where we have recognised A(T) to be the free energy of the high-symmetry phase, G 0 (t). The order parameter is then t 1/2, t 0, (3.

where we have recognised A(T) to be the free energy of the high-symmetry phase, G 0 (t). The order parameter is then [ η = ± B ] 1/2 1t t 1/2, t 0, (3.1) 2C 0 giving β = 1/2, as in mean-field theory. In Fig. 3.1 the evolution of the free energy difference G = G G 0 between the ordered and disordered phases with respect to the order parameter η, as the temperature is decreased from a high value, is shown. It exhibits the usual evolution with two minima arising below some critical temperature, separated by a free-energy barrier. In fact Landau theory is a mean-field theory and all critical exponents derived from it are classical. All mean-field theories can be cast in Landau form near the critical phase transition; for example, the Bragg-Williams free energy for an Ising model can be expanded about m = 0 close to the transition (for H = 0) and of course the same conclusions as to the qualitative behaviour of m and, in particular, the value of the critical exponent β, can be obtained. This is again because only the mean value of the order parameter is dealt with, fluctuations being neglected altogether. Figure 3.1: Free energy difference G = G G 0 between ordered and disordered phases with respect to η as temperature is decreased from a high value, with T > T c, to a low value, with T < T c. The situation at the critical temperature, T = T c, is also shown. 3.7.1 First-order phase transitions At a first-order phase transition first derivatives of the free energy undergo discontinuous changes. The order parameter is also discontinuous at the transition. In principle, the 88

expansion (1.134), which assumes continuity of the free energy, ceases to be valid. Even then, the Landau theory is commonly used in these situations. However, it is necessary to modify the theory in order to incorporate the possibility of having a metastable phase, i.e. a local mininum in the free energy. This can be realised in two ways. Depending on the problem, the invariance under the sign inversion η η may or may not hold. Let us consider the first case. Expansion in even powers of η. We add an additional term in the expansion: G(T; η) = G 0 (t) + B(t)η 2 + C 0 η 4 + D 0 η 6. (3.2) Clearly, D 0 > 0 to ensure stability. Differentiating with respect to η: B(t)η + 2C 0 η 3 + 3D 0 η 5 = 0. (3.3) We again take B(t) = B 1 t. The solution η = 0 (high-symmetry phase) is always possible. The other roots are given by η 2 = 1 ( ) C 0 ± C0 2 3B 1 td 0. (3.4) 3D 0 In order to have real solutions we choose C 0 < 0. Then: [ 1 ( 1/2 η = ± C0 2 3B 1 td 0 + C 0 )]. (3.5) 3D 0 At very high temperatures the discriminant C0 2 3B 1 td 0 < 0 and the only real solution is η = 0 (disordered phase). As t is lowered the (doubly degenerate) metastable ordered phase appears for the first time at a temperature t (1) c given by C0 2 3B 1t (1) c D 0 = 0, i.e. t (1) c = C2 0 3B 1 D 0. (3.6) As t is further decreased the metastable states become as stable as the disordered state with η = 0, and the first-order transition takes place. Further reduction of the temperature makes the states with η 0 more stable, until a temperature t = t (2) c = 0 where the disordered phase becomes unstable (a maximum of G). Fig. 3.2(a) sums up the evolution of G with t. An external field may be included in the expansion; the method of solution parallels the one shown. Expansion in odd powers of η. In case the invariance under the operation η η does not hold, the more general expansion up to fourth order in the order parameter is G(T; η) = G 0 (t) + B 1 tη 2 + γη 3 + C 0 η 4, (3.7) with C 0 > 0. In this case a secondary minimum for t > 0 and η 0 is only possible if γ < 0 (η 0 is not essential but is the preferred choice). Differentiating: 2B 1 tη 3 γ η 2 + 4C 0 η 3 = 0, (3.8) 89

which again gives the disordered phase as the solution η = 0, plus two additional roots, provided η = 1 ( ) 3 γ ± 9γ 8C 2 32C 0 B 1 t, (3.9) 0 9γ 2 32C 0 B 1 t 0 t t (1) c = 9γ2 32C 0 B 1. (3.10) One of these two solutions (the one with the sign) can be seen to be a maximum while the other corresponds to the metastable phase. As t decreases the free energy of this metastable state eventually becomes equal to that of the disordered phase, at which value the first-order transition occurs. As t is further decreased, the disordered phase becomes metastable and finally, at t = t (2) c = 0, becomes unstable (maximum in G at η = 0). Fig. 3.2(b) summarises the situation. 3.7.2 Application to phase transitions in liquid crystals Liquid crystals are experimental systems with an extremely rich phase behaviour. As a consequence, they provide a useful test bed for theories. From a practical point of view their properties are very useful in a variety of applications, including graphical display technologies and others. That is why it is very important to understand these properties, and Landau theory provides a very useful starting point. Liquid crystals are materials made of elongated (and to some degree flexible) organic molecules, that can orient collectively in space along a common direction, the director, to form the nematic phase. This usually happens below some temperature, the so-called clearing point, though some mixed systems are sensitive to changes in concentration. The transition from the disordered phase (called isotropic), stable at high temperature, to the ordered phase, stable at lower temperature, the nematic, where a privileged direction in space appears, is of first-order in three dimensions. Even though the nematic phase has positional disorder, as the isotropic phase, there usually appears, at even lower temperatures, another phase, the smectic, which is orientationally ordered but additionally possesses translational order along the direction of the director: it is a one-dimensional solid along the director, but fluid in the perpendicular directions. More complicated liquid-crystalline phases may stabilise at lower temperatures before the crystal phase (or a glass) may become stable. It is this intermediate order of liquid crystals that make them interesting from the theoretical and practical points of view. See Fig. 3.3 for a pictorial representation of the liquid-crystal phases. The choice of order parameter for the nematic phase is more tricky than it seems at first. In fact it is a traceless 3 3 tensor order parameter, with two independent eigenvalues which are s, the uniaxial order parameter, and P, the biaxial order parameter, measuring, respectively, the amount of order around the director, and the departures from isotropicity in this order. For uniaxial molecules, and in the absence of any external orienting field, 90

Figure 3.2: Difference G between free energies of ordered and disordered phases as a function of a scalar order parameter for various temperatures in the case of a first-order transition, as given by Landau theory. (a) Case where symmetry η η holds. (1) High temperature, with only disordered phase being stable; (2) system about to create a critical (spinodal) point where a metastable phase appears with η 0, at t = t (1) c ; (3) coexistence between ordered and disordered phases separated by a free-energy barrier; (4) system close to second spinodal point at t (2) c = 0, where disordered phase ceases to be metastable. In all cases the ordered phase is doubly degenerate. (b) Case without η η. (1) High temperature, with only disordered phase being stable; (2) system about to create a spinodal point where a metastable phase appears with η 0, at t = t (1) c ; (3) coexistence between ordered and disordered phases separated by a free-energy barrier; (4) disordered phase less stable that ordered phase but already metastable; (5) second spinodal point at t (2) c = 0 where disordered phase ceases to be metastable and becomes completely unstable. only s can be different from zero. The nematic uniaxial order parameter can be defined in terms of microscopic properties as s = d ˆΩf(ˆΩ)P 2 (cosθ), (3.11) where f(ˆω) = f(θ, φ) is the orientational distribution function, giving the probability of having the symmetry axis of a uniaxial particle oriented along the unit vector ˆΩ = (θ, φ) with respect to some laboratory-fixed reference frame [obviously f(θ, φ) is normalised over the whole solid angle]. Isotropic-nematic transition In the isotropic phase the order parameter is zero, because any orientation is equally probable, so f(ˆω) = 1/4π and it follows that s = 0 from Eqn. (3.11); whereas in the nematic phase s 0 [its maximum value is unity, when order is perfect, and f(ˆω) = δ(ˆω) gives s = 1]. An important observation is that configurations with different sign of s are not equivalent: for example, s = 1/2 implies that all molecules have θ = π/2 (i.e. they 91

Figure 3.3: (Colour on the Web). Left: nematic phase, with molecules pointing on average along a common direction, the director, given by unit vector ˆn. Right: smectic phase (in the so-called A variety), where molecules arrange themselves in layers along a second unit vector ˆk ˆn. lie in the plane perpendicular to the director) whilst s = 1/2 represents a considerable amount of order along the director. This means that, in the expansion of G in s about s = 0, a cubic term must exist: G(T; s) = B 1 ts 2 γ s 3 + C 0 s 4, t = T T. (3.12) T As seen previously, this corresponds to a first-order phase transition. T is the temperature at which the isotropic phase becomes unstable; the transition temperature, T NI, must be obtained by demanding that the two minima of G have the same value G = 0. Nematic-smectic transition Smectic order, involving a density modulation along the direction of the director, requires an additional order parameter. The simplest way to define such an order parameter is via a Fourier expansion of the density distribution function (we take z as the direction of modulation): ρ(z) = ρ 0 + ρ 1 cos (q 1 z ϕ) + (3.13) We only consider the first amplitude, ρ 1, such that ρ 1 = 0 corresponds to the nematic phase, whereas ρ 1 0 implies smectic order. q 1 is a wavevector related to the smectic layer spacing d 0 by q 1 = 2π/d 0, and ϕ is a phase which gives the location of the layers within one smectic period (changing ϕ involves displacing the smectic layers along the z direction as a whole). In order to describe the nematic-smectic transition we have to write a Landau expansion. As a first step, we will only consider the smectic order parameter ρ 1 (which amounts to considering the uniaxial nematic order parameter to be equal to one; the nematic order parameter s, which should in principle be explicitely included since it is also involved in the transition, will be incorporated later). Note that the free energy is invariant with respect to a change of sign ρ 1 1ρ 1, since this simply amounts to a solid displacement of the smectic sample by d 0 /2. Then we write G(T; ρ 1 ) = B(t)ρ 2 1 + C 0ρ 4 1, B(t) = B 1t, t = (T T NS) T NS ). (3.14) 92

In this case G = G G 0 is the free energy with respect to the nematic phase, which is here the disordered (with respect to spatial order) phase. To ensure overall stability at large ρ 1, we require C 0 > 0. Note that the expansion is only up to fourth order in ρ 1 so that with this form the nematic-smectic transition is predicted to be of second order always. T is the transition temperature, at which the nematic phase becomes unstable. NS However, experimentally it is known that the nematic-smectic transition can be of first or second order, depending on the particular system, or may even change its nature from second to first as one parameter is varied. In order to describe this phenomenology, we have to improve Landaus theory. The key is to realise that the nematic order parameter is different in the nematic and the smectic phases, and therefore it should be involved in the transition. Let s 0 be the value of the nematic order parameter of the nematic phase at the continuous transition, and s the corresponding parameter of the smectic phase at an arbitrary temperature. We define the difference δs = s s 0 as a new order parameter, which will be zero at T = T T. It is expected that δs 0 in the smectic phase. We NS augment the Landau expansion with two terms: one giving the energy cost associated with increasing s over s 0, the other taking care of the coupling between the two order parameters, ρ 1 and δs: G(T; ρ 1, δs) = 1 2ζ(t) δs2 + B 1 tρ 2 1 + C 0ρ 4 1 Cδsρ2 1. (3.15) The coefficient C > 0 since the coupling must be such that the free energy decreases with respect to not considering any coupling at all. ζ(t) says how the free energy of the smectic phase changes when the orientational order parameter is changed (it is technically a response function). Now our situation is rather unusual, since our free energy depends on two order parameters, not just one. But the assumption we make is the same: the equilibrium state follows from minimisation, this time with respect to both order parameters. In order to make things more transparent, we first minimise with respect to, say, δs: 1 ζ(t) δs Cρ2 1 = 0 δs = Cζρ 2 1, (3.16) which provides a relation between δs and ρ 1, allowing one of them to be eliminated. We substitute δs into Eqn. (3.15), and obtain an effective free energy depending only on ρ 1, but which contains implicitely the effect of the nematic order parameter: G(T; ρ 1 ) = G 0 (t) + B 1 tρ 2 1 + u(t)ρ 4 1, u(t) = C 0 1 2 C2 ζ(t). (3.17) Now the sign of u, the coefficient proportional to ρ 4 1, may not always be positive. We can distinguish the following cases: C 2 ζ(t) < 2C 0. Then u(t) > 0 and the transition is predicted to be of second order. C 2 ζ(t) > 2C 0. Then u(t) < 0 and an additional term, D 0 ρ 6 1, is required in order for the free energy to be bounded from below. Here the transition is predicted to be of first order. 93

C 2 ζ(t) = 2C 0. Then u(t) = 0 and again a positive term proportional to ρ 6 1 is needed. This is a tricritical point, separating first- from second-order behaviour. A system can actually pass through a tricritical point as temperature is varied, because u(t) depends on temperature. 3.8 Monte Carlo simulation of phase transitions At the end of last chapter we briefly mentioned the two techniques that are used to analyse the properties of liquids on a computer. We said that this topic is too specialised to be developed here in any depth; one of the courses in the Master covers this material. However, we give here a very brief introduction to the Monte Carlo method as applied to the spin-1/2 Ising model, in order to demonstrate the power of this technique in the theory of phase transitions. In a Monte Carlo simulation we directly compute the canonical average of a given quantity, say the energy E, as a simple average over a subset of the full set of possible spin configurations {s k }. These configurations are weighted by a probability p({s k }) = e βh({s k}), (3.18) Q where H is given by (??) and Q is the partition function. Therefore, the required average is E = H = H({s k })p({s k }) = [ ] e βh({s k }) H({s k }). (3.19) {s k } {s k Q } Two actions are taken in Monte Carlo simulation. One is that only a reduced number of configurations, out of the total number of 2 N configurations (where N is the number of spins), is generated to compute the average. The problem with this is that in a large (and not so large) system only a very small fraction of the possible configurations have a statistically significant weight [i.e. a large value of p({s k })]. So, how can we choose the relevant configurations? The trick is to generate configurations distributed according to the Boltzmann probability function p({sk}). So one does not simply visit configurations and then multiplies the energy of that configuration by the corresponding Boltzmann factor, but generates configurations that have a large value of this factor. Using statistical theory, it can be demonstrated that, in this case, the correct average is E = 1 M {s k } p H({s k }), (3.20) where M is the number of configurations generated and the subscript p indicates that the configurations already are distributed according to the probability function p({s k }). All that remains to be done is to generate configurations. This is usually done using the socalled Metropolis et al. algorithm. This algorithm provides a sequence of configurations, one after the other, such that one configuration results from the previous one (a Markov 94

chain) and guarantees that, asymptotically, the sequence becomes distributed exactly as p({s k }). What is more important, the algorithm only depends on the Boltzmann exponential, and not on the value of the partition function, which is a great advantage since Q is impossible to evaluate in practical terms. The Metropolis et al. algorithm, as applied to a spin-1/2 Ising model with N spins, generates a sequence of spin configurations {s k } (0), {s k } (1), {s k } (2), (3.21) where the first conguration is imposed on the system (for example, all spins up ). The n + 1-th conguration is obtained from the n-th conguration by means of the following construction: We choose a spin i at random and make a spin flip (turn the spin up if it was down or the other way round). Then compute the energy change E involved in the spin flip. If E < 0 the spin flip is accepted right away. If E > 0, the spin flip is accepted with probability e β E. (3.22) In case the spin flip is not accepted, the i-th spin remains as it was originally This process is repeated N times, each time choosing a spin at random After the N iterations, we have generated a new configuration of spins (a Monte Carlo step), which we use to perform all necessary averages [for instance the energy, using Eqn. (3.20)] Following this process, we perform M steps (i.e. generate M configurations) Since the process tends asymptotically to the actual Boltzmann probability, it is always necessary to generate some initial configurations (the warming-up interval) that are discarded and not included in any average. Typical quantities obtained in a Monte Carlo simulation of the Ising model are the energy, the magnetisation, or the correlation function. Fig. 3.4 shows the magnetisation per spin m = M/N and energy per spin e = E/N as a function of reduced temperature kt/j. For each value of temperature a separate simulation was performed (more technical issues, such as the required number of Monte Carlo configurations, how the initial configuration is chosen, etc., will not be discussed here). We see that the magnetisation goes to zero close to the reduced temperature corresponding to the theoretical phase transition, kt c /J = 2.269. Since the system is finite, the derivative of m(t) at T = T c is not singular, as it should be; therefore, it is not clear how the location of the phase transition can be identify quantitatively. This is one of the typical problems that have to be solved in Monte Carlo simulation using special tricks. Of course there are many other issues in computer simulation that pose difficulties and make the technique an almost separate discipline in statistical physics. Despite these problems, computer simulation is a very powerful tool in the theory of phase transitions, and critical phenomena can also be analysed: values of critical exponents, complete phase diagrams, etc. 95

Figure 3.4: Energy and magnetisation of the spin-1/2 Ising model as obtained from Monte Carlo simulations on a 20 20 lattice. 3.9 Ginzburg-Landau theory of fluctuations We know that, in the critical region, fluctuations are very important but, as we have realised from the previous discussion, what is missing in mean-field theory is a proper treatment of fluctuations. We already discussed that, as the fluid approaches a critical point, larger and larger density fluctuations grow in the system; the size of these fluctuations is controlled by the correlation length, ξ. A possible framework to deal with these fluctuations is the Ginzburg-Landau (GL) theory. Unlike the simple Landau theory, which assumes the order parameter η to be constant everywhere in the system, GL theory considers the order parameter to be a spatially varying field η(r) with a typical length much larger than any atomic dimension. The free-energy price paid due to these fluctuations is made to be given by a simple gradient and a curvature parameter κ: G([η];T) = G 0 (T) + dr { b(t) [η(r)] 2 + c(t) [η(r)] 4 + κ(t) η(r) 2}. (3.23) Here we use the simplest Landau form for the local contributions [those coming from η(r) evaluated at the point r; the gradient term, by contrast, depends on values of η at r and its neighbourhood]. b(t) = B(t)/V and c(t) = C(t)/V are coefficients per unit volume. The basic assumption here is that the order parameter varies sufficiently slowly that a square-gradient approximation is valid. That is, η a 1, where a is for example the mean atomic distance. Assume that the value of the order-parameter field is given by η(r) = η + δη(r), with drδη(r) = 0, (3.24) where η is the value of the order parameter in the particular phase we are dealing with (in the disordered phase η = 0 while in the disordered phase η 0 and δη(r) is a local 96

fluctuation. The change in free energy involved in this fluctuation with respect to the uniform-η state is: G([η];T) = G 0 (T) + dr { b(t) [ η + δη(r)] 2 + c(t) [ η + δη(r)] 4 + κ(t) δη(r) 2}. (3.25) Expanding, collecting terms in powers of δη, eliminating terms in odd powers, and taking only the leading contributions, we are left with G = G(T, η) G(T, η) = dr {[ b(t) + 6c(t) η 2] [δη(r)] 2 + κ(t) δη(r) 2}. (3.26) Fourier-transforming with δη(r) = δη q e iq r, δηq = δη q, δη(r) 2 = δη q δη q e i(q q ) r, (3.27) q q q and δη(r) 2 = iq 2 δη q e iq r = δη q δη q e i(q q ) r, (3.28) q q,q we have G = q [ δη q δη q b(t) + 6c(t) η 2 + κ(t) (q q ) ] dre i(q q ) r q [ b(t) + 6c(t) η 2 + κ(t)q 2] δη q 2. (3.29) = V q We can take G to operate like a Hamiltonian where the basic system s coordinates are the Fourier coefficients δη q that describe a fluctuation. Therefore the probability of such a fluctuation to appear in the system will be [ b(t) + 6c(t) η 2 + κ(t)q 2] δη q 2 w[δη] e β G = e βv q. (3.30) Since G is quadratic in δη q, the equipartition theorem applies, and we have so that V [ b(t) + 6c(t) η 2 + κ(t)q 2] δη q 2 = kt 2, (3.31) ( ) δηq 2 kt 1 = 2V b(t) + 6c(t) η 2 + κ(t)q2. (3.32) Here an average of a quantity A = A(δη q ), A, is to be understood as A = dδη q1 dδη q1 dδη q2 A(δη q )e β G(δηq) dδη q2 e β G(δηq) (3.33) 97

A correlation function G( r r ) in real space is introduced as: G( r r ) = δη(r)δη(r ) = [η(r) η] [η(r ) η]. (3.34) Let us calculate the Fourier transform of G: G(q) = dr dr G( r r )e iq (r r ) = dr dr δη(r)δη(r ) e iq (r r ) = q q δη q δη q [ ] [ dre i(q+q ) r dr e i(q q) r ] = V 2 q q δη q δη q δ q, q δ q,q = V 2 δη q δη q = V 2 δη q 2. (3.35) Now, taking B(T) = B 1 t and C(T) = C 0 > 0, we have G(q) = ( ktv 2κ ) ( ) 1 ktv (b 1 t + 6c 0 η 2 )/κ + q = 1 2 2κ a 1 + q2. (3.36) Here we have used a 1 b 1t + 6c 0 η 2 κ = b 1 t κ = b 1(T T c ) κt c, T T c, 4c 0 η 2 κ = 2b 1(T c T) κt c, T < T c, (3.37) together with Eqn. (3.1). From here we can calculate the correlation function in real space: G(r) = 1 (2π) 2 V dqg(q)e iq r = ( ) κt 1 dqe iq r 2κ (2π) 3 a 1 + q. 2 (3.38) Now we define the correlation length as ξ a 1/2. The integral can be calculated in the complex plane: G(r) = = ( ) κt 1 4π ( dqq sin qr κt 2κ (2π) 3 r 0 ξ 2 + q = 2 2κ ( ) κt 1 2π 2κ (2π) 3 r Im 2πiRes ( qe iqr ξ 2 + q 2 ) 1 (2π) 3 2π r Im ) q=iξ 1 = [ dqqe iqr ] ξ 2 + q 2 ( ) κt e r/ξ. (3.39) 8πκ r Therefore, correlations decay exponentially with distance, and are governed by a correlation length which diverges at the critical point T = T c, since ktc b 1 (T T c ), T T c, ξ = kt c 2b 1 (T c T), T < T c. 98 (3.40)

We define the critical exponents ν and ν as (T T c ) ν, T T c, ξ (T c T) ν, T < T c. (3.41) From (3.40), the values of ν, ν predicted by Landau (i.e. mean-field) theory are ν = ν = 1/2. The correlation length can be interpreted as the maximum distance over which particles are correlated, and therefore gives the maximum size of ordered domains in the system (liquid or gas regions, domains of positive or negative averaged magnetisation, etc.) Alternatively, it can be thought of as the distance where the order parameter changes significantly. Since ξ at the critical point, the correlation function behaves as G(r) 1/r. Therefore the generalised susceptibility is going to diverge at the critical point. These results are perfectly general and not specific for any particular system; they apply whenever the system possesses a continuous phase transition (critical point). We can see that the simplest Landau theory gives the equilibrium value of the free energy and the averaged order parameter η; the Ginzburg-Landau version then considers the free-energy cost involved in introducing slowly-varying fluctuations about this averaged value. The previous conclusions are valid provided the assumption made at the beginning of the section, η a 1, is valid. Translated into the correlation function, this condition is equivalent to saying that G(ξ) η 2 ( ) kt e 1 8πκ ξ b 1(T c T) 2c 0 T c. (3.42) Substituting the value of ξ, setting T = T c on the left-hand side, and rearranging: kt c c 0 2 4πeb 1/2 1 κ Tc T 1. (3.43) 3/2 T c The second inequality comes from the assumption that we should be close to the critical point in order for the Landau expansion (low order parameter) to be valid. Then, Eqn. (3.43), called the Ginzburg criterion 1 gives a temperature window (close to T c but not so close for large fluctuations to dominate the scene completely) where Landau-theory predictions are valid. For each type of system and transition one should know the constants T c, b 1, c 0 and κ(t c ) in order to quantify the validity of the theory. These constants are typically obtained experimentally from heat capacities and structure factors. In the case of superconductivity Ginzburg-Landau theory works quite well, but in other cases the window is more limited (or even there may be no window at all!). 1 This should (but is seldom) referred to as Levanyuk criterion since it was A. Levanyuk the first who derived it, in 1959; Ginzburg redid it one year later. 99

3.10 Critical exponents The rigorous mathematical definition of critical exponents is as follows. We write a thermodynamic quantity f as f(t) t λ, t = T T c T log f(t) λ = lim. (3.44) t 0 log t Note that the case λ = 0 may include cases where f is discontinuous at t = 0, or diverges logarithmically, or exhibits a cusp-like singularity. The table summarises the critical exponents that are normally defined for fluids and magnets. It turns out that critical exponents are related by inequalities. Some of them (those involving thermodynamic exponents, i.e. all except ν and η) can be derived from simple thermodynamic arguments. The inequalities are: α + 2β + γ 2 Rushbrook α + β(δ + 1) 2 Griffiths γ (δ + 1) (2 α )(δ 1) Griffiths γ β(δ 1) Griffiths d(δ 1) (2 η)(δ + 1) Buckingham-Gunton dγ (2 η)(2β + γ ) Fisher (2 η)ν γ Fisher dν 2 α, dν 2 α Josephson We only prove the first one for a magnet, since it is quite easy to obtain. It can be done from the following relation between heat capacities: C H C M = TV [( ) ] 2 M, (3.45) χ T T which is similar to the relation C p C V = T κ T H ( ) 2 V T p (3.46) for normal systems. To prove (3.46), we take S = S(p, T) [remember that (p, T) completely specify the system for a given N] and write ( ) ( ) S S ds = dp + dt C ( ) ( ) V S p p T T = + C p p T T. (3.47) T p T V 100

Exponent Definition Classical value Comment α C V ( t) α (fluid) 0 below T c C H ( t) α (magnet) α C V t α (fluid) 0 above T c C H t α (magnet) β ρ liq ρ gas ( t) β (fluid) 1/2 only below T c M ( t) β (magnet) γ κ T ( t) γ (fluid) 1 below T c χ T ( t) γ (magnet) γ κ T t γ (fluid) 1 above T c χ T t γ (magnet) δ p p c ρ ρ c δ sign(ρ ρ c ) (fluid) 3 on critical isotherm H M δ signm (magnet) ν ξ ( t) ν (fluid & magnet) 1/2 below T c (from GL theory) ν ξ t ν (fluid & magnet) 1/2 above T c (from GL theory) η G(r) r (d 2+η) 0 d =dimensionality (from GL theory) Now, by means of the Maxwell relation ( ) ( ) S V =, (3.48) p T T p we can write and using the relation C p C V = T ( ) ( ) ( ) V T p T p V p V T ( ) ( ) V p, (3.49) T T p V = 1, (3.50) the derivative ( p/ T) V can be eliminated to give ( ) ( ) ( ) V V p C p C V = T = T ( ) 2 V. (3.51) T T V V κ p p T T T p For a magnetic system χ T = ( M/ H) T, V M and p H: ( ) [( ) ( ) ] M M H C H C M = T = T [( ) ] 2 M. (3.52) T T M χ H H T T T H Now, since C M > 0, we have C H T χ T [( M T 101 ) H ] 2. (3.53)

As T T c, so that C H ( t) α, χ T ( t) γ, ( M T ) H ( t) β 1, (3.54) c 1 ( t) α c 2 T c ( t) γ ( t) 2(β 1) ( t) 2 (α +2β+γ ) c 2T c c 1, (3.55) where c 1, c 2 are constants (taking care of the amplitudes of all the thermodynamic factors). Taking logarithm on both sides (since log x is a monotonically increasing function the inequality is preserved): [2 (α + 2β + γ )]log ( t) log c 2T c c 1. (3.56) Dividing by log ( t) < 0 and taking the limit t 0 : 2 (α + 2β + γ ) 0 α + 2β + γ 2. (3.57) It is interesting to note that, in mean-field approximation, the first four inequalities are satisfied as equalities, but the Buckingham-Gunton, first Fisher and both Josephsonss inequalities are not satisfied. Experimentally some relations are seen to be satisfied as inequalities and others as equalities. 3.10.1 Experimental values of critical exponents and universality classes The following table collects some experimental measurements of critical exponents in various types of three-dimensional experimental systems. We see that these very different systems all seem to follow the same values for the exponents. Experimental systems and theoretical models that share the same set of exponents are said to belong to the same universality class. The systems above, together with the 3D Ising model and other models, belong to the same class. The values of the exponents that define a class seem to be determined by: (i) the dimensionality of the system, (ii) the range of the interaction (whether short- or long-ranged), and (iii) the dimensionality of the order parameter. 3.10.2 Exact solution of the 2D Ising model A good reference model to check the values of the exponents is the Ising model in two dimensions, which can be solved exactly [KramersWannier (1941), Onsager (1944)] in the absence of a magnetic field. There are various methods of solution, none of which is simple. The first approaches were based on transfer matrices (as applied in the 1D Ising model), but this is quite complicated in 2D. Subsequently other simpler methods were devised. Here we only give a rough indication of one of these methods and then comment 102

Exp. Magnets Gas-liquid Binary mixtures Binary alloys Ferroelectrics α, α 0.0 0.2 0.1 0.2 0.05 0.15 β 0.30 0.36 0.32 0.35 0.30 0.34 0.305 ± 0.005 0.33 0.34 γ 1.2 1.4 1.2 1.3 1.2 1.4 1.24 ± 0.015 1.0 ± 0.2 γ 1.0 1.2 1.1 1.2 1.23 ± 0.025 1.23 ± 0.02 δ 4.2 4.8 4.6 5.0 4.0 5.0 ν 0.62 0.68 0.65 ± 0.02 0.5 0.8 η 0.03 0.15 0.03 0.06 on the final expressions. The method is based on a comparison between the low- and high-temperature expansions of the partition function and the concept of dual lattice and duality (see more details in the referenced textbooks). For the square lattice this leads to an exact expression for the partition function Q(T): where log Q(T) N = log ( ) 1 π/2 ( ) 2cosh 2βJ + dφ log 1 + 1 κ π 2 sin 2 φ, (3.58) 0 κ = The critical temperature T c is given by sinh 2β c J = 1, so that kt c J = 1 β c J = arcsinh 1 2 2 sinh 2βJ cosh 2 2βJ. (3.59) = 2.2691 (3.60) to be compared with 4 (mean-field approx.) and 2.8851 (Bethe approximation). The energy per spin is: E N = kt 2 log Q/N T = 2J tanh2βj + 1 ( κ dκ ) π/2 π dβ 0 [ ] = J coth 2βJ 1 + 2κ π K 1(κ), K 1 (κ) = ( 1 + 1 κ 2 sin 2 φ π/2 0 dφ sin 2 φ ) 1 κ 2 sin 2 φ dφ, (3.61) 1 κ 2 sin 2 φ with K 1 (φ) the complete elliptic integral of the first kind, and κ = 2 tanh2βj 1 which satisfies κ 2 + κ 2 = 1. The specic heat at zero field C(T) = C H=0 (T) can be obtained by noting that K 1 (κ) = 1 [ E1 (κ) κ 2 K κ 2 1 (κ) ] π/2, E 1 (κ) = dφ 1 κ κ 2 sin 2 φ, (3.62) 0 where E 1 (κ) is the complete elliptic integral of the second kind. With this, C(T) Nk = 2 [ βj coth 2 2βJ ] 2 { [ π 2 [K 1 (κ) E 1 (κ)] (1 κ ) π 2 + κ K 1 (κ)]}. (3.63) 103

This expression has a logarithmic divergence at κ = 1 (exactly at the transition since κ = 1 is equivalent to sinh 2βJ = 1); this comes from the divergence of K 1 (κ): K 1 (κ) log 4 κ (3.64) (note that E 1 (κ) behaves correctly in κ = 1 and that this condition is equivalent to = 0 in view of the relation κ 2 + κ 2 = 1). Therefore α = α = 0 (with logarithmic divergence), which constrasts with the mean-field result [discontinuous jump in C(T)]. Another quantity of interest is the order parameter, which turns out to be given by the simple expression: ( 1 sinh 4 2βJ ) 1/8, T Tc [ m(t) = 16β c J ( 2 1 T )] 1/8, (3.65) T c 0, T > T c so that β = 1/8. Further analysis of the model indicates that ν = ν = 1 and η = 1/4. The magnetic susceptibility was obtained more recently, furnishing the γ exponents. Assembling all the values, we have: α = α = 0 (log), β = 1 8, γ = γ = 7 4, ν = ν = 1, η = 1 4. (3.66) (remember that in mean-field theory β = 1/2, γ = 1, ν = 1/2 and η = 0). One can see that all relations between critical exponents (given before as inequalities) involving the exponents α, α, β, γ, γ, ν, ν and η are satisfied by the Ising model as equalities; for example, α + 2β + γ = 0 + 2 1 8 + 7 4 = 2, etc. (3.67) The Ising model with H 0 has not been solved exactly yet; therefore no direct value for the δ exponent is obtained. However, assuming all the relations between exponents to be equalities, γ = β(δ 1), which gives δ = 1 + γ /β = 1 + 8 7/4 = 15 (to be compared with δ = 3 from mean field). 3.10.3 Scaling theory The modern theory of critical phenomena began in 1965, when Widom made some assumptions about how the free energy G = G(T, H) should scale with T and H. From these arguments he showed that α = α and γ = γ, and that the first four inequalities should be equalities (actually as predicted by mean-field theory). A year later Kadanoff gave a more sound reason for why the scaling form of Widom should hold; it involved a scaling form for the correlation function, and it allowed him to show that ν = ν and that in fact all inequalities should be equalities. Let us briefly discuss the basis of these ideas. Consider the Landau free energy with a magnetic field: G(H, T; m) = B 1 tm 2 + C 0 m 4 Hm, 2B 1 tm + 4C 0 m 3 = H (3.68) 104

(we use m as order parameter to avoid confusion with the η exponent, but the arguments are otherwise general). The second expression holds at equilibrium. At zero field the spontaneous magnetisation is m(t) = (B 1 t /2C 0 ) 1/2 ; let us write the second of Eqns. (3.68) in the complicated form: H = B3/2 1 t 3/2 C 1/2 0 2sign(t) C1/2 0 m B 1/2 1 t 1/2 + 4 3 C1/2 0 m. (3.69) B 1/2 1 t 1/2 If we were able to solve for m analytically, we would obtain something like m = B1/2 1 t 1/2 f HC1/2 0, (3.70) C 1/2 0 B 3/2 1 t 3/2 with f(x) some universal (for systems following Landau theory) function of its argument. Because of the sign(t), the function f(x) has two branches, f ± (x) for t > 0 and t < 0. Now the free energy can also be expressed as: = B2 1 t2 C 0 = B2 1 t2 C 0 G(H, T; m) = B 1 tm 2 + C 0 m 4 Hm g HC1/2 0 B 3/2 1 t 3/2 C1/2 0 m B 1/2 1 t 1/2 + sign(t) C1/2 0 m B 1/2 1 t 1/2 2 + 4 C1/2 0 m B 1/2 1 t 1/2 HC1/2 0, (3.71) B 3/2 1 t 3/2 where again g(x) is a universal function with two branches, g ± (x); the last equality was obtained by substituting (3.70). The important thing to notice is that we have passed from three variables, (H, T, m), to only two, (m/ t 1/2, H/ t 3/2 ). Then, we generalise and write ( ) G(H, T) = α 0 t 2 α α1 H g. (3.72) Here we expect α 0 and α 1 to be two non-universal constants, specific to the problem in question, but we also expect g(x) to be a universal function, and α, to be two universal exponents (within the given universality class). α is chosen to appear in the combination 2 α since we want α to be the exponent governing the behaviour of the specific heat, and C ( 2 G/ T 2 ). Also, as (α, ) are the only exponents, we expect that the others can be derived from these two, as we now demonstrate. The specic heat at zero field is: ( 2 ) G g + (0), t > 0 C H (t, H = 0) = = α T 2 0 (2 α)(1 α) t α (3.73) H=0 g (0), t < 0. From here α = α [since the difference in limits, t 0 + or t 0, is contained in g(0)]. The spontaneous magnetisation is: ( ) G g m(t) = = α 0 α 1 t 2 α + (0), t > 0 (3.74) H T g (0), t < 0. 105

Then β = 2 α. The susceptibility at zero field is: χ T = ( ) m = α 0 α1 2 H t 2 α 2 T,H=0 g + (0), t > 0 (3.75) g (0), t < 0. Therefore γ = γ = α + 2 2. Combining the last two equations we get α + 2β +γ = 2. Now we discuss the equation of state: ( ) m(t) = α 0 α 1 t 2 α g α1 H t. (3.76) Since g (x) is an unspecified function, we are entitled to write g (x) = x β/ n(x), with n(x) another unspecified function and β a new exponent. Then: m(t) = α 0 α 1 t 2 α α β/ 1 H β/ t β Solving for t in terms of a new function q(x): t α 1/ 1 H 1/ q α 0α 1+β/ m n ( ) α1 H t = α 0 α 1+β/ 1 H β/ n 1 H β/ ( ) α1 H t. (3.77). (3.78) On the critical isotherm, t = 0, and close to the critical point, H 1, the function q(x) must be zero, so that its argument must be a constant; therefore m H β/, δ = β = 2 α β. (3.79) β From here δβ + β = 2 α, which gives α + β(δ + 1) = 2. The previous equality can be combined with the first (α+2β+γ = 2) to give γ = β(δ 1). And the latter and the former give γ(δ +1) = (2 α)(δ 1). We then see that the simple scaling argument is sufficient to obtain the thermodynamically derived inequalities, but this time as equalities. Let us now turn to the correlation function. In Ginzburg-Landau (mean-field) theory we obtained, at the critical temperature T = T c, G(r) 1/r in 3D. If we had done the calculation in d dimensions, we would have obtained G(r) 1/r d 2. But in the 2D Ising model the behaviour is G(r) 1/r 1/4, which obviously does not adapt to this scheme. It then seems that the long-range decay of the correlation function depends on the model and the dimensionality of space. To combine these facts one writes G(r) 1 r d 2+η, T = T c, (3.80) where η is a new exponent (Ginzburg-Landau theory predicts η = 0, whereas in the 2D Ising model η = 1/4). Let us write a scaling form for G(r), valid a little outside of the critical point. We know that H should come in the combination H/ t, but what about 106

r? Here we introduce an important assumption: close to the critical point, the distance r scales with the correlation length ξ t ν. This idea will be exploited to its full consequences later. We then write: Its Fourier transform is: G(r; H, t) G 0(rt ν, H/ t r d 2+η. (3.81) G(q; H, t) G 0 (qt ν, H/ t ) q 2 η (3.82) [note that, on taking Fourier transform, the volume element dr d q d dx (with x a dimensionless variable) and r d+2 η q d 2+η ]. At the critical point H = 0, t = 0, G(q) 1/q 2 η, so that a nonvanishing value of η produces a correction to the 1/q 2 law for G(q) at q 0. In practice η is very small (as measured experimentally). The susceptibility at zero field is: χ T dr dg 0(rt ν, 0) r d 2+η t dν dx d G 0 (x, 0) t ν(d 2+η) x d 2+η t ν(2 η) (3.83) (here we made the change of variable rt ν x). From here we get γ = ν(2 η). This equality is used in practice to calculate the η exponent as it is difficult to measure experimentally. The results of scaling theory are impressive, but why does is it work? This question will have to be answered by a more sophisticated theory, which also provides a systematic way to compute the values of the critical exponents: the renormalisation group. 3.11 Renormalisation group (RG) We are already familiar with the fact that the correlation length ξ diverges at the critical point. Let us examine the pictures in Fig. 3.5, which correspond to typical configurations of the 2D Ising model with ferromagnetic interactions at various temperatures (obtained from Monte Carlo simulations; see 3.8). Black dots indicate up spin, white down spin. Can you sort out the pictures for increasing temperature? The answer is, from low to high temperature: down-middle, up-right, up-middle, down-left, up-left and down-right. The fact is that the top-middle configuration corresponds to the critical temperature, T = T c : domains are large, in fact as large as the system size, and the configuration is a fractal, showing self-similarity. What is happening is that the correlation length diverges at the critical point, while above or below it remains finite (of the order of the typical domain size, Fig. 3.6). The idea is that of scale invariance: since ξ, all microscopic lengths of the system (atomic or molecular sizes, typical intermolecular distances, etc.) become irrelevant at the critical point, which means that all properties of the system must remain the same if we apply a scale transformation, i.e. the system becomes insensitive to such a transformation. Outside of the critical point the system will be more and more sensitive as the departure gets larger and larger. To be more precise, we will consider the d-dimensional Ising model in all what follows. Suppose a is the lattice constant and we are close to the critical point, t 1 and H 1. A scale transformation can be realised in different ways. We mention two: 107

Figure 3.5: Six spin congurations of the 2D Ising model at various temperatures. Decimation: we consider the partition function, Q(H, t) = {s i } e βh N({s j };H). (3.84) Now we sum over some subset of N N spins, and hope to be able to write a partition function for the system with the same form as before: Q(H, t) = e βhn ({s j ;H), (3.85) {s i } where the sum is now over N spins. Fig. 3.7 suggests a possible choice of the N N spins on a square lattice. Block tranformation: we consider blocks of l d neighbour spins (see Fig. 3.8, where blocks contain four spins), and define some procedure to assign a spin to the block, S b = ±1 (for example, S b = +1 if i s i 0 in the block). Obviously a new lattice constant a = la (l > 1) emerges. The number of spins has been reduced from N to N = N/l d. In any case, if ξ is the correlation length of the original system measured in units of a, the correlation length in the transformed system, measured in units of a, will be ξ = ξ/l: the system has reduced its effective correlation length, so it is as if the thermodynamic conditions had changed to (H, t ). But, if ξ is large, the system is expected to be insensitive to this scale transformation, in the sense that the free energy per spin ψ(h, t) = G(H, t)/n satisfies: Nψ(H, t) = N ψ(h, t ) ψ(h, t) = l d ψ(h, t ). (3.86) 108

Figure 3.6: Correlation length below, at and above the critical temperature. Figure 3.7: (Colour on the Web). Possible process to renormalise the Ising 2D lattice by decimation. Now one assumes (H, t) to be related to (H, t ) linearly via a power of l: H = l x H, t = l y t, (3.87) so that ψ(h, t) = l d ψ(l x H, l y t). (3.88) But in order for G to satisfy this generalised homogeneity property, it can be seen 2 that we must have the scaling form ( ) H ψ(h, t) = t d/y ψ, (3.89) t 2 Take the simplest case of energy, which is a homogeneous function of degree one, meaning that, for arbitrary λ, E(λN, λv, λs) = λe(n, V, S). 109

Figure 3.8: (Colour on the Web). Possible process to renormalise the Ising 2D lattice by Kadanoff s blocking. where = x/y is some exponent. This can be checked as follows: ψ(l x H, l y t) = l y t d/y ψ ( l x H l y t ) = l d t d/y ψ ( ) H = l d ψ(h, t). (3.90) t lx y If this scale invariance were true, we could iterate this procedure, in each step reducing ξ; this can be thought of as an increase or decrease of T, depending on whether initially T < T c or T > T c (see Fig. 3.9) so we would end up at T = 0 or T =, both cases of which are trivial since ξ = 0 (i.e. interactions are not relevant). Figure 3.9: (Colour on the Web). Flow of the system on applying a RG transformation. Therefore, the scaled form (3.72), written intuitively from the Landau expression, is here obtained more rigorously using the hypothesis of scale invariance. Knowing x and y we could obtain all the critical exponents: from (3.72) and (3.89) it follows that α = 2 d y, β = 2 α = d x, y Taking λ = 1/N, this implies that E(N, V, S) = NE ( 1, V N, S ) ( V = Ne N N, S ), N where e is the energy per particle (we could also define an energy per unit volume or unit entropy). Our case is more general since it involves different exponents in the variables. 110

γ = α + 2 2 = 2x d, δ = y β = x d x. (3.91) Concerning the ν exponent, we may write ξ t ν, ξ t ν, and since t = l y t: ξ ( t ) ν ξ = = l νy ν = 1 t y. (3.92) Now since α = 2 d/y, we obtain dν = 2 α. A relation involving the η exponents can also be obtained. In summary, the derivation of the relations among exponents is a strong indication that the hypothesis of invariance under scale transformations close to the critical point may be correct. The question is: can we obtain values for x and y? 3.11.1 Application to the 1D Ising model Here we use the 1D Ising model to present the basic ideas of the RG, with no intention whatsoever at any quantitative calculations (detailed applications can be seen in the textbooks). With K 0 = 0, K 1 = βj, K 2 = βµ B H, and assuming N to be even, the partition function can be written as: Z N = {s i } e N i=1 [ K 0 + K 1 s i s i+1 + 1 ] 2 K 2(s i + s i+1 ) = N e K 0 + K 1 s i s i+1 + 1 2 K 2(s i + s i+1 ) {s i } i=1 = N/2 e 2K 0 + K 1 (s 2j 1 s 2j + s 2j s 2j+1 ) + 1 2 K 2(s 2j 1 + 2s 2j + s 2j+1 ). (3.93) {s i } j=1 The sums over s 2j (with even index) are done easily: s 2j =±1 e s 2j [K 1 (s 2j 1 + s 2j+1 ) + K 2 ] = 2 cosh [K1 (s 2j 1 + s 2j+1 ) + K 2 ], (3.94) so that, changing notation for the remaining, unsummed-over spins, from s 2j 1 to s j : Z N = {s N/2 j } j=1 2e 2K 0 cosh [K 1 (s j + s j+1 ) + K 2]e K 2 2 (s j + s j+1 ). (3.95) The sum now runs over half of the spins, N/2 (we have decimated once). Now we would like to write the above in the form of Eqn. (3.93), in order to preserve the identity of the system (i.e. that the system be described by the same model), and for this to be possible new coupling constants K 0, K 1 and K 2 will be needed: Z N = {s j } e N/2 j=1 [ K 0 + K 1s js j+1 + 1 ( ) ] 2 K 2 s j + s j+1. (3.96) 111

This requires that, for all possible values of two interacting spins s j = ±1 and s j+1 = ±1, e N/2 j=1 [ K 0 + K 1 s j s j+1 + 1 ( ) ] 2 K 2 s j + s j+1 which gives the three equations = 2e 2K 0 cosh [ K 1 ( s j + s j+1 e e K 0 +K 1 +K 2 = 2e 2K 0 +K 2 cosh (2K 1 + K 2 ), K 2 2 ) + K2 ] ( s j + s j+1), (3.97) e K 0 +K 1 K 2 = 2e 2K 0 K 2 cosh (2K 1 K 2 ), e K 0 K 1 = 2e 2K 0 cosh K 2, (3.98) from which we get {K 0, K 1, K 2 } in terms of {K 0, K 1, K 2 }: e K 0 = 2e 2K 0 [ cosh (2K1 + K 2 ) cosh (2K 1 K 2 ) cosh 2 K 2 ] 1/4, e K 1 = [ cosh (2K1 + K 2 )cosh (2K 1 K 2 )/ cosh 2 K 2 ] 1/4, e K 2 = e K 2 [cosh (2K 1 + K 2 )/ cosh (2K 1 K 2 )] 1/4. (3.99) From here it is obvious that the parameter K 0 is needed since otherwise the form (3.96) cannot be obtained (and, for notational consistency, the parameter K 0 = 0 is introduced for the original system). But K 0 does not play an essential role since the partition functions of the original and decimated systems are related by Z N (K 1, K 2 ) = e N K 0 QN (K 1, K 2), (3.100) i.e. it simply gets a prefactor that appears as a nonsingular (and therefore irrelevant) term in the free energy 3 The other two equations, G N (K 1, K 2 ) = ktn K 0 + G N (K 1, K 2 ). (3.101) K 1 = 1 4 log [cosh (2K 1 + K 2 )cosh (2K 1 K 2 )] 1 2 log cosh K 2, K 2 = K 2 + 1 2 log [cosh (2K 1 + K 2 )/ cosh (2K 1 K 2 )], (3.102) 3 Incidentally, the free energy of the system at infinite temperature can be obtained as follows. The free energy per spin is: ψ(k 1, K 2 ) = 1 2 ktk 0 + 1 2 ψ(k 1, K 2 ). Now in T, where K 1 = 0, K 2 = 0 [which implies, from (3.99), that K 1 = 0, K 2 = 0, K 0 = log 2, ψ(0, 0) = 1 2 ktk 0 + 1 2 ψ(0, 0) G(H = 0, T = ) = ψ(0, 0) = ktk 0 = kt log 2, as it should be. 112

define a procedure by which the Ks change (or flow) in a twodimensional space K = (K 1, K 2 ) as the RG equations, given by (3.102), are applied iteratively. The RG transformation can be written in abstract terms as K = R(K). (3.103) The transformation may have fixed points, K, such that K = R(K). At these points, which will appear isolated in K-space, the RG transformation has no effect; in particular, the correlation length ξ will not change. But, since ξ is reduced by a factor l at each step, we must have ξ = 0 or at a fixed point; the first possibility is trivial (trivial fixed point), so we focus on the second. In fact all initial points going to the nontrivial fixed point must also have ξ = : they form a surface of critical points in parameter space (a line in two dimensions). The actual critical point K c, corresponding to particular choices for the values of the coupling constants, is a point on this surface. In general, each physical critical point has an associated critical surface with an associated fixed point. The set of all points on the critical surface corresponds to the set of all systems (with different values of the coupling constants) that can be described by the model Hamiltonian. Figure 3.10: Flow of the system in a two-dimensional parameter space. The critical point corresponds to a particular system with particular values of the coupling constants. The situation is depicted qualitatively in Fig. 3.10. The critical line separates the T < T c (ordered) region from the T > T c (disordered) region. On the critical line ξ = so flow cannot cross it; trajectories generally get away from this line. Flow along the critical line goes towards the fixed point. The fact is that the critical properties and the values of critical exponents will depend on the properties of the relevant fixed point. To see this one 113