5 Phase Transitions and Critical Phenomena

5.1 The Ising Model

Ferromagnetism is an important phenomenon in the study of solids. Certainmetals, including iron, spontaneously develop a finite magnetization at ordinarytemperatures. Above the so-called Curie temperature, however, these systemsexhibit randomly oriented spins. Only below this temperature is a permanent,spontaneous magnet formed. It is also found that the heat capacity of suchmagnetic systems diverges near the Curie temperature.

The Ising model is a seemingly very simple model that was developed tounderstand such behavior. Although developed specifically in connection withferromagnetism, the Ising model has proven to be a very model that can alsobe applied to such diverse systems as interacting gases, simple binary liquidmixtures, and alloys. It also turns out to be far from simple to solve! We’lldiscuss one approach to solve the Ising model, as well as to understand phasetransitions in general.

The following Hamiltonian represents a simple model for a paramagnet, amaterial that does not exhibit a spontaneous magnetization in the absence ofa magnetic field, but which responds by developing a magnetization in thepresence of a field:

H = −h

N∑

i

si, (5.1)

where si = ±1 for “up” and “down” magnetic spins. The number of microstateswith a given magnetization M = N↑ −N↓ is then

Ω =N !

N↑!N↓!. (5.2)

The entropy is then

S = Nk

log 2−1

2(1 + s) log (1 + s)−

1

2(1− s) log (1− s)

, (5.3)

where we have used a version of Sterling’s approximation

logN ! ≃ N logN −N, (5.4)

and the fact that N↑,↓ = N (1± s) /2. Here, s = M/N is the average spin.For a given magnetic field h, the energy of a single up/down spin is ǫ↑,↓ = ∓h.

Since these are the only two possibilities, the probability that a single spin isup/down is given by

e±βh/Q, (5.5)

where Q = eβh + e−βh, and β = 1/(kT ). Thus, the average spin

〈s〉 =eβh − e−βh

Q= tanh(βh). (5.6)

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The important physics that this simple model leaves out, of course, is the factthat the spins do interact. One must take interactions into account in order tounderstand transitions between phases. The interactions in a ferromagnet tendto align the a single spin with its neighbors. In the Ising model, we simply addto the Hamiltonian above a term that favors alignment of neighboring spins:

H = −h∑

i

si − J∑

i,j

sisj . (5.7)

Here, we have assigned an energy ±J for each pair of neighboring spins, de-pending on alignment. For J > 0, the energy is lower for spins that are aligned.This is what we expect for ferromagnets. The sum in this expression is over allpairs of neighboring spins (“bonds”). Note that each pair of spins that interactshould only be counted once. For a regular square lattice in two dimensions,there are four nearest neighbor spins, corresponding to spins to the left andright, as well as above and below a single spin. In d dimensions, there will bez = 2d neighbors.

One approach to solving the Ising model is to consider the average or meaneffect of the z spins neighboring a single spin si. This approach is known asmean field theory. We can write the Hamiltonian in the following way,

H = −h∑

i

si −1

2J∑

i

si

(

∑

nn

snn

)

, (5.8)

where nn refers to the z nearest neighbors of spin si. The factor of 1/2 here isintroduced to correct for double counting pairs of spins, since each spin that isincluded in the first sum will also appear in the second, although each pair ofspins should only be counted once. So far, no approximation has been made.The idea behind mean field theory is to treat the neighboring spins only in anaverage way. The part of the energy above that involves an individual spin sican be written

Hi = −hsi − Jsi

(

∑

nn

snn

)

. (5.9)

5-2

Note that H 6=∑

Hi, due to double counting of spin pairs. We approximatethe term

∑

nn snn by z〈s〉, since there are z nearest neighbors. This is a seriousapproximation, but we might expect that it is not a bad approximation if thenumber of neighbors is large, so the number of terms in

∑

nn snn is large. Theresult looks just like the model for a paramagnet, except with an effective mag-netic field heff = h+ zJ〈s〉. In other words, the effect of the surrounding spinsis similar to a molecular magnetic field acting on spin si due its neighbors.

With this approach, we obtain the average spin self-consistently using Eq.(5.6).

〈s〉 = tanh (β [h+ zJ〈s〉]) . (5.10)

This can be solved graphically, with the result that the only solution for h = 0and βzJ < 1 is 〈s〉 = 0. In other words, there is no spontaneous magnetization.For kT < zJ , however, nonzero solutions exist. Physically, these solutionscorrespond to states for which there is a spontaneous magnetization.

We can solve this model in another way, that is more instructive. Withinthe mean field approximation, the energy per spin is approximately

u = −hs−1

2zJs2, (5.11)

where we have dropped the brackets denoting the average spin. What aboutthe entropy? This we know from our solution of the paramagnet above. Thefree energy per spin in this approximation is therefore approximately given by

f = −hs+kT − zJ

2s2 +

kT

12s4, (5.12)

for small s.In the absence of a magnetic field h, the minimum of this free energy with

respect to the average spin s occurs for s = 0 when kT > zJ . For kT <zJ = kTc, there are two degenerate minima for s > 0 and s < 0. These statescorrespond to spontaneous magnetization “up” and “down”. These minimaoccur for ∂f

∂s = 0. Furthermore, near the transition,

s2 ≃ 3(Tc − T )/T. (5.13)

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Thus, near the transition the magnetization is given approximately by

M ≃ ±N√

3(1− T/Tc), (5.14)

provided that T < Tc.

Near the transition, the heat capacity can also be calculated, since

U ≃ −N

2zJs2 ≃ −

3N

2zJ (1− T/Tc) . (5.15)

The heat capacity is finite, with a value C = 3Nk/2 at the transition. Thenumerical solution of Eqs. (5.10) and (5.15) yields a heat capacity that decreasesto zero for T < Tc.

5-4

It is also instructive to calculate the susceptibility within the mean fieldapproximation as well. Above the transition (i.e., for T > Tc), we find for smallapplied magnetic fields h that

s =h

kT − zJ. (5.16)

This comes from minimizing f in Eq. (5.12) with respect to s for small h. Then,the susceptibility diverges above the transition as

χ ≡∂M

∂h=

N

kT − zJ∼ |T − Tc|

−γ , (5.17)

where γ = 1.

Although we may, and should question the validity of the mean-field ap-proximation above, the results derived above are illustrative of rather general

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behavior of systems near critical points. In particular, the continuous power-lawgrowth of magnetization below the transition, as well as the power-law diver-gence of the susceptibility are characteristic of critical points. Nevertheless, itis worth thinking ahead about what important physics this kind of approachleaves out. By assuming that the environment of each spin si can be treated by asingle number representing the same average field, we are clearly leaving out anypossibility of fluctuations. This can be an important effect, since a fluctuationthat gives more up spins in one region will tend be self-reinforcing by makingneighboring spins also flip up. In fact, we might expect that this reinforcingeffect becomes most important near the critical point, since the susceptibilityχ, which measures how easy it is to change the magnetization, diverges! Moresubtly, we will also find that precisely near the critical point, the spatial extentof correlations in the magnetization gets very large. I.e., the fluctuations involvemore and more spins the closer one gets to the critical point. With hindsight, itis, in fact, surprising that mean-field theory works as well as it does. It practice,it is still the first approach that most people take in dealing with a new problemin phase transitions.

5.2 Landau Theory

Arguably the first reasonably successful and general approach to understandingphase transitions goes back to Landau in the 1930’s. The basic idea here issimple enough; one assumes that that one can write the free energy (e.g., fabove) as a smooth function of a measurable parameter (the so-called orderparameter) describing the transition. In the case of the Ising model, the naturalorder parameter is the magnetization M , or equivalently, the average spin 〈s〉.This is a particularly convenient choice, since it is zero in the disordered phase(the paramagnet) and non-zero in the ordered phase (the ferromagnet).

By assuming that the free energy is smooth, we expect to be able to expandthis free energy in a Taylor series in the order parameter, which we will denoteas m. So, for instance, for a magnet at temperature T in the absence of amagnetic field, we expect to have

f(T,m) ≃ a(T ) +1

2b(T )m2 +

1

4c(T )m4 + · · · , (5.18)

were we have used the fact that the Ising model is symmetric under flippingall spins (changing the sign of m), even if the equilibrium phase may develop aspontaneous magnetization that breaks this symmetry. The free energy of anystate should be the same if m → −m, unless we have a non-zero magnetic fieldthat breaks this symmetry. This is, of course, the (approximate!) form of thefree energy we derived above for the Ising model. Here, the various coefficientsin the Taylor series are unknown functions of the remaining variable T . We donot know what they are, but by taking a cue from the Ising model above, wesee that if c(T ) > 0, then the transition will occur when b(T ) changes sign. So,if we now imagine that b(T ) can also be expressed as a Taylor series about Tc,

5-6

at which it changes sign, then we find

b(T ) ≃ b0(T − Tc) + · · · , (5.19)

and similarly,c(T ) = c0 + c1(T − Tc) + · · · . (5.20)

Near the critical point, provided that c0 > 0, we can find the equilibrium byminimizing the free energy to find

∂f

∂m= 0 = bm+ cm3 + · · · , (5.21)

which givesm = ±

√

b0/c0|T − Tc|β , (5.22)

where β = 1/2. This is valid for temperatures just slightly below Tc. (Apologiesfor the use of β for two things. But, there is simply no way around this: both usesare too deeply imbedded in the statistical physics literature to do otherwise.)

We could also calculate the behavior of both the heat capacity C and thesusceptibility χ within this model (assuming an additional term −hm in the freeenergy for small external magnetic fields). We would find essentially the sameresults as before, in that C remains finite with a discontinuity, and χ divergeswith the same exponent γ as above. Since all we have really assumed here isthat the free energy is smooth (analytic) as a function of the order parameter(here, m), this suggests that all phase transitions (with the same symmetry asthe magnet) should have the same behavior, e.g., in m, C, and χ, at least neara critical point, where the order parameter is small, allowing for Taylor seriesexpansions similar to those above.

You will probably not be surprised to hear that this is false, in that β 6= 1/2and γ 6= 1; we have, after all, made very strong assumptions/approximationsabove. What you probably will be more surprised to hear is that experimentson phase transitions in many different materials show THE SAME values of βand γ. So, while Landau theory may fail in detail (e.g., in predicting values ofβ and γ), it seems to get one very important qualitative feature right: criticalbehavior is remarkably universal from one system to another. This universalityin itself begs an explanation. We will begin to see why this is true. And, in theprocess, we will also learn something very deep about the nature of condensedphases of matter with many degrees of freedom: most of the microscopic details,such as the precise way in which molecules interact with each other, becomelargely irrelevant in determining the way systems behave at the macroscopicscale. In the development of the statistical mechanics approach to try to derivethermodynamic properties, this is simply something that we hope for. Criticalphenomena provide a wonderful window into the origins of the emergence ofmacroscopic simplicity out of microscopic complexity—i.e., the fact that only afew thermodynamic parameters are enough to describe the macroscopic behaviorof systems that are horribly complicated in microscopic detail. This is one ofthe reasons we will spend so much time studying relatively rare critical points!

5-7

Before we try to go beyond Landau theory, it is worth noting a few morethings about it. First of all, it is a sort of mean-field theory, in that we haveonly accounted for a single, average value of the order parameter m, and havenot allowed for this order parameter to fluctuate, either from point to pointwithin the sample, or from time to time at the same point. We can improve onthis limitation by allowing for a spatially varying order parameter m(x). Thiswill not, in the end, cure the above failures of Landau theory, although we willlearn something useful about spatial correlations.

There is another important success of Landau theory. It shows us thatsymmetry alone can determine whether a phase transition is first-order (likethe melting of ice, in which there is a latent heat) or second order (like thedevelopment of ferromagnetism below the Curie temperature). Notice that ifwe do not have the m → −m symmetry of the magnet, we could not assume allodd terms in the Taylor series expansion of f vanish above. You can check foryourselves, for instance, that with a free energy of the form

f(T,m) ≃ a(T ) +1

2b(T )m2 +

1

3c(T )m3 +

1

4d(T )m4 · · · , (5.23)

the order parameter jumps discontinuously from one value to another as b(t)decreases below the value 2c2/(9d). (Here, d is assumed to be positive forstability.) While this may seem like a curiosity, it explains why the transitionbetween isotropic (I) and nematic (N) phases of liquid crystals is first order: thesymmetry of liquid crystals is different from magnets, and permits odd termsin the Landau theory (Chaikin and Lubensky 2000).

Even when the symmetry of the problem does not permit odd terms in thefree energy, it is still possible to have a first-order transition. An example ofsuch a situation is what happens in the model above when the 4th order termis not positive:

f(T,m) ≃ a(T ) +1

2b(T )m2 +

1

4c(T )m4 +

1

6d(T )m6 + · · · . (5.24)

Here, we assume that d > 0, which ensures that the model is stable (m doesnot diverge.) By plotting this function for various values of the parameters b,c, and d > 0, you can convince yourself that there can be a transition witha discontinuous jump in m (sign of a first-order phase transition). The otherinteresting thing about this model is that it exhibits a line of critical pointsthat becomes a first-order phase boundary. This so-called, tricritical behavioris seen in the superfluid transition of 4He-3He mixtures. As 3He is added to4He, the continuous (critical) transition to a superfluid still occurs, but at alower temperature. As more 3He is added, however, this transition becomesfirst-order (discontinuous), with coexisting 4He-rich superfluid (S) and 3He-richnormal fluid (N). The dotted line indicates a line of critical points.

5-8

These examples show why Landau theory is so useful, even if it is not rigor-ous or accurate in its predictions of critical behavior. We will later return to theorigins of this failure, and why it is particularly apparent at the critical point.But, it is worth reflecting one of the key assumptions of Landau theory, thatthe free energy is analytic. Although this seems like a very innocent assump-tion, it is actually dangerous. Imagine building up a description of increasinglycomplicated systems beginning a particle at a time. The kinetic energy of aparticle is certainly analytic as a function of the momentum. Let’s assume thatthe interaction potential describing the way particles interact is also analytic.As we add more and more particles, it would seem that the Hamiltonian for thesystem will be analytic for any finite number of particles. It might get very hardor even impossible to solve this problem in practice for more than a few particles(like 3, even). But, how can the free energy not be analytic? It will turn outthat this failure of analyticity occurs (in general) in the thermodynamic limit,N → ∞. In fact, in a sense, there IS not real critical behavior for any finitesystem. More on this later. Of course, in practice, NA is a big number!

5.3 Critical Phenomena and Scaling

In the middle of the 20th century, a number of groups were performing ever moreprecise measurements of the behavior of magnets and other systems exhibitingcritical behavior. These experiments kept pushing to temperatures closer andcloser to the critical temperature, from above (t ≡ (T−Tc)/Tc → 0+) and below(t → 0−). In fact, these experiments can go to very low reduced temperaturest of order 10−3, or even 10−5 in some cases.

These experiments consistently showed behavior very similar to what mean-

5-9

field theory or Landau theory predicted. Specifically, it was seen that

Ch=0,t→0+ ∼ |t|−α

Ch=0,t→0− ∼ |t|−α′

mh=0,t→0− ∼ |t|β

χh→0,t→0+ ∼ |t|−γ

χh→0,t→0− ∼ |t|−γ′

mh→0,t=0 ∼ |h|1/δsign(h).

These are written in terms of magnetic properties. But, this generalizes easily toother systems, such as the liquid-gas transition, where, for instance, the densitydifference between liquid and gas plays the role of the order parameter m, whilepressure is analogous to the magnetic field. Then, for instance, δ expresses thedegree of pressure as a function of volume along the critical isotherm.

Mean-field theory predicts that that α = α′ = 0, β = 1/2, γ = γ′ = 1, andδ = 3. Experiments on a variety of systems shows that, indeed, the exponentsabove and below the transition seem to be the same, but that the values arevery different from those predicted by mean-field theory. For instance, α ≃ 0.1(i.e., the heat capacity diverges!), β ≃ 0.3, γ ≃ 1.3, and δ ≃ 4− 5.

In order to understand this, let’s begin by trying to fix Landau theory. Let’swrite the free energy in a simple, canonical form:

f(t, h) = −hm+1

2btm2 +

1

4cm4, (5.25)

where we shall treat b and c as constants. The equation of state can be obtainedby differentiation with respect to m:

h = btm+ cm3 = m(bt+ cm2). (5.26)

As t → 0+, we find that m = h/(bt), which means that

χ = χ+t−γ , (5.27)

where χ+ = 1/b, and γ = 1. Also, as t → 0−, m2 = b|t|/c, so that

|m| =√

b/c|t|β , (5.28)

where β = 1/2. Clearly, it is the exponent 2 in Eq. (5.15) that leads directly toβ = 1/2. Perhaps we should try

h = btm+ cm3 = m(bt+ cm1/β) (5.29)

as an equation of state. Obviously, unless 1/β happens to be an even integer,this cannot make any sense unless we replace m1/β by |m|1/β . Likewise, perhapswe could correct the exponent γ 6= 1 by the equation of state

h = m(btγ + c|m|1/β). (5.30)

5-10

This would give us the “right” divergence of χ as t → 0+. But, again, if thisis to make sense, we must take the absolute value of t here in order to describething below the transition. In that case, however, we need to also account forthe sign of t. Hence, we presumably should try

h = m(±b|t|γ + c|m|1/β). (5.31)

Unfortunately, there are real problems with all these attempts to “correct”Landau theory. Not only do we have nonanalytic behavior at the critical pointt = h = 0, but we even have nonanalytic behavior away from the critical pointalong the critical isotherm (t = 0)! While divergences and even nonanalyticbehavior are expected AT the transition, there is NO experimental evidence ofany singularities on the critical isotherm for finite h.

There is another way to do this that DOES work. Beginning again withthe original, mean-field equation of state, we can rescale the whole equation byb3/2|t|3/2/c1/2:

Dh

|t|∆=

m

B|t|β

(

±1 +

(

|m|

B|t|β

)1/β)

, (5.32)

where B =√

b/c, D = c1/2/b3/2, β = 1/2, and ∆ = 3/2. As strange as thismay appear, this CAN be generalized to arbitrary values of the two exponentsβ and ∆ in a consistent way, without introducing any singularities away fromthe critical point. For general values of these exponents, as t → 0+,

Dh

|t|∆≃

m

B|t|β. (5.33)

Thus, the susceptibility scales according to χ ∼ |t|β−∆, or γ = ∆− β.Equation (5.32) can be inverted to obtain the “reduced” or normalized mag-

netization m = mB|t|β

as a function of the reduced field h = D h|t|∆ . This can

be done graphically for the mean-field case of β = 1/2, for which we can solve

h = m(

±1 + m2)

.

5-11

In general, we write

m = W±

(

h)

, (5.34)

where ± refer to the t > 0 and t < 0 branches of these solutions of m in termsof h. I.e.,

m

B|t|β= W±

(

Dh

|t|∆

)

(5.35)

This suggests that if we plot m/|t|β versus h/|t|∆, we should see all of the datafor T > Tc collapse onto a single curve defining the function W+, while all ofthe data for T < Tc should collapse onto another curve defining W−, providedthat we choose the right exponents β and ∆. In fact, this works remarkablywell. Starting with raw data of the form shown below,

rescaling the data in this way results in two curves similar to the following.

You should try this yourself with classic data obtained by Weiss and Forrerin 1926 (next page).

5-12

T (K) h (oe) M (emu/g)678.86 4,160 0.649678.86 10,070 1.577678.86 17,775 2.776653.31 2,290 0.845653.31 4,160 1.535653.31 6,015 2.215653.31 10,070 3.665653.31 14,210 5.034653.31 17,775 6.145653.31 21,315 7.211629.53 1,355 6.099629.53 3,230 9.287629.53 6,015 11.793629.53 10,070 13.945629.53 14,210 15.405629.53 17,775 16.474614.41 1,820 19.43614.41 3,230 20614.41 6,015 20.77614.41 10,070 21.67614.41 14,210 22.36614.41 17,775 22.89601.14 3,230 25.34601.14 6,015 25.83601.14 10,070 26.4601.14 14,210 26.83601.14 17,775 27.21

Data taken from Statistical Mechanics, K. Huang (Wiley, New York, 1987).Original data taken from P. Weiss and R. Forrer, Ann. Phys. (Paris) 5, 153(1926). These data were reanalyzed by Kouvel and Fisher (J.S. Kouvel andM.E. Fisher, Phys. Rev. 136, A1626 (1964)), who found γ = 1.35 and δ = 4.22.The transition temperature is Tc = 627.2.

Unfortunately, this is not the full data set. These data have been analyzedand re-analyzed many times in the literature. One such example is the follow-ing plot of scaled field versus scaled magnetization (Arrott and Noakes, Phys.Rev. Lett. 19, 786 (1967)). Although these particular data were obtainedfor Nickel, when experiments are performed on other ferromagnets, the sameexponents AND scaling functions W± are found, apart from possibly differentscaling factors B and D! If that is not surprising enough, when experiments areperformed on fluids (liquid-liquid phase separation, or liguid-gas transitions),not only is similar scaling behavior found, but again, the exponents and evenscaling functions W± are the same as for magnets!

5-13

The fact that our scaling hypothesis (actually, due to Widom) works hasimportant implications for other exponents, such as γ. Since we can write min terms of h and one of two universal functions W±, means that we can nowcalculate the susceptibility. For t > 0, in particular,

χ =

(

∂m

∂h

)

h→0±∼ |t|β−∆W ′

+ (0) . (5.36)

Here, W ′+ (0) refers to the derivative of W+. Thus, γ must be ∆− β. In other

words,∆ = β + γ. (5.37)

A similar argument for t < 0 tells us also that

γ′ = γ. (5.38)

What about the critical isotherm, where t = 0? As we noted above, it is wellestablished that the magnetization and magnetic field are both well-defined and

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smooth away from the critical point, along the critical isotherm. But here, bothm and h appear to be ill-defined (infinite!) at t = 0. Thus, the critical isothermcorresponds to the asymptotic regime of very large arguments of the functionsW±. How do these functions behave for large arguments? What if W (x) itselfexhibits a power-law here? If W (x) ≃ W∞xλ as x → ∞, then

m ∼ |t|βW∞hλ/|t|λ∆. (5.39)

But, if this is to make any sense as t → 0, then the t-dependence in the numer-ator and denominator must just cancel. I.e., λ must be β/∆. We can then findthe relationship between m and h along the critical isotherm: m ∼ hλ, so that

δ = 1/λ = 1 + γ/β. (5.40)

Moreover, the fact that the data collapse for m and h for the same ∆ above andbelow the critical temperature, we must also have that the exponent λ, as wellas the amplitude W∞ must be the same for W+ and W−. In other words, thefunctions W+ and W− must converge for large arguments.

What we learn from all of this is that basic thermodynamics imposes con-straints on some of the various exponents. Specifically, there only appear to betwo independent exponents so far. In fact, the exponent α is also completelydetermined by β and γ. In order to see this, we note that m and h are conjugatethermodynamic variables, like volume and pressure for a gas. Thus

−∂f

∂h= m ∼ |t|βW±

(

Dh/|t|∆)

. (5.41)

We can integrate this (with respect to h), to find that

f = −

∫

mdh ∼ −|t|β+∆

∫

mdh ∼ −|t|β+∆. (5.42)

Since

C = −T∂2f

∂T 2, (5.43)

we find thatC ∼ |t|β+∆−2, (5.44)

meaning thatα+ 2β + γ = 2. (5.45)

It can also be shown thatα′ = α. (5.46)

Thus, there appear to be at most just two independent critical exponents, sayβ and γ, from which the other four exponents can be derived. In other words,although the various critical exponents are not so simple as mean-field wouldpredict, there seems to be some order among these exponents.

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5.4 Why does scaling theory work?

We are left with the question of why our basic scaling hypothesis works. Onevery basic observation we can make from the fact that we observe power-lawbehavior, e.g., the divergence of χ near t = 0, is that things look essentiallythe same as one moves closer to the critical point, i.e., as t decreases towardzero. This property is especially clear from Widom’s approach to scaling: amagnet close to the critical point behaves just like one far from the criticalpoint, apart from a simple rescaling of various physical quantities like h andM . This mathematical property, known as scale invariance, appears to be akey physical property of systems near critical points. In fact, scale invariance isa very important property of many physical systems, and scaling theories andapproaches such as we have employed above can be applied rather widely.

Consider a simple mathematical property of a function such as the suscepti-bility χ as a function of t. The fact that we observe a power-law dependence tellsus that there is no intrinsic scale, e.g., for t. Contrast this with, for instance,an exponential dependence, for which there IS a characteristic scale. If χ wereexponential in t, we would necessarily have a characteristic scale t0 for t:

χ ∼ et/t0 .

Instead, χ is a homogeneous function of t of degree −γ:

χ(Λt) = Λ−γχ(t), (5.47)

for Λ > 0. Our magnetic system, however, depends on two thermodynamicvariables, t and h. What if the free energy f itself is also homogenous in bothof these variables? We’ll examine the consequences of

f(Λat,Λbh) = Λf(t, h). (5.48)

The reason for assuming this form is that, since our critical point is at t = h = 0,for any finite t or h, Λ > 1 moves us farther away from the critical point (atleast for a, b > 0). But, it may do so at different rates for the t and h. Theassumption we are making here is that, apart from an overall scaling factor,the free energy is the unchanged. (By the way, our choice of exponent 1 for Λon the right-hand side of this equation is not essential. If we replace this byanother exponent c 6= 0, we could redefine a and b so that the equation aboveis valid. There are, thus, only two independent exponents above, and we havechosen this particular form without any loss of generality.)

The free energy can then be expressed as

f(t, h) = Λ−1f(Λat,Λbh). (5.49)

From this, we obtain the entropy

S(t, h) = −∂F

∂T∼ −

∂f

∂t∼ Λ−1+af1,0(Λ

at,Λbh), (5.50)

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where fi,j represents the i-th derivative of f with respect to its first argument,and j-th derivative with respect to its second argument. The heat capacity isthen

C(t, h) = −T∂2F

∂T 2∼ −

∂2f

∂t2∼ Λ−1+2af2,0(Λ

at,Λbh). (5.51)

The magnetization is given by

m(t, h) = −∂f

∂h∼ Λ−1+bf0,1(Λ

at,Λbh). (5.52)

The susceptibility involves one more derivative with respect to h:

χ(t, h) =∂m

∂h∼ Λ−1+2bf0,2(Λ

at,Λbh). (5.53)

These relations can be used to find the various critical exponents as follows.For h = 0, if we let Λ = |t|−1/a, then

C(t, 0) ∼ |t|−(−1+2a)/af2,0(±1, 0), (5.54)

where ± refer to t > 0 and t < 0. Thus,

α = (2a− 1)/a. (5.55)

Similarly, for m and χ we find that

m(t, 0) ∼ |t|(1−b)/af0,1(−1, 0) for t < 0, (5.56)

andχ(t, 0) ∼ |t|(1−2b)/af0,2(±1, 0). (5.57)

This means thatβ = (1− b)/a (5.58)

andγ = (2b− 1)/a. (5.59)

Also, for t = 0, by considering Λ = |h|−1/b we find that

m(0, h) ∼ |h|(1−b)/bf0,1(0,±1). (5.60)

Thus,δ = b/(1− b). (5.61)

You should check that the above scaling relationships among these exponents(Eqs. 5.40, 5.45) are satisfied.

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5.4.1 The story so far...

So, we learn, yet again that there appear only to be two independent exponents,on which all the others depend. Beyond that, we now see that the number ofthese key independent parameters describing the critical behavior is connectedto the fact that we have two (intensive) physical parameters describing thesystem. For the magnet, these are the temperature (t) and magnetic field (h).For liquid-gas systems, the corresponding parameters are the temperature andthe pressure.

So far, I have made things sound a bit simpler than they really are. Inparticular, I implied that the critical behavior of magnets is the same as for allother systems. This is not the case. For instance, uniaxial magnets, for whichthe magnetic moment points preferentially along one axis, differ in their criticalbehavior (exponents and scaling functions) from Heisenberg-like magnets, inwhich spins can point in any direction. Nevertheless, it IS true (and very sur-prising) that uniaxial ferromagnets and fluid systems such as liquid-gas systemsshould behave in the same way! What is now known (and understood) is thatall systems with the same underlying symmetry have the same critical behavior.A liquid-gas system, for instance, can be described by the presence/absence of aparticle, just like an up/down spin in the Ising model for (uniaxial!) ferromag-netism. So, the critical exponents depend on the symmetry of the problem, butNOT on any details of how the components interact: what could be more dif-ferent in detail, after all, than magnetic spins and particles colliding! This lackof dependence on molecular details is a very striking aspect of critical behaviorthat begs for some explanation.

While we shall focus our attention on scaling theory as an approach, itis worth noting that some relationships between the various critical exponentswere found by other means. For instance, based on very general thermodynamicprinciples, Rushbrooke showed that

α′ + 2β + γ′ ≥ 2 (5.62)

must be true. Similarly, Griffiths showed that

α′ + β(1 + δ) ≥ 2. (5.63)

Predictions of scaling theory do not conflict with these general conditions. But,both of these inequalities become equalities within scaling theory. Not surpris-ingly, many people attempted to derive exact expressions for various exponentsdirectly from thermodynamics or rigorous statistical mechanics. Occasionally,people even reported derivations that were at odds with the above general in-equalities. But, these all suffered the same fate as various reports of violationsof the second law!

So, we are left with two burning questions: (1) why does scaling theory work;and (2) why is the behavior universal, in that molecular details do not matter.Perhaps the most important conceptual breakthrough that actually lead to asimultaneous answer to both of these questions was due to Kadanoff in mid-1960’s. The beauty of this idea lies in its simplicity and physical intuition. It

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does not require, for instance, complicated analysis and calculation. The down-side of this approach is that is is essentially conceptual. It was not possibleto turn it directly into a practical procedure to actually calculate an exponent.That had to wait for the work of Wilson in the 1970s, for which he (Wilson) wasawarded the Nobel prize in 1982. We shall NOT go through the latter approach.But, we can learn a lot from the insights of Kadanoff, which we can understandbased on just one more property of critical phenomena that we have not focusedon so far: fluctuations.

5.5 Critical Fluctuations

One of the most striking features of a critical point is the large fluctuationsthat become apparent there. This is often refered to as critical opalescence,because of the opalescent or milky appearance of otherwise transparent systemsat a critical point. For instance, in the case of the liquid-gas transition, (very!)near the critical temperature and pressure, the sample appears cloudy. Thisis because the system tends to fluctuate wildly between the two phases, sincethey are nearly indistinguishable near the critical point. These fluctuationsbetween phases amount to fluctuations in density. These fluctuations are bothlarge in magnitude (of density), as well as of large spatial extent. The densityfluctuations result in fluctuations in the index of refraction. This, together withthe large length scale of the fluctuations (i.e., comparable to the wavelength oflight) means that the sample scatters light strongly, as do clouds.

Not only are these fluctuations strong at the critical point, but in a sensethey provide the key to understanding the key features of critical behavior thatwe have mentioned so far: (1) the scaling behavior, (2) the anomalous (non-mean-field) exponents, and even (3) universality and the fact that microscopicdetails of the system become irrelevant.

Let’s try to generalize Landau theory to allow for fluctuations. The resultis usually known as Ginsburg-Landau theory. We consider a spatially varyingmagnetization m(x), and generalize our free energy to include a penalty forthese spatial variations. (If there were no penalty, why would we have anymacroscopically homogeneous phases at all?) We write the free energy as

F [m(x)] ∝

∫

ddx

[

1

2tm2(x) +

1

4um4 +

1

2ξ20 (∇m)

2

]

, (5.64)

where we have used a slightly simpler normalization of the various terms in theoriginal Landau theory. We have normalized by b and let u = c/b. We havealso introduced a new parameter with dimensions of a length, ξ0. This new freeenergy is actually a functional of the spatially varying m(x). Otherwise, this isstill in the spirit of Landau theory, in that we are expanding the free energy insmall quantities, now including gradients or derivatives of the field m. We arekeeping only the lowest order derivatives, with the assumption that these willdominate at long length scales.

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We are not so interested in a precise function m(x) so much as how themagnetization, or order parameter is correlated from one point to another. Withthis in mind, we begin with replacing m by its Fourier series/transform:

m(x) =∑

k

mkeikx. (5.65)

Furthermore, we shall focus on the case of t > 0, where we can ignore thefourth-order term, at least if m remains small enough. We can then write thefree energy

F ∝

∫

ddx1

2

∑

k,k′

(

t+ ξ20(ik)(ik′))

ei(k+k′)xmkmk′

= V∑

k

1

2

(

t+ ξ20k2)

|mk|2. (5.66)

Here, we have used the fact that∫

ddx ei(k+k′)x = V δk,k′ , (5.67)

and the fact that m−k is the complex conjugate of mk. By using the equiparti-tion theorem, we then find that

〈|mk|2〉 ∝

kT

t+ ξ20k2. (5.68)

We can measure the correlations of the fluctuating m(x) field by the corre-lation function

Γ (x) = 〈m(x+ x′)m(x′)〉. (5.69)

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This correlation function must vanish for large separations x, since 〈m〉 = 0.Furthermore, it must also be independent of x′. Thus,

〈m(x+ x′)m(x′)〉 = 〈∑

k,k′

mkeik(x+x′)mk′eikx

′

〉

=∑

k

〈|mk|2〉eikx. (5.70)

Thus, we can find Γ(x) by Fourier transform. You might recognize the corre-sponding problem with t = 0, since the Fourier transform of the electric potentialabout a point-charge is proportional to 1/k2. For t > 0, the result is

Γ (x) ∝e−|x|/ξ

|x|d−2. (5.71)

While writing most of the expressions above for one dimension (d = 1), thisfinal expression exhibits the expected inverse dependence on separation x thatis characteristic of the the electrostatic problem mentioned above. For finitet > 0, the correlation function also exhibits an exponential decay with distance|x|. The decay or screening length is ξ = ξ0|t|

−1/2. This correlation lengthrepresents the typical size of a fluctuating domain (with correlated value ofthe order parameter, such as a magnetic domain in magnets). This correlationlength diverges near the critical point with an exponent 1/2. Of course, this iswhat one would expect within Landau(-Ginsburg) theory. Real systems exhibita somewhat stronger dependence

ξ = ξ0|t|−ν , (5.72)

where the new (no pun intended) exponent ν ≃ 0.6. In fact, it is also observedthat the exponent in the denominator of Eq. 5.71 is not the predicted one fromLandau-Ginsburg theory:

Γ (x) ∝e−|x|/ξ

|x|d−2+η, (5.73)

where η is observed to be non-zero, but small.So, what do these fluctuations tell us about the above puzzles of critical

behavior? Well, the most important thing to note is that the correlation lengthnot only can be large, but strictly diverges at the critical point.

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5.6 Real-space Renormalization

Since the correlation length ξ diverges near the critical point, if we look at theIsing model near the critical point, we expect to see neighboring spins to behighly correlated. This suggests that we can consider larger blocks of spinsas effectively acting in concert. We introduce the concept of block spins, wheregroups of spins are thought of as acting like a single spin. Consider, for instance,a 2-d lattice of spins indicated by si, which take on values ±1. If we now considersquares of L spins on a side, which we define to be a block, then we can treatthe resulting lattice of block spins as a magnet. (In the figure, L = 2.) Wedefine these block spins SI = (

∑

i si) /L2, which takes on values between -1 and

1. Provided that L is small enough so that the block spins are still smaller thanξ, we expect that the system is not fundamentally altered, i.e., that it still lookslike an Ising model. This is because the spins in each block are highly correlated.This procedure amounts to a transformation of the original Hamiltonian from

H = −h∑

i

si − J∑

i,j

sisj (5.74)

toH ′ = −h′

∑

I

SI − J ′∑

I,J

SISJ . (5.75)

Here, we might expect that h′ ≃ L2h, since there are L2 spins in a single block.Note that on scaling up by a factor of L, this amounts to moving AWAY fromthe critical point. This also makes sense from the point of view of the correlationlength ξ, since under this rescaling of the system, ξ decreases by a factor of L:ξ → ξ′ = ξ/L. Given that ξ = ξ0|t|

−ν , this also corresponds to moving awayfrom the critical point, as characterized by t → t′ = L1/νt.

All of this can be generalized to d dimensions, by noting that upon rescalingthe system by a factor of L, we expect that the system behaves as though botht and h increase according to

t → t′ = Lxt, (5.76)

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andh → h′ = Lyh. (5.77)

Here, as noted above, we expect that x = 1/ν ≃ 2 and y ≃ d, the number ofdimensions.

This is an expression of the scale invariance we discussed above. We reallyhave not derived anything yet. We are simply arguing for a simple rescalingof the parameters t and h by factors that we might expect to be powers of L.We argued for this by appealing to the fact that near the critical point, sincethe range of correlations is large, we expect to be able to lump large groups ofspins together to form blocks, resulting in a simple geometric increase in theeffective field strength h. Likewise, the rescaling results in a simple decrease ofthe correlation length, as measured in the new lattice units. This is equivalentto an increase in t by a factor of L1/ν . These arguments really only apply nearthe critical point, where the correlation length ξ is sufficiently large that we stillhave correlated block spins after scaling the system up: ξ′ is still larger than alattice spacing.

We can also see that the scale factors relating t to t′ and h to h′ must besimple powers of L. This is really a consequence of scale invariance. If our systemreally is scale invariant, meaning that it still looks like an Ising ferromagnet nearthe critical point as we rescale our system by a factor of L, then we expect tohave the same form for the Hamiltonian H and free energy F after rescaling.Consider, for instance, a rescaling of the system by a factor of L1 followed byanother rescaling of the system by a factor of L2. Assuming that the effectivetemperature simply rescales by some factor φ that depends on L1:

t → t′ = φ(L1)t. (5.78)

If we then follow this with another rescaling of the system by a factor of L2,then

t′ → t′′ = φ(L2)t′ = φ(L2)φ(L1)t. (5.79)

But, this should be the same as if we simply rescale the original system by afactor of L1 × L2:

t′′ = φ(L2 × L1)t. (5.80)

This is satisfied by any simple power φ(L) = Lx. Likewise, we also expectthe scaling of h to involve a (possibly different) power Ly. This compositionproperty of these rescaling/renormalization transformations suggests a group-like structure, where following one transformation with another corresponds to asingle transformation by a product of the scale factors. In fact, these operationsdo NOT form a group in the mathematical sense, since these transformationcannot, in general, be inverted: we only lose information in a renormalizationtransformation.

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It is very interesting to think about how this loss of information happens.Let us say, for instance, that in our original system/lattice we have some freeenergy that depends on a collection of parameters: t, h1, h2, . . .. Then, thearguments above suggest that near the critical point, these all transform ashj → h′

j = Lyj upon rescaling. If the exponent here yj happens to be less thanzero, then the corresponding parameter becomes irrelevant near the criticalpoint. In fact, it becomes increasingly irrelevant near the transition, since thelarge correlation length means that we can rescale by a large factor, makingh′j small. This is one of the reasons why most of the microscopic details of

the system become increasingly irrelevant near the transition. We actually onlycare about those parameters for which y > 0. This also begins to explain whywe have universality, and why it is valid only close to the critical point.

So, what does this tell us about scaling? Consider the free energy f(t, h).Under a renormalization transformation, we have argued that t → Lxt andh → Lyh. Thus, we expect that f(t, h) → f(Lxt, Lyh). But, we have to becareful here. Since the free energy is an extensive quantity, depending on thesize of our sample, we must account for the fact that our sample effectivelyshrinks under this renormalization: there are fewer block spins than originalspins. Thus, it is more appropriate to describe the free energy per spin. Sincewe have Ld original spins in a block, the free energy per spin in the new latticemust be

f(Lxt, Lyh) = Ldf(t, h). (5.81)

This is just like Eq. (5.48) above, with Λ = Ld. Thus, a = x/d and b = y/d.Using Eqs. (5.55), (5.58), (5.59) and (5.61), we also obtain the various criticalexponents in terms of x and y. We also have, from above, that

ν = 1/x, (5.82)

which also means thatα = 2− dν. (5.83)

This illustrates once again that there are only two independent critical ex-ponents. This is even true if we consider the exponent η above, since it can beshown that γ = (2− η)ν.

As it stands, we now have a concrete theoretical procedure/approach toquantitatively understand critical behavior. If we can learn about how Hamil-tonians and free energies transform under the rescaling/renormalization trans-formations outlined above, then we can, for instance, extract x and y. This is

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more or less what is done. But, this procedure is complicated to do in prac-tice (one needs to consider transformations in the space of Hamiltonians), andthe so-called real-space procedure of Kadanoff is not actually a practical one.Nevertheless, we have at least conceptually identified the origin of scaling, andclosely associated aspects of critical phenomena such as universality. It remainsnow to explain just why/when mean-field theory fails, and when/if it can everbe valid.

Before doing so, however, we an already learn something about this fromwhat we have just derived. In particular, there is something very odd about thescaling relation in Eq. (5.83). It depends on dimensionality. All of the mean-fieldexponents are independent of dimension. On the one hand, this would appearto tell us that mean-field theory CANNOT be right. On the other hand, we alsolearn that there is a special dimension, d = 4, for which Eq. (5.83) is satisfiedfor mean-field theory. This is an important point that we shall return to: fourdimensions is a special case. In fact, for any dimension d > 4, mean-field theoryworks! We are either incredibly unfortunate, or lucky that we live in a worldwith funny critical exponents. I would argue that it is LUCK. The developmentsof phase transitions and critical phenomena would have concluded with Landautheory, and we would not have learned anything about scaling, renormalization,etc..

5.7 The Breakdown of Mean-field Theory

We have seen that fluctuations play an important role in the behavior of systemsnear a critical point. These fluctuations are also the cause of the breakdown ofmean-field theory. We can see this by making an estimate of the size of, e.g.,magnetization fluctuations in a magnet. If these fluctuations are larger than orcomparable to the average value of the magnetization, then clearly the wholeidea behind mean-field theory is questionable at best.

In order to estimate the magnitude of the fluctuations, we consider the caseof t < 0, where we expect to find a non-zero average of the order parameter m.From the Ginsburg-Landau theory in Eq. (5.64), for t < 0 we expect to find〈|m2|〉 = m2

0 = −t/u = |t|/u. About this value, we expect to find fluctuationsδm(x). The free-energy can then be expressed as

F [m(x)] ∝ F0 +

∫

ddx

[

|t||δm(x)|2 +1

2ξ20 (∇m)2

]

, (5.84)

where we have expanded the integrand about the minimum at m = ±m0. Wehave also indicated by F0 the mean-field value of the free energy, i.e., the freeenergy for a uniform m(x) = ±m0. Note that

d2

dm2

[

1

2tm2 +

1

4um4

]

m=±m0

= 2|t|. (5.85)

We can estimate the remaining integral by taking very seriously (of course,a little too seriously in detail) the idea of the correlation length ξ, and the figure

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above. We’ll assume that at any given time, the sample can be decomposed intoregions of size ξ that fluctuate independently. Because the field δm is stronglycorrelated over distances smaller than ξ, we will treat the field as uniform withina single region. Furthermore, these regions effectively behave independently ofeach other since the the field δm is weakly correlated over distances larger thanξ. Thus, we can calculate the contribution to the free energy above of one ofthese regions as

∆F ≃ ξd|t|δm2. (5.86)

We have neglected the gradient term in this free energy, but you can convinceyourself that this is of the same order as the above, since |∇m|2 ≃ δm2/ξ2.

This means that the probability distribution of δm is given by the exponen-tial of −∆F/kT . In fact, we have actually normalized the free energy by kT .Thus,

〈δm2〉 ∼ |t|−1ξ−d ∼ |t|dν−1ξ−d0 . (5.87)

But, for mean-field theory to make sense, we have to have that 〈δm2〉/m20 ≪ 1.

Thus, we must have|t|dν−2 ≪ ξd0/u, (5.88)

since m20 = |t|/u. Of course, ν = 1/2 within mean-field theory, so that this

condition becomes|t|d−4 ≪ ξ2d0 /u2. (5.89)

This can always be satisfied close to the critical point for d > 4, but cannot besatisfied close to the critical point for d < 4. In three dimensions, this can onlybe satisfied for |t| > t0, there

t0 ∼ (u/ξd0 )2. (5.90)

This shows us that mean-field theory cannot be valid arbitrarily close to thethe critical point. As t → 0, the fluctuations will always dominate the average,

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or mean field. But, depending on the system in question, it may be possible tosatisfy the inequality above for even very small |t|. This can happen especially ifthe bare correlation length ξ0 is large enough. This is the case, for instance, forthe transition to superconductivity in many metals. This is because the naturalsmall distance, and therefore ξ0, in this problem is the size of a Cooper pair,which can be much larger than the atomic/molecular scale.

5.8 Finite-size Scaling

We have found that physical quantities such as the heat capacity and suscep-tibility diverge, seemingly to infinity at the critical point. It is natural to ask,in what sense can these things be infinite. In fact, in any finite system, theheat capacity and susceptibility remain finite and continuous through the tran-sition. The divergence is, strictly speaking, only in the thermodynamic limitthat N → ∞, where N is the number of degrees of freedom. But, Avagadro’snumber NA is BIG. For all practical purposes in most experiments, these quan-tities appear to get ever larger near the critical point.

If one were to perform experiments on much smaller samples, say with N ≃106 or less, one would see that these divergences are rounded off. One placewhere one definitely deals with small systems is numerical simulation. Here,one must deal with systems that have many fewer degrees of freedom thanNA. In fact, simulation systems are usually so limited in size it is even hardto recognize what would be critical phenomena in much larger systems. Thisraises a challenge for numerical simulations to see critical phenomena. This isa particular problem with second-order phase transitions, since the correlationlength ξ diverges. When ξ becomes comparable to the size of the system, thebehavior begins to deviate strongly from the thermodynamic limit.

One can actually use these finite-size limitations to ones advantage. Sincethe correlation length diverges as ξ ≃ ξ0|t|

−ν , for a finite system of linear di-

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mension L, once ξ becomes comparable to L, the system cannot get any morecritical than it already is. Again, we identify the divergence of ξ as the definingcharacteristic of criticality, and once ξ reaches the full system size, we have cor-related fluctuations that span the whole system: ξ is as big as we can measure!We expect this to happen when |t| ≃ tL = (L/ξ0)

−1/ν . In practice, this willmean that the divergence of χ is rounded off and rendered finite in a region oforder tL on either side of the transition. Thus, we expect that for χ ≃ χ0|t|

−γ inan infinite system, one should observe a maximum value χmax ∝ Lγ/ν. This issomething that can easily be checked in numerical simulations on different size“samples.” What is more, if we consider the normalized susceptibility χ/Lγ/ν,it should approach a constant, independent both of t and the size of the system,below |t| of order tL. Only for t > tL should we see the critical behavior, andhere χ should be independent of L. Thus, χ(t, L) should satisfy

L−γ/νχ = φ(

L1/νt)

, (5.91)

where φ(x) is a function that approaches a constant for small x and must varyas x−γ for large x. Thus, by plotting L−γ/νχ versus L1/ν |t|, one should see acollapse of data obtained for different values of L onto a single scaling curveφ (actually, two curves φ± for t > 0 and t < 0), provided that γ and ν arechosen appropriately. A very similar approach is also possible with the orderparameter m. If Lβ/νm is plotted versus L1/ν |t| (for t < 0), one obtains datacollapse onto a universal function that approaches a constant for small argumentand increases as a power law with exponent β for large arguments.

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This approach of finite-size scaling, and its generalizations, provides a veryuseful tool for studying critical phenomena in numerical simulations, which nec-essarily deal with small systems. In fact, simulations have played an importantrole in the field of critical phenomena over the past few decades. The particu-larities of the the critical point, and especially the divergent correlation lengthhave helped to make this possible.

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