Top Qs
Timeline
Chat
Perspective

Universality class

Collection of models with the same renormalization group flow limit From Wikipedia, the free encyclopedia

Remove ads

In statistical mechanics, a 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 behavior 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: below the lower critical dimension, the universality class becomes degenerate (this dimension is 2 for the Ising model, or for directed percolation, but 1 for undirected percolation), and above the upper critical dimension the critical exponents stabilize and can be calculated by an analog of mean-field theory (this dimension is 4 for Ising or for directed percolation, and 6 for undirected percolation).

Remove ads

List of critical exponents

Summarize
Perspective

Critical exponents are defined in terms of the variation of certain physical properties of the system near its phase transition point. These physical properties will include its reduced temperature , its order parameter measuring how much of the system is in the "ordered" phase, the specific heat, and so on.

  • The exponent is the exponent relating the specific heat C to the reduced temperature: we have . The specific heat will usually be singular at the critical point, but the minus sign in the definition of allows it to remain positive.
  • The exponent relates the order parameter to the temperature. Unlike most critical exponents it is assumed positive, since the order parameter will usually be zero at the critical point. So we have .
  • The exponent relates the temperature with the system's response to an external driving force, or source field. We have , with J the driving force.
  • The exponent relates the order parameter to the source field at the critical temperature, where this relationship becomes nonlinear. We have (hence ), with the same meanings as before.
  • The exponent 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 . We have .
  • The exponent measures the size of correlations at the critical temperature. It is defined so that the correlation function of the order parameter scales as .
  • The exponent , used in percolation theory, measures the size of the largest clusters (roughly, the largest ordered blocks) at 'temperatures' (connection probabilities) below the critical point. So .
  • The exponent , also from percolation theory, measures the number of size s clusters far from (or the number of clusters at criticality): , with the factor removed at critical probability.

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 and 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.

More information , ...

From the quantum field theory point of view, the critical exponents can be expressed in terms of scaling dimensions of the local operators of the conformal field theory describing the phase transition[1] (In the Ginzburg–Landau description, these are the operators normally called .) 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:

More information , ...

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.[2][3][4][5][6][7][8] These are the values reported in the tables. Renormalization group methods,[9][10][11][12] Monte-Carlo simulations,[13] and the fuzzy sphere regulator[14] give results in agreement with the conformal bootstrap, but are several orders of magnitude less accurate.

Others

For symmetries, the group listed gives the symmetry of the order parameter. The group is the dihedral group, the symmetry group of the n-gon, is the n-element symmetric group, is the octahedral group, and is the orthogonal group in n dimensions. 1 is the trivial group.

More information , ...
Remove ads

References

Further reading

Loading related searches...

Wikiwand - on

Seamless Wikipedia browsing. On steroids.

Remove ads