

Binary fluid Concentration CCl4-C7F14 301.78mixture of one fluid

Binary alloy Density of one Cu − Zn 739kind on a sub-lattice

Ferroelectric Polarization Triglycine 322.5sulfate

The cases in Table I are examples of so-called 2nd order phase transitions, according to thenaming scheme introduced by P. Eherenfest. More generally, an nth order phase transition is onewhere, in addition to the free energies, (n − 1) derivatives of the free energies are continuous atthe transition. Since the first derivatives of the free energy give entropy and volume, all of thefreezing and sublimation, and most of the liquid-vapor line would be classified as first-order tran-sition lines; only at the critical point does it become second order. Also note that not all phasetransitions can be second order; in some cases, symmetry demands that the transition be firstorder.

At a second order phase transition, we continuously go from one phase to another. Whatdifferentiates being in a liquid or gas phase? Clearly, both have the same symmetries, so whatquantitative measurement would tell us which phase we are in? We will call this quantity (orquantities) an order parameter, and adopt the convention that it is zero in the one phase region of

3S.K. Ma, Modern Theory of Critical Phenomena, (W.A. Benjamin, Inc., 1976), p. 6.

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the phase diagram. In some cases there is a symmetry difference between the phases and thismakes the identification of the order parameters simpler, in others, there is no obvious uniquechoice, although we will show later that for many questions, the choice doesn’t matter.

For example, at the liquid-gas critical point the density (molar volume) difference betweenthe liquid and vapor phases vanishes, cf. Fig. 2, and the density difference between the twophases is often used as the order parameter. Second order transitions are also observed in ferro-magnetic or ferroelectric materials, where the magnetization (degree or spin alignment) or polar-ization (degree of dipole moment alignment) continuously vanishes as the critical point isapproached, and we will use these, respectively, as the order parameters. Other examples aregiven in Table 1.

At a second order critical point, many quantities vanish (e.g., the order parameter) whileothers can diverge (e.g., the isothermal compressibility, −V −1(∂V /∂P)T ,N cf. Fig. 2). In order toquantify this behavior, we introduce the idea of a critical exponent. For example, consider a fer-romagnetic system. As we just mentioned, the magnetization vanishes at the critical point (here,this means at the critical temperature and in the absence of any externally applied magnetic field,H), thus near the critical point we might expect that the magnetization, m might vanish like

m∝|Tc − T |β , when H = 0, (1.2)

or at the critical temperature, in the presence of a magnetic field,

m∝H1/δ . (1.3)

The exponents β and δ are examples of critical exponents and are sometimes referred to as theorder parameter and equation of state exponents, respectively; we expect both of these to be posi-tive. Other thermodynamic quantities have their own exponents; for example, the constant mag-netic field heat capacity (or CP in the liquid-gas system) can be written as

CH ∝|Tc − T |−α , (1.4)

while the magnetic susceptibility, χ , (analogous to the compressibility) becomes

χ ∝|Tc − T |−γ . (1.5)

Non-thermodynamic quantities can also exhibit critical behavior similar to Eqs. (1.2)−(1.5).Perhaps the most important of these is the scattering intensity measured in light or neutron scat-tering experiments. As you learned in statistical mechanics (or will see again later in thiscourse), the elastic scattering intensity at scattering wav e-vector q is proportional to the staticstructure factor

NS(q) ≡< |N (q)|2 > , (1.6)

where N (q) is the spatial Fourier transform of the density (or magnetization density), and < . . . >denotes an average in the grand canonical ensemble. In general, the susceptibility or compress-ibility and the q → 0 limit of the structure factor4 are proportional, and thus, we expect the scat-tered intensity to diverge with exponent γ as the critical point, cf. Eq. (1.5). This is indeedobserved in the phenomena called critical opalescence. At T = Tc and non-zero wav e-vectors,we write,

4See, e.g., http://ronispc.chem.mcgill.ca/ronis/chem593/structure_factor.1.html.

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S(q)∝1

q2−η . (1.7)

Finally, we introduce one last exponent, one that characterizes the range of molecular corre-lations in our systems. The correlation-length is called ξ , and we expect that

ξ ∝|Tc − T |−ν , (1.8)

where we shall see later that the correlation-length exponent, ν > 0.

Some experimental values for these exponents for ferromagnets are given in Table 2. Theprimes on the exponents denote measurements approaching the critical point from the two-phaseregion (in principle, different values could be observed). What is interesting, is that even thoughthe materials are comprised of different atoms, have different symmetries and transition tempera-tures, the same critical exponents are observed, to within the experimental uncertainty.

TABLE 2. Exponents at ferromagnetic critical points5

Material Symmetry T (oK ) α , α ′ β γ , γ ′ δ ηFe Isotropic 1044.0 α = α ′ = 0. 120 0.34 1.333 0.07

±0.01 ±0.02 ±0.015 ±0.07

Ni Isotropic 631.58 α = α ′ = 0. 10 0.33 1.32 4.2±0.03 ±0.03 ±0.02 ±0.1

EuO Isotropic 69.33 α = α ′ = 0. 09±0.01

YFeO3 Uniaxial 643 0.354 γ = 1. 33±0.005 ±0.04

γ ′ = 0. 7±0.1

Gd Anisotropic 292.5 γ = 1. 33 4.0±0.1

Of course, this behavior might not be unexpected. After all, these are all ferromagnetictransitions; a phase transition where "all" that happens is that the spins align. What is more inter-esting are the examples shown in Table 3. Clearly, the phase transitions are very different physi-cally; nonetheless, universal values for the critical exponents seem to emerge.

5S.K. Ma, op. cit., p. 12.

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TABLE 3. Exponents for various critical points6

Critical

PointsMaterial Symmetry Tc(oK ) α , α ′ β γ , γ ′ δ η

Antiferro- CoCl2 ⋅ 6H2O Uniaxial 2.29 α ≤ 0. 11 0.23

magnetic α ′ ≤ 0. 19 ± 0. 02

FeF2 Uniaxial 78.26 α = α ′ = 0. 112

± 0. 044

RbMnF3 Isotropic 83.05 α = α ′ = −0. 139 0.316 γ = 1. 397 0.067

± 0. 007 ± 0. 008 ± 0. 034 ± 0. 01

Liquid-gas CO2 n = 1 304.16 α ∼1/8 0.3447 γ = γ ′ = 1. 20 4.2

± 0. 0007 ± 0. 02

Xe 289.74 α = α ′ = 0. 08 0.344 γ = γ ′ = 1. 203 4.4

± 0. 02 ± 0. 003 ± 0. 002 ±0. 4

3 He 3.3105 α ≤ 0. 3 0.361 γ = γ ′ = 1. 15

α ′ ≤ 0. 2 ±0. 001 ±0. 03

4 He 5.1885 α = 0. 127 0.3554 γ = γ ′ = 1. 17

α ′ = 0. 159 ±0. 0028 ±0. 0005

Super-fluid 4 He 1.8-2.1 0. 04 ≤ α = α ′ < 0

Binary CCl4 − C7F14 n = 1 301.78 0.335 γ = 1. 2 ∼ 4

Mixture ±0. 02

Binary Co − Zn n = 1 739 0.305 γ = 1. 25

alloy ±0. 005 ±0. 02

Ferro- Triglycine n = 1 322.6 γ = γ ′ = 1. 00

electric sulfate ±0. 05

Our goals in these lectures are as follows:

1. To come up with some simple theory that results in phase transitions in general, and secondorder phase transitions in particular.

2. To show how universal critical exponents result.

3. To be able to predict the correct values for the critical exponents.

It turns out the 1. and 2. are relatively easily accomplished; 3. is not and Kenneth G. Wilson, wonthe 1982 physics Nobel Prize for showing how to calculate the critical exponents.

2. Thermodynamic Approach

2.1. General Considerations

Other than the already mentioned Clapeyron equation, cf. Eq. (1.1), and its generalizationsto higher order phase transitions (not discussed), thermodynamics has relatively little to sayabout the critical exponents. One class of inequalities can be obtained by using thermodynamicstability requirements (e.g., that arise by requirements that the free energy be a minimum at equi-librium). As an example of how this works, recall the well known relationship between the heatcapacities CP and CV , namely,

6S.K. Ma, op. cit., pp. 24-25.

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CP = CV +TVγ 2

T

γ P

, (2.1)

where γT ≡ V −1(∂V /∂T )P,N is the thermal expansion coefficient and γ P ≡ −V −1(∂V /∂P)T ,N is theisothermal compressibility. Since thermodynamic stability requires that CV and γ P be positive, itfollows that

CP ≥TVγ 2

T

γ P

. (2.2)

By using the different exponent expressions, Eqs. (1.2) (1.4) and (1.5), this last inequality impliesthat, as T → Tc,

τ −α ≥ positive constant × τ 2(β −1)+γ , (2.3)

where τ ≡ |T − Tc |/Tc. The inequality will hold at Tc only if

α + 2β + γ ≥ 2. (2.4)

This is known as the Rushbrook inequality. If you check some of the experimental data given inTables S2 and 3, you will see that in most of the cases, α + 2β + γ ≈ 2, and to within the experi-mental error, the inequality becomes an equality. This is no accident!

2.2. Landau-Ginzburg Free Energy

We now try to come up with the simplest model for a free energy or equation of state thatcaptures some of the physical phenomena introduced above. For example, we could analyze thewell known van der Waals equation near the critical point. It turns out however, that a modelproposed by Landau and Ginzburg is even simpler and in a very general manner shows many ofthe features of systems near their critical points. Specifically, they modeled free energy differ-ence between the ordered and disordered phases as

∆G ≡ −HΨ +A

2Ψ2 +

B

3Ψ3 +

C

4Ψ4+. . . , (2.5)

where A, B, C, etc., depend on the material and on temperature, and where H plays the role ofan external field (e.g., magnetic or electric or pressure).

In some cases, symmetry can be used to eliminate some of the terms in ∆G; for example, insystems with inversion or reflection symmetry (magnets), in the absence of an external fieldeither Ψ or −Ψ must give the same free energy. This means that the free energy must be an evenfunction of Ψ in the absence of an external field, and from Eq. (2.5) we see that this implies thatB = 0. Examples of the Landau free energy for ferromagnets are shown in Figs. 3 and 4.

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Fig. 3. The Landau-Ginzberg free energy (cf. Eq.(2.5)) for ferromagnets (B = 0. 0) at zero externalmagnetic field, and C = 1. 0.

Fig. 4. The Landau-Ginzberg free energy, cf. Eq.(2.5), for ferromagnets (B = 0. 0) at non-zero externalmagnetic field (H = 1. 0), and C = 1. 0.

The minima of the free energy correspond to the stable and metastable thermodynamicequilibrium states. In general, we see that an external field induces order (i.e., the free energyhas a minimum with Ψ ≠ 0 when H ≠ 0) and that multiple minima occur for A < 0. When theexternal field is zero, there are a pair of degenerate minima when A < 0. This is like the behaviorseen at the critical point, where we go from a one- to two-phase region of the phase diagram, cf.Fig. 2. To make this more quantitative, we assume that

A∝T − Tc, as T → Tc, (2.7)

with a positive proportionality constant, while the other parameters are assumed to be roughlyconstant in temperature near Tc.

In order to extract the critical exponents, the equilibrium must be analyzed more carefully.The equilibrium state minimizes the free energy, and hence, Eq. (2.5) gives:

H = AΨ + BΨ2 + CΨ3. (2.8)

For ferromagnets with no external field, B = 0, and Eq. (2.8) is easily solved, giving

Ψ = 0 (2.9a)

and

Ψ = ±√ −A

C. (2.9b)

Clearly, the latter makes physical sense only if A < 0, i.e., according to the preceding discussion,when T < Tc. Indeed, for A < 0 the it is easy to see that the nonzero roots correspond to the min-ima shown in Fig. 3, while Ψ = 0 is just the maximum separating them, and is thus not the equi-librium state.

With the assumed temperature dependence of A, cf. Eq. (2.7), we can easily obtain the thecritical exponents. For example, from Eqs. (2.7) and (2.9b), it follows that Ψ∝(Tc − T )1/2, and

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thus, β = 1/2. In the absence of a magnetic field, the free energy difference in the equilibriumstate is easily shown to be

∆G =

0, when A > 0 (T > Tc)

−A2

4C, otherwise.

(2.10)

Since

CH = T

∂S

∂T

H ,N

= −T

∂2G

∂T 2

H ,N

, (2.11)

it follows that the critical contribution to the heat capacity is independent of temperature, andhence, α = α ′ = 0.

An equation for the susceptibility can be obtained by differentiating both sides of Eq. (2.8)with respect to magnetic field and solving for χ ≡ (∂Ψ/∂H)T ,H=0. This gives:

χ =1

A + 2BΨ + 3CΨ2=

1

A, for T > Tc

1

2|A|, for T < Tc,

(2.12)

which shows, cf. Eq. (1.5), that γ = γ ′ = 1, and also shows that the amplitude of the divergenceof the susceptibility is different above and below Tc.

Finally, by comparing Eq. (2.8) at T = Tc (A = 0) with Eq. (1.3) we see that δ = 3. Theseresults are summarized in Table 4. Note that the Rushbrook inequality is satisfied as an equality,cf. Eq. (2.4).

Table 4. Mean-Field Critical Exponents

Quantity Exponent Value

Heat Capacity α 0Order Parameter β 1/2Susceptibility γ 1Eq. of State at Tc δ 3Correlation length ν 1/2Correlation function η 0

The table also shows the results for the exponents η and ν , which strictly speaking, don’t arisefrom our simple analysis. They can be obtained from a slightly more complicated version of thefree energy we’ve just discussed, one that allows for thermal fluctuations and spatially nonuni-form states. This is beyond the scope of present discussion and will not be pursued further here.

Where do we stand? The good news is that this simple analysis predicts universal valuesfor the critical exponents. We’ve found values for them independent of the material parameters.Unfortunately, while they are in the right ball-park compared to what is seen experimentally, theyare all quantitatively incorrect. In addition, the Landau-Ginzburg model is completely phe-nomenological and sheds no light on the physical or microscopic origin of the phase transition.

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3. Weiss Mean-Field Theory

The first microscopic approach to phase transitions was given by Weiss for ferromagnets.As is well known, a spin in an external magnetic field has a Zeeman energy given by

E = −γ h− H ⋅ S, (3.1)

where S is the spin operator, γ is called the gyromagnetic ratio, and H is the magnetic field at thespin (which we use to define the z axis of our system).

First consider a system of non-interacting spins in an external field. This is a simple prob-lem in statistical thermodynamics. If the total spin is S, the molecular partition function, q, isgiven by

q =S

Sz=−SΣ eα Sz =

sinh[α (S + 1/2)]

sinh(α /2), (3.2)

where α ≡ γ h− H /(kBT ), kB is Boltzmann’s constant, and where the second equality is obtainedby realizing that the sum is just a geometric series.

With the partition function in hand it is straightforward, albeit messy, to work out variousthermodynamic quantities. For example, the average spin per atom, < s >, is easily shown to begiven by

< s >=∂ ln q

∂α= BS(α ), (3.3)

where

BS(α ) ≡ (S + 1/2)coth[α (S + 1/2)] − coth(α /2)/2 (3.4)

is known as the Brillouin function. The average energy per spin is just −γ h− H < s >, while theHelmholtz free energy per spin is −kBT ln q, as usual. The spin contribution to the heat capacityis obtained by taking the temperature derivative of the energy and becomes:

CH

NkB

= α 2

1

4 sinh2(α /2)−

(S + 1/2)2

sinh2[(S + 1/2)α ]

. (3.5)

Other thermodynamic quantities are obtained in a similar manner. The magnetization and spincontributions to the heat capacity are shown in Figs. 5 and 6.

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Fig. 5. Spin polarization of an ideal spin in an exter-nal magnetic field.

Fig. 6. Spin contribution to the constant magneticfield heat capacity, CH .

While this simple model gets many aspects of a spin system correct (e.g., the saturation val-ues of the magnetization and the high temperature behavior of the magnetic susceptibilities), itclearly doesn’t describe any phase transition. The magnetization vanishes when the field isturned off and the susceptibility is finite at any finite temperature. Of course, the model didn’tinclude interactions between the spins, so no ordered phase should arise.

Weiss included magnetic interactions between the spins by realizing that the magnetic fieldwas made up of two parts: the external magnetic field and a local field that is the net magneticfield associated with the spins on the atoms surrounding the spin in question. In a disorderedsystem (i.e., one with T > Tc and no applied field) the neighboring spins are more or less ran-domly oriented and the resulting net field vanishes, on the other hand, in a spin aligned systemthe neighboring spins are ordered and the net field won’t cancel out. To be more specific, Weissassumed that

H = Hext + λ < s > , (3.6)

where Hext is the externally applied field and λ is a parameter that mainly depends on the crystallattice. In ferromagnets the field of the neighboring atoms tends to further polarize the spin, andthus, λ > 0 (it is negative in anti-ferromagnetic materials). Note that the mean field that goesinto the partition function depends on the average order parameter, which must be determinedself-consistently.

When Weiss’s expression for the magnetic field is used in Eqs. (3.3) and (3.4) a transcen-dental equation is obtained, i.e.,

< s >= BS((γ h− (Hext + λ < s >) / (kBT ))). (3.7)

In general, while it is easy to show that there are at most three real solutions and a critical point,Eq. (3.7) must be solved graphically or numerically. Nonetheless, it can be analyzed analyticallyclose to the critical point since there < s > and Hext are small as is α . We can use this by notingthe Taylor series expansion,

coth(x) =1

x+

x

3−

x3

45+. . . , (3.8)

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which when used in Eq. (3.7) gives

< s >=α3

[(S + 1/2)2 − (1 / 2)2] −α 3

45[(S + 1/2)4 − (1 / 2)4]+. . . . (3.9)

If the higher order terms are omitted, Eq. (3.9) is easily solved. For example, when Hext = 0, wesee that in addition to the root < s >= 0, we have:

< s >= ±√ 45T 2(Tc − T )

[(S + 1/2)4 − (1 / 2)4](γ h− λ /kB)3, (3.10)

where the critical temperature (known as the Curie temperature in ferromagnets) is

Tc ≡γ h− λ S(S + 1)

3kB

. (3.11)

When T < Tc the state with the nonzero value of < s > has the lower free energy. Thus we’vebeen able to show that the Weiss theory has a critical point and have come up with a microscopicexpression for the critical temperature. By repeating the analysis of the preceding section, onecan easily obtain expressions for the other common thermodynamic functions.

The Weiss mean field theory is the simplest theory of ferro-magnetism, and over the yearsmany refinements to the approach have been proposed that better estimate the critical tempera-ture. Unfortunately, they all fail in one key prediction, namely, the critical exponents are exactlythe same as those obtained in preceding section, e.g., compare Eqs. (2.9b) and (3.10). Thisshouldn’t be too surprising, given the similarity between Eqs. (2.8) and (3.9), and thus, whilewe’ve been able to answer some of our questions, the matter of the critical exponents stillremains.

4. The Scaling Hypothesis

When introducing the critical exponents, cf. Table 4, we mentioned the exponent ν associ-ated with the correlation length, that is ξ ∼|T − Tc |−ν . What exactly does a diverging correlationlength mean? Basically, it is the length over which the order parameter is strongly correlated; forexample, in a ferromagnet above its Curie temperature, if we find a part of the sample where thespins are aligned and pointing up, then it is very likely that all the neighboring spins out to a dis-tance ξ will have the same alignment.

In the disordered phase, far from the critical point the correlation-length is microscopic,typically a few molecular diameters in size. At these scales, all of the molecular details areimportant. What happens as we approach the critical point and the correlation length grows?

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Fig. 7. A snapshot of the spin configuration in a computer simulation of the 2D Isingmodel of a ferromagnet slightly above its critical temperature. Dark and light regionscorrespond to spin down and spin up, respectively.

Figure 7 shows the spin configuration obtained from a Monte Carlo simulation of the Isingspin system (S = 1/2 with nearest-neighbor interactions) close to its critical point. We see largeinterconnected domains of spin up and spin down, each containing roughly 103 − 104 spins. Ifthis is the case more generally, what determines the free energy and other thermodynamic quanti-ties? Clearly, two very different contributions will arise. One is associated with the short-rangeinteractions between the aligned spins within any giv en domain, while the other involves theinteractions between the ever larger (as T → Tc) aligned domains. The former should becomeroughly independent of temperature once the correlation length is much larger than the molecularlengths and should not contain any of the singularities characteristic of the critical point. The lat-ter, then, is responsible for the critical phenomena and describes the interactions between largealigned domains. As such, it shouldn’t depend strongly on the microscopic details of the interac-tions, and universal behavior should be observed.

The next question is how do these observations help us determine the structure of the quan-tities measured in thermodynamic or scattering experiments? First consider the scattering inten-sity or structure factor, S(q), introduced in Eq. (1.6). The scattering wav e-vector, q, probes the

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length-scales present in the density or magnetization fluctuations. From the discussion of thepreceding paragraph, the only length scale that is relevant near the critical point (at least forquantities that exhibit critical behavior) is the correlation length ξ ; hence, we should be able towrite

S(q, T )∝ξ γ /ν F(qξ ), (4.1)

where the factor of ξ γ /ν was introduced in order to capture the divergence in the scattering inten-sity at q = 0 associated with the susceptibility, cf. Eqs. (1.5) and (1.8). The function F(x) is arbi-trary, except for two properties: 1) F(0) is nonzero; and 2) F(x)∼1/x2−η as x → ∞. The formerimplies that there is a nonzero susceptibility, while the latter is necessary if the behavior given inEq. (1.7) is to be recovered. Strictly speaking, Eq. (1.7) holds only at the critical point where ξ isinfinite, and hence, the factors of ξ must cancel in Eq. (4.1); this only happens if

γ = ν (2 − η). (4.2)

This sort of relationship between the exponents is known as a scaling law, and seems to hold towithin the experimental accuracy of the measurements.

Widom7 formalized these ideas by assuming that the critical parts of the thermodynamicfunctions were generalized homogeneous functions. For example, for the critical part of themolar free energy, a function of temperature and external field, this means that

G(λ pτ , λ q H) = λG(τ , H) (4.3)

for any λ , and where recall that τ ≡ (T − Tc)/Tc. All the remaining critical exponents can begiven in terms of p and q.

We showed in Eq. (2.11) that the heat capacity is obtained from two temperature derivativesof the free energy. From Eq. (4.3) this implies that at H = 0,

λ2pCH (λ pτ ) = λCH (τ ). (4.4a)

Since λ is arbitrary, we set it to τ −1/ p and rewrite Eq. (4.4a) as

CH (τ ) = τ −(2 p−1) / pCH (1), (4.4b)

which gives

α = 2 −1

p. (4.5)

Similarly, the magnetization is obtained by taking the derivative of the free energy with respect toH . Thus, Eq. (4.3) gives

M(τ , H) = λ q−1 M(λ pτ , λ q H). (4.6a)

When H = 0 we set λ = τ −1/ p, as before, and find that

M = τ (1−q)/ p M(1, 0), (4.6b)

or

β =1 − q

p. (4.7)

7B. Widom, J. Chem. Phys., 43, 3898 (1965).

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At the critical temperature (τ = 0) we let λ = H−1/q, and rewrite Eq.(4.6a) as

M = H (1−q)/q M(0, 1), (4.8)

giving

δ =q

1 − q. (4.9)

Finally, the susceptibility is obtained from the derivative of the magnetization with respect tofield. By repeating the steps leading to the exponent β , we can easily show that

γ =2q − 1

p. (4.10)

All four exponents, α , β , δ and γ , hav e been expressed in terms of p and q, and thus twoscaling laws can be obtained. For example, by using Eqs. (4.7), (4.9) and (4.10) it follows that

γ = β (δ − 1), (4.11)

while by using Eqs. (4.5), (4.7) and (4.10) we recover the Rushbrook inequality (as an equality),cf. Eq. (2.4).

5. Kadanoff Transformation and The Renormalization Group

The discussion of the scaling hypothesis given in the preceding section is ad hoc to say theleast. Moreover, even if it is correct, it still doesn’t tell us how to calculate the independent expo-nents, p and q. Kadanoff8 has given a very physical interpretation of what scaling really means,and has shown how to apply it to the remaining problem.

In order to introduce the ideas, consider the Hamiltonian for a ferromagnet:

H = −J<n.n.>Σ si s j − H

iΣ si , (5.1)

where < n. n. > denotes a sum over nearest neighbor pairs on the lattice and si ≡ ±1 is a spin vari-able (scaled perhaps by 2) for the atom on the i’th lattice site. This is known as the Ising modeland, with appropriate reinterpretations of the spin variables, can be used to model liquids (e.g.,si = ±1 for empty or filled sites, respectively) solutions, surface adsorption, polymers etc. It canalso be generalized to allow for more complicated interactions (e.g., between triplets of spins ornon-nearest-neighbors) or to allow for more states per site. Note that J > 0 favors alignment(ferromagnetic order).

For a system of N spins, the exact canonical partition function, Q, is

Q =1

s1=−1Σ

1

s2=−1Σ . . .

1

sN =−1Σ e−H /kBT (5.2)

In general, the sums cannot be performed exactly; nonetheless, consider what happens if we wereto split up them up in the following way:

1. Divide up the crystal into blocks, each containing Ld spins (d is the dimension of space), cfFig. 8

8L. Kadanoff, Physics 2, 263 (1966).

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2. Fix each block’s spin. There is no unique way to do this. Usually we fix the total spin ofthe block, i.e.,

SL ≡ Zi∈block

Σ Si (5.3)

where Z ≈ 1/Ld is introduced to make SL ≈ ±1. Alternately, for Ld odd, we could assignSL ≡ ±1 depending on whether the majority of the spins in the block had spin ±1. As longas the block size is comparable to or smaller than the correlation length, these two choicesshould give the same answer (why?).

3. Average over the internal configurations of each block and calculate the mean interactionpotential between different blocks.

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Fig. 8. An example of the block transformation on a square lattice. Here L = 2.

With these, the partition function can be rewritten as

Q =SL

Σ e−W /kBT , (5.4)

where

e−W /kBT ≡1

{si}=−1Σ ′e−H /kBT , (5.5)

and where the prime on the spin sums means to only include those configurations that are consis-tent with the block-spin configuration being summed. The effective potential, W , is analogous tothe potential of mean force encountered in statistical mechanics and is just the reversible workneeded to bring the system into a configuration given by the SL’s.

What will W look like as a function of the block spin configuration? Clearly it toodescribes the interactions between spins (now blocks of spins) and should look something likethe original spin Hamiltonian introduced in Eq. (5.1), perhaps with some of the additional termsdiscussed above. Thus, we expect that

W = −JL<n.n. blocks>

Σ Si S j − HLiΣ Si+. . . , (5.6)

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where . . . represents the extra terms, and where note that the coupling constant and magneticfield have changed.

Equation (5.6) is just the Hamiltonian for a spin system with N /Ld spins. Hence, if we useit to carry out the remaining sums in Eq. (5.4) we see that the original Helmholtz free energy,−kBT ln Q is equal to the Helmholtz free energy of a system with fewer spins and different valuesfor the parameters appearing in the Hamiltonian; nonetheless, it is still the free energy of a spinsystem and we conclude that the free energy per spin,

A(J , H , . . . ) = L−d A(JL , HL , . . . ), (5.7)

where . . . denotes the parameters that appear in the extra terms.

A key issue is to understand what the block transformation does to the parameters in theHamiltonian. If we denote the latter by a column vector µ then our procedure allows us to write

µL = RL(µ) (5.8)

and Eq. (5.7) becomes

A(µ) = L−d A((RL(µ))). (5.9)

Obviously we could have done the block transformation in more than one step, and hence,

RL RL′ = RLL′, (5.10)

which some of you may realize is an operator multiplication rule, and has led to the characteriza-tion of the entire procedure as a group called the renormalization-group (RG). (Actually it is onlya semi-group since the inverse operations don’t exist).

In general, the renormalized problem will appear less critical, since the correlation lengthwill be smaller on the re-blocked lattice (remember, we’re simply playing games with how wecarry out the sums, the real system is the same). One exception to this observation is at the criti-cal point, where the correlation length is infinite to begin with. In order that the renormalizedappear as critical as the original one, the Hamiltonians before and after the block transformationmust describe critical systems, or equivalently, the renormalized parameters will turn out to bethe same as the original critical ones; i.e.,

µ = RL(µ). (5.11)

This is known as the fixed point of the RG transformation, and we will denote the special valuesof the parameters at the fixed point as µ*.

Suppose we’re near the critical point and we write µ = µ * +δ µ, where δ µ is not too large.By using Eqs. (5.8) and (5.11) we can write

δ µL = RL(µ* + δ µ) − RL(µ*) ≈ KLδ µ, (5.12)

where

KL ≡

∂RL(µ)

∂µ

µ=µ*

(5.13)

is a matrix that characterizes the linearized RG transformation at the fixed point.

Rather than use the parameters directly, it is useful to rewrite δ µ in terms of the normalizedeigenvectors of the matrix KL . These are defined by

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KLui = λ i(L)ui , (5.14)

with ui ⋅ ui = 1, as usual. It turns out that the eigenvalues must have a very simple dependenceon L. From the multiplication property of the RG transformation, cf. Eq. (5.10), we see that

λ i(LL′) = λ i(L)λ i(L′), (5.15)

and in turn this implies that

λ i(L) = L yi , (5.16)

where the exponent is obtained from the eigenvalue as

yi =ln((λ i(L)))

ln(L). (5.17)

We use this eigenanalysis by writing

δ µ =iΣ ciui , (5.18)

where we’ve assumed that the ui’s form a basis, and thus the ci’s linear combinations of the δ µ’s.Clearly, the ci’s are just as good at describing the Hamiltonian as the original µ’s. If the RGtransformation is applied to δ µ, and the result expressed in terms of the ci’s, cf. Eqs. (5.12),(5.14), and (5.16), it follows that

ci,L = L yi ci , (5.19)

which no longer involves matrices and has a very simple dependence on L. Indeed, if we goback to our discussion of the free energy associated with the RG transformation, cf. Eq. (5.7),and express the parameters in terms of the ci’s, we see that

A(c1, c2, . . . ) = L−d A(L y1 c1, L y2 c2, . . . ), (5.20)

which is a generalization of the scaling form assumed by Widom, cf. Eq. (4.3); moreover, we canrepeat the analysis of the preceding section to express the experimental exponents in terms of theyi’s. Before doing so, however, it is useful to look at some of the qualitative properties of Eq.(5.20). The notation is slightly different, but nonetheless, the scaling analysis given above can berepeated, and the results are summarized in Table 5.

Table 5: Critical exponents in terms of the yi’s†

Exponent In terms of the yi’s Numerical Value*

α (2y1 − d)/y1 - 0.2671β (d − y2)/y1 0.7152γ (2y2 − d)/y1 0.8367δ y2/(d − y2) 2.1699ν 1/y1 1.1335

†We’v e assumed that c1∝τ and that c2∝H .

*These are for the analysis of the 2d triangular lattice ferromagnet presented in the next

section.

First, as was noted above, the RG transformation may not exactly preserve the form of themicroscopic Hamiltonian, e.g., Eq. (5.1); as such, there will be more than two parameters

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generated by the RG iterations. How does this agree with the thermodynamics, which says thatthe critical point in a ferromagnetic or liquid-gas system is determined solely by temperature(coupling constant) and magnetic field? What about the other parameters? Whatever they are, itis inconceivable that all phase transitions of the same universality class and materials have thesame values for these, and thus, how can universal exponents arise? The way out of this problemis for the other parameters to have neg ative yi exponents. If this is the case, then the scalinganalysis (where we write L in terms of the reduced temperature or magnetic field) gives L → ∞as the critical point is approached, and these variables naturally assume their fixed point values,i.e., δ ci → 0. These kind of quantities are called irrelevant variables, since they will adopt theirfixed-point values irrespective of their initial ones. In other words, irrespective of the actualparameter values, as determined by the microscopic nature of the material and phases under con-sideration, close enough to the critical point the systems will all behave like the one with theparameters set to those that characterize the fixed point.

Second, quantities that have positive exponents are called relevant variables, and describethings like temperature and magnetic field. We know that different materials have different criti-cal temperatures, pressures etc., and thus we expect that their values are important. Explicit cal-culations show that only two relevant variables arise for the class of problems under discussion,and so the thermodynamics of our model is consistent with experiment.

Taken together, these two observations explain why universal behavior is observed at thecritical point and how scaling laws arise. Moreover, we hav e the blueprint for the calculation ofthe critical exponents. All we have to do is to compute the eigenvalues of the linearized RG

transformation and carry out some simple algebra. As we will now see, this isn’t as easy as itsounds.

6. An Example

As the simplest (although not very accurate) example9 of the RG approach consider the twodimensional triangular lattice depicted in Fig. 9.

9Th. Neimeijer and J.M.J. van Leeuwen, Phys. Rev. Lett. 31, 1411 (1973); Physica 71, 17 (1974).

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Fig. 9. A portion of a triangular lattice with blocking scheme withL = √3 as indicated. The arrows show the spins that interact on anadjacent pair of blocks.

We will perform the block transformation in blocks of three as indicated (L = √3) using a major-ity rule to assign the block spin. Each block of three can assume 8 spin configurations, 4 willhave the majority spin up (↑↑↑, ↑↑↓, ↑↓↑, ↓↑↑) and 4 spin down (↓↓↓, ↓↓↑, ↓↑↓, ↑↓↓).

It turns out to be difficult to evaluate the restricted sum in Eq. (5.5), and we will use pertur-bation theory to get an approximate expression; specifically, we will treat the interactionsbetween the blocks and the external field as a perturbation. Hence, to leading order the blocksare uncoupled and the partition function can be easily evaluated by explicitly summing the con-figurations. This gives:

e−W (0)

= (e3J + 3e−J )N /3, (6.1)

where we absorb the factors of kBT into J and H . Since our answer doesn’t depend on the val-ues of the block spins or the magnetic field, it’s not particularly interesting.

By expanding W and the Hamiltonian in the perturbing terms and comparing the results, it’seasy to show that to first order,10

W (1) =< H (1) >0 . (6.2)

10More generally, one must carry out what is known as a cumulant expansion; namely,

< eλ A >= exp

∞

j=1Σ λ j

j!<< A j >>

,

where << A j >> is known as a cumulant average. It is expressed in terms of the usual moment averages,< An >, by expanding both sides of the equation in a series in λ and comparing terms. To first order, itturns out that << A >>=< A >, which gives Eq. (6.2). It is also easy to show that<< A2 >>=< (A− < A >)2 > is just the variance. For more information on cumulants and moments, see,e.g., R. Kubo, Proc. Phys. Soc. Japan 17, 1100, 1962.

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When the average is performed we find terms that are linear in the block spin variables, the coef-ficient of which is the new magnetic field. Hence,

HL = 3H

e3J + e−J

e3J + 3e−J

. (6.3)

In addition, Eq. (6.2) will give terms that are products of the spin variables on adjacent blocks,the coefficient of these gives the new coupling constant, and

JL = 2J

e3J + e−J

e3J + 3e−J

2

. (6.4)

At the fixed point, JL = J and HL = H . From Eq. (6.3) we see that H = 0 at the fixed point; i.e.,the ferromagnetic critical point occurs at zero external magnetic field, as expected. Equation(6.4) shows that there are actually two fixed points: one with J = 0 and the other with

J =1

4ln(1 + 2√2) = 0. 335614. . . . (6.5)

The fixed point with J = 0 has no interactions and will not be considered further (actually, itdescribes the infinite temperature limit of the theory, and will yield mean-field behavior if ana-lyzed carefully). The other fixed point describes the finite-temperature critical point.

The linearized RG transformation, cf. Eq. (5.13), turns out to be diagonal with eigenvalues

λ H = 3

e3J + e−J

e3J + 3e−J

(6.6)

and

λ J =2(e4J + 1)(e8J + 16Je4J + 4e4J + 3)

(e4J + 3)3(6.7)

which gives the numerical values shown in Table 6. The exponents were obtained from Eq.

(5.17), remembering that L = √3.

Table 6. Some Numerical Values

Quantity J* = ln(1 + 2√2) / 4 Exact‡

λ H 3/√2 = 2. 1213 2.80yH 1.3691 15/16

λ J 1.6235 √3 = 1. 73yJ 0.8822 0.5

α - 0.2671 0β 0.7152 0.125δ 2.1699 15

‡The 2d Ising model has an exact solution, first given by Onsager.

The agreement with the exact results isn’t great, the error mainly coming form λ H , and ismainly due to our use of first order perturbation theory for the spin potential of mean force, W .

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While it is conceptually simple to carry out the perturbation calculation to higher order, at theexpense of a lot of algebra, to do so requires that we consider some of the extra terms in theHamiltonian. The higher order calculation automatically generates interactions beyond nearestneighbors and beyond pairwise additive ones, and some of these must be considered if goodagreement is to be obtained. When this is done, the correct exponents and thermodynamic func-tions are found (to about 5% accuracy). An example of some of the better results is shown inFig. 10.

Fig. 10. Results of higher order numerical RG calculation for the 2dlattice of Nieuhuis and Nauenberg11. The solid curve is the exact CH ,the dashed curve is the free energy, and the dot-dashed curve is theenergy, all from Onsager’s exact solution of the model. The points arethe numerical results. K is J in the text.

11B. Nienhuis and M. Nauenberg, Phys. Rev. B11, 4152, 1975.

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7. Concluding Remarks

We hav e accomplished the goals set out in Sec. 1. We now see how phase transitions arise,why universal behavior is expected, and perhaps, most important, have provided a framework inwhich to calculate the critical exponents. Several important issues have not been dealt with.

For example, other than the fact that the simple mean field approaches didn’t giv e the cor-rect experimental answer, we still don’t really understand why they failed, especially since manyof the qualitative features of a phase transition were described correctly. There is a consistent,albeit complicated, way in which to do perturbation theory on a partition function, which givesmean field theory as the leading order result. If we were to examine the next corrections, wewould see that they become large as the critical point is reached, thereby signaling the break-down of mean field theory. What is more interesting, is that the dimensionality of space plays akey role in this breakdown; in fact, perturbation theory doesn’t fail for spatial dimensions greaterthan four.

The dependence on the dimensionality of space plays a key role in Wilson’s work on criticalphenomena; in short, he’s (along with some key collaborators) have shown how to use ε ≡ 4 − d

as a small parameter in order to consistently move between mean field and non-mean fieldbehavior.

These, and other, issues require better tools for performing the perturbative analysis and forconsidering very general models with complicated sets of interactions. This will not be pursuedhere, but the interested reader should have a look at the books by Ma3 or by Amit12 for a discus-sion of the more advanced topics.

12D.J Amit, Field Theory, the Renormalization Group, and Critical Phenomena (McGraw-Hill, Inc.,1978).

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