2 1 Introduction
of these systems require extensive numerical work, some of the essential features
can often be captured by these analytic techniques. The confrontation between
calculations and real life experiments in such cases provides a good measure of the
worth of the theory. In this introductory chapter, our focus will be on providing a
glimpse of some of the more interesting comparisons of theory and experiment in
one of the systems we wish to study - the second order phase transitions.
The thing to stress, as the theme of this book, is the common platform of theo-
retical techniques suited for diverse problems - ‘universal’ techniques to borrow a
much used word. Universal was the key word that triggered a tremendous amount
of interest in critical phenomena in the nineteen sixties and seventies. Critical phe-
nomena exhibited universality and hence it was worth the effort spent to get to the
bottom of it. We will begin by explaining what universality meant in that context.
The two most common second order phase transitions or critical phenomena
are the paramagnet to ferromagnet transition at the Curie point and the gas liquid
transition at the critical point. These are second order transitions since they require
no latent heat.
If a material like iron is taken and kept at a temperature signiﬁcantly higher
than the room temperature (but lower than its melting point, needless to say) then
it will not show any spontaneous magnetization. Spontaneous magnetization is the
magnetization that one obtains in the absence of any external ﬁeld. Experimentally
it is determined by switching on a small external ﬁeld, ﬁnding the magnetization
at different values of the ﬁeld and taking the limit of the external ﬁeld going to
zero. If we cool the piece of iron, then at a certain temperature T
c
, the critical
temperature or the Curie temperature, the system undergoes a phase transition. For
temperatures above the critical temperature T
c
the spontaneous magnetization is
zero, while below T
c
the iron exhibits non-zero spontaneous magnetization. The
magnitude of the magnetization rises automatically from its value of zero at T
c
as
the temperature is lowered. A typical measurement yields the kind of behaviour
shown in Fig. 1.1.
It is the region very close to T
c
which is interesting. Here the magnetization
has the behaviour
M ∝(T
c
−T)
β
for
T
c
−T
T
c
1 (1.0.1)
The quantity β is called a critical exponent and for this magnetic transition, β
is independent of what material one is using for the experiments. It could be a
transition metal like iron, nickel or manganese, or it could be a rare earth compound
like EuO or EuS. For different materials the actual values of magnetization are
different but the exponent β remains the same. This is universality.
Large ﬂuctuations characterize the critical point. This can be seen from a study
of the linear response functions. In the case of the magnet the response function is
the zero ﬁeld susceptibility. One applies an inﬁnitesimal ﬁeld H and determines the
corresponding magnetization M. The isothermal susceptibility is deﬁned as χ
T
=
∂M
∂H
)
T
. It is the zero external ﬁeld that concerns us. Whether for T>T
c
or T<T
c
,