English: This tartan-like graph shows the Ising model probability density ${\displaystyle P(\sigma )}$ for the two-sided lattice using the dyadic mapping.

That is, a lattice configuration of length ${\displaystyle 2N}$

${\displaystyle \sigma =(\sigma _{-N-1},\cdots ,\sigma _{-2},\sigma _{-1},\sigma _{0},\sigma _{1},\sigma _{2},\cdots ,\sigma _{N})}$

is understood to consist of a sequence of "spins" ${\displaystyle \sigma _{k}=\pm 1}$. This sequence may be represented by two real numbers ${\displaystyle 0\leq x,y\leq 1}$ with

${\displaystyle x(\sigma )=\sum _{k=0}^{N}\left({\frac {\sigma _{k}+1}{2}}\right)2^{-(k+1)}}$

and

${\displaystyle y(\sigma )=\sum _{k=0}^{N}\left({\frac {\sigma _{-k-1}+1}{2}}\right)2^{-(k+1)}}$

The energy of a given configuration ${\displaystyle \sigma }$ is computed using the classical Hamiltonian,

${\displaystyle H(\sigma )=\sum _{k=-N}^{N}V(\tau ^{k}\sigma )}$

Here, ${\displaystyle \tau }$ is the shift operator, acting on the lattice by shifting all spins over by one position:

${\displaystyle \tau (\cdots ,\sigma _{0},\sigma _{1},\sigma _{2},\cdots )=(\cdots ,\sigma _{-1},\sigma _{0},\sigma _{1},\cdots )}$

The interaction potential ${\displaystyle V}$ is given by the Ising model interaction

${\displaystyle V(\sigma )=J\sigma _{0}\sigma _{1}+B\sigma _{0}}$

Here, the constant ${\displaystyle J}$ is the interaction strength between two neighboring spins ${\displaystyle \sigma _{0}}$ and ${\displaystyle \sigma _{1}}$, while the constant ${\displaystyle B}$ may be interpreted as the strength of the interaction between the magnetic field and the magnetic moment of the spin.

The set of all possible configurations ${\displaystyle \Omega =\{\sigma \}}$ form a canonical ensemble, with each different configuration occurring with a probability ${\displaystyle P(\sigma )}$ given by the Boltzmann distribution

${\displaystyle P(\sigma )={\frac {1}{Z(T)}}e^{-H(\sigma )/k_{B}T}}$

where ${\displaystyle k_{B}}$ is Boltzmann's constant, ${\displaystyle T}$ is the temperature, and ${\displaystyle Z(T)}$ is the partition function. The partition function is defined to be such that the sum over all probabilities adds up to one; that is, so that

${\displaystyle Z(T)=\sum _{\sigma \in \Omega }e^{-H(\sigma )/k_{B}T}}$

Image details

The image here shows ${\displaystyle P(\sigma )=P(x,y)}$ for the Ising model, with ${\displaystyle J=0.3}$, ${\displaystyle B=0}$ and temperature ${\displaystyle T=1/k_{B}}$. The lattice is finite sized, with ${\displaystyle N=10}$, so that all ${\displaystyle 1024\times 1024=2^{N}\times 2^{N}}$ lattice configurations are represented, each configuration denoted by one pixel. The color choices here are such that black represents values where ${\displaystyle P(\sigma )=P(x,y)}$ are zero, blue are small values, with yellow and red being progressively larger values.

As an invariant measure

This fractal tartan is invariant under the Baker's map. The shift operator ${\displaystyle \tau }$ on the lattice has an action on the unit square with the following representation:

${\displaystyle \tau (x,y)=\left({\frac {x+\left\lfloor 2y\right\rfloor }{2}}\,,\,2y-\left\lfloor 2y\right\rfloor \right)}$

This map (up to a reflection/rotation around the 45-degree axis) is essentially the Baker's map or equivalently the Horseshoe map. As the article on the Horseshoe map explains, the invariant sets have such a tartan pattern (an appropriately deformed Sierpinski carpet). In this case, the invariance arises from the translation invariance of the Gibbs states of the Ising model: that is, the energy ${\displaystyle H(\sigma )}$ associated with the state ${\displaystyle \sigma }$ is invariant under the action of ${\displaystyle \tau }$:

${\displaystyle H\left(\tau ^{k}\sigma \right)=H(\sigma )}$

for all integers ${\displaystyle k}$. Similarly, the probability density is invariant as well:

${\displaystyle P\left(\tau ^{k}\sigma \right)=P(\sigma )}$

The naive classical treatment given here suffers from conceptual difficulties in the ${\displaystyle N\to \infty }$ limit. These problems can be remedied by using a more appropriate topology on the set of states that make up the configuration space. This topology is the cylinder set topology, and using it allows one to construct a sigma algebra and thus a measure on the set of states. With this topology, the probability density can be understood to be a translation-invariant measure on the topology. Indeed, there is a certain sense in which the seemingly fractal patterns generated by the iterated Baker's map or horseshoe map can be understood with a conventional and well-behaved topology on a lattice model.

Created by Linas Vepstas User:Linas on 24 September 2006