In statistical mechanics, an universality class is a set of mathematical models which share a scale-invariant limit under renormalization group flow. While the models within a class may differ at finite scales, their behaviour become increasingly similar as the limit scale is approached. In particular, asymptotic phenomena such as critical exponents are the same for all models in the class.

Well-studied examples include the universality classes of the Ising model or the percolation theory at their respective phase transition points; these are both families of classes, one for each lattice dimension. Typically, a family of universality classes has a lower and upper critical dimension the critical exponents stabilize and can be calculated by an analog of mean-field theory for Ising or for directed percolation, and 6 for undirected percolation).

Definition of critical exponents

Critical exponents characterize the variation of certain physical properties of the system as the control parameter approaches the critical point. For temperature-driven transitions, one usually defines the reduced temperature:

• is the exponent relating the specific heat allows it to remain positive.

• .

• , with J the driving force.

• ), with the same meanings as before.

• relates the size of correlations (i.e. patches of the ordered phase) to the temperature; away from the critical point these are characterized by a correlation length.

• measures the size of correlations at the critical temperature. It is defined so that the correlation function of the order parameter.

• , used in percolation theory.

• , also from percolation theory, measures the number of size s factor removed at critical probability.

The critical exponents are independent of microscopic details of the model, but depend on dimensionality, symmetry, and range of interactions (i.e., only depend on the universality class). In rare cases, the critical exponents governing the behaviors below and above the critical point are not the same.

List of critical exponents

is the n-element symmetric group is the orthogonal group in n is the cyclic group of order 2 (parity, or Ising symmetry), and 1 is the trivial group. Mean-field theory result is indicated with (MF). In the two-dimensional Ashkin-Teller model, the exponents in general depend continuously on a parameter along the critical line. symmetry.

Ising model

This section lists the critical exponents of the ferromagnetic transition in the Ising model. In statistical physics, the Ising model is the simplest system exhibiting a continuous phase transition with a scalar order parameter symmetry. The critical exponents of the transition are universal values and characterize the singular properties of physical quantities. The ferromagnetic transition of the Ising model establishes an important universality class, which contains a variety of phase transitions as different as ferromagnetism close to the Curie point and critical opalescence of liquid near its critical point.From the quantum field theory point of view, the critical exponents can be expressed in terms of scaling dimensions of the conformal field theory describing the phase transition (In the Ginzburg–Landau.) These expressions are given in the last column of the above table, and were used to calculate the values of the critical exponents using the operator dimensions values from the following table:In d=2, the two-dimensional critical Ising model's critical exponents can be computed exactly using the minimal model. In d=4, it is the free massless scalar theory (also referred to as mean field theory). These two theories are exactly solved, and the exact solutions give values reported in the table.

The d=3 theory is not yet exactly solved. The most accurate results come from the conformal bootstrap. These are the values reported in the tables. Renormalization group methods, Monte-Carlo simulations, and the fuzzy sphere regulator give results in agreement with the conformal bootstrap, but are several orders of magnitude less accurate.

Berezinskii-Kosterlitz-Thouless universality class

The phase transition present in the two-dimensional XY model and superconductors is governed by a distinct universality class, the Berezinskii–Kosterlitz–Thouless transition. The disordered phase (high-temperature phase) contains free vortices, while the ordered phase (low-temperature phase) contains bound vortices. At the phase transition, the free energy and all its derivatives are continuous, hence it is an infinite-order transition in the Ehrenfest classification.

.

Growth phenomena

In epitaxial growth,) stands between the Edwards-Wilkinson and Kardar-Parisi-Zhang equations analogously to the place of the spherical model between mean-field-theory and the Ising model.For a list of experimentally measured values of these exponents, one might consult .

References

Further reading

• Universality classes from Sklogwiki