Universality class

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 \tau = (T-T_c)/T_c, and for small |\tau| various observables follow power laws of \tau:

• The exponent \alpha is the exponent relating the specific heat C to the reduced temperature: we have C = \tau^{-\alpha}. The specific heat will usually be singular at the critical point, but the minus sign in the definition of \alpha allows it to remain positive.
• The exponent \beta relates the order parameter \Psi to the temperature. Unlike most critical exponents, it is assumed to be positive, since the order parameter will usually be zero at the critical point. So we have \Psi = |\tau|^{\beta}.
• The exponent \gamma relates the temperature with the system's response to an external driving force, or source field. We have d\Psi/dJ = \tau^{-\gamma}, with J the driving force.
• The exponent \delta relates the order parameter to the source field at the critical temperature, where this relationship becomes nonlinear. We have J = \Psi^\delta (hence \Psi = J^{1/\delta}), with the same meanings as before.
• The exponent \nu 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 \xi. We have \xi = \tau^{-\nu}.
• The exponent \eta measures the size of correlations at the critical temperature. It is defined so that the correlation function of the order parameter scales as r^{-d+2-\eta}.
• The exponent \sigma, 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 s_{\max} \sim (p_c - p)^{-1/\sigma}.
• The exponent \tau, also from percolation theory, measures the number of size s clusters far from s_{\max} (or the number of clusters at criticality): n_s \sim s^{-\tau} f(s/s_{\max}), with the f 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

For symmetries, the group listed gives the symmetry of the order parameter. The group \mathrm{S}_n is the n-element symmetric group, O(n) is the orthogonal group in n dimensions, \mathbb{Z}_2 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).

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 \mathbb{Z}_2 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 local operators \sigma,\epsilon,\epsilon' of the conformal field theory describing the phase transition (In the Ginzburg–Landau description, these are the operators normally called \phi,\phi^2,\phi^4.) 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 M_{3,4}. 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.{{cite web | author-link = Leo Kadanoff | access-date = July 18, 2015 | archive-url = https://web.archive.org/web/20150722062827/http://www.condmatjournalclub.org/?p=2384 | archive-date = July 22, 2015 | url-status = usurped

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.

The thermodynamic quantities do not show power-law singularities, as they do in second-order phase transitions. Instead, above the critical point (T T_c), the correlation length scales as \xi \sim \exp(b |T-T_c|^{-\nu}), where b is a constant and \nu = 1/2. Susceptibility is then \chi \sim \xi^{2-\eta(T)}, where \eta(T) depends on the temperature (and \eta(T_c)=1/4). Specific heat is finite at T_c. The two-point correlation function scales as G(r) \sim r^{-\eta(T)} for T, while it behaves as G(r) \sim \exp(-r/\xi) for TT_c.

Growth phenomena

In epitaxial growth,{{cite book|first1=A.-L. |last1=Barbási |first2=H.E.|last2=Stanley|title=Fractal Concepts in Surface Growth

References

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