Theory of Phase Transitions and Critical Phenomena

When a system changes its symmetiy under the effect of an external parameter such as temperature, pressure, applied magnetic or electric field, we say it undergoes a phase transition. Phase transitions in systems of interacting particles are subjects of intensive investigations in modern physics.

In the present chapter, we show the theory of phase transition using some systems of interacting spins. Note that various systems of different nature can be mapped into spin systems and solved using the spin language. There are many approximations and methods with various degrees of precision for the study of phase transitions. Among these approximations, the mean-field approximation is by far the simplest one as we have seen in Chapter 2. However, the mean-field theory gives some artifacts at low dimensions and cannot determine with precision the critical exponents which characterize the nature of a phase transition (see Section 5.1.5). These critical exponents are intimately related to the microscopic interaction between the particles and the symmetry of the system.

In the following, we first present Beth's approximation and the Landau-Ginzburg theory, which improve the mean-field theory.

Physics of Magnetic Thin Films: Theory and Simulation Hung T. Diep

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The concept of the renormalization group is presented next. The renormalization group was formulated by K. G. Wilson [361, 362] in the early 1970s. It provides an accurate insight into the mechanism of a phase transition and has been applied with success in many fields of physics ranging from condensed matter to quantum field theory. Among the most remarkable results, we can mention the notion of universality. Other well-known successes in condensed matter include the Kondo problem [363] and the Kosterlitz-Thouless transition [191]. The chapter ends with the presentation of some other methods dealing with the phase transition.

Introduction

We present here some fundamental notions necessary to understand a phase transition. There exist a huge number of reviews and books specialized in this field [10, 54,95, 380].

All systems do not have obligatorily a phase transition. In general, the existence of a phase transition depends on a few general parameters such as the space dimension, the nature of the interaction between particles and the system symmetry. For spin systems, one can give a brief summary here. In one dimension, in general there is no phase transition at a non-zero temperature for systems of short-range interactions regardless of the spin model. The long-range ordering at Г = 0 is destroyed as soon as Г ф 0. However, in two dimensions discrete spin models such as the Ising and Potts models have a phase transition at a finite temperature Tc, while continuous spin models such as the Heisenberg model do not have a transition at a finite temperature [231]. The XY spin model in two dimensions is a very particular case: In spite of the absence of a long-range order at finite temperatures, there is a phase transition of a special kind called the "Kosterlitz-Thouless" transition [191]. In three dimensions, all known spin models have in general a phase transition at Tc Ф 0. Note that this summary is for non-frustrated systems. Frustrated spin systems have to be separately considered, they often do not follow these observations (see reviews in Ref. [85]).

Symmetry Breaking: Order Parameter

A transition from one phase to another may take place when an external parameter varies. Such a parameter can be the temperature or an external applied magnetic field. At the transition point, the system changes from one symmetry to another. The most studied type of transition is no doubt the order-disorder transition. A well- known transition of this kind is the loss of magnetic attraction of a permanent magnet with increasing temperature: This is a transition from a magnetically ordered phase to a magnetically disordered (or paramagnetic) phase.

In order to measure the degree of ordering, one defines an order parameter which depends on the system symmetry. A good order parameter should be non-zero in one phase and zero in the other phase, signaling thus the symmetry breaking when the system changes its phase. Let us present in the following the order parameters defined for some systems.

For a ferromagnetic system of N Ising spins, the order parameter is defined as

where P = 1, and in the disordered state (or paramagnetic state) where there is a random mixing of up and down spins with the same number, one has P = 0.

For an antiferromagnetic system, the order parameter is the so- called staggered magnetization. For example, in one dimension with the Ising model, the staggered magnetization is given by

where (—1)' is the parity of the site /. One can verify that in the antiferromagnetic ground state one has P = 1, and in the disordered phase P = 0.

For the ferromagnetic Potts model with q states, namely ; = 1,,q, the order parameter is defined as

where

with

where the sum on j is performed over all sites of the system. It is observed that in the ground state where there is only one kind of a j one has P = 1. In the disordered phase where all values of spin are equally present, namely Mi = N/q for any / = 1,,q, one has P = 0.

For the Heisenberg spins of amplitude 1 in a ferromagnet, the order parameter is the magnetization defined by

One can verify that in the ground state, where all spins are parallel, M is 1 whatever the orientation of spins with respect to the crystal axes. In the disordered state, each spin has a random spatial orientation so that J2i S/ = 0, namely M = 0.

Order of a Phase Transition

A phase transition takes place when physical quantities of the system undergo an anomaly. In order to define properly a phase transition, one should examine various physical quantities at the transition temperature Tc. If physical quantities which are second derivatives of the free energy F, such as the specific heat Cv and the susceptibility y, diverge (see Appendix A), then the corresponding phase transition is a "phase transition of second order.” In this case, the first derivatives of F such as the average energy E and the average magnetization M are continuous functions at Tc. On the other hand, in a first-order phase transition, these first-derivative quantities undergo a discontinuity at the transition point. The reader is referred to Section 6.3 for schematic illustrations of these two kinds of phase transition.

Correlation Function: Correlation Length

An important function in the study of phase transitions is the correlation function defined by

where S(0) is the spin at a site chosen as the origin of the coordinates, S(r) the spin at the position r, and < • • ■ > denotes the thermal average. In an isotropic system, G(r) depends on r.

In a phase where S(0) and S(r) are independent, namely their fluctuations are not correlated, G(r) is zero due to the thermal average. This is the case of a point in the paramagnetic phase well above the transition temperature Tc and at a large distance r.

When the transition temperature Tc is approached from the high-temperature side, a correlation resulting from the interaction between spins sets in, G(r) becomes non-zero for spins at short distances. One can define the "correlation length” £ as the distance beyond which G(r) is no more significant. The fluctuations of two spins at a distance r < £ are said "correlated." The correlation length £ is written as

where d is the space dimension and A a constant. In a second-order transition, the correlation length diverges at the transition, namely all spins are correlated at the transition regardless of their distance. On the contrary, at a first-order transition, the correlation length is finite and there is a coexistence of the two phases at the transition point.

Critical Exponents

When the transition is of second order, one can define in the vicinity of Tc the following critical exponents:

Note that the same a is defined for T > Tc and T < Tc but the coefficient A is different for each side of Tc. This is also the case for y. However, ft is defined only for T < Tc because M = 0 for T > Tc. The definition of <5 is valid only at T = Tc when the system is under an applied magnetic field of amplitude H. Finally, at T = Tc, one defines exponent ij of the correlation function by

We note that there is another exponent called "dynamic exponent” z defined via the relaxation time r of the spin system for T > Tc:

Only the six exponents a, p, y, 8, v and /7 are critical exponents. It is observed below that there are four relations between them (see Section 5.4). Therefore, only two of them are to be determined.

It is obvious that the above exponents are not defined in a first-order transition because there is no divergence of physical quantities at Tc. For this reason, one says that first-order transitions are not critical. To be precise, we shall call "critical temperature” for second-order transitions and "transition temperature” for first- order transitions.

Universality Class

Phase transitions of second order are distinguished by their "universality class.” Phase transitions having the same values of critical exponents belong to the same university class. Renormalization group analysis shows that the universality class depends only on a few very general parameters such as the space dimension, the symmetiy of the order parameter and the nature of the interaction. So, for example, Ising spin systems with short-range ferromagnetic interaction in two dimensions belong to the same universality class whatever the lattice structure is. Of course, the critical temperature Tc is not the same for square, hexagonal, rectangular, honeycomb, ..., lattices, but Tc is not a universal quantity. It depends on the interaction value, the coordination number, ... but these quantities do not affect the values of the critical exponents.

Table 5.1 shows the critical exponents of some known universality classes.

Table 5.1 Critical exponents of some known universality classes

Class

Symmetry

2d Ising

0

1/8

7/4

1

1/4

2d Potts (q = 3)

1/3

1/9

13/9

5/6

4/15

2d Potts (q = 4)

2/3

1/12

7/6

2/3

1/4

3d Ising

0.11

0.325

1.241

0.63

0.031

3d XY

0(2)

-0.007

0.345

1.316

0.669

0.033

3d Heisenberg

0(3)

0.115

0.3645

1.386

0.705

0.033

Mean-field

0

1/2

1

1/2

0

Note that when a system is invariant by the following local transformation (y ->• -/, S, ->■ —S,), where / is the interaction of S, with its neighbors, the universality class of the new system does not change. This is understood immediately if one looks at the partition function: Such a local transformation does not change the argument of the exponential of the partition function. By consequence, physical properties do not change. As an example, let us consider a ferromagnetic square lattice. If one operates the local transformation on one spin out of eveiy two in the square lattice (see Fig. 5.1), the system changes from a ferromagnetic crystal into an antiferromagnetic one. However, as said above, this local transformation does not change the physical properties of the system: One concludes that a ferromagnetic ciystal and its

Local transformation J -*■ —), S, -> —S, operated on one spin out of every two changes the square ferromagnetic lattice

Figure 5.1 Local transformation J -*■ —), S, -> —S, operated on one spin out of every two changes the square ferromagnetic lattice (left) into an antiferromagnetic lattice (right). White and black circles indicate f spins and i spins, respectively.

antiferromagnetic counterpart have the same critical temperature and the same critical exponents.

We emphasize that there exist many systems in which it is impossible to operate local transformation without changing the argument of the partition function. One of these systems is the triangular lattice: It is impossible to find a spin configuration to satisfy all interactions if one changes J into — J everywhere. The energy of the system is not conserved, so the partition function changes, giving rise to a new system with different properties. Such systems are called "frustrated systems" which are discussed in Section 5.7.3 and studied in Ref. [85].

In Chapter 2, we have presented the mean-field theory by using the Heisenberg model for ferromagnets, antiferromagnets and ferrimagnets. The mean-field theory neglects instantaneous fluctuations of spins. Due to this approximation, it overestimates the critical temperature Tc. Fluctuations favor disorder, so when taken into account, fluctuations cause a transition at a temperature lower than Tc given by the mean-field theory [see (2.24)], or even destroy the magnetic long-range order at any finite temperature in low dimensions d = 1 and d = 2. The mean-field theory can be improved by several methods. In the following, we present the method proposed by Bethe. The treatment of fluctuations with the Landau-Ginzburg theoiy is presented in Section 5.3 where the mean-field theory is shown to be exact for dimension d > 4.

 
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