Multicritical points are special points in the parameter space of thermodynamic or
other systems with a continuous phase transition

Phase transition

A phase transition is the transformation of a thermodynamic system from one phase or state of matter to another.A phase of a thermodynamic system and the states of matter have uniform physical properties....

. 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 different from the "normal" universality class.

A more detailed definition requires concepts from the theory of critical phenomena

Critical phenomena

In physics, critical phenomena is the collective name associated with thephysics of critical points. Most of them stem from the divergence of thecorrelation length, but also the dynamics slows down...

,
a branch of physics

Physics

Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

 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

Manifold

In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

.

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 than the universality class 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.

Tricritical Point and Multicritical Points of Higher Order

To reach a tricritical point

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 whichthree-phase coexistence terminates...

 the parameters must be tuned in such a way that the renormalized counterpart of the -term of the Hamiltonian vanishes.

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.

Lifshitz Tricritical Point

This multicritical point is simultaneously tricritical and Lifshitz. Three parameters must be adjusted to reach
a Lifshitz tricritical point.

Renormalization Group

The renormalization group

Renormalization group

In theoretical physics, the renormalization group refers to a mathematical apparatus that allows systematic investigation of the changes of a physical system as viewed at different distance scales...

provides a detailed and quantitative explanation of critical phenomena.