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 breaking in the ordered phase. So, we will arbitrarily take any solution in the phase.
| Keys | |
|---|---|
| order parameter (e.g. for the liquid-gas critical point, magnetization for the Curie point,etc.) | |
| specific free energy | |
| specific heat; | |
| source field (e.g. where P is the pressure and Pc the critical pressure for the liquid-gas critical point, reduced chemical potential, the magnetic field H for the Curie point ) | |
| the susceptibility/compressibility/etc.; | |
| correlation length | |
| the number of spatial dimensions | |
| the correlation function | |
| 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 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.
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