Critical exponent
Encyclopedia
Critical exponents describe the behaviour of physical quantities near continuous phase transitions. It is believed, though not proven, that they are universal, i.e. they do not depend on the details of the physical system, but only on
These properties of critical exponents are supported by experimental data. The experimental results can be theoretically achieved in mean field theory
for higher-dimensional systems (4 or more dimensions). The theoretical treatment of lower-dimensional systems (1 or 2 dimensions) is more difficult and requires the renormalization group
.
Phase transitions and critical exponents appear also in percolation systems.
. We want to describe the behaviour of a physical quantity 
in terms of a power law
around the critical temperature. So we introduce the reduced temperature
, which is zero at the phase transition
, and define the critical exponent
.
This results in the power law we were looking for.
It is important to remember that this only represents the asymptotic behavior of the function
as 
.
More generally one might expect
the system has two different phases characterized by an order parameter 
, which vanishes at and above 
.
Let us consider the disordered phase (
> 0), ordered phase (
< 0 ) and critical temperature (
= 0) phases separately. Following the standard convention, the critical exponents related to the ordered phase are primed. It's also another standard convention to use the super/subscript +(-) for the disordered(ordered) state. We have spontaneous symmetry breaking
in the ordered phase. So, we will arbitrarily take any solution in the phase.
The following entries are evaluated at
(except for the 
entry)
The critical exponents can be derived from the specific free energy
as a function of the source and temperature. The correlation length can be derived from the functional
.
These relations are accurate close to the critical point in two- and three-dimensional systems. In four dimensions, however, the power laws are modified by logarithmic factors.
This problem does not appear in 3.99 dimensions
, though.
(aka mean field theory
) values for a scalar field are
If we add derivative terms turning it into a mean field Ginzburg–Landau theory, we get
One of the major discoveries in the study of critical points is that mean field theory is completely wrong in the vicinity of critical points in two and three dimensions. In four dimensions, we have logarithmic corrections.
is −0.0127 for the phase transition of superfluid helium (the so-called lambda transition
). The value was measured in a satellite to minimize pressure differences in the sample (see here). This result agrees with theoretical prediction obtained by variational perturbation theory (see here or here).
The origin of scaling functions can be seen from the renormalization group. The critical point is an infrared fixed point
. In a sufficiently small neighborhood of the critical point, we may linearize the action of the renormalization group. This basically means that rescaling the system by a factor of a will be equivalent to rescaling operators and source fields by a factor of
for some Δ. So, we may reparameterize all quantities in terms of rescaled scale independent quantities.
Thus, the exponents above and below the critical temperature, respectively, have identical values. This is understandable, since the respective scaling functions,
, originally defined for 
, should become identical if extrapolated to 
Critical exponents are denoted by Greek letters. They fall into universality classes and obey scaling relations such as
and a lot of similar relations, which implies that there are only two independent exponents, e.g.,
and All this follows from the theory of the renormalization group
.
s, at the border or on intersections of critical manifolds.
, of a system diverges as 
, with a dynamical exponent 
. Moreover, the large static universality classes of equivalent models with identical static critical exponents decompose into smaller dynamical universality classes, if one demands that also the dynamical exponents are identical.
The critical exponents can be computed from conformal field theory
.
See also anomalous scaling dimension
.
and heat conductivity.
s.
- the dimension of the system,
- the range of the interaction,
- the spinSpin (physics)In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
dimension.
These properties of critical exponents are supported by experimental data. The experimental results can be theoretically achieved in mean field theory
Mean field theory
Mean field theory is a method to analyse physical systems with multiple bodies. A many-body system with interactions is generally very difficult to solve exactly, except for extremely simple cases . The n-body system is replaced by a 1-body problem with a chosen good external field...
for higher-dimensional systems (4 or more dimensions). The theoretical treatment of lower-dimensional systems (1 or 2 dimensions) is more difficult and requires 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...
.
Phase transitions and critical exponents appear also in percolation systems.
Definition
Phase transitions occur at a certain temperature, called the critical temperature. We want to describe the behaviour of a physical quantity in terms of a power lawPower law
A power law is a special kind of mathematical relationship between two quantities. When the frequency of an event varies as a power of some attribute of that event , the frequency is said to follow a power law. For instance, the number of cities having a certain population size is found to vary...
around the critical temperature. So we introduce the reduced temperature
, which is zero at the phase transitionPhase 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....
, and define the critical exponent
.This results in the power law we were looking for.
It is important to remember that this only represents the asymptotic behavior of the function
as .More generally one might expect
The most important critical exponents
Above and below the system has two different phases characterized by an order parameter , which vanishes at and above .Let us consider the disordered phase (
> 0), ordered phase ( < 0 ) and critical temperature ( = 0) phases separately. Following the standard convention, the critical exponents related to the ordered phase are primed. It's also another standard convention to use the super/subscript +(-) for the disordered(ordered) state. We have spontaneous symmetry breakingSpontaneous symmetry breaking
Spontaneous symmetry breaking is the process by which a system described in a theoretically symmetrical way ends up in an apparently asymmetric state....
in the ordered phase. So, we will arbitrarily take any solution in the phase.
| Keys | |
|---|---|
 | order parameter (  for the liquid-gas critical point, magnetizationMagnetization In classical electromagnetism, magnetization or magnetic polarization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material... for the Curie point Curie point In physics and materials science, the Curie temperature , or Curie point, is the temperature at which a ferromagnetic or a ferrimagnetic material becomes paramagnetic on heating; the effect is reversible. A magnet will lose its magnetism if heated above the Curie temperature... ,etc.) |
 |  |
 | specific free energy Thermodynamic free energy The thermodynamic free energy is the amount of work that a thermodynamic system can perform. The concept is useful in the thermodynamics of chemical or thermal processes in engineering and science. The free energy is the internal energy of a system less the amount of energy that cannot be used to... |
 | specific heat;  |
 | source field (e.g.  where P is the pressurePressure Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :... and Pc the critical pressure for the liquid-gas critical point, reduced chemical potential Chemical potential Chemical potential, symbolized by μ, is a measure first described by the American engineer, chemist and mathematical physicist Josiah Willard Gibbs. It is the potential that a substance has to produce in order to alter a system... , the magnetic field Magnetic field A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;... H for the Curie point Curie point In physics and materials science, the Curie temperature , or Curie point, is the temperature at which a ferromagnetic or a ferrimagnetic material becomes paramagnetic on heating; the effect is reversible. A magnet will lose its magnetism if heated above the Curie temperature... ) |
 | the susceptibility Susceptibility *In physics, the susceptibility of a material or substance describes its response to an applied field. There are many kinds of susceptibilities, for example:These two susceptibilities are particular examples of a linear response function;... /compressibility/etc.;  |
 | correlation length |
 | the number of spatial dimension Dimension In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it... s |
 | the correlation function Correlation function (statistical mechanics) In statistical mechanics, the correlation function is a measure of the order in a system, as characterized by a mathematical correlation function, and describes how microscopic variables at different positions are correlated.... |
 | spatial distance |
The following entries are evaluated at
(except for the entry)Critical exponents for  > 0 (disordered phase) | |
|---|---|
| Greek letter | relation |
 |  |
 |  |
 |  |
Critical exponents for  < 0 (ordered phase) | |
|---|---|
| Greek letter | relation |
 |  |
 |  |
 |  |
 |  |
Critical exponents for  = 0 | |
|---|---|
 |  |
 |  |
The critical exponents can be derived from the specific free energy
as a function of the source and temperature. The correlation length can be derived from the functionalFunctional
Generally, functional refers to something able to fulfill its purpose or function.*Functionalism and Functional form, movements in architectural design*Functional group, certain atomic combinations that occur in various molecules, e.g...
.These relations are accurate close to the critical point in two- and three-dimensional systems. In four dimensions, however, the power laws are modified by logarithmic factors.
This problem does not appear in 3.99 dimensions
Dimensional regularization
In theoretical physics, dimensional regularization is a method introduced by Giambiagi and Bollini for regularizing integrals in the evaluation of Feynman diagrams; in other words, assigning values to them that are meromorphic functions of an auxiliary complex parameter d, called the...
, though.
Mean field theory
The classical Landau theoryLandau theory
Landau theory in physics was introduced by Lev Landau in an attempt to formulate a general theory of second-order phase transitions. He was motivated to suggest that the free energy of any system should obey two conditions: that the free energy is analytic, and that it obeys the symmetry of the...
(aka mean field theory
Mean field theory
Mean field theory is a method to analyse physical systems with multiple bodies. A many-body system with interactions is generally very difficult to solve exactly, except for extremely simple cases . The n-body system is replaced by a 1-body problem with a chosen good external field...
) values for a scalar field are
If we add derivative terms turning it into a mean field Ginzburg–Landau theory, we get
One of the major discoveries in the study of critical points is that mean field theory is completely wrong in the vicinity of critical points in two and three dimensions. In four dimensions, we have logarithmic corrections.
Experimental values
The most accurately measured value of is −0.0127 for the phase transition of superfluid helium (the so-called lambda transitionLambda transition
The λ universality class is probably the most important group in condensed matter physics. It regroups several systems possessing strong analogies, namely, superfluids, superconductors and smectics...
). The value was measured in a satellite to minimize pressure differences in the sample (see here). This result agrees with theoretical prediction obtained by variational perturbation theory (see here or here).
Scaling functions
In light of the critical scalings, we can reexpress all thermodynamic quantities in terms of dimensionless quantities. Close enough to the critical point, everything can be reexpressed in terms of certain ratios of the powers of the reduced quantities. These are the scaling functions.The origin of scaling functions can be seen from the renormalization group. The critical point is an infrared fixed point
Infrared fixed point
In physics, an infrared fixed point is a set of coupling constants, or other parameters that evolve from initial values at very high energies , to fixed stable values, usually predictable, at low energies...
. In a sufficiently small neighborhood of the critical point, we may linearize the action of the renormalization group. This basically means that rescaling the system by a factor of a will be equivalent to rescaling operators and source fields by a factor of
for some Δ. So, we may reparameterize all quantities in terms of rescaled scale independent quantities.Scaling relations
Thus, the exponents above and below the critical temperature, respectively, have identical values. This is understandable, since the respective scaling functions,
, originally defined for , should become identical if extrapolated to Critical exponents are denoted by Greek letters. They fall into universality classes and obey scaling relations such as
and a lot of similar relations, which implies that there are only two independent exponents, e.g.,
and All this follows from the theory of the renormalization groupRenormalization 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...
.
Anisotropy
There are some anisotropic systems where the correlation length is direction dependent.Multicritical points
More complex behaviour may occur at multicritical pointMulticritical point
Multicritical points are special points in the parameter space of thermodynamic orother systems with a continuous phase transition. At least two thermodynamic or otherparameters must be adjusted to reach a multicritical point...
s, at the border or on intersections of critical manifolds.
Static versus dynamic properties
The above examples exclusively refer to the static properties of a critical system. However dynamic properties of the system may become critical, too. Especially, the characteristic time,, of a system diverges as , with a dynamical exponent . Moreover, the large static universality classes of equivalent models with identical static critical exponents decompose into smaller dynamical universality classes, if one demands that also the dynamical exponents are identical.The critical exponents can be computed from conformal field theory
Conformal field theory
A conformal field theory is a quantum field theory that is invariant under conformal transformations...
.
See also anomalous scaling dimension
Anomalous scaling dimension
In theoretical physics, by anomaly one usually means that the symmetry remains broken when the symmetry-breaking factor goes to zero.When the symmetry which is broken is scale invariance, then true power laws usually cannot be found from dimensional reasoning like in turbulence or quantum field...
.
Transport properties
Critical exponents also exist for transport quantities like viscosityViscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...
and heat conductivity.
Self-organized criticality
Critical exponents also exist for self organized criticality for dissipative systemDissipative system
A dissipative system is a thermodynamically open system which is operating out of, and often far from, thermodynamic equilibrium in an environment with which it exchanges energy and matter....
s.
Percolation Theory
Phase transitions and critical exponents appear also in percolation processes where the concentration of occupied sites or links play the role of temperature.External links and literature
- Hagen KleinertHagen KleinertHagen Kleinert is Professor of Theoretical Physics at the Free University of Berlin, Germany , at theWest University of Timişoara, at thein Bishkek. He is also of the...
and Verena Schulte-Frohlinde, Critical Properties of φ4-Theories, World Scientific (Singapore, 2001); Paperback ISBN 981-02-4658-7 (Read online at http://www.physik.fu-berlin.de/~kleinert/b8) - Toda, M., Kubo, R., N. Saito, Statistical Physics I, Springer-Verlag (Berlin, 1983); Hardcover ISBN 3-540-11460-2
- J.M.Yeomans, Statistical Mechanics of Phase Transitions, Oxford Clarendon Press
- H. E. StanleyH. Eugene StanleyHarry Eugene Stanley is an American physicist and University Professor at Boston University. He has made seminal contributions to statistical physics and is one of the pioneers of interdisciplinary science...
Introduction to Phase Transitions and Critical Phenomena, Oxford University Press, 1971 - A. Bunde and S. HavlinShlomo HavlinShlomo Havlin is a Professor in the Department of Physics at Bar-Ilan University, Ramat-Gan, Israel. He served as President of the Israel Physical Society , Dean of Faculty of Exact Sciences , Chairman, Department of Physics .-Biography:Professor Shlomo Havlin was born in Jerusalem, Israel...
(editors), Fractals in Science, Springer, 1995 - A. Bunde and S. HavlinShlomo HavlinShlomo Havlin is a Professor in the Department of Physics at Bar-Ilan University, Ramat-Gan, Israel. He served as President of the Israel Physical Society , Dean of Faculty of Exact Sciences , Chairman, Department of Physics .-Biography:Professor Shlomo Havlin was born in Jerusalem, Israel...
(editors), Fractals and Disordered Systems, Springer, 1996