Improved Mean-Field Theory: Bethe’s Approximation

We consider the case of Ising spins. The mean-field theoiy can be improved by using the partition function of an approximate Hamiltonian constructed as follows. We separate the whole Hamiltonian into two parts: a cluster containing a spin, say 0 and its z neighbors, and the remaining crystal. The method consists in treating exactly the interactions of 0 with its surrounding z neighbors, but treating the interaction of the cluster with the remaining crystal by the use of the mean-field theory. The Hamiltonian is written as

where В is an applied magnetic field. The spin a₀ and its z surrounding nearest neighbors H is the molecular field acting on (z - 1) neighbors outside of the cluster. H is thus given by H = (z — 1)/ < о > //л_в. The mean-field equation will be obtained at the end by setting < (T₀ >=< оi >=< . One has

where a = J /k_BT, b = ц_вВ/k_BT and с = ц_вН / k_BT = [z. - 1)/ < о > / k_B T. Summing on бт₀ = ±1 and factorizing the other sums, one gets

The averaged < 0 > and < o, > (/ = 1,..., z) are given by

Setting < (т₀ >=< <т,- >=< a >, one has Replacing Z± by (5.18), one obtains

This equation is used to determine self-consistently c, namely

< a >. In zero field (B = 0), one has b = 0 so that

It is observed that c = 0 is a solution of the last equation. However, there is a non-zero solution by making an expansion at small c (small

< a >) to the third order:

from which one has

Since the left-hand side is positive, this equation admits a non-zero solution if [tanh a - > 0. This means that the non-zero solution

exists if T < T_c where T_c is given by

It is interesting to note that for z — 1, namely a cluster of two spins, there is no solution for T_c, and for z = 2 (one-dimensional chain) one has T_c = 0. This corresponds to the results from the exact solutions (see paragraphs 5.4.2 and 5.6, and Problems 1). We recall that the mean-field theory incorrectly yields T_c Ф 0 for one dimension [see Eq. (2.23)].

Landau–Ginzburg Theory

It has been shown above that the mean-field theory suffers a serious flaw due to the fact that instantaneous local spin fluctuations have been neglected in this theory. For example, the mean-field theory can give rise to a phase transition where there is none in low dimensions. In addition, in the critical region neglecting fluctuations modifies the behavior of the phase transition as we will see when comparing the mean-field critical exponents with the exact ones. There exist several more efficient theories such as the high- and low- temperature series expansions [95], the Landau-Ginzburg theory and the renormalization group. In what follows, we present the Landau-Ginzburg theory and the concepts of the renormalization group.

The Landau-Ginzburg theory is an extension of the mean-field theory which includes a great part of fluctuations so far neglected near the transition. The main idea is to start from an expansion of the free energy per spin / in the vicinity of T_c when the magnetization m is sufficiently small:

where С, A, В and D are constants, h is an applied magnetic field and the suffix MF denotes quantities coming from the mean-field theory. The form of this expansion and the sign of В reflect the system symmetry. In the case where h = 0 and В > 0, /_Mr presents a minimum at m = 0 for T > T_C^MF and two symmetric minima at ±m₀ for T < T_C^MF (see Fig. 5.2), indicating two degenerate ordered states. This degeneracy is removed when h ^ 0 as shown in Fig. 5.3.

Figure 5.2 Mean-field free energy at Г > T_c (left) and at Г < T_c (right).

Figure 5.3 Mean-field free energy at Г < T_c in an applied magnetic field.

When В < 0, a first-order transition is possible. At T = T_c, there are three equivalent minima of f_MF at 0, ±m₀ (see Fig. 5.4) contrary to the case В > 0. This means that at the transition the three phases m = 0 (paramagnetic state) and ±m₀ (ordered states) coexist. The energy distribution at T = T_c is thus bimodal, the peak at low

Figure 5.4 Mean-field free energy at a first-order transition.

energy corresponds to the energy of the ordered phase while that at high energy corresponds to the energy of the disordered state. The distance between the two peaks is the latent heat which is observed at a first-order transition.

When an m³ term is present in the expansion (5.28), the transition is always of first order.

Mean-Field Critical Exponents

The critical exponents calculated by the mean-field theory are (see Chapter 2) = 1/2 [see (2.24)], у = 1 [see (2.38)]. One can find

them again here by using (5.28). Putting t = (Г — T_C^MF)/ T_C^MF, one has

• When t < 0 and h = 0, /mf is minimum at mo oc (—£)?, so that

P = 1/2.

• • When h Ф 0 and t > 0, one has m oc h/t so that / a t^-1, hence У = 1-
• • At t = 0, /mf is minimum at m ос (^)^1/3, hence 5 = 3.
• • For f > 0, the minimum of /mf is equal to C and for t < 0, it is equal to C + 0(^t²). One has Cy oc which is 0 for f > 0 and equal to ^ for t < 0. Cy is thus independent of t. The discontinuity of Cy at t = 0 is an artifact of the mean-field theory. When fluctuations are included, one finds a = 0 as will be seen below.

Correlation Function

This subsection considers a system of Ising spins for simplicity. A simplified notation has been used in this paragraph: r instead of r. The correct notation will be recovered when necessary. The Hamiltonian is given by

The correlation function is calculated as follows:

where one has taken the spin at the origin 5(0) = 1. The trace was taken over all configurations. In the mean-field spirit, one replaces 5(r') on the right-hand side of (5.30) by its average value < S(r') >, then one makes an expansion around T_c when < 5(r') > is small, one obtains

using the notation m(r) =< 5(r) >. The Fourier transform of this equation gives

where C is a constant. For small k, one has

where the sum on the term J (r)r of the second equality is zero because ] (r)r is an odd function of r [J (r) is an even function: J (r) = J (—r)]. / is defined by

and R² is defined by

R² is thus the order of the interaction range. One obtains from (5.32) and (5.33)

One recalls here that in the mean-field theoiy k_BT_c = /. One writes thus

where one has taken T ~ T_c. Putting

and using m(r) = G(r) of (5.30), one finally arrives at

The inverse Fourier transform of (5.39) gives

This form of G(r) justifies the fact that £ is called the "correlation length" in the mean-field theoiy. The expression (5.39) is called "Ornstein-Zernike correlation function.”

From (5.38), one sees that £ a t^-1/2; therefore, v = 1/2 in this mean-field theory. In addition, at t = 0, £ tends to oc (see the following paragraph), G(k) of (5.39) is then proportional to 1/k². The inverse Fourier transform of this function is ^ for large r, indicating therefore that exponent i; of the mean-field theoiy is equal to 0 [see definition (5.14)].

Corrections to Mean-Field Theory

One decomposes the following term [see (12.16)]:

where the average values of the linear terms in <5S_r and 6S,' are zero by symmetiy (rotating vectors in the xy plane). In the last equality, one has neglected, in the mean-field spirit, the following term:

This term, however, when taken into account, will improve the mean-field theory as seen below. Using the correlation function

one writes (5.42) as

One is interested now in a long-distance behavior, namely at small k. Using the Fourier transform of (5.39) for small k, one writes

where the integral is performed in the first Brillouin zone. One decomposes this integral as follows:

The first integral does not depend on £, namely independent of T. It contributes to a shift of the value of T_c calculated by the mean-field theory. The second integral depends on £ thus on T: It converges if к = ±n/a -> oo, namely a -*■ 0 (continuum limit), and if d (space dimension) < 4. By a simple dimension analysis, one sees that this integral is proportional to $²~^d at the limit £ -*■ oo. The second term is thus proportional to This term has been neglected before in the mean-field theory because it was wrongly considered as always small with respect to the term coming from the mean field. Let us examine the condition when it can be neglected:

where t = ДД. One knows that£ = Rt~^1/2 [see (5.38)], expression

(5.46) can be thus rewritten as

This condition for neglecting the second term of (5.45) is called "Ginzburg's criterion." One sees that when d < 4 this criterion is not satisfied near T_c where f -> oo. By consequence, the mean-field theory which neglects fluctuations characterized by £^4-d is not valid in the critical region for d < 4. The dimension d = 4 is called "upper critical dimension” for the Ising model with short-range interaction.

Renormalization Group

Transformation of the Renormalization Group: Fixed Point

The central idea of the renormalization group is to replace the set of parameters which define the system by another set of parameters while conserving the essential physical ingredients, in particular the system symmetry. This set of parameter is simpler to deal with.

In the study of the phase transition, this transformation consists in dividing the system into blocks of spins and replacing each block by a single spin. This "new” spin interacts with the others by the renormalized interactions calculated while decimating the block. The distances between the new spins are measured with a new lattice constant. This procedure is repeated for the system of new spins to obtain the next generation of spins which is used to generate the following generation, and so on. At each iteration, also called decimation or transformation, one has a relation between the new interaction K' = (5]' and the previous one К = pj :

The new correlation length £ which is a function of K' is equal to the previous correlation length divided by the factor called "dilatation” b defined as the ratio between the new and old lattice constants. One has

At the phase transition, £ becomes infinite, the measuring distance unit is no more important. The interaction constants K' and К become identical К' = К = К*. This point is called the “fixed point” in the renormalization group language. The fixed point is thus determined by

The one-dimensional case is simple to proceed as seen in paragraph 5.4.2 below. However, in the case of a general dimension d > 1, relation (5.48) is rather complicated. It is often impossible to find a solution of (5.50). One then has to take into account physical considerations in order to find appropriate approximations.

One considers a point К in the proximity of the fixed point K*. If the iteration process takes К away from the fixed point, K* is an unstable fixed point. This is a “run away” case. On the other hand, in the case where К tends to K* by iteration, К* is a stable fixed point (see Problem 5.4.2 below). The map of these trajectories near a fixed point with indicated moving directions is called a "flow diagram." An example is shown in Fig. 5.5. This figure corresponds to the case of a square lattice of Ising spins with interactions K = ^ in the x direction and K₂ = in the у direction. P is the fixed point. For a given ratio K₂/Ki, namely one follows the discontinued line: Intersection Pi with the line separating regions of different flows is the critical point corresponding to that ratio K₂/K_x. The line of flow separation shown by the heavy solid line is the critical line. One sees that Pi runs toward P on the critical line; therefore, Pi and P belong to the same universality class: The universality class does not thus depend on the ratio K₂/K.

One can calculate the critical exponents using the renormalization group if one knows how K' depends on К in the vicinity of a fixed point K*. One makes then an expansion of their relation (5.48)

Figure 5.5 Flow diagram for a square lattice of Ising spins with nearest- neighbor interactions K_y along the x axis and K₂ along the у axis. See text for comments.

around К*:

where one has replaced f(K*) by К” using (5.50), and one has used the following notation:

Now, near K* one knows that t- v (definition of u). Therefore, by using (5.49) and (5.51) one obtains

By identifying the two sides of the above equation, one gets

This example shows that the critical exponent v is a derivative of the renormalization group equation (5.48). The other critical exponents, defined in (5.9), (5.10), (5.11), (5.13) and (5.14), are calculated by the use of the free energy. Details are given elsewhere, for example, in Refs. [10, 54, 380]. They are connected by the following scaling relations:

In addition, the two exponents concerning the spin-spin correlation, v and )/, are connected via the following "hyperscaling relations”:

In summary, there are six critical exponents and four relations between them. It suffices to determine two among six exponents. The other four can be then calculated using the above four relations.

To close this section, one emphasizes that, in addition to the results shown above, another important result of the renormalization group is the relations connecting the system size to the critical exponents: Physical quantities calculated at a finite system size are shown to depend on powers of the linear system size. These powers are simple functions of critical exponents. They are veiy useful for the determination of critical exponents by Monte Carlo simulation (see, for example, Refs. [33, 87,199]).

Renormalization Group Applied to an Ising-Spin Chain

One applies in the following the renormalization group to the case of a chain of Ising spins with a ferromagnetic interaction between nearest neighbors. One will show that there is no phase transition at finite temperature.

One considers the following Hamiltonian:

where К = J /кдТ, J being a ferromagnetic interaction between nearest neighbors. The partition function is given by

To study this spin chain, one uses the decimation method. One divides the system into three-spin blocks as shown in Fig. 5.6.

One writes for the two blocks in the figure the corresponding factors in Z

Using the following equality for the case o„ = ±1:

Figure 5.6 Blocks of three spins used for the decimation.

where x = tanh K, one rewrites (5.60) as

Writing explicitly each term of the product and making the sum on 3 = ±1 and 4 = ±1 (decimation of two spins at the block border), one sees that the odd terms of these variables give zero contributions. There remains

One can rewrite it as

where C is a constant. If one forgets C, the right-hand side is similar to the initial Hamiltonian with a new interaction K' between the remaining spins a₂ and 0-5. To calculate K' one writes

By identifying this with the left-hand side of (5.63), one obtains and

One renumbers the spins after the first decimation as follows: = 2i o'z = 05, • ■ • (every three old spins). The new Hamiltonian is thus

Figure 5.7 Flow diagram of a chain of Ising spins.

where N/3 is the number of three-spin blocks (IV: initial number of spins).

The equation of the renormalization group is thus

From the above equation, one sees that К' = К if К = 0 and

K = oo:

• (1) At high T, К —*■ 0⁺, tanh³ К < 1, hence К’ -> 0 after successive decimations. The fixed point at Г = oo (К = 0) is thus stable.
• (2) At low T, К -*■ oo, tanh³ К -*■ 1^_, hence K' —*■ 0 after successive decimations, namely a "run away" flow. The fixed point at Г = 0 (К = oo) is thus unstable.

The flow diagram is shown in Fig. 5.7. Any point between К = 0 and К = oo moves to К = 0 after successive decimations. The nature of any point between these limits is therefore the same as that of К = 0 (Г = oo), namely it belongs to the paramagnetic phase. The one-dimensional chain is thus disordered at any finite temperature.