40
2 Phase Transitions
2.2.2 Order Parameter
In some phase transitions, we can identify φ that is null (φ 0 = 0) in one but finite
(φ 0 = 0) in another phase involved in the phase transition. This quantity is called the
order parameter relevant to the transition. For example, the amplitude of a periodic
density modulation (density wave) may serve as an order parameter for the appearance of a layered structure like a smectic liquid crystal. Upon the formation of the
layer, the translational symmetry by arbitrary length is broken. In this sense, the
phase with null amplitude (φ 0 = 0) is more symmetric than the one with finite φ 0
( = 0).
In terms of the group-theoretical notions, an order parameter belongs to an irreducible representation of the symmetry of the more symmetric phase, and the expansion (Eq. 2.23) is totally symmetric. If the irreducible representation is higher than
one dimensional, the expansion of thermodynamic potential is forced to contain
multiple, yet equivalent, order parameters.
For investigations describing possible orders, it is an easy task to identify the order
parameter. The main difficulty lies in writing appropriate expansion of thermodynamic potential compatible with the symmetry of a system considered. In contrast,
the identification itself is an essential step for understanding a real phase transition based on experimental observations, some of which are crucially important and
others not so.
2.2.3 Simplest Case: Continuous Transition
We first consider the case where the relevant order parameter is a scalar and belongs
to a one-dimensional representation. The physical equivalence between two states
with ±φ is also assumed (though two states are distinct). The previous example of
the formation of a one-dimensional density wave is an example of this case. Other
examples include the uniaxial alignment of magnetic spins or dipoles. In this setting,
a continuous phase transition between two phases with φ 0 = 0 [symmetric (S) phase]
and φ 0 = 0 [symmetry-broken (B) phase] is deduced. We assume that the change in
temperature induces the transition, and the S phase is stable at T > T 0 while the B
phase is so at T < T 0 .
As far as we consider states with small φ, we can truncate the expansion at a few
first terms. Note that the order of the last term should be even, and its coefficient
must be positive to guarantee the thermodynamic stability of the symmetry-broken
phase. In the present case, the expansion is
ΔF(φ, τ ) = a 2 φ
2
+ a 4 φ
4
.
(2.26)
with a 4 > 0 around T 0 .
2 Phase Transitions
2.2.2 Order Parameter
In some phase transitions, we can identify φ that is null (φ 0 = 0) in one but finite
(φ 0 = 0) in another phase involved in the phase transition. This quantity is called the
order parameter relevant to the transition. For example, the amplitude of a periodic
density modulation (density wave) may serve as an order parameter for the appearance of a layered structure like a smectic liquid crystal. Upon the formation of the
layer, the translational symmetry by arbitrary length is broken. In this sense, the
phase with null amplitude (φ 0 = 0) is more symmetric than the one with finite φ 0
( = 0).
In terms of the group-theoretical notions, an order parameter belongs to an irreducible representation of the symmetry of the more symmetric phase, and the expansion (Eq. 2.23) is totally symmetric. If the irreducible representation is higher than
one dimensional, the expansion of thermodynamic potential is forced to contain
multiple, yet equivalent, order parameters.
For investigations describing possible orders, it is an easy task to identify the order
parameter. The main difficulty lies in writing appropriate expansion of thermodynamic potential compatible with the symmetry of a system considered. In contrast,
the identification itself is an essential step for understanding a real phase transition based on experimental observations, some of which are crucially important and
others not so.
2.2.3 Simplest Case: Continuous Transition
We first consider the case where the relevant order parameter is a scalar and belongs
to a one-dimensional representation. The physical equivalence between two states
with ±φ is also assumed (though two states are distinct). The previous example of
the formation of a one-dimensional density wave is an example of this case. Other
examples include the uniaxial alignment of magnetic spins or dipoles. In this setting,
a continuous phase transition between two phases with φ 0 = 0 [symmetric (S) phase]
and φ 0 = 0 [symmetry-broken (B) phase] is deduced. We assume that the change in
temperature induces the transition, and the S phase is stable at T > T 0 while the B
phase is so at T < T 0 .
As far as we consider states with small φ, we can truncate the expansion at a few
first terms. Note that the order of the last term should be even, and its coefficient
must be positive to guarantee the thermodynamic stability of the symmetry-broken
phase. In the present case, the expansion is
ΔF(φ, τ ) = a 2 φ
2
+ a 4 φ
4
.
(2.26)
with a 4 > 0 around T 0 .