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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]] different from the "normal" universality class. 


A more detailed definition requires concepts from the theory of [[critical phenomena]],
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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]].  
 
[[Image:multicritical_end.png|thumb|right|A critical curve terminating at a multicritical point (schematic).]]
 
As an example consider a substance [[ferromagnetic]] below a
transition temperature <math>T_{c}</math>, and paramagnetic above <math>T_c</math>. The parameter space here is
the temperature axis, and the critical manifold consists of the point <math>T_c</math>. Now add
hydrostatic pressure <math>P</math> to the parameter space. Under hydrostatic pressure the substance
normally still becomes ferromagnetic below a temperature <math>T_{c}</math>(<math>P</math>).  
 
This leads to a
critical curve in the (<math>T,P</math>) plane - a <math>1</math>-dimensional critical manifold. Also taking into account
shear stress <math>K</math> as a thermodynamic parameter leads to a critical surface <math>T_c</math>(<math>P,K</math>) in the
(<math>T,P,K</math>) parameter space - a <math>2</math>-dimensional critical manifold.
Critical manifolds of dimension <math>d > 1</math> and <math>d > 2</math> may have physically reachable borders of dimension
<math>d-1</math> which in turn may have borders of dimension <math>d-2</math>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 <math>2</math>-dimensional critical manifold may have two <math>1</math>-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 <math>P</math>(<math>T</math>) 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]] the parameters must be tuned in such a way that the renormalized counterpart of the <math>\phi^4</math>-term of the Hamiltonian vanishes. A well-known experimental realization is found in the mixture of [[Helium-3]] 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 <math>\left(\nabla\phi\right)^2</math>-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]].
 
=== 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-[[stochiometric]] ferroelectrics.
 
== Renormalization Group ==
The [[renormalization group]] provides a detailed and quantitative explanation of critical phenomena.
 
[[Category:Critical phenomena| ]]
[[Category:Renormalization group]]

Latest revision as of 17:59, 17 July 2014

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