Potts model
In
statistical mechanics, the
Potts model, a generalization of the
Ising model, is a model of interacting
spins on a
crystalline lattice. By studying the Potts model, one may gain insight into the behaviour of
ferromagnets and certain other phenomena of
solid state physics. The strength of the Potts model is not so much that it models these physical systems well; it is rather that the one-dimensional case is
exactly solvable, and that it has a rich mathematical formulation that has been studied extensively.The model is named after
Renfrey B. Potts who described the model near the end of his 1952 Ph.D. thesis. The model was related to the "planar Potts" or "clock model", which was suggested to him by his advisor
Cyril Domb. The Potts model is sometimes known as the
Ashkin-Teller model (after
Julius Ashkin and
Edward Teller), as they considered a four component version in 1943.The Potts model is related to, and generalized by, several other models, including the
XY model, the
Heisenberg model and the
N-vector model. The infinite-range Potts model is known as the
Kac model. When the spins are taken to interact in a
non-Abelian manner, the model is related to the
flux tube model, which is used to discuss confinement in
quantum chromodynamics. Generalizations of the Potts model have also been used to model
grain growth in metals and
coarsening in
foams. A further generalization of these methods by
James Glazier and
Francois Graner, known as the
Cellular Potts Model has been used to simulate static and kinetic phenomena in foam and biological
morphogenesis.
Physical description
The Potts model consists of
spins that are placed on a
lattice; the lattice is usually taken to be a two-dimensional rectangular
Euclidean lattice, but is often generalized to other dimensions or other lattices. Domb originally suggested that the spin take one of
q possible values, distributed uniformly about the
circle, at angles
and that the interaction
Hamiltonian be given by
left( theta_{s_i} - theta_{s_j} right)with the sum running over the nearest neighbor pairs
(ij)
over all lattice sites. The site
colors si
take on values, ranging from
1lderiv(⋅)s q
. Here,
Jc
is a coupling constant, determining the interaction strength. This model is now known as the
vector Potts model or the
clock model. Potts provided a solution for two dimensions, for
q=2,3 and 4. In the limit as q approaches infinity, this becomes the
XY model.What is now known as the standard
Potts model was suggested by Potts in the course of the solution above, and uses a simpler Hamiltonian, given by:
where
δ(sisj)
is the
Kronecker delta, which equals one whenever
si=sj
and zero otherwise. The
q=2 standard Potts model is equivalent to the 2D
Ising model and the 2-state vector Potts model, with
J(=-2Jc
. The
q=3 standard Potts model is equivalent to the three-state vector Potts model, with
J(=-3Jc/2
.A common generalization is to introduce an external "magnetic field" term
h
, and moving the parameters inside the sums and allowing them to vary across the model:
&bη; Hg = - &bη; Σ(ij)Jij δ(sisj) - Σi hi si
where
&bη;=1/kT
the
inverse temperature,
k the
Boltzmann constant and
T the
temperature. The summation may run over more distant neighbors on the lattice, or may in fact be an infinite-range force. Different papers may adopt slightly different conventions, which can alter
H
and the associated
partition function by additive or multiplicative constants.
Discussion
Despite its simplicity as a model of a physical system, the Potts model is useful as a model system for the study of
phase transitions. For example, two dimensional lattices with
J>0
exhibit a first order transition if
q>4
. When
q≤q 4
a continuous transition is observed, as in the Ising model where
q=2
. Further use is found through the model's relation to percolation problems and the Tutte and chromatic polynomials found in combinatorics.The model has a close relation to the
Fortuin-Kasteleyn random cluster model, another model in statistical mechanics. Understanding this relationship has helped develop efficient
Markov chain Monte Carlo methods for numerical exploration of the model at small
q
.
Measure theoretic description
The one dimensional Potts model may be expressed in terms of a
subshift of finite type, and thus gains access to all of the mathematical techniques associated with this formalism. In particular, it can be solved exactly using the techniques of
transfer operators. (However,
Ernst Ising used combinatorial methods to solve the
Ising model, which is the "ancestor" of the Potts model, in his 1925 PhD thesis). This section develops the mathematical formalism, based on
measure theory, behind this solution.While the example below is developed for the one-dimensional case, many of the arguments, and almost all of the notation, generalizes easily to any number of dimensions. Some of the formalism is also broad enough to handle related models, such as the
XY model, the
Heisenberg model and the
N-vector model.
Topology of the space of states
Let
Q=12cderiv(⋅)sq
be a finite set of symbols, and let
s_k in Q ; forall k in mathbb{Z} }be the set of all bi-infinite strings of values from the set
Q. This set is called a
full shift. For defining the Potts model, either this whole space, or a certain subset of it, a
subshift of finite type, may be used. Shifts get this name because there exists a natural operator on this space, the
shift operator τ:QZ→ QZ
, acting as
This set has a natural
product topology; the
base for this topology are the
cylinder sets
s_m = xi_0, ldots ,s_{m+k} = xi_k }that is, the set of all possible strings where
k+1 spins match up exactly to a given, specific set of values
ξ0 lderiv(⋅)s ξk
. Explicit representations for the cylinder sets can be gotten by noting that the string of values corresponds to a
q-adic number, and thus, intuitively, the product topology resembles that of the
real number line.
Interaction energy
The interaction between the spins is then given by a
continuous function V:QZ→R
on this topology.
Any continuous function will do; for example
will be seen to describe the interaction between nearest neighbors. Of course, different functions give different interactions; so a function of
s0
,
s1
and
s2
will describe a next-nearest neighbor interaction. A function
V gives interaction energy between a set of spins; it is
not the Hamiltonian, but is used to build it. The argument to the function
V is an element
s∈ QZ
, that is, an infinite string of spins. In the above example, the function
V just picked out two spins out of the infinite string: the values
s0
and
s1
. In general, the function
V may depend on some or all of the spins; currently, only those that depend on a finite number are exactly solvable.Define the function
Hn:QZ→R
as
This function can be seen to consist of two parts: the self-energy of a configuration
[s0 s1 lderiv(⋅)ssn]
of spins, plus the interaction energy of this set and all the other spins in the lattice. The
n→&∈f∈;
limit of this function is the Hamiltonian of the system; for finite
n, these are sometimes called the
finite state Hamiltonians.
Partition function and measure
The corresponding finite-state
partition function is given by
exp -beta H_n(C_0[s_0,s_1,ldots,s_n])
with
C0
being the cylinder sets defined above. Here, β=1/
kT, where
k is
Boltzmann's constant, and
T is the
temperature. It is very common in mathematical treatments to set β=1, as it is easily regained by rescaling the interaction energy. This partition function is written as a function of the interaction
V to emphasize that it is only a function of the interaction, and not of any specific configuration of spins. The partition function, together with the Hamiltonian, are used to define a
measure on the Borel σ-algebra in the following way: The measure of a cylinder set, i.e. an element of the base, is given by
frac{1}{Z_n(V)} exp -beta H_n (C_k[s_0,s_1,ldots,s_n])One can then extend by countable additivity to the full σ-algebra. This measure is a
probability measure; it gives the likelihood of a given configuration occurring in the
configuration space QZ
. By endowing the configuration space with a probability measure built from a Hamiltonian in this way, the configuration space turns into a
canonical ensemble.Most thermodynamic properties can be expressed directly in terms of the partition function. Thus, for example, the
Helmholtz free energy is given by
Another important related quantity is the
topological pressure, defined as
P(V) = limn→&∈f∈; 1/n log Zn(V)
which will show up as the logarithm of the leading eigenvalue of the
transfer operator of the solution.
Free field solution
The simplest model is the model where there is no interaction at all, and so
V=0
and
Hn=0
. The partition function becomes
Zn(0) = e-&bη; Σs0lderiv(⋅)ssn ∈ Q 1
If all states are allowed, that is, the underlying set of states is given by a
full shift, then the sum may be trivially evaluated as
If neighboring spins are only allowed in certain specific configurations, then the state space is given by a
subshift of finite type. The partition function may then be written as
e^{-beta} mbox{Tr} A^nwhere card is the
cardinality or count of a set, and Fix is the set of
fixed points of the iterated shift function:
{ s in Q^mathbb{Z} : tau^n s = s }The
q ⋅ q
matrix
A
is the
adjacency matrix specifying which neighboring spin values are allowed.
Interacting model
The simplest case of the interacting model is the
Ising model, where the spin can only take on one of two values,
sn ∈ -1+1
and only nearest neighbor spins interact. The interaction potential is given by
This potential can be captured in a
2 ⋅ 2
matrix with matrix elements
Mσ σ' = ex( (( &bη; J( σ σ' &nbs(;))
with the index
σσ' ∈ -1+1
. The partition function is then given by
The general solution for an arbitrary number of spins, and an arbitrary finite-range interaction, is given by the same general form. In this case, the precise expression for the matrix
M is a bit more complex.The goal of solving a model such as the Potts model is to give the an exact
closed-form expression for the partition function (which we've done) and an expression for the
Gibbs states or
equilibrium states in the limit of
n→&∈f∈;
, the
thermodynamic limit.
References
- Julius Ashkin, Edward Teller (1943); Statistics of Two-Dimensional Lattices With Four Components, Physical Review, 64, pp. 178–184
- Renfrey B. Potts, (1952); Some Generalized Order-Disorder Transformations, Proceedings of the Cambridge Philosophical Society, Vol. 48, pp. 106−109
- François Graner and James A. Glazier (1992); "Simulation of Biological Cell Sorting Using a Two-Dimensional Extended Potts Model", Physical Review Letters 69, 2013-2016
- Fred Y. Wu (1982); The Potts model, Reviews of Modern Physics, Vo. 54, pp. 235–268
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