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Chapter 5

Mean-Field Theory

5.1 Mean-Field Treatments of the Ising Model

In this Section, we are discussing various approaches to obtain a mean-field solu-tion to the Ising model. In fact, several of the approaches will yield exactly iden-tical results. The reason they are presented is that they highlight different ways ofcarrying out the approximation(s) that are commonly referred to as “mean-fieldapproximations”. Essentially, they differ by whether one (i) neglects spin fluctua-tions around the mean or (ii) considers spins to behave statistically independentlyand by which part of the system one treats exactly (Bethe-Peierls mean-field the-ory).

5.1.1 Weiss Molecular Field Theory

We start by identifying the order parameter of the magnet which distinguishesthe ordered (magnetic) from the disordered (nonmagnetic) phase. For describ-ing a paramagnetic to ferromagnetic transition, the obvious choice is the (local)magnetization

m = 〈si〉. (5.1)

We now focus on a single spin, s0, which on a hypercubic lattice in d dimensionshas z = 2d nearest neighbours, which we label s1, . . . s2d. The scenario consid-ered is shown for the two-dimensional case in Fig. 5.1

The part of the Ising Hamiltonian containing spin s0 reads as follows, wherewe decompose the spins sj into their mean value (the magnetization), and fluctu-

87

88 CHAPTER 5. MEAN-FIELD THEORY

Figure 5.1: A spin s0 on a two-dimensional square lattice interacts with its nearestneighbours s1, . . . , s4.

ations around it, sj = m+ (sj −m):

Hs0 = −s0

(J

z∑j=1

sj +H

)

= −s0(zJm+H)− Js0

z∑j=1

(sj −m)

The fundamental assumption of mean-field theory is now to set the fluctuations to0, (sj −m) = (sj − 〈sj〉)→ 0, such that the resulting Hamiltonian reads:

H0s0

= −s0(zJm+H) (5.2)

This represents a non-interacting spin in an effective field Heff = zJm+H . ThisHamiltonian allows us to calculate 〈s0〉. The second step of mean-field theory isto argue that the chosen spin is not special at all, hence its mean must be identicalto the magnetization. This gives us a self-consistency condition

〈s0〉!

= m (5.3)

As

〈s0〉 =tr s0eβs0(zJm+H)

tr eβs0(zJm+H)

=e+β(zJm+H) − e−β(zJm+H)

e+β(zJm+H) + e−β(zJm+H)

= tanh β(zJm+H)

we obtain the following self-consistent equation for the magnetization:

m = tanh[β(zJm+H)] (5.4)

5.1. MEAN-FIELD TREATMENTS OF THE ISING MODEL 89

Figure 5.2: Graphical representation of the self-consistency equation for the mag-netization in the Weiss mean field approximation.

While this equation cannot be solved analytically, the main features can be ex-tracted anyways. For a finite field H 6= 0, we see for example that solutionschange sign with the field: m(T,H) = −m(T,−H).

The field-free case H = 0 is more interesting. First of all, m = 0 is alwaysa solution, which corresponds to the paramagnetic regime. But if we draw bothsides of the equation m = tanh βzJm as a function of m (see Fig. 5.2), we seethat there are two more solutions ±m0 if the slope of the hyperbolic tangent atm = 0 exceeds slope 1. Now the slope of the hyperbolic tangent is given by

d

dmtanh βzJm

∣∣∣∣m=0

=1

cosh2 βzJm(βzJ)

∣∣∣∣m=0

= βzJ.

Hence, if βzJ > 1, there are 3 solutions, m = 0 and m = ±m0, which corre-sponds to the two Z2-symmetry breaking solutions of an Ising magnet. The criticaltemperature is given by

βczJ = 1 kBTc = zJ (5.5)

For the hypercubic lattices, this means:

• There is a phase transition at kBTc = 2J in the one-dimensional case. Infact, we will see from an entropy argument due to Peierls and from an exactcalculation in Chapter 7 that this is wrong; there is no phase transition in1D. Of course, omitting all fluctuations is a serious approximation, but wewill have to see why our theory fails even qualitatively.

• There is a phase transition at kBTc = 4J in the two-dimensional case. Wewill see from exact calculations in Chapter 7 that there is indeed a phase


transition, so our mean field theory is qualitatively right, but it is quantita-tively wrong: the true transition temperature is at kBTc = 2.269J , henceoverestimated. This is a typical feature of mean-field theories, because theirneglect of fluctuations makes them overestimate the tendency to order.

• In three and more transitions, mean-field theory continues to predict (cor-rectly) the existence of phase transitions, and the estimates for Tc get in-creasingly better (kBTc = 6J versus numerical kBTc = 4.511J). Thiscan be understood qualitatively: in higher dimensions, spin s0 is coupledto more and more neighbours, whose fluctuations around the mean (i.e. themagnetization) will increasingly tend to cancel each other (cf. the growth as1/√N of the relative fluctuations for N independent identically distributed

random variables with the proviso that the neighbouring spins are not act-ing independently). This means that on average s0 indeed is coupled to aneffective field zJm+H .

5.1.2 Mean-Field Theory and Bogolyubov Inequality

In this approach the essential approximation will consist in neglecting the statisti-cal interdependence of the orientation of spins, and explore an inequality on freeenergies.

Bogolyubov Inequality

Consider a classical Hamiltonian that can be decomposed into two parts,

H = H0 +H1. (5.6)

The decomposition is arbitrary but to make sense in the following it should bechosen such that the statistical physics ofH0 can be calculated more or less easilyexactly, whereasH1 contains all the difficult parts of the Hamiltonian. If we thinkof an Ising-like model, the partition functions are given as

Z =∑{s}

e−βH(s) Z0 =∑{s}

e−βH0(s) (5.7)

where we assume that we can calculate Z0, but not Z. Then

Z

Z0

=

∑{s} e−β(H0+H1)∑{s} e−βH0

=∑{s}

p0(s)e−βH1(s) = 〈e−βH1〉0, (5.8)


where the expectation value is taken with respect to the probability distributiongenerated by H0. We now invoke the convexity inequality which says that for afunction f(x) with f ′(x) > 0 and f ′′(x) > 0 we have

f(〈x〉p) ≤ 〈f(x)〉p, (5.9)

where the expectation value is taken with respect to some probability density p(x).Then, setting x equal toH1, we have

〈e−βH1〉0 ≥ e−β〈H1〉0 . (5.10)

But the left-hand side is just Z/Z0, hence

Z

Z0

≥ e−β〈H1〉0 . (5.11)

Taking the logarithm and using G = −kBT lnZ (both for Z and Z0), we obtainthe Bogolyubov inequality:

G ≤ G0 + 〈H1〉0 (5.12)

The same expression also holds for F ; we anticipate that we work with G(T,H)rather than F (T,M), i.e. subtract HM from the interaction energy of the Isingmodel. This expression can be reformulated a bit if we insert

G0 = 〈H0〉0 − TS0 (5.13)

where the entropy S0 is given by

S0 = −kB∑{s}

p0(s) ln p0(s). (5.14)

Then the Bogolyubov inequality takes the form

G ≤ 〈H〉0 − TS0 (5.15)

This shows that we can give a rigorous upper bound on the energy.

Bogolyubov Inequality and Mean-Field Theory

We split the Hamiltonian H of the Ising model in a way which depends on somecontrol parameter λ, where the goal will be to choose it such as to minimize thefree energy:

H(λ)0 = −λ

∑i

si H(λ)1 = −J

∑〈i,j〉

sisj + (λ−H)∑i

si (5.16)


AsH(λ)0 is a sum of single-spin terms, Z0 will be a product of single-spin partition

functions e−βλ + e+βλ = 2 cosh βλ:

Z0 = (2 cosh βλ)N . (5.17)

Then G0 = −kBT lnZ. The expectation values of the spins are independent ofeach other and do not depend on the spin index:

〈si〉0 = tanh βλ. (5.18)

Due to independence, the expectation value ofH(λ)1 reads

H(λ)1 = −1

2NJz tanh2 βλ+N(λ−H) tanh βλ, (5.19)

with z/2 nearest-neighbour bonds. Then we have as variational ansatz fromEq. (5.12)

G(λ) = N [−β−1 ln(2 cosh βλ)− 1

2Jz tanh2 βλ+ (λ−H) tanh βλ] (5.20)

which we seek to minimize to provide an upper bound on the free energy: differ-entiating with respect to λ, and setting dG/dλ = 0, we have

λmin −H = Jz tanh βλmin, (5.21)

which upon insertion into G(λ) gives as variationally minimal free energy

G = −NkBT ln(2 cosh βλmin) +N(λmin −H)2

2zJ. (5.22)

The magnetization per spin is then given by

m = − 1

N

dG

dH= − 1

N

(∂G

∂H+∂G

∂λ

∂λ

∂H

)λmin

=λmin −H

Jz. (5.23)

Here we have used that ∂G∂λ

= 0 for λ = λmin. Inserting Eq. (5.21), we revert tothe previously established relation

m = tanh β(Jzm+H), (5.24)

from which the analysis proceeds as before.


5.1.3 Bragg-Williams Mean-Field TheoryIn this approach, the approximation made will again lead to an effective single-spin problem; instead of solving a self-consistency equation, we will minimizethe resulting approximate free energy. The appeal of this approach is not anyimproved result, but a first insight into a very generic form of mean-field freeenergies.

We start by rewriting the Hamiltonian in terms of the number of nearest-neighbour bonds with both spins up (N++), both spins down (N−−) and one upand one down (N+−); for a hypercubic lattice in d dimensions with z = 2d nearestneighbours, the total number of bonds for N sites is dN = Nz/2 (assuming pe-riodic boundary conditions). We also use N±, the number of up and down spins;N = N+ +N−. Then the Hamiltonian can be expressed exactly as

H = −J(N++ +N−− −N+−)−H(N+ −N−). (5.25)

As magnetization per site m = (N+ − N−)/N , therefore magnetization M =Nm = N+ −N−, we can express N± in terms of m:

N± =N(1±m)

2. (5.26)

The approximation now consists in a decoupling assumption: the spin states ondifferent sites are statistically independent, and we have site-independent proba-bilities for s = ±1 as

p± =N±N. (5.27)

We can now construct a free energy G = U − TS. Using the von Neumann ex-pression S = −kB

∑k pk ln pk and that entropy is simply additive for independent

subsystems (i.e. individual spins), we have

S = −kBN(N+

NlnN+

N+N−N

lnN−N

). (5.28)

Under the decoupling assumption, N++, N−−, N+− can be traced back to N±.There are Nz/2 bonds. The probability that both spins are up is p2

+ due to statis-tical independence. Then

N++ =Nz

2p2

+ = zN2

+

2NN−− = z

N2−

2NN+− = z

N+N−N

. (5.29)

We can now express the internal energy and hence the Gibbs free energyG(T,H) =〈H〉 − TS as:

G(T,H) = − Jz2N

(N2++N2

−−2N+N−)−H(N+−N−)+kBTN

(N+

NlnN+

N+N−N

lnN−N

),

(5.30)


which can be expressed in terms of magnetization per site m as

G(T,H) = −JzN2

m2−NHm+kBTN

(1 +m

2ln

1 +m

2+

1−m2

ln1−m

2

).

(5.31)The dependence on T is implicit in m. Passing to the intensive Gibbs free energyper site g = G/N , we now determine m by minimizing g, setting ∂g/∂m = 0.The result is

0 = −Jzm−H +1

2kBT ln

1 +m

1−m(5.32)

This looks new, but isn’t, because arctanh x = 12

ln 1+x1−x . Hence, we recover m =

tanh β(Jzm + H), and the same predictions for Tc and m(T ). The advantage ofthis approach is that we get an idea of the form of the free energy expressed in theorder parameter m.

Let us focus on the field-free case H = 0. Then

g(T, 0) = −Jz2m2+

1

2kBT [(1+m) ln(1+m)+(1−m) ln(1−m)−2 ln 2]. (5.33)

Expanding the logarithm up to fourth order,

ln(1± x) ' ±x− x2

2± x3

3− x4

4,

we find that

g(T, 0) = −kBT ln 2 +1

2(kBT − Jz)m2 +

1

12kBTm

4 +O(m6). (5.34)

Looking at the structure of the expansion of the logarithm, one sees that there areonly even powers of m present. This is as it should be, as the Hamiltonian isinvariant under reflection si → −si, hence the sign of m may not appear. One canalso see from the prefactors of the powers of m in the expansion that all higherorders m6, m8, etc. will have positive prefactors.

Using a more generic form and recognizing that (1/2)(kBT−Jz) = (kB/2)(T−Tc), we have

g(T, 0) = a(T ) +b(T )

2m2 +

c(T )

4m4 +O(m6) (5.35)

where

• a(T ) is an irrelevant constant

• b(T ) > 0 for T > Tc, b(T ) < 0 for T < Tc: a sign change occurs at T = Tc


Figure 5.3: Gibbs free energy g from Eq. (5.33) as a function of m (|m| ≤ 1)for temperatures above, at and below the critical temperature. Dots indicate themagnetization that minimizes energy. At T = Tc, at the minimum the second (andthird) derivative(s) vanish too, making the energy curve very flat.

• b(T ) is linear in (T − Tc)

• c(T ) > 0

We will encounter this form again in the framework of Landau’s theory of phasetransitions, where it will emerge as the minimum instance of the description ofa continuous phase transition. The prefactors are for later convenience. Fig. 5.3shows g above, at and below Tc. For symmetry reasons, g(m) = g(−m). Theshape depends on the sign of b(T ): above Tc it is positive, so there is a globalminimum at m = 0. For T < Tc it is negative, such that a double-well structureemerges, where the minima sit at finite magnetization.

5.1.4 Bethe Mean-Field TheoryLet us conclude by looking at one slightly more refined variety of mean-field the-ory due to Hans Bethe [H. Bethe, Proc. Roy. Soc. London A150, 552 (1935)]. Itwill show that there is room for qualitative and quantitative improvement, but alsothat certain features of mean-field theory are independent of the precise details ofapproximation. Last but not least, we will get a feeling how much more compli-cated calculations will become if one pushes the ideas of this section further.

In our original mean-field theory treatment due to Pierre Weiss (1865 – 1940),we considered uncorrelated spins. Obviously, the correlations will be strongest


Figure 5.4: Smallest cluster on a square lattice: The interactions of a central spins0 with its nearest neighbours s1 through s4 are treated exactly, whereas the sur-face spins of the cluster see only an effective field as far as their interactions withtheir other 3 neighbours are concerned.

between spins that are within each others immediate vicinity, and we can improvesystematically by considering uncorrelated clusters of spins of increasing size.

The smallest cluster imaginable is a spin s0 together with its z nearest neigh-bours sj , hence a cluster of z + 1 spins, for example 5 spins on a square lattice(Fig. 5.4). Then s0 can interact exactly with its 4 neighbours. Those 4 neighboursinteract exactly with s0, of course, but see their own further three neighbours onlythrough an effective field as in Weiss theory.

The Hamiltonian for the cluster can be written as

Hc = −Js0

z∑j=1

sj −Hs0 −H ′z∑j=1

sj. (5.36)

The last term contains the effective field interaction and the external field H , andwill be related to the magnetization of the surrounding spins. As in simple Weisstheory, the field has to be determined self-consistently. Obviously, spin s0 in thecenter is treated on a different footing than the cluster surface spins sj . But in atranslationally invariant system there is no preferred spin site, and the magnetiza-tion of all spins must be identical. Hence the self-consistency condition reads

〈s0〉 = 〈sj〉, (5.37)

and this will be achieved by a proper choice of the effective field H ′. To calculate


the magnetizations, we need

Zc =∑{sj}

∑s0=±1

eβHs0eβJs0∑j sjeβH

′∑j sj

=∑{sj}

(e+βH

∏j

eβ(J+H′)sj + e−βH∏j

eβ(−J+H′)sj

)= e+βH(2 cosh β(J +H ′))z + e−βH(2 cosh β(J −H ′))z

These steps can be rerun to calculate 〈s0〉 and 〈sj〉.

Zc〈s0〉 = e+βH(2 cosh β(J +H ′))z − e−βH(2 cosh β(J −H ′))z (5.38)Zc〈sj〉 = 2e+βH sinh β(J +H ′)(2 cosh β(J +H ′))z−1

−2e−βH sinh β(J −H ′)(2 cosh β(J −H ′))z−1 (5.39)

Specializing to H = 0, the self-consistency condition (5.37) reduces to

(cosh β(J +H ′))z − (cosh β(J −H ′))z (5.40)= sinh β(J +H ′)(cosh β(J +H ′))z−1 − sinh β(J −H ′)(cosh β(J −H ′))z−1,

which can be rewritten as(cosh β(J +H ′)

cosh β(J −H ′)

)z−1

= e2βH′ (5.41)

Obviously, H ′ = 0 is always a solution. But again there are further solutions,because the right hand side diverges with H ′ → ∞, whereas the left hand sideconverges to e2β(z−1)J . So, if the slope of the left hand side exceeds that of theright hand side for H ′ = 0, there must be a further solution H ′0 and by symmetryalso −H ′0.

The derivative of the right hand side atH ′ = 0 is 2βe2βH′ = 2β. The derivativeof the left hand side is a but more involved, but in the end is just 2β(z−1) tanh βJ .Hence, the transition temperature is reached when these expressions become iden-tical, 1

!= (z − 1) tanh βcJ or

coth βcJ = z − 1 (5.42)

As arccoth x = 12

ln 1−x1+x

, we obtain

kBTc =2J

ln( zz−2

)(5.43)

This leads to the following predictions:


• For a one-dimensional chain, z = 2, hence Tc = 0: there is no phasetransition at finite temperature, which is a correct prediction not obtained inthe simpler mean-field theory.

• For a two-dimensional square lattice z = 4, hence

kBTc =2J

ln 2= 2.885J, (5.44)

which is still far from the exact 2.269J , but already much better than theoriginal 4J prediction.

One could now consider larger and larger clusters. Approximations will becomebetter, but at the same time the calculational effort will explode. Ultimately, invain: correlations will diverge at the transition temperature: some response func-tion diverges at the transition, but this can be related to diverging two-point cor-relators due to a suitable fluctuation-dissipation theorem. No finite cluster will beable to account for this, so even these improved approaches will fail at the phasetransition itself. Of course, this does not exclude reasonable results far away fromthe transition.

5.1.5 Construction of a Free EnergyQuite independent of these approximation schemes, we can construct a free energyas follows. We decompose a Hamiltonian into a non-interacting (hence easy) andinteracting part,Hλ = H0 + λV . In the case of the Ising model,

H0 = −H∑i

si V = −∑〈i,j〉

sisj λ = J. (5.45)

We now take λ ∈ [0, J ] and introduce

Gλ = −kBT lnZλ = −kBT ln tr e−βH0−βλV . (5.46)

Then G0 is the free energy of the simple non-interacting model, and G = GJ isthe free energy of the Ising model. With the notation

〈O〉λ =1

ZλtrOe−β(H+λV) (5.47)

we find thatdG

dλ= 〈V〉λ, (5.48)

hence the exact result

G = GJ = G0 +

∫ J

0

dλ 〈V〉λ. (5.49)

The approximation comes in through 〈V〉λ, which one can evaluate at one of theseveral levels of approximation introduced here.


Figure 5.5: Power-law behaviour of the magnetization in mean-field prediction.Critical exponent β = 1/2.

5.1.6 Critical Exponents in Mean-Field TheoryCritical behaviour is characterized by singularities in various response functions.

Magnetization

For T < Tc close to Tc, magnetization m0 → 0. In this limit, we can expandtanhx = x− x3/3 +O(x5) to obtain

m0 = βzJm0 −1

3(βzJ)3m3

0 + . . . (5.50)

Dividing by m0, we can solve as

m0 = ±√

3

(kBT

zJ

)3/2(zJ

kBT− 1

)1/2

. (5.51)

Using Tc = zJ/kB we obtain

m0 = ±√

3

(T

Tc

)3/2(TcT− 1

)1/2

(5.52)

The first factor depends only weakly on T around Tc; the dominant contributioncomes from the second factor which is singular at T = Tc (see Fig. 5.5):

m0(T ) ∼ (Tc − T )1/2 (T → T−c ) (5.53)

Mean-field theory (correctly) predicts that the magnetization vanishes as apower law with temperature. The critical exponent, which is conventionally called


β, is predicted to be 12

(which we will find to be incorrect; it is rather ≈ 0.31):

m(T ) ∼ (Tc − T )β β =1

2(5.54)

Susceptibility

In the magnetic case, we consider isothermal susceptibility per spin

χ(T,H) =∂m

∂H

∣∣∣∣T

, (5.55)

which we obtain by taking the variation of the self-consistency equation m =tanh β(zJm+H) in m and H . From

δm =1

cosh2 β(zJm+H)βzJδm+

1

cosh2 β(zJm+H)βδh (5.56)

susceptibility can be read off directly. In the case H = 0 it reads

χ(T, 0) =β

cosh2 β(zJm+H)− βzJ(5.57)

We now have to insert m as obtained in the mean-field approximation.

• T > Tc: In this case, m = 0 and

χ(T, 0) =β

1− βzJ=

β

βczJ − βzJ=

1

kB(T − Tc)(5.58)

Susceptibility diverges as (T − Tc)−1 for T → T+c .

• T < Tc: Close to the transition, m will be small and we can expandcosh2 ε ≈ 1+ε2 to lowest non-trivial order. Then we obtain using Eq. (5.50)

χ(T, 0) =β

1 + (βzJ)2m2 − (βzJ)=

β

1 + 3(βzJ − 1)− (βzJ)=

1

2kB(Tc − T ).

(5.59)

The susceptibility diverges with a power law both above and below the criticaltemperature Tc. By convention one uses the exponent(s) γ±:

χ(T, 0) ∼ A±|T − Tc|−γ± γ± = 1 A+ = 2A− (5.60)


Heat Capacity

The heat capacity (which we consider at zero field H = 0) is given by the deriva-tive of the internal energy U with respect to temperature T . In the Weiss approxi-mation U is given as

U = 〈H〉 = −J∑〈i,j〉

〈si〉〈sj〉 = −JNz2m2. (5.61)

At this level of approximation m2 is given by Eq. (5.52) as

m2 =

{3(TTc

)3 (TcT− 1)

= 3T−3c T 2(Tc − T ) (T < Tc)

0 (T > Tc)(5.62)

Upon insertion, we find that C = ∂U/∂T = 0 for T > Tc. For T < Tc we obtain

∂U

∂T= −J 3Nz

2T−3c T (2Tc − 3T ). (5.63)

Approaching Tc from below, we obtain with Tc = Jz/kB

C =3

2NkB. (5.64)

Hence, in the Weiss mean-field theory the heat capacity does not exhibit a power-law divergence as C ∼ B±|T−Tc|−α, but it has a discontinuous jump at the phasetransition. This is not confirmed by either exact calculation where available or ex-periment. More advanced mean-field schemes do not yield the correct qualitativebehaviour, either.

Critical Isotherm

The critical isotherm m(Tc, H) traces the dependence of m on H at the criticaltemperature Tc. To obtain it, we expand the Weiss consistency equation to thirdorder in the hyperbolic tangent,

m = β(Jzm+H)− 1

3[β(Jzm+H)]3.

Using βczJ = 1 at criticality, this turns into

βcH =1

3(m+ βcH)3.


In the limit of very small field H → 0 we approximate the right hand side inlowest order to obtain

βcH =m3

3H ∼ mδ δ = 3 (5.65)

Here we have assumed H,m > 0 for simplicity of the expression and called theexponent δ by convention.

5.2 Landau TheoryIn 1936, Lev Landau (1908 – 1968) developed a very simple theory of phasetransitions that nevertheless manages to give a phenomenological description ofboth first order and second order phase transitions and their interplay.

The fundamental paradigm of Landau theory is that phase transitions are de-scribed by an order parameter which describes some local property of the systemwhich is finite on the “ordered” side of the transition and zero on the “disordered”side. Its identification is the first crucial step: for Ising magnets it is obviouslymagnetization. In what follows we will first assume that the order parameter is ascalar, but in general it will be a n-dimensional vector.

The fundamental assumption of Landau theory is that the Gibbs free energycan be expanded in a power series of the order parameter m, where the only con-straint are the underlying symmetries of the system, that should be respected. Forexample, an inversion symmetry of the system implies that only even powers ofthe order parameter may appear. We take m to be intensive, but nothing changesif we switch to the extensive M = Nm as order parameter.

The problem with this assumption is that the free energy cannot be truly ex-panded in a power series at the phase transition, because the phase transition isassociated with some non-analyticity in some derivative of the free energy, whichtherefore should not exist. But we can describe phase transitions in Landau the-ory, and the reason for this is that the actual free energy is always some minimumof the free energy function. The minimum operation is not necessarily analytic,and we will see how it brings about non-analyticities.

Let us consider an Ising magnet at H = 0. The free energy should not dependon the sign of m:

G(T,m) = G(T,−m) (5.66)

Therefore, only even powers of m should appear, and we expand as

G(m,T ) = a(T ) +1

2b(T )m2 +

1

4c(T )m4 +

1

6d(T )m6 + . . . (5.67)

5.2. LANDAU THEORY 103

The prefactors could of course be absorbed into the coefficients b(T ), . . ., but areconvenient once we take derivatives. But this varies from author to author, andin Chapter 6 on an extension of Landau theory, Ginzburg-Landau theory, we willfind it convenient to choose another convention.

The important question is whether the functions b(T ), c(T ), . . . change sign,and if so, at which temperature? We will discuss various scenarios and see thatthey lead to continuous and first-order phase transitions as well as a new phe-nomenon, tricritical points.

5.2.1 Continuous TransitionsIn the first scenario, we assume c, d, e > 0 and

b(T ) = b0(T − Tc), (5.68)

i.e. a Taylor expansion of b(T ) up to linear order, where we assume b0 > 0. Manyphysical systems can be represented like this; we have already seen one examplein Section 5.1.3.

Magnetization

Under the provisions just made, we can identify the value of the order parametervia

0!

=∂G

∂m

∣∣∣∣T

= bm+ cm3 + dm5 + . . . (5.69)

Obviously, m = 0 is a solution, corresponding to a disordered phase. Othersolutions are given by 0 = b(T ) + c(T )m2 + d(T )m4 + . . .. Truncating after thesecond term for simplicity, we find m0 = ±

√−b(T )/c(T ) if b(T ) < 0, hence

T < Tc. Close to Tc, we can approximate c(T ) ≈ c(Tc) > 0, such that we obtain

m0 ≈ ±

√b0

c(Tc)

√Tc − T T → T−c (5.70)

Graphically, we can represent three typical forms of the free energy for T > Tc,T = Tc and T < Tc (Fig. 5.6). If temperature drops below Tc, instead of oneextremum (more specifically, minimum) at m = 0, there are now three at m =0,±m0. As the free energy ultimately is monotonically growing for |m| large, theadditional extrema must be minima, and the central extremum at m = 0 must be amaximum. Hence, the order parameter is now given by m0. This can be checkedby evaluating the free energy for all three extrema explicitly. The order parameterm changes continuously at all times, hence there is no first order phase transition.


Figure 5.6: Shapes of the free energy versus order parameter m in Landau theory(scenario for a continuous phase transition, c, d, e > 0): above Tc, there is oneglobal minimum at m = 0. Positive prefactors of the higher powers of m ensurestability, i.e. prevent a minimum of the free energy at some unphysical divergingvalue of m. At Tc, the free energy becomes flat at m = 0, i.e. the derivativedevelops three zeros. Two of the extrema separate away at T < Tc and are minimadue to symmetry and the stability requirement. Their positions give the new valuesof the order parameter ±m0.


Heat Capacity

In order to see how a non-analyticity is generated at Tc, we consider the heatcapacity C = T (∂S/∂T ). Entropy S itself is a first derivative,

S = −∂G∂T

= −a′ − b′

2m2 − c′

4m4 − . . .− b

2(m2)′ − c

4(m4)′, (5.71)

where the primes indicate derivatives with respect to temperature. Taking anotherderivative, we obtain

C = −Ta′′−Tb′(m2)′−Tb2

(m2)′′−Tc′′

4m4−Tc

′

2(m4)′−Tc

4(m4)′′−. . . (5.72)

in orders up to m4. b′′ = 0 as b is linear in T .For T > Tc, m = 0; so are all derivatives of m. Hence

C = −Ta′′ (T > Tc) (5.73)

For T < Tc, we considerm2 = (b0/c)(Tc−T ). Then (m2)′ = −b0/c and (m2)′′ =0. At the same time m4 = (b2

0/c2)(Tc − T )2, hence (m4)′ = −(2b2

0/c2)(Tc − T )

and (m4)′′ = 2b20/c

2.Close to the transition, m is very small but changes rapidly. Therefore we can

assumem4 < (m4)′ � (m4)′′ (5.74)

and the dominant contributions to C read for T → T−c :

C = −Ta′′ − Tb′(m2)′ − (Tc/4)(m4)′′ = −Ta′′ + Tb20

2c(5.75)

Hence, we have a drop (discontinuity) in C right at Tc, going towards highertemperatures:

∆C =Tcb

20

2c(5.76)

This scenario of Landau theory therefore describes a continuous phase transition.The critical exponent of the order parameter is β = 1/2, the critical exponent ofthe heat capacity is by convention set to α = 0.

5.2.2 First-Order TransitionsThe situation changes if we assume that c(T ) has a sign change, too, which weassume to happen at T ∗: c(T ) < 0 for T < T ∗ and c(T ) > 0 for T > T ∗. Ifc(T ) < 0, the stability of the system has to be ensured by d(T ), e(T ) > 0. Theconsequences of the sign change of c(T ) depend on which scenario prevails:


(1) T ∗ < Tc will give again a continuous phase transition

(2) T ∗ > Tc will give a first-order phase transition

(3) T ∗ = Tc will give a tricritical point

Continuous Phase Transition

Let us briefly discuss the first case (1). For T > T ∗ we have exactly the samescenario as before, c(T ) > 0, and the sign change of b(T ) at Tc generates acontinuous phase transition. The sign change in c(T ) at the lower temperatureT ∗ does not generate any additional minima. To see this, consider

0 =∂G

∂m

∣∣∣∣T

= bm+ cm3 + dm5 = m(b+ cm2 + dm4). (5.77)

m = 0 is always a solution. For the alternative solutions, we focus on the positivesolutions of the quadratic equation 0 = b + cx + dx2, because x = m2 ≥ 0. Thetwo solutions read

x± = − c

2d± 1

2d

√c2 − 4db. (5.78)

If we are below Tc, b < 0. Hence, the square root is greater than |c|, and there isonly one positive solution

m2 = x+ = − c

2d+

1

2d

√c2 − 4db > 0, (5.79)

independent of the sign of c.

First-Order Phase Transition

Let us now turn to the more exciting case (2). We reconsider the analysis of thesolutions of 0 = m(b + cm2 + dm4). m = 0 is always a solution. The solutionsof 0 = b+ cx+ dx2 with x = m2 read again

x± = − c

2d± 1

2d

√c2 − 4db, (5.80)

but the different pattern of sign changes leads to a substantially different outcome.

• For T > T ∗ > Tc, b, c, d > 0. Then the square root is either imaginary (noreal solution for x) or less than c (no positive solution for x). m = 0 remainsthe only solution, as would also be clear from a graphical inspection.


• For T ∗ > T > Tc, b, d > 0 but c < 0. Then −c/(2d) > 0, and there mightbe 2 positive solutions for x, hence 4 additional solutions form0 (symmetricabout m = 0). If c2 − 4db < 0, there are again no positive solutions, henceonly m = 0. If c2 − 4db > 0, there are now two positive solutions for x,hence m2. There must be a temperature T̃ with Tc < T̃ < T ∗ where thishappens, as b→ 0 as T → T+

c . This corresponds to 4 new solutions for theorder parameter, ±√x+ and ±√x−. Due to stability and symmetry, the 5extrema must be ordered min – max – min – max – min. What the physicalsystem now does is a competition between the minimum at m = 0 and

the two minima at m0 = ±√− c

2d+ 1

2d

√c2 − 4db: which one is a global

minimum of the free energy? The phase transition will happen if they allhave the same free energy at some temperature T1.

We therefore solve for

G(m0)−G(0) =1

2b(T1)m2

0 +1

4c(T1)m4

0 +1

6d(T1)m6

0!

= 0 (5.81)

One solution would obviously be provided by m0 = 0. Of interest to us isthe alternative solution

1

2b(T1) +

1

4c(T1)m2

0 +1

6d(T1)m4

0 = 0, (5.82)

where we insert our previous result for m0, namely b(T1) = −c(T1)m20 −

d(T1)m40. Then

0 = −1

3d(T1)m2

0 −1

4c(T1) (5.83)

which we can solve trivially for

m20 = −3c(T1)

4d(T1)(5.84)

as the value where the phase transition happens. As there is a sudden jumpin the value of the order parameter, the phase transition at T = T1 is afirst-order phase transition.

To check that this transition happens above Tc, hence preempts the continu-ous phase transition, we determine

b(T1) = −d(T1)m40 − c(T1)m2

0 =3c(T1)2

16d(T1)> 0. (5.85)

If b(T1) > 0, then T1 > Tc.


Figure 5.7: Shapes of the free energy versus order parameter m in Landau theory(scenario for a first-order phase transition: c changes sign at a higher temperatureT ∗ than b (at Tc)): we descend with temperature. Above T ∗, there is one globalminimum atm = 0. Positive prefactors of the higher powers ofm ensure stability,i.e. prevent a minimum of the free energy at some unphysical diverging value ofm. At some T̃ , the free energy develops five extrema, with minima for the largestvalues of m and m = 0. At T1 > Tc, the two outlying minima turn from local intoglobal minima, a first-order phase transition happens. At T = Tc the two maximamerge into the central minimum, which becomes a maximum. At T < Tc, thereis a maximum separating two minima.

• For T < Tc < T ∗, nothing special happens. If b turns negative, the second(smaller) solution of the quadratic equation turns negative, and we are backto three solutions for m, namely m = 0 and m = ±m0. The latter twomust then be minima, which they already were. The two vanishing max-ima merge at T = Tc into the central minimum, which then turns into amaximum.

The shapes of the free energy are shown for illustrative purposes in Fig. 5.7.

5.2.3 Tricritical PointsThe situation that T ∗ = Tc, that both b(T ) and c(T ) change sign at exactly thesame temperature, might seem such a rare coincidence that it is of no practicalrelevance. This is maybe true if temperature is the only control parameter in thesystem. As soon as there is at least a second such “tuning” parameter, it may be


Figure 5.8: Tricritical phase transition: The dashed-dotted line is the line of po-tential locations of continuous phase transitions, as b = 0. They however onlyhappen in the dashed part, because they are preempted by a first-order transitionline. The matching point of the first-order and continuous PT (phase transition)lines is the tricritical point (Tt,∆t). The slopes of both curves are identical at(Tt,∆t).

possible to engineer exactly this situation. A good example is provided by theBlume-Emery-Griffiths model for mixtures of 3He and 4He. Here, we will not bevery specific about this tuning parameter ∆; except that we take it to be intensive.Still assuming symmetry with respect to the sign of the order parameter, the freeenergy is given in Landau theory as

G(T,m,∆) = a(T,∆) +1

2b(T,∆)m2 +

1

4c(T,∆)m4 +

1

6d(T,∆)m6. (5.86)

Consider Fig. 5.8. b(T,∆) = 0 defines a line on which a continuous phase tran-sition will happen, unless it is preempted (as just discussed) by a first-order phasetransition, which will happen if c changes sign for a higher temperature than b. As-sume that we can control c(T,∆) in the following way. Calling the temperaturesof sign changes again Tc for b and T ∗ for c,

• ∆ > ∆t: T ∗ > Tc

• ∆ < ∆t: T ∗ < Tc

This gives the scenario shown in the Fig. 5.8. The matching point of the first-orderand continuous PT lines is the tricritical point (Tt,∆t). Let us now discuss thebehaviour of the transition lines at the tricritical point. To simplify notation, weintroduce bT and b∆ (and similarly for c) for the partial derivatives ∂b/∂T |∆ and∂b/∂∆|T .

The location of the first-order transitions is given by [Eq. (5.85)]

b(T,∆) =3c2(T,∆)

16d(T,∆), (5.87)


and the location of the (possible) continuous phase transitions by

b(T,∆) = 0. (5.88)

The slope of this critical line is given by

d∆

dT

∣∣∣∣crit

= − bTb∆

. (5.89)

At the same time, we can take the variation of the first-order line to turn Eq. (5.87)into

b∆δ∆ + bT δT −6c(cT δT + c∆δ∆)

16d+

3c2

16d2(dT∆T + d∆δ∆) = 0. (5.90)

This gives a slope of the first-order line as

d∆

dT

∣∣∣∣first order

= − bT − (3/8)(ccT/d) + (3c2/16d2)dTb∆ − (3/8)(cc∆/d) + (3c2/16d2)d∆

= − bT − (3/8)ccT + bdTb∆ − (3/8)cc∆ + bd∆

.

(5.91)At the tricritical point, b→ 0, c→ 0, hence

d∆

dT

∣∣∣∣first order, tricritical

= − bTb∆

. (5.92)

The slopes of (5.89) and (5.92) are therefore identical, there is no kink where thetransition lines join.

To analyze the behaviour right at the tricritical point, we remember that inthermodynamics external control parameters come with conjugate quantities suchthat in energy expressions terms like −pdV or µdN appear. Let us call ζ theintensive conjugate quantity to ∆. Then, if g = G/N , we have

ζ = − ∂g

∂∆

∣∣∣∣T

, (5.93)

where the sign is an arbitrary convention. Along the first-order transition line, wehave, using the values for m at both sides of the transition, to lowest order

ζ+ ≈ −a∆ ζ− ≈ −a∆ +3

8d(b∆c+ c∆b). (5.94)

Thenδζ = ζ+ − ζ− ≈ −

3

8d(b∆c+ c∆b). (5.95)

(The difference was written such as to make δζ > 0.) In the T -ζ plane, the phasediagram therefore looks somewhat like in Fig. 5.9: there is a coexistence regionopening up below Tt.


Figure 5.9: Coexistence region below Tt in a T -ζ plane, where ζ is conjugate tothe control parameter ∆.

As previously discussed, the order parameter at the tricritical point must obey

m2 = − c

2d+√c2/4d2 − b/d. (5.96)

Now we approach the tricritical point, hence b→ 0 and c→ 0. If both are small,then it is reasonable to assume (at first) that |b| � |c2| and |b/d| � c2/4d2. If thisholds, we can neglect the contributions from c/2d and obtain

m(T ) = (b/d)1/4 ∼ (b0/d)1/4(Tt − T )1/4 β =1

4(tricritical) (5.97)

for temperatures below Tt. But if we approach the tricritical point along the criticalline b = 0, |b/d| < c2/4d2 in a narrow region around it. In this narrow region, wefind that m2 ∝ c. Linearizing c around its zero at T = Tt, we obtain the “normal”critical behaviour

m(T ) = (Tt − T )1/2 β =1

2(critical) (5.98)

for temperatures below Tt.The crossover line between the two types of criticality is defined by b/d ≈

c2/4d2. With additional calculations, one can determine all critical and tricriticalexponents as

βc = 12

γc = 2 αc = −1βc = 1

4γc = 1 αc = 1

2

(5.99)

A sketch of the regions is given in Fig. 5.10.

5.2.4 Symmetry and First-Order TransitionsSo far, we have assumed a fundamental invariance of G(T,m) with respect to asign change in m. Let us now consider the case

G(T,m) 6= G(T,−m). (5.100)


Figure 5.10: Critical and tricritical regions in the vicinity of the tricritical point(Tt,∆t) in a T -∆ diagram.

This opens up the possibility of odd powers of m appearing in the free energyexpansion. It is easy to show that by a suitable transformation m → m + m̃ onecan make the linear term always disappear. The most general form of the freeenergy therefore reads

G(T,m) = a(T ) +1

2b(T )m2 − 1

3c(T )m3 +

1

4d(T )m4. (5.101)

The sign in front of the m3 term is of course arbitrary; it is chosen such thatthe interesting physics happens for c(T ) > 0 (for all T ). As before, b(T ) 'b0(T − Tc). Then the finite order parameter m0 is given by

0!

=∂G

∂m

∣∣∣∣m0

= bm0 − cm20 + dm3

0, (5.102)

hence 0 = b− cm0 + dm20. The transition happens if G(T1,m0) = G(T1, 0), this

determinesm0 =

2c

3d. (5.103)

A simple calculation gives that at the transition

b =2c2

9d> 0, (5.104)

hence there is a first-order transition at T1 > Tc. A sketch of the free energies atvarious temperatures is shown in Fig. 5.11.

In this case, the first-order transition is not due to a sign change in c, but tothe appearance of a cubic term due to the breaking of the inversion symmetry!Had we reversed the sign of the m3 term, m0 < 0 instead of m0 > 0, which


Figure 5.11: Free energies in the presence of a cubic term: At some tempera-ture, a local minimum at finite order parameter m0 turns into a global minimum,replacing m = 0 and generating a first-order phase transition.

now is different physics. Indeed, if we associate a continuous phase transitionwith symmetry breaking, there can’t be a continuous phase transition for lack of abreakable symmetry.

A classical example for such a symmetry-driven first-order transition is pro-vided by the q-state Potts model

H = −J∑〈i,j〉

δsi,sj si = 0, 1, . . . , q − 1 (5.105)

for q = 3 (for q = 2 we revert to the Ising model).


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