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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).
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.
For symmetries, the group listed gives the symmetry of the order parameter. The group is the n-element symmetric group, is the orthogonal group in n dimensions, and 1 is the trivial group. Mean-field theory result is indicated with (MF).
Class | Dimension | Symmetry | ||||||
---|---|---|---|---|---|---|---|---|
3-state Potts | 2 | 1/3 | 1/9 | 13/9 | 14 | 5/6 | 4/15 | |
Ashkin–Teller (4-state Potts) | 2 | 2/3 | 1/12 | 7/6 | 15 | 2/3 | 1/4 | |
Ordinary percolation | 1 | 1 | 1 | 0 | 1 | 1 | 1 | |
2 | 1 | −2/3 | 5/36 | 43/18 | 91/5 | 4/3 | 5/24 | |
3 | 1 | −0.625(3) | 0.4181(8) | 1.793(3) | 5.29(6) | 0.87619(12) | 0.46(8) or 0.59(9) | |
4 | 1 | −0.756(40) | 0.657(9) | 1.422(16) | 3.9 or 3.198(6) | 0.689(10) | −0.0944(28) | |
5 | 1 | ≈ −0.85 | 0.830(10) | 1.185(5) | 3.0 | 0.569(5) | −0.075(20) or −0.0565 | |
6+ (MF) | 1 | −1 | 1 | 1 | 2 | 1/2 | 0 | |
Directed percolation | 1 | 1 | 0.159464(6) | 0.276486(8) | 2.277730(5) | 0.159464(6) | 1.096854(4) | 0.313686(8) |
2 | 1 | 0.451 | 0.536(3) | 1.60 | 0.451 | 0.733(8) | 0.230 | |
3 | 1 | 0.73 | 0.813(9) | 1.25 | 0.73 | 0.584(5) | 0.12 | |
4+ (MF) | 1 | 1 | 1 | 1 | 1 | 1/2 | 0 | |
Conserved directed percolation (Manna, or "local linear interface") | 1 | 1 | 0.28(1) | 0.14(1) | 1.11(2) [1] | 0.34(2) [1] | ||
2 | 1 | 0.64(1) | 1.59(3) | 0.50(5) | 1.29(8) | 0.29(5) | ||
3 | 1 | 0.84(2) | 1.23(4) | 0.90(3) | 1.12(8) | 0.16(5) | ||
4+ (MF) | 1 | 1 | 1 | 1 | 1 | 0 | ||
Protected percolation | 2 [2] | 1 | 5/41 | 86/41 | ||||
3 [2] | 1 | 0.28871(15) | 1.3066(19) | |||||
Ising | 2 | 0 | 1/8 | 7/4 | 15 | 1 | 1/4 | |
3 [3] | 0.11008708(35) | 0.32641871(75) | 1.23707551(26) | 4.78984254(27) | 0.62997097(12) | 0.036297612(48) | ||
4+ (MF) | 0 | 1/2 | 1 | 3 | 1/2 | 0 | ||
XY | 2 | — | Berezinskii-Kosterlitz-Thouless universality class | |||||
3 [4] | −0.01526(30) | 0.34869(7) | 1.3179(2) | 4.77937(25) | 0.67175(10) | 0.038176(44) | ||
4+ (MF) | 0 | 1/2 | 1 | 3 | 1/2 | 0 | ||
Heisenberg | 3 [5] | −0.1336(15) | 0.3689(3) | 1.3960(9) | 4.783(3) | 0.7112(5) | 0.0375(5) | |
4+ (MF) | 0 | 1/2 | 1 | 3 | 1/2 | 0 | ||
Self-avoiding walk | 1 | 1 | 1 | 0 | 1 | 1 | 1 | |
2 | 1 | 1/2 | 5/64 | 43/32 | 91/5 | 3/4 | 5/24 | |
3 | 1 | 0.2372090(12) | 0.3029190(8) | 1.1569530(10) [6] | 4.819348(15) | 0.5875970(4) [7] | 0.0310434(21) | |
4+ (MF) | 1 | 0 | 1/2 | 1 | 3 | 1/2 | 0 |
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.
d=2 | d=3 | d=4 | general expression | |
---|---|---|---|---|
α | 0 | 0.11008708(35) | 0 | |
β | 1/8 | 0.32641871(75) | 1/2 | |
γ | 7/4 | 1.23707551(26) | 1 | |
δ | 15 | 4.78984254(27) | 3 | |
η | 1/4 | 0.036297612(48) | 0 | |
ν | 1 | 0.62997097(12) | 1/2 | |
ω | 2 | 0.82966(9) | 0 |
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 [8] (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:
d=2 | d=3 | d=4 | |
---|---|---|---|
1/8 | 0.518148806(24) [3] | 1 | |
1 | 1.41262528(29) [3] | 2 | |
4 | 3.82966(9) [9] [10] | 4 |
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. [3] [9] [10] [11] [12] [13] [14] These are the values reported in the tables. Renormalization group methods, [15] [16] [17] [18] Monte-Carlo simulations, [19] and the fuzzy sphere regulator [20] give results in agreement with the conformal bootstrap, but are several orders of magnitude less accurate.
The phase transition present in the two-dimensional XY model and superconductors is governed by a distinct universality class, the Berezinskii–Kosterlitz–Thouless transition [21] . 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 , the correlation length scales as , where is a constant and . Susceptibility is then , where depends on the temperature (and ). Specific heat is finite at . The two-point correlation function scales as for , while it behaves as for .
In epitaxial growth [22] [23] , there is a change in the roughness of surfaces, from atomically flat to rough. The root mean square fluctuation in the evolving surface height (which characterizes roughness) increases as initially, and eventually saturates at a size-dependent value . is called the growth exponent, and is the roughness exponent. The crossover time between the two regimes depends on the system size as , where is the dynamical exponent obeying the scaling law .
class | dimensionality | |||
---|---|---|---|---|
Edwards-Wilkinson (EW) | ||||
Kardar-Parisi-Zhang (KPZ) [24] | ||||
Mullins-Herring (MH) | ||||
Molecular-beam epitaxy (MBE) |
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