Table of Contents:
Migdal–Kadanoff Decimation Method: Migdal–Kadanoff Bond-Moving Approximation
In the same spirit as the renormalization concept presented above, the Migdal-Kadanoff decimation method consists of decimating one spin out of every two on a chain to reduce the number of spins. The scaling parameter is thus b = 2. Let us describe how it works.
One writes the partition function for an Ising spin chain in an applied magnetic field h:
where the first sum runs over all spin configurations. Now, instead of dividing the chain into blocks as in the renormalization group described in the previous section, one decimates one spin out of every two as follows: One considers three consecutive spins /, к and j and one writes the corresponding part of the Hamiltonian with the sum on the middle spin ak
where in the last line one has assumed a small h. Since af = 1, one has the following expressions:
With these, one can write for any even and odd functions of K[(jj +
Using these expressions, one gets from (5.68) where
Note that all quantities independent of spins have been omitted since they do not affect expectation values of physical quantities. Equation (5.71) has the same form as if the spins
One sees that К varies very slowly starting with the first iteration where K = K0=J/T^>1[T ~ 0): Between two iterations, К diminishes by dK = — On the other hand, the scaling parameter after n iterations is b = 2" so that In b = n In 2. For one step, one has d(ln b) = In 2. Therefore,
This yields an exponential law as seen below, instead of the power law in the case where there is a fixed point at a finite K. Integrating the above equation, one has
Figure 5.8 Moving one horizontal line and one vertical line out of every two lines as indicated by arrows: The left lattice becomes the right lattice. Black circles denote the spins.
The correlation length is measured in unit of b: One has the following scaling relation:
where, under renormalization, Кф) tends to zero. Since £(0) ~ 1 (paramagnetic state), one has £(K) ~ b. Equation (5.75) becomes at the limit Кф) ~ 0
The Migdal-Kadanoff decimation shown above is exact for one dimension.
In two dimensions, one uses the so-called bond-moving approximation which consists of moving one horizontal bond line and one vertical bond line every two lines as shown in Fig. 5.8. In doing so, the number of lattice cells is reduced to a half and the spins left behind are free spins, they do not participate in the collective properties of the system. Each remaining bond has a new strength 2/. Note that the spins on the new bonds (not at the crossings) have each two neighbors: One can decimate these spins by the Migdal- Kadanoff decimation shown above with the result
The fact that К becomes 2 К changes a lot of things: The equation for the fixed points ф = 0) is
which admits a solution at a finite value of K. One sees this by examining the two limits:
From this one sees that the function К — |lncosh(4f6) should change its sign somewhere between 0 and oc. The phase transition occurs thus at a finite temperature. In the case of the square lattice considered here, one has /6* ~ 0.30469, or Tc/J ~ 3.282 which is below the mean-field value 4, but above the exact value 2.2692. Our conclusion is that the bond moving improves the mean-field theory but the moving procedure is not justified. In spite of this, the bond- moving approximation gives the values of critical exponents not very bad.
To compute the critical exponents, one expands the recursion relation (5.78) around its fixed point. Putting К = К* + t where t is the reduced temperature, one has t' = 1.67861 = b l,t where one has used b = 2 and
This value is better than the mean-field value v = 0.5 but still smaller than the exact value v = 1. Note that the bond moving approximation becomes exact for some hierarchical lattices which correspond to fractal spatial dimensions .
The transfer-matrix method is very useful when the system can be divided into subsystems, each of which interacts only with its adjacent nearest neighboring subsystems. For example, the simple cubic lattice can be considered as composed of planes each of which interacts only with its neighboring planes. In the case of periodic boundaiy conditions, the partition function can be written as a product of partition functions of its N subsystems:
where И) is the "transfer matrix” of dimension nxn representing the interaction connection between two adjacent blocks (subsystems). The trace of Z is the sum of the eigenvalues of Z. If the system is homogeneous, then all Щ are identical: Each eigenvalue of the product of identical matrices Щ is equal the product of the corresponding eigenvalue of 14) (properties of trace). Let Z, z2, • ••, z„ be the eigenvalues of Щ, one writes Z = zf + z% + • • • + z^. If N -> oo, then Z = Zj^ax where zmax is the largest eigenvalue among zb z2, ■ • • , z„.
One applies in the following the transfer-matrix method to the case of a chain of Ising spins using the periodic boundary condition. Let N be the total number of spins. The Hamiltonian is given by
where the last term expresses the periodic boundaiy condition. One has ai = ±1. One can define new variables an = anan+i. a„ takes the values ±1 as
The partition function is then
This result is the same as that obtained by the exact method shown in Problem 1. The average energy is calculated by E = -9 In Z/д() [see (A.10)]:
One obtains the following heat capacity:
In an applied magnetic field H, one proceeds as follows:
where Ho is given by (5.80). The partition function is Z = Пп=1 V» where
The matrix elements Vn, of dimension 2x2, depend on
The matrix Vn is called "transfer matrix." Note that all Vn (n = 1, ■ ■ ■ , N) have the same elements, say V. One thus has Z = Tr7w. Let zx and z2 be the eigenvalues of V obtained by diagonalizing V, using (5.88). One has
One obtains then
where Z denotes the larger eigenvalue. When N —>■ oc, one has Z = zf.
The susceptibility is calculated by x = [dM/dH)H^о where M = —dF/dH with F = -квТ In Z. One obtains
This result shows that there is no phase transition in one dimension (absence of anomaly of / with varying T as seen in Fig. 5.9), confirming the results shown in the two previous sections.
Figure 5.9 x versus T [Eq. (5.90)].
Phase Transition in Particular Systems
One has seen so far various methods used to study phase transitions and critical phenomena. In general, the nature of a phase transition depends on the symmetry of the order parameter, the spatial dimension and the nature of the interaction (short or long range). Standard methods presented above can be used to determine it with satisfactory precision. However, in some particular systems one needs special methods. Some of these remarkable systems are presented hereafter.
Exactly Solved Spin Systems
There are several families of systems in one or two dimensions with short-range non-crossing interactions which can be exactly solved. The spin models in those solvable systems are often Ising and Potts models. One needs exact solutions in such simple systems to test approximations conceived for more complicated systems or systems in three dimensions. Methods for searching exact solutions are lengthy to present here. The reader is referred to the book by R. J. Baxter  for the general methods and examples of exactly solved models. For some exactly solved frustrated spin systems, the reader is referred to the review by Diep and Giacomini . To summarize, to find an exact solution, the most frequently used method is to transform the system under study into a vertex model where the solutions for the critical surfaces are known. Among the most popular models, one can mention the 8-, 16- and 32-vertex models .
One considers the XY spins on a two-dimensional lattice with a ferromagnetic interaction between nearest neighbors. In the ground state, the spin configuration is a perfect ferromagnetic state, namely all spins are parallel. However, this system does not have a normal order-disorder transition at a finite temperature: There is no long- range ordering as soon as the temperature is not zero, following the Mermin-Wagner theorem  valid for two-dimensional systems with continuous spins (see discussion in Chapter 3). Kosterlitz and Thouless  have shown that this system has a special phase transition due to the unbinding of vortex-antivortex pairs at a finite temperature below (above) which the correlation function decays as a power law (exponential law) with increasing distance. This transition, called Kosterlitz-Thouless (KT) or Kosterlitz- Thouless-Berezinskii transition, is of infinite order. For the reader interested in this special transition, an appendix in Ref.  gives the main points explaining the mechanism lying behind the KT transition.
Frustrated Spin Systems
A system is said "frustrated" when the interaction bonds between a spin with its neighbors cannot be fully satisfied. An example is the triangular antiferromagnet: (i) in the case of Ising spin model, the three spins on a triangle cannot find orientations to satisfy the three antiferromagnetic bonds, ii) in the case of XY or Heisenberg spin models, the spins make a "compromise" to form a non-collinear configuration in order to partially satisfy each bond as shown in Fig. 18.6 in Problems 6 and 7 of Chapter 3. The ground-state spin configuration of a helimagnet has been given in Section 3.4.
Effects due to the frustration are numerous and spectacular. One can mention a few of them: (i) high ground-state degeneracy, (ii) non-collinear spin configuration, (hi) multiple phase transitions, (iv) reentrance phenomenon, (v) disorder lines, (vi) partial ordering at equilibrium, (vii) difficulty in determining the nature of phase transitions in several systems, etc. Some of these spectacular effects
(iii)-(vi) have been observed in exactly solved two-dimensional systems [16, 67,68, 76, 86]. It is believed that these effects persist in three-dimensional systems and in other more complicated unsolved models. For advanced reviews on frustrated systems, the reader is referred to Refs. [85, 87].
In this chapter, we introduced basic notions as well as some fundamental methods which are widely used in the field of phase transitions. The mean-field theoiy presented in Chapter 2 paves the way for other improving methods such as the Bethe's approximation and the Ginzburg's criterion shown in this chapter. The renormalization group has been shown with simple examples to illustrate its concepts. In particular, the notion of universality class and the relations between the critical exponents have been discussed. The Migdal-Kadanoff decimation method and bond- moving technique have been explained. An example of the transfer- matrix method has been treated, and some complementary methods such as canonical and micro-canonical methods are also introduced as problems which are given below. More advanced methods, such as quantum phase transitions of low-dimensional systems [202, 235, 301] and the Hubbard model  are not included here to keep the contents of the book suitable for lectures in a graduate course.
Problem 1. Chain of Ising spins by exact method:
Consider a chain of N Ising spins with a ferromagnetic interaction between nearest neighbors, maintained at temperature T, with the following Hamiltonian:
One supposes the periodic boundaiy condition ст/v+i = <*- Calculate exactly the partition function of the system. Find the free energy, the average energy and the heat capacity, as functions of T. Show that there is no phase transition at finite temperature.
Problem 2. Chain of Ising spins by micro-canonical method:
Consider a chain of N Ising spins interacting with each other via a nearest-neighbor coupling J > 0. The system is isolated with the Hamiltonian
One supposes that the periodic boundary condition aN+i = a applies and N is even.
Problem 3. Chain of Ising spins by canonical method:
Consider again the system in the preceding problem but put it now in the canonical situation at temperature T.
Problem 4. Low- and high-temperature expansions of the Ising model on the square lattice:
The low- and high-temperature expansions are useful not only for studying physical properties of a spin system in these temperature regions, but also for introducing a new concept called duality which allows to map a system of weak coupling into a system of strong coupling, as seen in the problem below.
Consider N Ising spins on a square lattice with the Hamiltonian
where the sum is performed over nearest neighbors and o-,Q) = ±1. The periodic boundary conditions are used.
where Nb is the total number of links and К = J/[kBT).
(d) High-temperature expansion: Using Eq. (18.118) or Eq. (5.61) show that the partition function is written as
Expand the product in the last equation and show that
(e) Duality: The partition function Z in (5.93) and (5.95) has the same structure: the prefactors are non-singular, the summations over the paths determine the singularity of Z. Show that the two Z have the same critical behavior if
where K* corresponds to the low-Г phase and К to the high-Г phase. The relation (5.96) is called the "duality” condition which connects the low- and high-Г phases.
(П Deduce the critical temperature of the Ising model on the square lattice.
Problem 5. Critical temperatures of the triangular lattice and the honeycomb lattice by duality:
Consider the triangular lattice with Ising spins with a ferromagnetic interaction between nearest neighbors. Construct its dual lattice. Calculate the partition functions of the two lattices. Deduce the critical temperature of each of them by following the method outlined in the previous problem.