Introductory Notes on Critical Phenomena

1. Introduction In these lectures some of the general features of the phenomena of phase transitions in matter will be examined. Wewill ﬁrst reviewsome of the experimental phenomena. We then turn to a discussion of simple thermodynamic and so-called mean-ﬁeld theoretical approaches to the problem of phase transitions in general and critical phenomena in particular,showing what theyget right and what theyget wrong. Finally,wewill examine modern aspects of the problem, the scaling 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 isotherms near the critical point.2 Consider the twowell known phase diagrams shown in Figs. 1 and 2. Along anyofthe coexistence lines, thermodynamics requires that the chemical potentials in the coexisting phases be equal, and this in turn givesthe well known Clapeyron equation:   ∆ dP = H   ∆ ,(1.1) dT coexistence T V where ∆H and ∆V are molar enthalpyand volume changes, respectively,and T is the temperature. Manyofthe qualitative features of a phase diagram can be understood simply by using the Clapeyron equation, and knowing the relative magnitudes and signs of the enthalpyand volume changes. Nonetheless, there are points on the phase diagram where the Clapeyron equation cannot be applied naively,namely at the critical point where ∆V vanishes. The existence of critical points was controversial when it was ﬁrst considered in the 19th century because it means that you can continuously transform a material from one phase (e.g., a liquid) into another (e.g., a gas). Wenow hav e manyexperimental examples of systems that 1G. W.Castellan, Physical Chemistry,3rd ed.,(Benjamin Pub.Co., 1983), p. 266. 2R.J. Silbeyand R.A. Alberty, Physical Chemistry,3rd ed.,(John Wiley&Sons, Inc. 2001) p. 16.

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have 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 viewofthe qualitative symmetries of the phases involved).

TABLE 1. Examples of critical points and their order parameters3 Critical Order Example T (o K) Point Parameter c

Liquid-gas Density H2O 647.05

Ferromagnetic Magnetization Fe 1044.0

Anti-ferromagnetic Sub-lattice FeF2 78.26 magnetization

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

Super-Electron pair Pb 7.19 conductivity amplitude

Binary ﬂuid Concentration CCl4-C7 F14 301.78 mixture of one ﬂuid

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

Ferroelectric Polarization Triglycine 322.5 sulfate

The cases in Table I are examples of so-called 2nd order phase transitions, according to the naming scheme introduced by P.Eherenfest. More generally,annth order phase transition is one where, in addition to the free energies, (n − 1) derivativesofthe free energies are continuous at the transition. Since the first derivativesofthe free energy give entropyand volume, all of the freezing and sublimation, and most of the liquid-vapor line would be classified as first-order transition lines; only at the critical point does it become second order.Also note that not all phase transitions can be second order; in some cases, symmetry demands that the transition be first order. At a second order phase transition, we continuously go from one phase to another.What differentiates being in a liquid or gas phase? Clearly,both have the same symmetries, so what quantitative measurement would tell us which phase we are in? We will call this quantity (or quantities) an order parameter,and adopt the convention that it is zero in the one phase region of the phase diagram. In some cases there is a symmetry difference between the phases and this makes the identification of the order parameters simpler,inothers, there is no obvious unique choice, although we will showlater that for manyquestions, the choice doesn’tmatter.

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

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Forexample, at the liquid-gas critical point the density (molar volume) difference between the liquid and vapor phases vanishes, cf. Fig. 2, and the density difference between the two phases is often used as the order parameter.Second order transitions are also observed in ferromagnetic or ferroelectric materials, where the magnetization (degree or spin alignment) or polarization (degree of dipole moment alignment) continuously vanishes as the critical point is approached, and we will use these, respectively,asthe order parameters. Other examples are giveninTable 1. At a second order critical point, manyquantities vanish (e.g., the order parameter) while − −1 ∂ ∂ others can diverge(e.g., the isothermal compressibility, V ( V/ P)T,N cf. Fig. 2). In order to quantify this behavior,weintroduce the idea of a critical exponent.For example, consider a ferromagnetic system. As we just mentioned, the magnetization vanishes at the critical point (here, this means at the critical temperature and in the absence of anyexternally applied magnetic field, H), thus near the critical point we might expect that the magnetization, m might vanish like β m∝|Tc − T| ,when H = 0, (1.2) or at the critical temperature, in the presence of a magnetic field, m∝H 1/δ .(1.3) The exponents β and δ are examples of critical exponents and are sometimes referred to as the order parameter and equation of state exponents, respectively; we expect both of these to be positive.Other thermodynamic quantities have their own exponents; for example, the constant magnetic field heat capacity (or CP in the liquid-gas system) can be written as −α CH ∝|Tc − T| ,(1.4) while the magnetic susceptibility, χ ,(analogous to the compressibility) becomes −γ χ ∝|Tc − T| .(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 scattering experiments. As you learned in statistical mechanics (or will see again later in this course), the elastic scattering intensity at scattering wav e-vector q is proportional to the static structure factor NS(q) ≡<|N(q)|2 >, (1.6) where N(q)isthe spatial Fourier transform of the density (or magnetization density), and < ...> denotes an average in the grand canonical ensemble. In general, the susceptibility or compressibility and the q → 0limit of the structure factor4 are proportional, and thus, we expect the scat- tered intensity to divergewith exponent γ as the critical point, cf. Eq. (1.5). This is indeed observed in the phenomena called critical opalescence. At T = Tc and non-zero wav e-vectors, we write, 1 S(q)∝ .(1.7) q2−η

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

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Finally,weintroduce one last exponent, one that characterizes the range of molecular cor- relations in our systems. The correlation-length is called ξ ,and we expect that −ν ξ ∝|Tc − T| ,(1.8) where we shall see later that the correlation-length exponent, ν >0. Some experimental values for these exponents for ferromagnets are giveninTable 2. The primes on the exponents denote measurements approaching the critical point from the two-phase region (in principle, different values could be observed). What is interesting, is that eventhough the materials are comprised of different atoms, have different symmetries and transition temperatures, the same critical exponents are observed, to within the experimental uncertainty. TABLE 2. Exponents at ferromagnetic critical points5 Material Symmetry T(o K) α , α ′ βγ, γ ′ δη Fe 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

γ = YFeO3 Uniaxial 643 0.354 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 ferromagnetic transitions; a phase transition where "all" that happens is that the spins align. What is more interesting 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 o Material Symmetry Tc( K) α , α ′ β γ , γ ′ δ η Points ⋅ α ≤ Antiferro- CoCl2 6H2O Uniaxial 2.29 0. 11 0.23 magnetic α ′≤0. 19 ± 0. 02 α = α ′= FeF2 Uniaxial 78.26 0. 112 ± 0. 044 α = α ′=− γ = RbMnF3 Isotropic 83.05 0. 139 0.316 1. 397 0.067 ± 0. 007 ± 0. 008 ± 0. 034 ± 0. 01 = α ∼ γ = γ ′= Liquid-gas CO2 n 1304.16 1/8 0.3447 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 − = γ = ∼ Binary CCl4 C7 F14 n 1301.78 0.335 1. 2 4 Mixture ±0. 02 Binary Co − Zn n = 1739 0.305 γ = 1. 25 alloy ±0. 005 ±0. 02 Ferro- Triglycine n = 1322.6 γ = γ ′=1. 00 electric sulfate ±0. 05

Our goals in these lectures are as follows: 1. Tocome up with some simple theory that results in phase transitions in general, and second order phase transitions in particular. 2. Toshowhow universal critical exponents result. 3. Tobeable 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, won the 1982 physics Nobel Prize for showing howtocalculate the critical exponents.

2. Thermodynamic Approach

2.1. General Considerations Other than the already mentioned Clapeyron equation, cf. Eq. (1.1), and its generalizations to higher order phase transitions (not discussed), thermodynamics has relatively little to say about the critical exponents. One class of inequalities can be obtained by using thermodynamic stability requirements (e.g., that arise by requirements that the free energy be a minimum at equilibrium). As an example of howthis works, recall the well known relationship between the heat capacities CP and CV ,namely,

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

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γ 2 TV T CP = CV + ,(2.1) γ P γ ≡ −1 ∂ ∂ γ ≡− −1 ∂ ∂ where T V ( V/ T)P,N is the thermal expansion coefﬁcient and P V ( V/ P)T,N is the isothermal compressibility.Since thermodynamic stability requires that CV and γ P be positive,it follows that γ 2 TV T CP ≥ .(2.2) γ P By using the different exponent expressions, Eqs. (1.2) (1.4) and (1.5), this last inequality implies that, as T → Tc, −α β − +γ τ ≥ positive constant × τ 2( 1) ,(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.Ifyou check some of the experimental data givenin Tables S2 and 3, you will see that in most of the cases, α + 2β + γ ≈ 2, and to within the experimental error,the inequality becomes an equality.This is no accident!

2.2. Landau-GinzburgFreeEnergy We now try to come up with the simplest model for a free energy or equation of state that captures some of the physical phenomena introduced above.For example, we could analyze the well known van der Waals equation near the critical point. It turns out however, that a model proposed by Landau and Ginzburgisevensimpler and in a very general manner shows manyof the features of systems near their critical points. Specifically,theymodeled free energy difference between the ordered and disordered phases as A B C ∆G ≡−HΨ+ Ψ2 + Ψ3 + Ψ4+..., (2.5) 2 3 4 where A, B, C,etc., depend on the material and on temperature, and where H plays the role of an 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, in systems with inversion or reflection symmetry (magnets), in the absence of an external field either Ψ or −Ψ must give the same free energy.This means that the free energy must be an even function of Ψ in the absence of an external field, and from Eq. (2.5) we see that this implies that B = 0. Examples of the Landau free energy for ferromagnets are shown in Figs. 3 and 4.

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FIG. 4.3. TheTheLandau-GinzberLandau-Ginzbergfgfreeree ener energygy,c f.(cf. Eq. Eq. (2.5), (2.5)) for forferromagnets ferromagnets (B =(B0. =0)0.at 0) non-zeroat zero external magnetic field,field (andH =C1.= 0),1. and0. C = 1. 0. The minima of the free energy correspond to the stable and metastable thermodynamic equilibrium states. In general, we see that an external field induces order (i.e., the free energy has a minimum with Ψ≠0when H ≠ 0) and that multiple minima occur for A <0.When the external field is zero, there are a pair of degenerate minima when A <0.This is likethe behavior seen at the critical point, where we go from a one- to two-phase region of the phase diagram, cf. Fig. 2. To makethis more quantitative,weassume that

A∝T − Tc,as T → Tc,(2.7) with a positive proportionality constant, while the other parameters are assumed to be roughly constant 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) Forferromagnets with no external ﬁeld, B = 0, and Eq. (2.8) is easily solved, giving Ψ=0(2.9a) and A Ψ=± − .(2.9b) √ √  C Clearly,the latter makes physical sense only if A <0,i.e., according to the preceding discussion, when T < Tc.Indeed, for A <0the it is easy to see that the nonzero roots correspond to the minima shown in Fig. 3, while Ψ=0isjust the maximum separating them, and is thus not the equilibrium state. With the assumed temperature dependence of A,cf. Eq. (2.7), we can easily obtain the the 1/2 critical exponents. For example, from Eqs. (2.7) and (2.9b), it follows that Ψ∝(Tc − T) ,and thus, β = 1/2.Inthe absence of a magnetic ﬁeld, the free energy difference in the equilibrium

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state is easily shown to be  0, when A >0(T > Tc) ∆ = G  A2 (2.10) − ,otherwise.  4C Since  ∂S   ∂2G  C = T =−T ,(2.11) H  ∂   ∂ 2  T H,N T H,N it follows that the critical contribution to the heat capacity is independent of temperature, and hence, α = α ′=0. An equation for the susceptibility can be obtained by differentiating both sides of Eq. (2.8) χ ≡ ∂Ψ ∂ with respect to magnetic ﬁeld and solving for ( / H)T,H=0.This gives:  1 1  ,for T > Tc χ = =  A (2.12) A + 2BΨ+3CΨ2 1  ,for T < Tc, 2|A| which shows, cf. Eq. (1.5), that γ = γ ′=1, and also shows that the amplitude of the divergence of the susceptibility is different above and below Tc.

Finally,bycomparing Eq. (2.8) at T = Tc (A = 0) with Eq. (1.3) we see that δ = 3. These results 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 0 Order Parameter β 1/2 Susceptibility γ 1 Eq. of State at Tc δ 3 Correlation length ν 1/2 Correlation function η 0

The table also shows the results for the exponents η and ν ,which strictly speaking, don’tarise from our simple analysis. Theycan be obtained from a slightly more complicated version of the free 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 values for the critical exponents. We’ve found values for them independent of the material parameters. Unfortunately,while theyare in the right ball-park compared to what is seen experimentally,they are all quantitatively incorrect. In addition, the Landau-Ginzburgmodel 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 givenbyWeiss for ferromagnets. As is well known, a spin in an external magnetic field has a Zeeman energy givenby − 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 the spin (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 problem in statistical thermodynamics. If the total spin is S, the molecular partition function, q,is givenby S α sinh[α (S + 1/2)] q = Σ e Sz = ,(3.2) α Sz=−S sinh( /2) − where α ≡ γ h H/(kBT), kB is Boltzmann’sconstant, and where the second equality is obtained by realizing that the sum is just a geometric series. With the partition function in hand it is straightforward, albeit messy,towork out various thermodynamic quantities. Forexample, the average spin per atom, < s >, is easily shown to be givenby ∂ ln q < s >= = B (α ), (3.3) ∂α S 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 the Helmholtz free energy per spin is −kBT ln q,asusual. The spin contribution to the heat capacity is obtained by taking the temperature derivative ofthe energy and becomes: C  1 (S + 1/2)2  H = α 2 −  2 2 .(3.5) NkB 4sinh (α /2) sinh [(S + 1/2)α ] Other thermodynamic quantities are obtained in a similar manner.The magnetization and spin contributions to the heat capacity are shown in Figs. 5 and 6.

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C FIG. 5.6. Spin polarizationcontribution ofto anthe ideal constant spin magneticin an external ﬁeld magneticheat capacity, ﬁeld. H .

While this simple model gets manyaspects of a spin system correct (e.g., the saturation values of the magnetization and the high temperature behavior of the magnetic susceptibilities), it clearly doesn’tdescribe anyphase transition. The magnetization vanishes when the field is turned offand the susceptibility is finite at anyfinite temperature. Of course, the model didn’t include interactions between the spins, so no ordered phase should arise. Weiss included magnetic interactions between the spins by realizing that the magnetic field wasmade up of twoparts: the external magnetic field and a local field that is the net magnetic field associated with the spins on the atoms surrounding the spin in question. In a disordered system (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 system the neighboring spins are ordered and the net field won’tcancel out. To bemore specific, Weiss assumed that

H = Hext + λ < s >, (3.6) where Hext is the externally applied field and λ is a parameter that mainly depends on the crystal lattice. In ferromagnets the field of the neighboring atoms tends to further polarize the spin, and thus, λ >0 (it is negative inanti-ferromagnetic materials). Note that the mean field that goes into the partition function depends on the average order parameter,which must be determined self-consistently. When Weiss’sexpression 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 showthat 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 analytically close to the critical point since there < s >and Hext are small as is α .Wecan use this by noting the Taylor series expansion, 1 x x3 coth(x) = + − +..., (3.8) x 3 45 which when used in Eq. (3.7) gives

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α α 3 < s >= [(S + 1/2)2 − (1 / 2)2] − [(S + 1/2)4 − (1 / 2)4]+... . (3.9) 3 45

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

45T 2(T − T) < s >=± c ,(3.10) √ √  + 4 − 4 γ − λ 3 [(S 1/2) (1 / 2) ]( h /kB) where the critical temperature (known as the Curie temperature in ferromagnets) is − γ h λ S(S + 1) Tc ≡ .(3.11) 3kB

When T < Tc the state with the nonzero value of < s >has the lower free energy.Thus we’ve been able to showthat the Weiss theory has a critical point and have come up with a microscopic expression for the critical temperature. By repeating the analysis of the preceding section, one can easily obtain expressions for the other common thermodynamic functions. The Weiss mean ﬁeld theory is the simplest theory of ferro-magnetism, and overthe years manyreﬁnements to the approach have been proposed that better estimate the critical temperature. Unfortunately,theyall fail in one key prediction, namely,the critical exponents are exactly the same as those obtained in preceding section, e.g., compare Eqs. (2.9b) and (3.10). This shouldn’tbetoo surprising, giventhe similarity between Eqs. (2.8) and (3.9), and thus, while we’ve been able to answer some of our questions, the matter of the critical exponents still remains.

4. The Scaling Hypothesis When introducing the critical exponents, cf. Table 4, we mentioned the exponent ν associ- −ν ated with the correlation length, that is ξ ∼|T − Tc| .What exactly does a diverging correlation length mean? Basically,itisthe length overwhich the order parameter is strongly correlated; for example, in a ferromagnet above its Curie temperature, if we ﬁnd a part of the sample where the spins 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 fewmolecular diameters in size. At these scales, all of the molecular details are important. 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 Ising model of aferromagnet slightly above its critical temperature. Dark and light regions correspond to spin down and spin up, respectively. Figure 7 shows the spin conﬁguration obtained from a Monte Carlo simulation of the Ising spin system (S = 1/2 with nearest-neighbor interactions) close to its critical point. We see large interconnected domains of spin up and spin down, each containing roughly 103 − 104 spins. If this is the case more generally,what determines the free energy and other thermodynamic quantities? Clearly,two very different contributions will arise. One is associated with the short-range interactions between the aligned spins within anygiv endomain, while the other involves the interactions between the everlarger (as T → Tc)aligned domains. The former should become roughly independent of temperature once the correlation length is much larger than the molecular lengths and should not contain anyofthe singularities characteristic of the critical point. The latter,then, is responsible for the critical phenomena and describes the interactions between large aligned domains. As such, it shouldn’tdepend strongly on the microscopic details of the interactions, and universal behavior should be observed. The next question is howdothese observations help us determine the structure of the quantities measured in thermodynamic or scattering experiments? First consider the scattering intensity or structure factor, S(q), introduced in Eq. (1.6). The scattering wav e-vector, q,probes

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the length-scales present in the density or magnetization fluctuations. From the discussion of the preceding paragraph, the only length scale that is relevant near the critical point (at least for quantities that exhibit critical behavior) is the correlation length ξ ;hence, we should be able to write S(q, T)∝ξ γ /ν F(qξ ), (4.1) where the factor of ξ γ /ν wasintroduced in order to capture the divergence in the scattering intensity at q = 0associated with the susceptibility,cf. Eqs. (1.5) and (1.8). The function F(x)isarbi- −η trary,except for twoproperties: 1) F(0) is nonzero; and 2) F(x)∼1/x2 as x → ∞.The former implies that there is a nonzero susceptibility,while the latter is necessary if the behavior givenin Eq. (1.7) is to be recovered. Strictly speaking, Eq. (1.7) holds only at the critical point where ξ is infinite, 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 to within the experimental accuracyofthe measurements. Widom7 formalized these ideas by assuming that the critical parts of the thermodynamic functions were generalized homogeneous functions.For example, for the critical part of the molar free energy,afunction 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 be giveninterms of p and q. We showed in Eq. (2.11) that the heat capacity is obtained from twotemperature derivativesofthe free energy.From Eq. (4.3) this implies that at H = 0, 2p p λ CH (λ τ ) = λCH (τ ). (4.4a) − Since λ is arbitrary,weset it to τ 1/p and rewrite Eq. (4.4a) as −(2p−1) /p CH (τ ) = τ CH (1), (4.4b) which gives 1 α = 2 − .(4.5) p Similarly,the magnetization is obtained by taking the derivative ofthe free energy with respect to H.Thus, Eq. (4.3) gives − M(τ , H) = λ q 1 M(λ pτ , λ q H). (4.6a) − When H = 0weset λ = τ 1/p,asbefore, and find that − M = τ (1 q)/p M(1, 0), (4.6b) or 1 − q β = .(4.7) p 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 q δ = .(4.9) 1 − q Finally,the susceptibility is obtained from the derivative ofthe magnetization with respect to ﬁeld. By repeating the steps leading to the exponent β ,wecan easily showthat 2q − 1 γ = .(4.10) p All four exponents, α , β , δ and γ ,hav e been expressed in terms of p and q,and thus two scaling laws can be obtained. Forexample, 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 recoverthe Rushbrook inequality (as an equality), cf. Eq. (2.4).

5. Kadanoff Transformation and The Renormalization Group The discussion of the scaling hypothesis giveninthe preceding section is ad hoc to say the least. Moreover, evenifitiscorrect, it still doesn’ttell us howtocalculate the independent exponents, p and q.Kadanoff8 has givenavery physical interpretation of what scaling really means, and has shown howtoapply it to the remaining problem. In order to introduce the ideas, consider the Hamiltonian for a ferromagnet:

H =−J Σ si s j − H Σ si,(5.1) i

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

1 1 1 − Q = Σ Σ ... Σ e H/kBT (5.2) =− =− =− s1 1 s2 1 sN 1 In general, the sums cannot be performed exactly; nonetheless, consider what happens if we were to 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), cf Fig. 8

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

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

SL ≡ Z Σ Si (5.3) i∈block d d where Z ≈ 1/L is introduced to make SL ≈±1. Alternately,for L odd, we could assign SL ≡±1depending on whether the majority of the spins in the block had spin ±1. As long as the block size is comparable to or smaller than the correlation length, these twochoices should give the same answer (why?). 3. Average overthe internal conﬁgurations of each block and calculate the mean interaction potential between different blocks.

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 = Σ e W/kBT ,(5.4) SL where

− 1 − e W/kBT ≡ Σ ′e H/kBT ,(5.5) {si}=−1 and where the prime on the spin sums means to only include those configurations that are consistent with the block-spin configuration being summed. The effective potential, W,isanalogous to the potential of mean force encountered in statistical mechanics and is just the reversible work needed to bring the system into a configuration givenbythe SL’s. What will W look likeasafunction of the block spin configuration? Clearly it too describes the interactions between spins (nowblocks of spins) and should look something like the original spin Hamiltonian introduced in Eq. (5.1), perhaps with some of the additional terms discussed above.Thus, we expect that

W =−JL Σ Si S j − HL Σ Si+..., (5.6) i where . .. represents the extra terms, and where note that the coupling constant and magnetic

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ﬁeld have changed. Equation (5.6) is just the Hamiltonian for a spin system with N/Ld spins. Hence, if we use it 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 values for the parameters appearing in the Hamiltonian; nonetheless, it is still the free energy of a spin system and we conclude that the free energy per spin, −d A(J, H,...)= L A(JL, HL,...),(5.7) where . .. denotes the parameters that appear in the extra terms. Akey issue is to understand what the block transformation does to the parameters in the Hamiltonian. 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 −d A(µ) = L 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 only asemi-group since the inverse operations don’texist). In general, the renormalized problem will appear less critical, since the correlation length will be smaller on the re-blocked lattice (remember,we’re simply playing games with howwe carry out the sums, the real system is the same). One exception to this observation is at the critical point, where the correlation length is inﬁnite to begin with. In order that the renormalized appear as critical as the original one, the Hamiltonians before and after the block transformation must describe critical systems, or equivalently,the renormalized parameters will turn out to be the 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 values of the parameters at the fixed point as µ*. Suppose we’re near the critical point and we write µ = µ*+δ µ, where δ µisnot 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  ∂  ≡ RL(µ) KL  ∂  (5.13) µ µ=µ* is a matrix that characterizes the linearized RG transformation at the fixed point. Rather than use the parameters directly,itisuseful to rewrite δ µinterms of the normalized eigenvectors of the matrix KL.These are defined by

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KL ui = λ i(L)ui,(5.14)

with ui ⋅ ui = 1, as usual. It turns out that the eigenvalues must have a very simple dependence on 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

yi λ i(L) = L ,(5.16) where the exponent is obtained from the eigenvalue as ln((λ (L))) y = i .(5.17) i ln(L) We use this eigenanalysis by writing

δ µ = Σ ciui,(5.18) i

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 RG transformation 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

= yi ci,L L ci,(5.19) which no longer involves matrices and has a very simple dependence on L. Indeed, if we go back 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,wesee that − = d y1 y2 A(c1, c2,...) L A(L c1, L c2,...),(5.20) whichisageneralization of the scaling form assumed by Widom,cf. Eq. (4.3); moreover, wecan repeat the analysis of the preceding section to express the experimental exponents in terms of the yi’s.Before doing so, however, itisuseful to look at some of the qualitative properties of Eq. (5.20). The notation is slightly different, but nonetheless, the scaling analysis givenabove can be repeated, 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 -0.2671 β − (d y2)/y1 0.7152 γ − (2y2 d)/y1 0.8367 δ − y2/(d y2)2.1699 ν 1/y1 1.1335 † ∝τ ∝ We’v e assumed that c1 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 preservethe form of the microscopic Hamiltonian, e.g., Eq. (5.1); as such, there will be more than twoparameters

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generated by the RG iterations. Howdoes this agree with the thermodynamics, which says that the critical point in a ferromagnetic or liquid-gas system is determined solely by temperature (coupling constant) and magnetic field? What about the other parameters? Whatevertheyare, it is inconceivable that all phase transitions of the same universality class and materials have the same values for these, and thus, howcan universal exponents arise? The way out of this problem is for the other parameters to have neg ative yi exponents. If this is the case, then the scaling analysis (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 theywill adopt their fixed-point values irrespective oftheir initial ones. In other words, irrespective ofthe actual parameter 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 likethe one with the parameters set to those that characterize the fixed point. Second, quantities that have positive exponents are called relevant variables,and describe things liketemperature and magnetic field. We knowthat different materials have different critical temperatures, pressures etc., and thus we expect that their values are important. Explicit cal- culations showthat only tworelevant variables arise for the class of problems under discussion, and so the thermodynamics of our model is consistent with experiment. Takentogether,these twoobservations explain whyuniversal behavior is observed at the critical point and howscaling laws arise. Moreover, wehav e the blueprint for the calculation of the critical exponents. All we have to do is to compute the eigenvalues of the linearized RG transformation and carry out some simple algebra.Aswewill nowsee, this isn’taseasy as it sounds.

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 with L = √ 3asindicated. The arrows showthe spins that interact on an adjacent pair of blocks. We will perform the block transformation in blocks of three as indicated (L = √ 3) using a majority rule to assign the block spin. Each block of three can assume 8 spin configurations, 4 will have 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 perturbation theory to get an approximate expression; specifically,wewill treat the interactions between the blocks and the external field as a perturbation. Hence, to leading order the blocks are uncoupled and the partition function can be easily evaluated by explicitly summing the configurations. This gives: − (0) − e W = (e3J + 3e J )N/3,(6.1)

where we absorb the factors of kBT into J and H.Since our answer doesn’tdepend on the values of the block spins or the magnetic ﬁeld, it’snot particularly interesting. By expanding W and the Hamiltonian in the perturbing terms and comparing the results, it’seasy to showthat to ﬁrst order,10 (1) = (1) W < H >0 .(6.2)

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

 ∞ λ j  < eλ A >= exp Σ << A j >> ,  j=1 j!  where << A j >> i sknown as a cumulant average. It is expressed in terms of the usual moment av erages, < 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 givesEq. (6.2). It is also easy to showthat << A2 >> =<(A− < A >)2 >isjust 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 coefficient of which is the newmagnetic field. Hence,  e3J + e−J  H = 3H .(6.3) L  e3J + 3e−J  In addition, Eq. (6.2) will give terms that are products of the spin variables on adjacent blocks, the coefficient of these givesthe newcoupling constant, and 2  e3J + e−J  J = 2J .(6.4) L  e3J + 3e−J 

At the fixed point, JL = J and HL = H.From Eq. (6.3) we see that H = 0atthe fixed point; i.e., the ferromagnetic critical point occurs at zero external magnetic field, as expected. Equation (6.4) shows that there are actually twofixedpoints: one with J = 0and the other with 1 J = ln(1 + 2√ 2) = 0. 335614. . . .(6.5) 4 The fixed point with J = 0has no interactions and will not be considered further (actually,it describes the infinite temperature limit of the theory,and will yield mean-field behavior if analyzed 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  e3J + e−J  λ = 3 (6.6) H  e3J + 3e−J  and 2(e4J + 1)(e8J + 16Je4J + 4e4J + 3) λ = (6.7) J (e4J + 3)3 which givesthe 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 2.80 yH 1.3691 15/16

λ J 1.6235 √ 3 = 1. 73 yJ 0.8822 0.5

α -0.2671 0 β 0.7152 0.125 δ 2.1699 15 ‡The 2d Ising model has an exact solution, ﬁrst givenbyOnsager.

The agreement with the exact results isn’tgreat, the error mainly coming form λ H ,and is mainly 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,atthe expense of a lot of algebra, to do so requires that we consider some of the extra terms in the Hamiltonian. The higher order calculation automatically generates interactions beyond nearest neighbors and beyond pairwise additive ones, and some of these must be considered if good agreement is to be obtained. When this is done, the correct exponents and thermodynamic functions are found (to about 5% accuracy). An example of some of the better results is shown in Fig. 10.

FIG. 10. Results of higher order numerical RG calculation for the 2d lattice of Nieuhuis and 11 Nauenberg .The solid curveisthe exact CH ,the dashed curveisthe free energy,and the dot- dashed curveisthe energy,all from Onsager’sexact solution of the model. The points are the numerical results. KisJinthe 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 howphase transitions arise, whyuniversal behavior is expected, and perhaps, most important, have provided a framework in which to calculate the critical exponents. Several important issues have not been dealt with. Forexample, other than the fact that the simple mean field approaches didn’tgiv e the correct experimental answer,westill don’treally understand whytheyfailed, especially since many of 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 gives mean field theory as the leading order result. If we were to examine the next corrections, we would see that theybecome large as the critical point is reached, thereby signaling the breakdown of mean field theory.What is more interesting, is that the dimensionality of space plays a keyrole in this breakdown; in fact, perturbation theory doesn’tfail for spatial dimensions greater than four. The dependence on the dimensionality of space plays a key role in Wilson’swork on critical phenomena; in short, he’s(along with some key collaborators) have shown howtouse ε ≡ 4 − d as a small parameter in order to consistently move between mean field and non-mean field behavior. These, and other,issues require better tools for performing the perturbative analysis and for considering very general models with complicated sets of interactions. This will not be pursued here, but the interested reader should have a look at the books by Ma3 or by Amit12 for a discussion of the more advanced topics.

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

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