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## Critical exponent
- the dimension of the system,
- the range of the interaction,
- the spin dimension.
## Additional recommended knowledgeThese properties of critical exponents were found in experiments. 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. ## DefinitionPhase transitions occour at a certain temperature, called the critical temperature This results in the power law we were looking for. ## The most important critical exponentsAbove and below Let us consider the disordered phase (τ > 0), ordered phase (τ < 0 ) and critical temperature (τ = 0) phases separately. Following the standard convention, the critical exponents related to the ordered phase are primed. 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
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. - α = α'
- γ = γ'
- ν = ν'
The classical (Landau theory aka mean field theory) values are - α = α' = 0
- β = 1/2
- γ = γ' = 1
- δ = 3
If we add derivative terms turning it into a mean field Landau-Ginzburg theory, we get - η = 0
- ν = 1 / 2
Critical exponents are denoted by Greek letters. They fall into universality classes and obey scaling relations such as The critical exponents can be computed from conformal field theory. See also anomalous scaling dimension. ## See also |
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This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Critical_exponent". A list of authors is available in Wikipedia. |