# Universality classes

**Universality classes** are groups of models that have the same set of critical exponents

dimension class 3-state Potts Ashkin-Teller Chiral Directed percolation 2 0 1/8 7/4 1 1/4 2D Ising 3 0.1096(5) 0.32653(10) 1.2373(2) 4.7893(8) 0.63012(16) 0.03639(15) 3D Ising Local linear interface any 0 1/2 1 3 1/2 0 Mean-field Molecular beam epitaxy Random-field 3 −0.0146(8) 0.3485(2) 1.3177(5) 4.780(2) 0.67155(27) 0.0380(4) XY

where

- is known as the heat capacity exponent
- is known as the magnetic order parameter exponent
- is known as the susceptibility exponent
- is known as the equation of state exponent
- is known as the correlation length exponent
- is known as the anomalous dimension in the critical correlation function.

## Contents

# Derivations[edit]

## 3-state Potts[edit]

## Ashkin-Teller[edit]

## Chiral[edit]

## Directed percolation[edit]

## Ising[edit]

The Hamiltonian of the Ising model is

where and the summation runs over the lattice sites.

The order parameter is

In two dimensions, Onsager obtained the exact solution in the absence of a external field, and the critical exponents are

(In fact, the specific heat diverges logarithmically with the critical temperature)

along with ^{[1]}:

In three dimensions, the critical exponents are not known exactly. However, Monte Carlo simulations and Renormalisation group analysis provide accurate estimates ^{[2]}:

with a critical temperature of ^{[3]}. In four and higher dimensions, the critical exponents are mean-field with logarithmic corrections.

## Local linear interface[edit]

## Mean-field[edit]

The critical exponents of are derived as follows ^{[4]}:

#### Heat capacity exponent: [edit]

(final result: )

#### Magnetic order parameter exponent: [edit]

(final result: )

#### Susceptibility exponent: [edit]

(final result: )

#### Equation of state exponent: [edit]

(final result: )

#### Correlation length exponent: [edit]

(final result: )

#### Correlation function exponent: [edit]

(final result: )

## Molecular beam epitaxy[edit]

## Random-field[edit]

## XY[edit]

For the three dimensional XY model one has the following critical exponents^{[5]}:

# References[edit]

- ↑ Michael E. Fisher "Rigorous Inequalities for Critical-Point Correlation Exponents", Physical Review
**180**pp. 594-600 (1969) - ↑ Massimo Campostrini, Andrea Pelissetto, Paolo Rossi, and Ettore Vicari "25th-order high-temperature expansion results for three-dimensional Ising-like systems on the simple-cubic lattice", Physical Review E
**65**066127 (2002) - ↑ A. L. Talapov and H. W. J Blöte "The magnetization of the 3D Ising model", Journal of Physics A: Mathematical and General
**29**pp. 5727-5733 (1996) - ↑ Linda E. Reichl "A Modern Course in Statistical Physics", Wiley-VCH, Berlin 3rd Edition (2009) ISBN 3-527-40782-0 § 4.9.4
- ↑ Massimo Campostrini, Martin Hasenbusch, Andrea Pelissetto, Paolo Rossi, and Ettore Vicari "Critical behavior of the three-dimensional XY universality class" Physical Review B
**63**214503 (2001)