2.3 Critical Phenomena and Universality
49
Landau’s phenomenology is its representative. The fluctuation strongly depends on
the dimension of the system. In fact, for example, a phase transition is impossible
for any one-dimensional system because significant fluctuation destroys any order
except for the absolute zero even if the one-dimensional lattice is a priori given [2].
When the system under study is low-dimensional, e.g., highly-anisotropic magnetic
or electronic systems, or surface phenomena, possible effects of fluctuation arising
from the low-dimensionality should adequately be taken into account.
It is usual to express singularities in physical properties near the critical point
using the reduced temperature
t =
T − T c
T c
.
(2.61)
as
χ ∝ |t|
−γ
(2.62)
C ∝ |t|
−α
(2.63)
ξ ∝ |t|
−ν
(2.64)
where γ, α, and ν are critical indices (or critical exponents) for susceptibility, heat
capacity, and correlation length ξ, respectively. The correlation length is defined as
the length that characterizes the decay of the spatial correlation expressed by the
correlation function of the (local) order parameter. The correlation function as a
function of distance r is defined as
G(r ) = =σ i σ i+r − −σ i σ i+r ,
(2.65)
where σ j is a spin at site j (r j ) and ·· means the average over the ensemble. G(r )
decays as
G(r ) ∝ r
−τ e
−r/ξ
(2.66)
at t = 0. That is, the decay is expressed by a product of a power-law (index, τ ) and
an exponential law. This decay form defines the correlation length ξ.
For t < 0, there is another critical index. Namely,
φ ∝ |t|
β
(2.67)
which defines a critical index for the order parameter. The critical indices that are
meaningful also for t > 0 are distinguished by adding a prime (like α
for one of
heat capacity) though it has widely been believed that those for t > 0 and t < 0 are
identical.
At t = 0, other critical indices are defined through
φ ∝ |h|
1/δ
(2.68)
G(r ) ∝ r
−d+2−η
(2.69)
49
Landau’s phenomenology is its representative. The fluctuation strongly depends on
the dimension of the system. In fact, for example, a phase transition is impossible
for any one-dimensional system because significant fluctuation destroys any order
except for the absolute zero even if the one-dimensional lattice is a priori given [2].
When the system under study is low-dimensional, e.g., highly-anisotropic magnetic
or electronic systems, or surface phenomena, possible effects of fluctuation arising
from the low-dimensionality should adequately be taken into account.
It is usual to express singularities in physical properties near the critical point
using the reduced temperature
t =
T − T c
T c
.
(2.61)
as
χ ∝ |t|
−γ
(2.62)
C ∝ |t|
−α
(2.63)
ξ ∝ |t|
−ν
(2.64)
where γ, α, and ν are critical indices (or critical exponents) for susceptibility, heat
capacity, and correlation length ξ, respectively. The correlation length is defined as
the length that characterizes the decay of the spatial correlation expressed by the
correlation function of the (local) order parameter. The correlation function as a
function of distance r is defined as
G(r ) = =σ i σ i+r − −σ i σ i+r ,
(2.65)
where σ j is a spin at site j (r j ) and ·· means the average over the ensemble. G(r )
decays as
G(r ) ∝ r
−τ e
−r/ξ
(2.66)
at t = 0. That is, the decay is expressed by a product of a power-law (index, τ ) and
an exponential law. This decay form defines the correlation length ξ.
For t < 0, there is another critical index. Namely,
φ ∝ |t|
β
(2.67)
which defines a critical index for the order parameter. The critical indices that are
meaningful also for t > 0 are distinguished by adding a prime (like α
for one of
heat capacity) though it has widely been believed that those for t > 0 and t < 0 are
identical.
At t = 0, other critical indices are defined through
φ ∝ |h|
1/δ
(2.68)
G(r ) ∝ r
−d+2−η
(2.69)