Multicritical points are special points in the parameter space of thermodynamic or
other systems with a continuous
phase transition. At least two thermodynamic or other
parameters must be adjusted to reach a multicritical point. At a multicritical point the
system belongs to a
universality class
In statistical mechanics, a universality class is a collection of mathematical models which share a single scale invariant limit under the process of renormalization group flow. While the models within a class may differ dramatically at finite s ...
different from the "normal" universality class.
A more detailed definition requires concepts from the theory of
critical phenomena
In physics, critical phenomena is the collective name associated with the
physics of critical points. Most of them stem from the divergence of the
correlation length, but also the dynamics slows down. Critical phenomena include scaling relation ...
,
a branch of
physics that reached a very satisfying state in the 1970s.
Definition
The union of all the points of the parameter space for which the system is critical is
called a critical
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
.
As an example consider a substance
ferromagnetic below a
transition temperature
, and paramagnetic above
. The parameter space here is
the temperature axis, and the critical manifold consists of the point
. Now add
hydrostatic pressure
to the parameter space. Under hydrostatic pressure the substance
normally still becomes ferromagnetic below a temperature
(
).
This leads to a
critical curve in the (
) plane - a
-dimensional critical manifold. Also taking into account
shear stress
as a thermodynamic parameter leads to a critical surface
(
) in the
(
) parameter space - a
-dimensional critical manifold.
Critical manifolds of dimension
and
may have physically reachable borders of dimension
which in turn may have borders of dimension
. The system still is critical at
these borders. However, criticality terminates for good reason, and the points on the
borders normally belong to another
universality class
In statistical mechanics, a universality class is a collection of mathematical models which share a single scale invariant limit under the process of renormalization group flow. While the models within a class may differ dramatically at finite s ...
than the
universality class
In statistical mechanics, a universality class is a collection of mathematical models which share a single scale invariant limit under the process of renormalization group flow. While the models within a class may differ dramatically at finite s ...
realized
within the critical manifold. All the points on the border of a critical manifold are
multicritical points.
Instead of terminating somewhere critical manifolds also may branch or intersect.
The points on the intersections or branch lines also are multicritical points.
At least two parameters must be adjusted to reach a multicritical point.
A
-dimensional critical manifold may have two
-dimensional borders intersecting at a point. Two parameters must be adjusted to reach such a border, three parameters must be adjusted to reach the intersection of the two borders. A system of this type represents up to four universality classes: one within the critical manifold, two on the borders and one on the intersection of the borders.
The gas-liquid critical point is not multicritical, because the phase transition at
the vapour pressure curve
(
) is discontinuous and the critical manifold thus consists of a single point.
Examples
Tricritical Point and Multicritical Points of Higher Order
To reach a
tricritical point In condensed matter physics, dealing with the macroscopic physical properties of matter, a tricritical point is a point in the phase diagram of a system at which three-phase coexistence terminates. This definition is clearly parallel to the definit ...
the parameters must be tuned in such a way that the renormalized counterpart of the
-term of the Hamiltonian vanishes. A well-known experimental realization is found in the mixture of
Helium-3
Helium-3 (3He see also helion) is a light, stable isotope of helium with two protons and one neutron (the most common isotope, helium-4, having two protons and two neutrons in contrast). Other than protium (ordinary hydrogen), helium-3 is the ...
and
Helium-4.
Lifshitz Point
To reach a Lifshitz point the parameters must be tuned in such a way that the renormalized counterpart of the
-term of the Hamiltonian vanishes. Consequently, at the Lifshitz point phases of uniform and modulated order meet the disordered phase. An experimental example is the
magnet
MnP. A Lifshitz point is realized in a prototypical way in the
ANNNI model
In statistical physics, the axial (or anisotropic) next-nearest neighbor Ising model, usually known as the ANNNI model, is a variant of the Ising model in which competing ferromagnetic and
antiferromagnetic exchange interactions couple spins at nea ...
. The Lifshitz point has been introduced by R.M. Hornreich,
S. Shtrikman and M. Luban in 1975, honoring the research of
Evgeny Lifshitz
Evgeny Mikhailovich Lifshitz (russian: Евге́ний Миха́йлович Ли́фшиц; February 21, 1915, Kharkiv, Russian Empire – October 29, 1985, Moscow, Russian SFSR) was a leading Soviet physicist and brother of the physicist ...
.
Lifshitz Tricritical Point
This multicritical point is simultaneously tricritical and Lifshitz. Three parameters must be adjusted to reach
a Lifshitz tricritical point. Such a point has been discussed to occur in non-
stoichiometric
Stoichiometry refers to the relationship between the quantities of reactants and products before, during, and following chemical reactions.
Stoichiometry is founded on the law of conservation of mass where the total mass of the reactants equal ...
ferroelectrics.
*
Renormalization group