# The Ising model - Introduction

The Ising model (in its one-dimensional version) was proposed by Ernst Ising in his PhD thesis in 1925 as a tool to describe the thermodynamic properties of magnetic systems from a microscopic point of view. Ising found (as we will shortly see) that in the case he considered the system does not exhibit any phase transition for ${\displaystyle T>0}$, and he incorrectly concluded that the whole model was not useful to describe such systems. However, this model has been later studied again and in different configurations and many important properties have been discovered; historically Ising model has been one of the most (if not the most itself) heavily studied model in statistical mechanics and it is often used as a testing ground when new theories or methods are developed. Another extremely important characteristic of the Ising model, that also justifies the effort we will make in order to study it, is that it does not only apply to magnetic systems: as we will see many other systems[1] can be shown to be equivalent to an appropriately defined Ising model.

The ${\displaystyle d}$-dimensional Ising model is defined as follows: consider a ${\displaystyle d}$-dimensional lattice with ${\displaystyle N}$ sites, each labelled by the index ${\displaystyle i=1,\dots ,N}$; in general the lattice is supposed to be hypercubic, but this is not necessary: in two dimensions, for example, we can consider triangular or "honeycomb" lattices, while in three dimensions we can have body-centered or face-centered cubic lattices[2]. What distinguishes one lattice from another is its coordination number${\displaystyle z}$, defined as the number of the nearest neighbours of a site. In the case of hypercubic lattices it can be easily seen (case by case) that ${\displaystyle z=2d}$, where ${\displaystyle d}$ is the dimensionality of the system. The degrees of freedom of the model are discrete variables ${\displaystyle S_{i}}$ defined on each site that can only assume the values ${\displaystyle +1}$ and ${\displaystyle -1}$; therefore, the number of the possible configurations of the system is ${\displaystyle 2^{N}}$.

In the original purpose of the Ising model the lattice represents the atomic lattice of a metal and the variables ${\displaystyle S_{i}}$ are the spins of the atoms, or rather their component along the vertical axis[3]; therefore, ${\displaystyle S_{i}=+1}$ corresponds to a spin pointing upwards while ${\displaystyle S_{i}=-1}$ downwards, and the study of this model should determine if and how all these spins can align so that the system can have a spontaneous net magnetization. However, since the Ising model can be used to describe completely different systems this interpretation is not the only possible one; even though, since this model has always been associated to magnets (but also for historical reasons) we will in the following continue to use a terminology proper only to magnetic systems.

Visually, for various dimensions the Ising model can be represented as follows:

Ising model in one dimension
Ising model in two dimensions
Ising model in three dimensions

The usefulness of the Ising model (but in general of lattice theories) goes much further that what can now be imagined (considering also the fact that this model can be used to describe systems different from magnetic ones). Lattice theories are in fact widely used in many areas of physics: just as an example, apart from "easily imaginable" applications of the Ising model in solid state physics, also quantum relativistic theories can be formulated in terms of lattices. QCD, for example, widely uses four-dimensional lattice models (with Minkowski spatio-temporal metric, of course) to describe quantum phenomena. We therefore see that also models with "strange" dimensionality (i.e. greater than ${\displaystyle d=3}$) can be actually useful.

In order for this model to be interesting, the degrees of freedom ${\displaystyle S_{i}}$ must not be independent[4]: we therefore assume that the spins interact with each other with exchange interactions that couple in general an arbitrary number of spins, and also with an external field ${\displaystyle H}$ that can change from site to site. Therefore, the most general form of the Hamiltonian of the Ising model for a given spin configuration is:

${\displaystyle {\mathcal {H}}=-\sum _{i}H_{i}S_{i}-\sum _{i,j}J_{ij}S_{i}S_{j}+\sum _{i,j,k}K_{ijk}S_{i}S_{j}S_{k}+\cdots }$
The first two minus signs are present because in general two adjacent magnetic moments tend to align, so for the system it is energetically convenient to have as many aligned spins (and aligned along the external field) as possible. For such systems the trace operator, being the sum over all the possible values of the degrees of freedom, takes the form:
${\displaystyle \operatorname {Tr} =\sum _{\lbrace S_{i}=\pm 1\rbrace }=\sum _{S_{1}=\pm 1}\sum _{S_{2}=\pm 1}\cdots \sum _{S_{N}=\pm 1}}$
As usual, the partition function will be:
${\displaystyle Z(T,\lbrace H_{i}\rbrace ,\lbrace J_{ij}\rbrace ,\dots )=\operatorname {Tr} e^{-\beta {\mathcal {H}}}}$
and the finite-size free energy:
${\displaystyle F(T,\lbrace H_{i}\rbrace ,\lbrace J_{ij}\rbrace ,\dots )=-k_{B}T\ln Z(T,\lbrace H_{i}\rbrace ,\lbrace J_{ij}\rbrace ,\dots )}$
Then, the thermodynamic properties of the system can be obtained taking the thermodynamic limit:
${\displaystyle f(T,\lbrace H_{i}\rbrace ,\lbrace J_{ij}\rbrace ,\dots )=\lim _{N\to \infty }{\frac {1}{N}}F(T,\lbrace H_{i}\rbrace ,\lbrace J_{ij}\rbrace ,\dots )}$
and then appropriately deriving ${\displaystyle f}$.

For the sake of simplicity, from now on we will always neglect interactions that couple more than two spins, and we will also consider the field ${\displaystyle H}$ as constant over the system[5].

Now, what about the existence of the thermodynamic limit for the Ising model? It has been shown that in general the thermodynamic limit exists if the two-spin interaction ${\displaystyle J_{ij}}$ satisfies:

${\displaystyle \sum _{i\neq j}|J_{ij}|<\infty }$
Therefore, we can see that what determines the existence of the thermodynamic limit is the dimensionality of the system and the range of the interactions. For example, if the interaction between two spins ${\displaystyle S_{i}}$ and ${\displaystyle S_{j}}$ at the positions ${\displaystyle {\vec {r}}_{i}}$ and ${\displaystyle {\vec {r}}_{j}}$ is of the form:
${\displaystyle J_{ij}=A|{\vec {r}}_{i}-{\vec {r}}_{j}|^{-\sigma }}$
then it can be shown that we must have ${\displaystyle \sigma >d}$ in order for the thermodynamic limit to exist.

1. Examples of such systems are fluids (see Ising model and fluids), binary alloys (see Ising model and binary alloys) and neural networks (see Ising model and neural networks).
2. The number of possible lattices is predicted by group theory, and amazingly all these mathematically possible lattices are found in nature.
3. Note that from quantum mechanics we know that in general the spin of an atom is proportional to a given fraction of ${\displaystyle \hbar }$: we are "ignoring" the magnitude of the spin, "reabsorbing" it in the definition of the coupling constants.
4. This is not really true: as we will see in Ising model and the ideal gas, we can learn something interesting also in the non-interacting case.
5. In the study of this situation, including the one in which there is no external field, we will use the method of sources: this consists in keeping always the term containing ${\displaystyle H_{i}}$ in the Hamiltonian, and at the end of the calculations (i.e. after the thermodynamic limit) set ${\displaystyle H_{i}}$ to any desired constant value, including zero.