Potts model

The Potts model is a mathematical model that generalizes the Ising model , which is often used in statistical physics . Instead of spins with only two states, as in the Ising model, there are variables with different states on a grid . In the simplest case, the interaction is limited to neighboring lattice sites. ${\displaystyle q}$

This model is used, among other things, not only in statistical physics (especially when studying phase transitions ), but also in computer science ( signal processing ) and biology ( neural networks ). The model was named after Renfrey Potts , who defined the model in his dissertation in 1951. Julius Ashkin and Edward Teller dealt with a special case as early as 1943 . An overview article by Fa-Yueh Wu from 1982 gives an overview of the history and analysis of the model.

definition

The distinction between planar and standard Potts models comes from Cyril Domb (1974).

Planar Potts model

The Potts model consists of a -dimensional grid graph , e.g. B. a two-dimensional rectangular grid, a set of node assignments (node ​​configurations) and a Hamilton operator on this set. Each node is assigned an element from the set ${\displaystyle d}$ ${\displaystyle L(V,E)}$ ${\displaystyle S'^{|V|}}$ ${\displaystyle H'}$

${\displaystyle S'=\{\theta _{s}={\frac {2\pi s}{q}},0\leq s\leq q\}\quad q\in \{2,3,\ldots \}}$

These can be interpreted as points on the 2-dimensional unit circle and are the directions that the "spins" can assume on the respective grid points.

The Hamilton operator is given in the planar Potts model (also vector Potts model or clock model ) by

${\displaystyle H'(\sigma )=J_{1}\sum _{\langle i,j\rangle \in E}\cos(\theta _{s_{i}}-\theta _{s_{j}}),\quad \sigma =(\theta _{s_{1}},\theta _{s_{2}},\dots )\in S'^{|V|}.}$

All neighboring nodes for . The coupling constant describes the interaction between spins on the neighboring nodes. ${\displaystyle \langle i,j\rangle }$ ${\displaystyle i,j\in V}$ ${\displaystyle J_{1}}$

Standard Potts model

As an alternative to the planar Potts model just described, there is the standard Potts model (or simply: Potts model). The nodes are occupied with elements from the set

${\displaystyle S=\{0,1,\dots ,q\}}$.

The Hamilton operator is given here by

${\displaystyle H(\sigma )=-J\sum _{\langle i,j\rangle \in E}\delta (s_{i},s_{j}),\quad \sigma =(s_{1},s_{2},\dots )\in S^{|V|},}$

where is the Kronecker delta . ${\displaystyle \delta }$

That means, if two neighboring nodes have different values ​​of the spins, the corresponding summand disappears. The negative sign of is a convention, motivated by the Ising model. The standard Potts model is ferromagnetic for and antiferromagnetic for . ${\displaystyle J}$ ${\displaystyle J>0}$ ${\displaystyle J<0}$

Relation to other statistical models

General version

A more general version of the Potts model can be defined on the grid graph with the amount of node occupancy : ${\displaystyle L(V,E)}$ ${\displaystyle \{1,2,\dots ,q-1\}^{|V|}}$

${\displaystyle H(\sigma )=-\sum _{\langle i,j\rangle \in E}J_{ij}\delta (s_{i},s_{j})-\beta ^{-1}\sum _{i\in \{1,2,\dots \,|V|\}}h_{i}s_{i},\quad \sigma =(s_{1},s_{2},\dots )\in S^{|V|}.}$

In contrast to the original model, the interaction between the neighboring nodes varies. An outer field can also be added:

${\displaystyle -\beta ^{-1}\sum _{i\in \{1,2,\dots ,|V|\}}h_{i}s_{i}}$

As usual , the Boltzmann constant and the temperature are used here . ${\displaystyle \beta ={\tfrac {1}{k_{\mathrm {B} }T}}}$ ${\displaystyle k_{\mathrm {B} }}$ ${\displaystyle T}$

The interactions do not have to be restricted to the next neighboring lattice sites. In diluted Potts models there are free grid positions ( grid gas ) or interactions of different strengths. With suitable boundary conditions , interesting effects such as B. wetting or interface adsorption , are induced. ${\displaystyle J_{ij}}$

The Ising model as a special case

If one sets , the Ising model follows from the Potts model . ${\displaystyle q=2}$

The XY model as a special case

For obtains the XY-model , which in turn a special case of the N-vector model with can be understood. If one considers the planar Potts model, the state space of the spins is not a finite subset of the unit circle, but the entire 2-dimensional unit circle. ${\displaystyle q\to \infty }$ ${\displaystyle N=2}$

Ashkin plate model

The Ashkin-Teller model is the planar Potts model with states. ${\displaystyle q=4}$

Others

There are also connections to the Heisenberg model , N-vector model, (ice-rule) vertex models and to percolation theory (first by PW Kasteleyn and CM Fortuin 1969 for bond percolation, later also for site percolation).

The Kirchhoff's law for networks of linear resistors is obtained as the limit of the Potts model (Kasteleyn, Fortuin 1972). ${\displaystyle q=0}$

discussion

Potts looked at the planar model and was able to determine the critical point for the rectangular grid and similar to the Ising model with Kramers-Wannier duality . At the end of his work he gave the critical point of the standard Potts model for everyone . ${\displaystyle q=2,3,4}$ ${\displaystyle q}$

The planar and the standard model are identical for (Ising model) with and for with . In addition, the planar model with can be reduced to the model with for any grid . For there are no obvious relationships between the planar and the standard model. ${\displaystyle q=2}$ ${\displaystyle J=2J_{1}}$ ${\displaystyle q=3}$ ${\displaystyle J={\tfrac {3}{2}}J_{1}}$ ${\displaystyle q=4}$ ${\displaystyle q=2}$ ${\displaystyle q>4}$

On a two-dimensional grid, the Potts model has a phase transition of the first order for and otherwise a continuous phase transition (second order) as in the Ising model ( ) ( Rodney Baxter 1973, 1978). Baxter used the identification of the two-dimensional Potts model with the ice-rule vertex model by Temperley and Elliott Lieb (1971 for a grid of squares). ${\displaystyle J>0}$ ${\displaystyle q>4}$ ${\displaystyle q=2}$

The one-dimensional Potts model can be solved exactly (with the help of the transfer matrix method) and so can the two-dimensional model with interactions between neighboring grid locations. In general, Monte Carlo simulations and renormalization group theory in particular provide reliable results.

Potts measure

With the Hamilton function as above

${\displaystyle H(\sigma )=-J\sum _{\langle i,j\rangle \in E}\delta (s_{i},s_{j}),\quad \sigma =(s_{1},s_{2},\dots )\in S^{|V|}.}$

and the usual definition of the state sum ${\displaystyle Z}$

${\displaystyle Z(\beta )=\sum _{\sigma \in S^{|V|}}e^{-\beta H(\sigma )},\quad \beta \in {]0,\infty [}.}$

one can define the Potts measure , which belongs to the Boltzmann distributions as a probability measure : ${\displaystyle \pi }$

${\displaystyle \pi (\sigma )=Z(\beta )^{-1}e^{-\beta H(\sigma )},\quad \sigma \in S^{|V|}.}$

The free energy is as usual:

${\displaystyle F(\beta )=-\beta ^{-1}\ln(Z(\beta )),\quad \beta \in {]0,\infty [}.}$

literature

• Julius Ashkin, Edward Teller: Statistics of Two-Dimensional Lattices With Four Components . In: Phys. Rev. . 64, No. 5-6, 1943, pp. 178-184. bibcode : 1943PhRv ... 64..178A . doi : 10.1103 / PhysRev.64.178 .
• Renfrey B. Potts: Some Generalized Order Disorder Transformations . In: Mathematical Proceedings of the Cambridge Philosophical Society . 48, No. 1, 1952, pp. 106-109. bibcode : 1952PCPS ... 48..106P . doi : 10.1017 / S0305004100027419 .
• Fa-Yueh Wu: The Potts model . In: Reviews of Modern Physics . 54, No. 1, 1982, pp. 235-268. bibcode : 1982RvMP ... 54..235W . doi : 10.1103 / RevModPhys.54.235 .

Individual evidence

1. ↑ C. Domb , J. Phys. A, Volume 7, 1974, p. 1335
2. ↑ S. Dietrich , in Phase transitions and critical phenomena (Eds. C. Domb and JL Lebowitz ), Volume 12, 1988
3. ↑ W. Selke , W. Pesch , Z. Phys. B, Volume 47, 1982, p. 335
4. ↑ PW Kasteleyn , CM Fortuin, J. Phys. Soc. Japan, Vol. 26 (Suppl.), 1969, p. 11
5. ↑ DPLandau , K.Binder , A Guide to Monte Carlo Simulations in Statistical Physics , 2014