Editing Potts model (section)

== 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 operator]]s.  (However, [[Ernst Ising]] used combinatorial methods to solve the [[Ising model]], which is the "ancestor" of the Potts model, in his 1924 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 (classical)|Heisenberg model]] and the [[N-vector model]].

=== Topology of the space of states ===
Let ''Q'' = {1, ..., ''q''} be a finite set of symbols, and let
: <math>Q^\mathbf{Z}=\{ s=(\ldots,s_{-1},s_0,s_1,\ldots) :  s_k \in Q \; \forall k \in \mathbf{Z} \}</math>
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 : ''Q''<sup>'''Z'''</sup> → ''Q''<sup>'''Z'''</sup>, acting as
: <math>\tau (s)_k = s_{k+1}</math>

This set has a natural [[product topology]]; the [[base (topology)|base]] for this topology are the [[cylinder set]]s
: <math>C_m[\xi_0, \ldots, \xi_k]= \{s \in Q^\mathbf{Z} : s_m = \xi_0, \ldots ,s_{m+k} = \xi_k \}</math>
that is, the set of all possible strings where ''k''+1 spins match up exactly to a given, specific set of values ξ<sub>0</sub>, ..., ξ<sub>''k''</sub>. Explicit representations for the cylinder sets can be gotten by noting that the string of values corresponds to a [[p-adic number|''q''-adic number]], however the natural topology of the q-adic numbers is finer than the above product topology.

=== Interaction energy ===
The interaction between the spins is then given by a [[continuous function (topology)|continuous function]] ''V'' : ''Q''<sup>'''Z'''</sup> → '''R''' on this topology. ''Any'' continuous function will do; for example
: <math>V(s) = -J\delta(s_0,s_1)</math>
will be seen to describe the interaction between nearest neighbors. Of course, different functions give different interactions; so a function of ''s''<sub>0</sub>,  ''s''<sub>1</sub> and ''s''<sub>2</sub> 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'' ∈ ''Q''<sup>'''Z'''</sup>, 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 ''s''<sub>0</sub> and ''s''<sub>1</sub>.  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 ''H<sub>n</sub>'' : ''Q''<sup>'''Z'''</sup> → '''R''' as
: <math>H_n(s)= \sum_{k=0}^n V(\tau^k s)</math>

This function can be seen to consist of two parts: the self-energy of a configuration [''s''<sub>0</sub>, ''s''<sub>1</sub>, ..., ''s<sub>n</sub>''] of spins, plus the interaction energy of this set and all the other spins in the lattice. The {{nowrap|''n'' → ∞}} 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 (statistical mechanics)|partition function]] is given by
: <math>Z_n(V) = \sum_{s_0,\ldots,s_n \in Q} \exp(-\beta H_n(C_0[s_0,s_1,\ldots,s_n]))</math>
with ''C''<sub>0</sub> being the cylinder sets defined above. Here, β = 1/''kT'', where ''k'' is the [[Boltzmann 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 (mathematics)|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
: <math>\mu (C_k[s_0,s_1,\ldots,s_n]) =  \frac{1}{Z_n(V)}  \exp(-\beta H_n (C_k[s_0,s_1,\ldots,s_n]))</math>

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 (physics)|configuration space]] ''Q''<sup>'''Z'''</sup>. 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
: <math>A_n(V)=-kT \log Z_n(V)</math>

Another important related quantity is the [[topological pressure]], defined as
: <math>P(V) = \lim_{n\to\infty} \frac{1}{n} \log Z_n(V)</math>
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'' = ''c'' and ''H<sub>n</sub>'' = ''c'' (with ''c'' constant and independent of any spin configuration).  The partition function becomes
: <math>Z_n(c) = e^{-c\beta} \sum_{s_0,\ldots,s_n \in Q} 1</math>

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
: <math>Z_n(c) = e^{-c\beta} q^{n+1}</math>

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
: <math>Z_n(c) = e^{-c\beta} |\mbox{Fix}\, \tau^n| =  e^{-c\beta} \mbox{Tr} A^n</math>
where card is the [[cardinality]] or count of a set, and Fix is the set of [[Fixed point (mathematics)|fixed points]] of the iterated shift function:
: <math>\mbox{Fix}\, \tau^n =  \{ s \in Q^\mathbf{Z} : \tau^n s = s \}</math>

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, ''s<sub>n</sub>'' ∈ {{mset|−1, 1}} and only nearest neighbor spins interact. The interaction potential is given by
: <math>V(\sigma) = -J_p s_0 s_1\,</math>

This potential can be captured in a 2 × 2 matrix with matrix elements
: <math>M_{\sigma \sigma'} = \exp \left( \beta J_p \sigma \sigma' \right)</math>
with the index σ, σ′ ∈ {−1, 1}.  The partition function is then given by
: <math>Z_n(V) = \mbox{Tr}\, M^n</math>

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 an exact [[closed-form expression]] for the partition function and an expression for the [[Gibbs state]]s or [[equilibrium state]]s in the limit of ''n'' → ∞, the [[thermodynamic limit]].