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3
Critical Phenomena and
Phase Transitions
INTRODUCTION
One of the most active areas of physics in the last decade has been
the subject of critical phenomena. Enormous progress was made
during the decade, both theoretically and experimentally, and research
in the field was honored with the award of the 1982 Nobel prize in
physics. It seems safe to predict that the study of critical phenomena
and closely related subjects will remain a major activity of condensed-
matter physics throughout the 1980s and that much further progress
will occur.
WHAT ARE CRITICAL PHENOMENA, AND WHY ARE THEY
INTERESTING TO PHYSICISTS?
The term critical phenomena refers to the peculiar behavior of a
substance when it is at or near the point of a continuous-phase
transition, or the critical point. A continuous-phase transition, in turn,
may be defined as a point at which a substance changes from one state
to another without a discontinuity or jump in its density, its internal
energy, its magnetization, or similar properties. The critical point or
continuous-phase transition may be contrasted with the more familiar
case of a first-order phase transition, where the above-mentioned
properties do jump discontinuously as the temperature or pressure
75

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76 A DECA DE OF CONDENSED-MA TTER PH YSI CS
passes through the transition point. Continuous-phase transitions in
many cases, but not all, are associated with a change of symmetry of
the system.
Although the critical point was first discovered more than 100 years
ago, a good understanding of behavior near a critical point has only
emerged recently. The peculiarities of the critical point arise because
there are, in each case, certain degrees of freedom of the system that
show anomalously large fluctuations on a long-wavelength scale,
compared with those of a normal substance far from a critical point.
These large fluctuations cause a breakdown of the normal macroscopic
laws of condensed-matter systems, in some dramatic ways and in some
subtle ways, and it has been a major challenge to learn what are the
new special laws that describe the systems at their critical points. The
challenge has been difficult for theorists because the large fluctuations
could not be handled by the old calculational schemes, which depended
implicitly on long-wavelength thermal fluctuations being small. The
challenge has been difficult for experimentalists, because in order to
make measurements sufficiently close to a critical point, to test existing
theoretical calculations, or to discover directly the laws of critical
behavior where no theory exists, it is necessary to have extremely
precise control over the sample temperature, and frequently over the
pressure and purity as well.
The study of critical phenomena has been rewarding in spite of its
difficulties, and the understanding gained has proved useful to the
understanding of other types of systems-including quantum field
theories in elementary-particle physics, analyses of phenomena in long
polymer chains, and the description of percolation in macroscopically
inhomogeneous systems in which fluctuations play an important and
subtle role but where precise direct experiments may be even more
difficult than in the case of critical phenomena. The techniques of
renormalization-group analysis, developed in the theory of critical
phenomena, have had a profound impact on an entire branch of
mathematics, for example in the study of iterative maps, which has
applications to economics, biology, and other sciences, as well as to
the study of nonlinear fluid dynamics and other problems in condensed-
matter physics.
Experimental research on critical phenomena has also had an impact
both inside and outside condensed-matter physics. The requirements
of experiments on critical phenomena have often stimulated the
synthesis of samples with a new degree of perfection and of materials
with special properties such as magnetic systems with anisotropic

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CRITICAL PHENOMENA AND PHASE TRANSITIONS 77
spin interactions. The precision measurement techniques developed
for the study of critical phenomena have also found application, for
example, in the study of the onset of fluid convection.
In the following sections, we further define the features of a critical
point, and we give some examples of properties that show critical-point
anomalies. We outline the progress that has been made in the field, and
we give a few selected examples of important problems that are still
unsolved.
EXAMPLES OF PHASE TRANSITIONS AND CRITICAL POINTS
Several examples may illustrate the difference between a first-order
transition and a continuous transition or critical point.
One example of a critical point is the Curie point of a ferromagnetic
substance such as iron (TC = 770°C for iron). At temperatures below
Tc, a single-domain sample of iron has a net magnetization M that
points arbitrarily along one of several directions that are energetically
equivalent, in the absence of an external orienting magnetic field. The
strength of the magnetization M(~ decreases with increasing temper-
ature until the Curie temperature Tc is reached. Above Tc the magne-
tization is zero in the absence of an applied magnetic field, and we say
the material is in a paramagnetic state. In most magnetic systems
(including iron) the magnetization M(D decreases continuously to zero
as the temperature approaches TC from below; then we say that there
is a continuous phase transition, or critical point, at Tc In some cases,
however, the magnetization of a substance approaches a finite, non-
zero value, as T approaches TC from below, and the magnetization
jumps discontinuously to zero, as the temperature passes through Tc
In these cases there is a first-order transition at Tc.
In the magnetic example there is a symmetry difference between the
phases involved, since the ferromagnetic phase has a lower symmetry
than the paramagnetic phase.
The familiar boiling transition, from liquid to vapor, is a first-order
transition. Thus, when water boils at 1 atmosphere pressure and a
temperature of 100°C, there is a decrease in density by a factor of 1700.
However, if the pressure is increased, the boiling temperature in-
creases, and the difference in density between the liquid and vapor
becomes smaller. There exists a critical pressure PC where the density
difference between liquid and vapor becomes zero; at this pressure the
phase transition is no longer first order, and the transition temperature
TC at pressure Pc is described as the gas-liquid critical point of the

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78 A DECADE OF CONDENSED-MATTER PHYSICS
substance. For pressures greater than Pc there is no distinction at all
between liquid and vapor. We may note that there is no symmetry
difference between the liquid and vapor phases.
In order to facilitate the comparison between critical points in
various systems, it has proved convenient to introduce the concept of
an order parameter associated with each phase transition. For systems
like the ferromagnet, where there is a broken symmetry below Tc' the
order parameter is a quantity like the magnetization, which measures
the amount of broken symmetry in the system. For systems without
broken symmetry, one chooses some quantity that is sensitive to the
difference between the two phases below the critical temperature and
measures the difference of this quantity from its value at the critical
point. For the liquid-vapor critical point, we may choose the order
parameter as the difference between the actual density of the fluid and
the density precisely at the critical point.
HISTORY
The earliest theories of critical phenomena, developed near the end
of the last century and at the beginning of this century, gave a good
qualitative description of the behavior of a system near its critical
point. However, it gradually became clear in the mid-twentieth century
that these classical theories were incorrect in important details.
A most important step in this realization occurred in the 1940s, when
Onsager found a remarkable exact solution of a model of a magnetic
system in two dimensions (known as the two-dimensional, or 2-D, Ising
model) and showed that its phase transition did not follow all the
predictions of the classical theories. In the 1960s, experiments on
actual three-dimensional (3-D) systems began to show more and more
clearly that their critical behavior was also different from that predicted
by the classical theories and different from that of the 2-D Ising model
as well. At the same time, there appeared a certain regularity to the
behavior of different 3-D systems, which was encouraging to the search
for some general theory of these transitions. Other evidence for this
viewpoint, and hints at the shape that the new theory must take, were
provided by various types of numerical calculations (one might call
them computer experiments), which included both computer simula-
tions of thermal fluctuations in simple magnetic models and also
numerical extrapolations of the properties of these magnetic systems
from temperatures far above the critical temperature, where accurate
calculations could be done.
An important step forward in our understanding of critical phenom

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CRITICAL PHENOMENA AND PHASE TRANSITIONS 79
ena occurred in the mid-1960s, with the development of a set of
empirical scaling laws, which were successful in describing certain
relations between different critical properties of a system, although
they could not predict all these properties from the beginning. The
concept of universality classes developed, as it appeared that systems
could be divided into certain broad classes, such that all members of a
given class had identical critical properties but that these same
properties varied from one class to another. One important factor that
affects the critical behavior is the spatial dimensionality of the sys-
tem- e.g., 3-D systems have different critical behavior than 2-D
systems but there are other factors that are relevant, including the
symmetry differences between the states at the phase transition, the
presence or absence of certain long-range interactions, and other
factors that will be discussed below. A proper understanding of the
factors that determine the universality class of a system had to await
the developments of the 1970s, however, and in fact, a classification in
the more difficult cases remains one of the tasks for the 1980s.
The most important theoretical advance of the 1970s was the
development of a set of mathematical methods known as renormaliza-
tion group techniques. These methods are not limited to critical
phenomena they are useful whenever one has to deal with fluctua-
tions that occur simultaneously over a large range of length scales (or
energy scales or time scales, for example). The methods proceed by
stages, in which one successively discards the remaining shortest-
wavelength fluctuations until only a few macroscopic degrees of
freedom remain. The effects of the short-wavelength fluctuations are
taken into account (approximately) at each stage by a renormalization
of the interactions among the remaining long-wavelength modes.
The renormalization group techniques have made possible a number
of achievements.
1. They have given us a justification for the scaling laws of the 1960s.
2. The renormalization group methods enable one to predict with
high reliability which features of a system are relevant to determining
its universality class for critical behavior and which features of the
microscopic description become irrelevant in the vicinity of the critical
point.
3. The renormalization group methods enable us to calculate prop-
erties of any given universality class. In the simpler cases, these critical
properties have been calculated with a high degree of accuracy; and
these predictions of the renormalization group have been confirmed in
turn by some beautiful experiments of high precision. In more compli

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80 A DECADE OF CONDENSED-MATTER PHYSICS
cased cases, the numerical accuracy of existing renormalization group
calculations is not high, and further improvements in them are badly
needed.
WHAT DOES ONE MEASURE?
In order to make more precise our discussion of critical phenomena,
it is useful to give some examples of quantities measured and to give
some examples of the laws that describe them.
Perhaps the most fundamental measurement to make in the vicinity
of a critical point is to determine the way in which the magnitude of the
order parameter approaches zero, as the critical point is approached
from the low-temperature side. According to the classical theories of
phase transitions, such as the van der Waals or mean-field theories, the
order parameter should approach zero as the square root of the
temperature difference from Tc. We may write this as
M= Mo(Tc- T)h,
(1)
where M is the order parameter on the coexistence curve (i.e., for a
ferromagnet, M is the magnetization in zero magnetic field; near the
gas-liquid critical point M is proportional to the density discontinuity
between liquid and vapor). Me is a constant that will vary from one
system to another, and the exponent ~ is equal to 1/2 for all critical
points, in the classical theories. Now the result of the modern theory of
critical phenomena is that the classical theory is not correct close to Tce
We can still write the temperature dependence of the order parameter
in the form of Eq. (1), but the value of the exponent ~ is not equal to
1/2. For the 2-D Ising model of magnetism, and for other 2-D systems
in the same universality class, the result is ,8 = 1/8, as given by the
Onsager solution. For the gas-liquid critical point in three dimensions,
as well as for the 3-D version of the Ising model, the result of the most
accurate experiments and renormalization group calculations is ,B =
0.325, with an estimated uncertainty of +0.001. Other 3-D systems may
belong to different universality classes, but their values of ~ are
typically in the range 0.3-0.4.
The forms of the power law Eq. (1), for various values of the
exponent is, are illustrated by the curves in Figure 3.1. The curves for
~ = 1/8, 0.325, and 1/2 are all qualitatively similar, and indeed the
quantitative differences appear small on this linear scale. The differ-
ences may actually be quite large, however, if precision measurements
are made sufficiently close to Tc. For example, if the constants Me are
chosen so that the various curves have unit magnetization at a

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CRITICAL PHENOMENA AND PHASE TRANSITIONS 81
\\'
M
\~,3~5 \
~\
in,
1_$
p=2i
Tc
FIGURE 3.1 Power-law M °° ~TC - To, for various values of the exponent p. Data
points are experimental measurements of the order parameter of the antiferromagnet
MnF2, in the range 1.8 degrees below the critical temperature TC = 67.336 K.
temperature 10 K below Tc, then the curves for ~ = 1/8, 0.325, and 1/2
take on the respective values M = 0.316, 0.050, and 0.010, at a
temperature 0.001 K below Tc. Thus there is a difference of a factor of
5 between the values in the last two cases.
The temperature variation of the order parameter on the coexistence
curve is certainly not the only quantity that can be studied with
experiments in critical phenomena. Another important quantity is the
order-parameter susceptibility, defined as the derivative (i.e., the rate
of change) of the order parameter with respect to a small change in the
field to which it is coupled, while the temperature is held constant. For
a magnetic system this quantity is the magnetic susceptibility (deriva-
tive of the magnetization with respect to magnetic field); for the
gas-liquid critical point the order-parameter susceptibility is the iso-
thermal compressibility (derivative of the density with respect to
pressure, at constant temperature). These quantities become extremely
large near the critical point, and we may write, for example, the
zero-field magnetic susceptibility as
X = Xo/~T- TO Y. (2)
where the exponent ~ is the same for all members of a universality
class. The coefficient Xo varies from one system to another, and it is

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82 A DECADE OF CONDENSED-MATTER PHYSICS
different above and below Tc; however, the ratio of its value above Tc
to its value below Tc is a universal number i.e., it is the same for all
members of a universality class.
Another important quantity is the specific heat, defined as the
derivative of the internal energy of the system with respect to a small
change in temperature. The specific heat is found to become infinite at
the critical point in some systems; for some other universality classes
one finds that the specific heat is finite but has a sharp cusplike
maximum at the critical point. In either case, one may define an
exponent ax that characterizes the anomalous behavior of the specific
heat at the critical point. An example of the specific heat behavior is
shown in Figure 3.2.
Although the critical exponents a, if, and fly defined above may be
independent in principle, they were found empirically, in the 1960s, to
obey a scaling law:
= 2-~-2~.
(3)
This scaling law is one of the consequences of the more recent
renormalization group theories.
80
60
o
E 40
:~. rig.
it.
Bow t.
O. T < Tc
0 a.
·Cb
.
20 _
o
~ ma
°.
° o
° °°oOOk
1 1 1 1 1 1
IO-6 10-5 10-4 10-3 IO-2 10 1
| T-TC |/TC
FIGURE 3.2 Specific heat Cs of liquid 4He at saturated vapor pressure, near the
temperature T`. = 2.172 K of the onset of superfluidity, as a function of IT - TO on a
logarithmic scale. A straight line on this plot would correspond to a logarithmic
temperature dependence for Cs or a critical exponent cx = 0. Careful analysis of the data
gives the result cx = -0.026 + 0.004.

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CRITICAL PHENOMENA AND PHASE TRANSITIONS 83
A property of great interest near the critical point is the statistical
correlation for fluctuations in the order-parameter density, at two
nearby points in space, as a function of the distance between the
points. This correlation function can be measured by neutron-
scattering experiments in magnetic systems, and it can be measured by
light-scattering experiments or small-angle x-ray-scattering experi-
ments near the liquid-vapor transition. Near to the critical point, the
correlation length, which characterizes the range of correlations for the
order-parameter fluctuations, becomes extremely large relative to the
typical spacing between atoms in the substance. This large correlation
length is directly related to the large amount of long-wavelength
fluctuations that were mentioned earlier and that give to critical
phenomena their special subtleties and complexities. Naturally, there
is great interest in studies of the variation of the correlation length with
temperature, pressure, and other parameters, near the critical point.
It should also be mentioned that the integrated order-parameter
correlation function, which can be directly measured by a scattering
experiment in the limit of small angles, is related by a theorem of
statistical mechanics to the order-parameter susceptibility X defined
above; thus a scattering experiment may be a convenient method of
measuring X. Also, for systems like an antiferromagnet, in which the
order parameter describes a quantity that oscillates as a function of
position in space, a scattering experiment may be the only direct way
of measuring the value of the order parameter M(1) in the broken-
symmetry phase below TC.
Many other experimental techniques have also been used to study
properties of various systems near critical points. For example, the
temperature coefficient of expansion of a solid, which can be measured
with great precision, has similar behavior to the specific heat near a
critical point; the index of refraction of a fluid has been used as a
measure of its density; the rotation of polarized light by a transparent
ferromagnet (Faraday effect) has been used to study the temperature
dependence of the magnetization.
The quantities discussed above are all equilibrium or static quanti-
ties; they can be measured in a time-independent experiment, under
conditions of thermal equilibrium, and any correlation functions in-
volved refer to the correlations of fluctuations at a single instant of
time. The majority of theoretical studies and of experiments on critical
phenomena are concerned with these static measurements, and the
usual division of systems into different universality classes is based on
these static phenomena. There are other properties of systems, known
as dynamic properties, which require a more detailed theoretical

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84 A DECADE OF CONDENSED-MATTER PHYSICS
analysis and which require a further subdivision of the universality
classes- i.e., two systems that belong to the same universality class for
their static properties may show quite different behaviors in their
dynamic properties. Examples of dynamic properties are various
relaxation rates when the system is slightly disturbed from equilibrium,
correlations involving fluctuations at two different instants of time, and
transport coefficients, such as the thermal and electrical conductivities.
Among the experiments used to study dynamic properties are mea-
surements of sound-wave attenuation and dispersion, widths of nuclear
or electron magnetic resonance lines, and inelastic-scattering experi-
ments, in which the energy change of the scattered particle is deter-
mined along with the scattering angle.
Typically, one finds that the relaxation rate of the order parameter
becomes anomalously slow at a critical point. Some other relaxation
rates are found to speed up, however, and transport coefficients
become large in a number of cases. In some cases, the results of a
dynamic experiment may be interpreted as an indirect measurement of
a static property of the system. In fact, some of the most precise
measurements of static critical properties have been obtained by
dynamic means. Examples here are measurements of the superfluid
properties of liquid helium, the low-frequency sound velocity of a fluid,
and the frequency of nuclear magnetic resonance in a magnetic system.
WHAT DETERMINES THE UNIVERSALITY CLASS?
We now know what determines the static and dynamic universality
classes in most cases, although there remain a number of difficult cases
that are not resolved. Since there is no simple rule that is completely
general, we shall describe here only a few simple cases and give a few
examples of factors that do or do not change the universality class of
these systems.
Consider an idealized magnetic system in which magnetic atoms sit
on the sites of an elementary periodic lattice such as a simple cubic
lattice. Each magnetic atom has an elementary magnetic moment or
spin, of fixed magnitude, which can point c' priori in one of several
directions of space.
In the using model, which we have referred to several times above,
we suppose that the atomic moments can point in only two directions-
either parallel or opposite to some fixed axis, which we shall take to be
the z axis. Then the microscopic state of this model is determined by
specifying for each atom whether the magnetic moment along the z axis
is positive or negative. In the Ising model we assume an interaction

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CRITICAL PHENOMENA AND PHASE TRANSITIONS 85
between nearest neighbor atoms, tending to align the moments in the
same direction. At temperatures above the critical temperature, the
aligning force cannot overcome the disordering tendency of random
thermal motion, and there are equally many positive and negative
moments. At temperatures below Tc' however, one of the two possible
directions acquires a majority of the moments, and there is a net
magnetization +M(I) that measures the size and direction of this
majority. Thus, the Ising model has a plus-minus symmetry that is
broken below TC.
Another model of magnetism is the Heisenberg model. Here it is
assumed that the atomic moments can point in any direction of space.
The aligning force between neighboring spins is assumed to be derived
from an energy that depends only on the angle between the two spins
and not on their orientation with respect to any fixed axis in space. The
energy of this model is unchanged if we rotate all the spins in the
system by the same angle, about any direction, and we say that the
model is symmetric under arbitrary rotations of the spins. This
symmetry is broken below Tc' when there is a magnetization that
spontaneously picks out some direction in space.
A system intermediate between the Heisenberg model and the Ising
model is the XY model, where the directions of the magnetic moments
are restricted to lie in a single plane (say the XY plane). Again, the
energy is taken to be unchanged if all spins are rotated by the same
angle in this plane. The Ising model, the XY model, and the Heisenberg
model may be said to have order parameters that are, respectively, a
1-D vector, a 2-D vector, and a 3-D vector.
We have already seen that spatial dimensionality is crucial in
determining the universality class of a system the Ising model on a
2-D lattice has different critical exponents from the 3-D Ising model, for
example. The dimensionality (or symmetry) of the order parameter
also turns out to be important. The critical exponents of the Ising
model, XY model, and Heisenberg model in three dimensions differ
from each other by a small but significant amount. (For example, the
exponent ~ defined above takes on the values 0.325, 0.346, and 0.365
in the three cases.) In two dimensions the differences are more
dramatic. We believe that the 2-D Heisenberg model has no phase
transition at all it remains paramagnetic (disordered) at all tempera-
tures other than zero. The 2-D XY model is believed to have a phase
transition of a peculiar type (see below), for which the critical
exponents ax, is, and my cannot be properly defined.
We may next ask what happens if a model has a 3-D vector order
parameter similar to the Heisenberg model, but the interactions give a

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86 A DECADE OF CONDENSED-MATTER PHYSICS
lower energy to spins in the z direction than to spins in the other
direction of space. In this case the order parameter has the symmetry
of the Ising model, and the critical exponents are those of the Ising
model rather than those of the Heisenberg model. If the energy
favoring the z direction is small compared to the Heisenberg-type
interactions favoring parallel alignment of spins without regard to the
particular direction in space, then we expect to observe a crossover
behavior: close to the critical point (say IT - TO ' 10-5 Tc) we will see
the critical exponents of the Ising model, but farther from TC there may
be a range of temperatures (say 10-2 TC > IT - TO > 10-4 Tc) where the
system appears to have the critical exponents of the Heisenberg model.
An experimentalist seeking to measure accurately the critical exponents
of some universality class will naturally try to avoid using systems that
have a crossover in the middle of the accessible temperature range.
In more complicated systems, with multicomponent order parame-
ters, there is a variety of possible higher-order symmetry breaking
terms, which may favor some discrete subset of the possible orienta-
tions of the order parameter. In some cases these terms lead to a
change in critical behavior; in some others they lead to a small
fluctuation-induced first-order transition, even though the classical
theory predicts a continuous transition.
There are also many factors that are known to be irrelevant to
deciding the universality class. The precise nature of the spatial lattice
is unimportant for example, an Ising model on a hexagonal lattice in
two dimensions will have the same critical exponents as on a square or
rectangular lattice. The exponents are also unaffected if the interac-
tions are stronger along one spatial direction than another.
The universality class of a magnetic model is unchanged if the
interaction between spins extends beyond nearest neighbors on the
lattice, provided that the interaction falls off sufficiently rapidly with
separation. In real magnetic systems, however, there is an important
long-range interaction that does not fall off rapidly with distance the
magnetic dipole interaction, which decreases only as the inverse cube
of the distance between atoms. This is sufficiently long range to change
the universality class of a ferromagnet. Particularly in an Ising system,
the dipole interaction has a drastic effect on the critical behavior. For
a system like iron, where the magnetic dipole interaction is weak
compared with the quantum-mechanical exchange interactions respon-
sible for the ferromagnetism of the material, the dipole interaction only
becomes important close to the critical point. However, there also
exist cases where the dipole interaction is large and one readily sees an
effect on the critical behavior. Neutron scattering and specific-heat
measurements on one of these systems (LiTbF4) have provided dra

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CRITICAL PHENOMENA AND PHASE TRANSITIONS 87
matic confirmation of the peculiar critical behavior predicted theoret-
ically for the Ising model with dipolar forces.
We may remark here that the long-range magnetic dipole interactions
are irrelevant to the critical behavior of antiferromagnets, because of
the cancellations arising from the alternating directions of the spins in
this case.
We have already noted that the liquid-vapor critical point has the
same critical exponents as the 3-D Ising model. The liquid-vapor order
parameter, which we take as the difference from the density at the
critical point, is a real quantity, which can be positive or negative like
the magnetization of the Ising model, but the fluid does not possess a
precise symmetry between positive and negative values of the order
parameter. It is a prediction of renormalization group calculations that
this remaining asymmetry is irrelevant for the critical behavior, and
indeed experiments confirm with high precision the identity of the
critical exponents for the fluid and Ising critical points.
In two-component fluid mixtures, there is often a critical point for
phase separation, which is closely analogous to the liquid-vapor critical
point. This critical point also falls in the Ising universality class, and it
has been studied in many experiments.
The critical behavior of the XY model is particularly interesting
because this model is predicted to fall in the same universality class as
the superfluid transition of liquid helium (4He). In the latter case the
order parameter is a complex number representing the quantum-
mechanical condensate wave function of the superfluid, and the
relation to the XY model results from the mathematical representation
of a complex number as a vector in the XY plane. Because liquid 4He
can be obtained with great purity, and because temperatures near the
superfluid transition can be controlled with high precision, critical
exponents have been measured with high accuracy in this system. The
excellent agreement with calculations for the XY model provide both a
confirmation of the modern theory of critical phenomena and an
important confirmation of the theory of superfluidity as well.
EXPERIMENTAL REALIZATIONS OF LOW-DIMENSIONAL
SYSTEMS
Although the world we live in is three dimensional, theoretical
studies of 2-D systems have direct applications to systems in nature.
For example, a transition between commensurate phases of a layer of
atoms adsorbed on a crystalline substrate, or the melting of a commen-
surate adsorbate phase, will generally fall into the same universality
class as some simple 2-D model with a discrete order parameter, such

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88 A DECADE OF CONDENSED-MATTER PHYSICS
as the 2-D Using model or the three-state Potts model (an Ising-like
model, realized by some gases adsorbed on graphoil, in which each
spin can take on three, rather than two, values; the energy of an
interacting pair of spins is lower if they have the same value, and higher
if they are different). Recent experimental developments, including
improved substrates, and the availability of synchrotron x-ray sources
have made possible new precise measurements of phase transitions in
adsorbed gas systems.
In general, a film that is extended in two dimensions, but thin in the
third dimension, will show the same critical behavior as some 2-D
models. Slightly thicker films may show a crossover behavior from 3-D
behavior to 2-D behavior as one gets closer to the critical point.
Two-dimensional behavior can also be studied in 3-D layered systems,
when the interactions between layers are sufficiently weak. In this
case, one typically sees a crossover from 2-D to 3-D behavior as one
gets closer to the critical point.
Interesting phenomena also occur in quasi-l-D materials such as
crystals with chains of magnetic atoms and only weak interactions
between chains, even though a true 1-D system does not show a phase
transition at finite temperature. There has been a variety of experi-
ments in quasi- 1 -D and quasi-2-D systems that demonstrates the
expected crossover behaviors.
MULTICRITICAL POINTS
Although the gas-liquid critical point of a pure fluid occurs at a single
point in the pressure-temperature plane, this same critical point
becomes a critical line in the three-parameter space of pressure,
temperature, and composition in the case of a two-component mixture.
The transition temperature of an antiferromagnet may also become a
line of critical points when a uniform applied magnetic field is included
as a parameter.
There exists in nature a variety of special multicritical points, where
several lines of critical points come together. Multicritical points have
been studied experimentally in multicomponent fluid mixtures, in
magnetic systems, and at the tricritical point of superfluidity and phase
separation in a liquid ~He-4He mixture.
SYSTEMS WITH ALMOST-BROKEN SYMMETRY
Some of the most interesting phase transitions involve systems in
which the low-temperature phase has a special type of order, where
there is almost, but not quite, a broken symmetry. It had been noted,

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CRITICA L PHk'NOMk'NA AND PHASE 7'RANS'/T/ONS' 89
as early as 1930, that thermally excited long-wavelength fluctuations
should have the eject of destroying the long-range order and the
broken symmetry of certain types of 2-D systems. (These include the
2-D Heisenberg and XY magnets and the 2-D superfluid.) It was noted
similarly that thermally excited long-wavelength vibrational modes
must destroy the periodic translational order of a 2-D crystal. Al-
though, for many years, it was believed that the absence of broken
symmetry implied, in turn, the absence of a phase transition in all those
cases, this conclusion began to be questioned in the 1960s. In partic-
ular, it was proposed that the XY magnet, the superfluid, and the
crystal might have a distinct low-temperature state, in two dimensions,
where the order parameter has a kind of quasi-order, in which there are
correlations over arbitrarily large distances that fall off only as a small
fractional power of the separation between two points. This behavior
has recently been proven with complete mathematical rigor, in the case
of the XY model at low temperatures. Since this power-law behavior is
different from the exponential fallow of the correlation function (short-
range correlations) that one finds at high temperatures in the same
systems, there must be a definite temperature separating these two
behaviors, which is by definition a phase transition temperature. (in the
2-D Heisenberg model, however, it is believed that there are only
short-range correlations at all temperatures above zero, and, hence,
there is no phase transition.)
Two-Dimensional Superfluid and XY Model
In the 1970s, there was developed a theory in which the transition to
short-range order in the 2-D X Y model and superfluid occurs as a contin-
uous transition (i.e. not first order) that can be described by the prolif-
eration of point-like topological defects in the order parameter of the
system. This theory makes a number of specific predictions about both
static and dynamic properties near the phase transition, which differ
significantly from the behaviors at other critical points. These predic-
tions have been confirmed to some extent by experiments on thin films
of superconductors and of superfluid helium and by numerical simulat-
ions of the 2-D XY model, but the accuracy of these comparisons is not
yet sufficient to be considered incontrovertible support for the theory.
Melting of a Two-Dimensional Crystal
An application of the point-defect mechanism to the melting of a 2-D
crystal again makes a number of striking predictions, the most inter-
esting of which is that there should be a new hexatic liquid-crystal

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90 A DECADE OF CONDENSED-MATTER PHYSICS
phase, existing in a narrow temperature region between the crystal and
the isotropic liquid. The hexatic phase would possess quasi-order in the
orientation of bonds between neighboring atoms but would have only
short-range correlations in the positions of atoms, as one finds in the
true liquid state. Many workers in the field believe, however, that some
other melting mechanism (perhaps grain boundaries) will necessarily
intervene and produce a first-order melting before the melting temper-
ature for the point defect (dislocation) mechanism is reached. A
first-order transition could make it impossible to reach the hexatic
liquid crystal phase. (In 3-D bulk systems, melting is always a
first-order transition.)
This question continues to generate controversy, as computer sim-
ulations tend to favor a first-order transition, while scattering experi-
ments on incommensurate crystalline layers of argon, xenon, or
methane, adsorbed on a graphite substrate, provide strong evidence for
a continuous melting transition, for some range of coverages. The
xenon experiments also provide evidence for the existence of a hexatic
phase. Further work is necessary, however, particularly to clarify the
possible erects of the crystal substrate on the melting transition.
In layered phases of certain organic molecules (smectic liquid
crystals), there are phase transitions arising from a change in the order
within a layer, which may be considered as generalizations of the 2-D
melting transition. A phase describable as a stack of hexatic layers has
been observed by x-ray experiments in several smectics.
Smectic A-to-Nematic Transition
Although the most elementary examples of systems with almost-
broken symmetry (quasi-order) are the 2-D systems discussed above,
the phenomenon also occurs in certain 3-D liquid-crystal phases. The
simplest of these is the smectic A phase, in which long organic
molecules are arranged with their axes parallel to a particular direction
of space, and, in addition, the centers of gravities of the molecules tend
to be arranged in a series of equally spaced layers perpendicular to the
molecular axes, i.e., there is a periodic modulation of the density in one
direction. In the directions parallel to the planes there is only short-
range, liquidlike order.
Because it costs little energy to excite long-wavelength bends in the
molecular layers, thermal fluctuations in these modes are large, and the
resulting displacements reduce the periodic translational order to
quasi-long-range order (power-law correlations), as in a 2-D solid.
When heated, a smectic A may lose its remaining translational order,

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CRITICAL PHENOMENA AND PHASE TRANSITIONS 91
while retaining the orientational order of the molecular axes. We then
say the material has undergone a transition to the nematic phase. The
theory of this phase transition is complicated by the coupling between
the translational- and orientational-order parameters and also by the
large differences in the microscopic properties of the nematic phase,
when measured along different directions.
Experimentally, well-defined critical exponents have been seen,
holding over several decades in the distance from the transition
temperature, for such properties as the translational correlation length,
measured by x-ray scattering, in the nematic phase. However, the
critical exponents are different in the directions parallel and perpen-
dicular to the molecular axis and also vary from one material to
another.
QUENCHED DISORDER
The discussion, until this point, has focused on systems in thermal
equilibrium, where the important fluctuations arise from the intrinsic
thermal population of excitations, required by the laws of statistical
mechanics. In many solid-state systems of interest, however, there
may be additional, frozen-in disorder quenched into the system on
formation of the sample. Depending on the nature of the phase
transition, the nature of the quenched disorder, and the way in which
the quenched disorder couples to the order parameter of the phase
transition, the critical behavior may or may not be changed by the
disorder. In the most extreme cases, the phase transition may be
smeared out or eliminated entirely. The effects of quenched disorder
appear to be well understood in many cases, but there remain others
that are poorly understood and are the subject of active investigation.
One particularly interesting case occurs when the quenched disorder
couples linearly to an Ising-like order parameter, as would be the case
if there were a local magnetic field of random sign on each site of the
Ising ferromagnet. (This situation has been realized experimentally in
an Ising-like antiferromagnet, with a uniform magnetic field and
randomly missing magnetic atoms.) Different theoretical approaches
have led to opposite expectations for whether or not there should be a
sharp phase transition in this case, and experimental measurements
have not yet resolved the issue in a satisfactory manner. The question
may also have implications for elementary-particle physics, because
one of the theoretical approaches takes advantages of a close mathe-
matical analogy between the random magnetic-field problem and
so-called supersymmetric models in quantum field theory.

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92 A DECADE OF CONDENSED-MATTER PHYSICS
Extreme examples of systems with quenched disorder are the
spin-glass phases, discussed in Chapter 4. There are many open ques-
tions related to phase transitions in these systems.
PERCOLATION AND THE METAL-INSULATOR TRANSITION
IN DISORDERED SYSTEMS
There are a number of problems in condensed-matter physics that
bear a qualitative resemblance to systems at a continuous phase
transition and that may indeed be understood by methods of analysis
similar to those used in the theory of critical phenomena but where the
source of disorder is entirely quenched randomness and not thermal
fluctuations. Among these are various problems concerned with geom-
etry and transport in disordered systems, including metal-insulator
transitions in disordered systems where quantum mechanics plays a
critical role, as well as the classical problem of percolation in a mixture
of macroscopic conducting and insulating particles.
NONEQUILIBRIUM SYSTEMS
A number of problems resembling critical phenomena have been
observed in systems out of equilibrium. As one example, there have
been experimental and theoretical studies of the phase-separation
critical point of a binary fluid mixture, under strong shear flow.
The renormalization group concept has already had a profound
impact on the theoretical ideas and mathematical techniques (e.g.,
iterative maps) used to describe changes of the state of motion in
nonequilibrium fluids, at moderate Reynolds numbers (Chapter 111.
Turbulence in fluids at high Reynolds numbers has certain features of
universality and scaling that suggest that theoretical techniques used to
understand critical phenomena may also have a bearing here.
.. . . .
FIRST-ORDER TRANSITIONS
In general, one does not find at a first-order phase transition the rich
variety of phenomena that one finds at a critical point, and first-order
transitions have consequently received relatively less attention in
recent years. Several classes of universal phenomena associated with
first-order transitions do deserve mention, however, because they have
been the subject of continuing research, and because they still contain
outstanding puzzles. These include the areas of nucleation phenomena,
limits of superheating and supercooling, spinodal decomposition, and

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CRITICAL PHENOMENA AND PHASE TRANSITIONS 93
mathematical questions concerning the nature of the singularities in
thermodynamic functions at a first-order transition.
The past few years have seen significant advances in research on the
dynamics of phase transitions in fluid mixtures, but many basic
questions remain unanswered. It is now clear that the transition from
spinodal decomposition (from an unstable region) to nucleation and
growth (from a metastable regions is a gradual one; there is no abrupt
changeover at a spinodal curve. Some progress has been made in
providing a general theoretical description of the phase separation
process, and machine and laboratory experiments have provided some
clues about the underlying physics. A global scaling procedure, first
recognized in computer simulations, has been described by simple
models whose validity has been demonstrated experimentally. A
long-standing doubt about the validity of nucleation theory in the
neighborhood of critical points has apparently been laid to rest by a
theory that demonstrates that the anomalous behavior is the result of
critical slowing down of growth and by experiments that are consistent
with the theory.
Of course, there are many important open questions of a nonuniver-
sal nature concerning first-order transitions, as there are for continuous
transitions. These include such matters as understanding the micro-
scopic mechanisms for various transitions and calculations of the
location of the transition and of the sizes of the discontinuities of
various physical quantities. Such questions are discussed elsewhere in
this report, in the chapter appropriate to the particular transition.
Indeed, the reader will find that phase transitions are featured in
virtually every chapter of the report.
OUTLOOK
Work on critical phenomena and related problems, in the 1980s,
should lead to progress in a number of directions, among which we may
expect the following:
1. There will be more precise experimental tests of the predictions of
the renormalization group theories and of some of its consequences
(e.g., scaling laws among exponents and the universality of certain
relations among absolute values of properties near Tc). This is neces-
sary, because it is important to test thoroughly the underpinnings of a
theoretical approach that is seeing such widespread application in
modern physics.
2. There will be an extension of our understanding to some of the

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94 A DECADE OF CONDENSED-MA TTER PHYSICS
more complicated types of critical points. We may also expect a better
understanding of crossover phenomena and, more generally, of the
corrections to simple power-law behavior that are necessary to under-
stand measurements that are some distance away from the critical
point. Combinations of renormalization-group approximations and
accurate microscopic models will be used increasingly to calculate
entire phase diagrams, including the locations of first-order transitions,
in a variety of systems.
3. Some of the outstanding problems mentioned above may be
solved, perhaps by means of some important new calculational meth-
ods, or perhaps by the development of some new physical ideas, or by
refinements in experimental techniques. For example, it seems likely
that in the 1980s considerable progress will be made in our understand-
ing of 2-D melting and related phenomena, through experiments on a
variety of systems, including liquid crystals, both in bulk and in
suspended films of several layers thickness; adsorbed layers; the
electron crystal on the surface of liquid helium; and perhaps synthetic
systems, such as a film containing colloidal polystyrene spheres. The
role of the substrate in the transition, in the case of adsorbed layers,
will be investigated. The construction of a theory of the smectic
A-to-nematic transition is one of the most challenging unsolved prob-
lems in critical phenomena. Interest in this problem is heightened by its
possible connection to phase transitions in idealized superconductors
and in certain quantum-mechanical models of interest to elementary-
particle theories. A variety of other phase transitions among other
liquid-crystal phases is also poorly understood at present and will
undoubtedly be the subject of major investigations in the next few
years. Problems of disordered systems in which the disorder is
quenched into the system during its formation, and is not due to
thermal fluctuations, remain an important area where new ideas are
necessary, and the theoretical methods of critical phenomena need
further development.
4. The theoretical and experimental methods used to study the
problems of critical phenomena will be applied to the study of other
problems in condensed-matter physics.