# The role of interaction range

In order to see how the range of the interactions between the degrees of freedom of the system affects its properties, let us consider a one-dimensional Ising model with infinite-ranged interactions:

(note that the sum in the interaction term is not restricted, and the factor has been introduced for later convenience). This model can be solved with the technique of

*Hubbard transformation*, also called

*auxiliary field method*. First, we must note that can't be a constant independent of the dimension of the system because the sum contains a number of terms of the order of and so in the thermodynamic limit it would diverge; we must therefore use the so called

*Kac prescription*, setting so that the thermodynamic limit exists. Under these assumptions the partition function of the system is:

Since the double sum is not restricted, we have:

If we now call and , we can use the

*Hubbard-Stratonovich identity*

^{[1]}:

where . The advantage of this approach is that the variable , which contains all the degrees of freedom of the system, is linear and not quadratic in the exponential; however we have "paid" the price of having introduced another field, (the

*auxiliary field*from which this method takes its name). The partition function then becomes:

Physically this can be interpreted as the "mean value" of the partition functions of non interacting Ising models subjected to an external field whose component is distributed along a Gaussian.

We can therefore write:

and can be easily computed factorizing the sum, similarly to what we have done for the Ising model in Bulk free energy, thermodynamic limit and absence of phase transitions:

Therefore:

Now, since the exponent in the integral that defines is extensive ( doesn't depend on ) and is large, we can compute it using the

*saddle point approximation*(see appendix The saddle point approximation). This consists in approximating the integral with the largest value of the integrand, namely:

where is the maximum of , thus given by the condition:

which yields:

Since it must be a maximum, we also must have:

Now, we can see that the physical meaning of is the magnetization of the system in the thermodynamic limit. In fact:

Since we are interested in determining if the system can exhibit a

*spontaneous*magnetization, we consider the case ; therefore we will have:

which is a transcendent equation, so it can't be solved analytically.
However, we can solve it graphically:

From these figures we can see that there are three possible cases (remembering that by definition ):

- for there are three solutions, one for and two at
- for these three solutions coincide

- for there is only one solution:

This means that two possible non null solutions appear when:

Let us see which of these solutions are acceptable, i.e. which of these solutions are maxima of . Still in the case , we have:

and so:

- :in this case , so:

so is indeed a maximum for

- :if then:

so this is not a maximum for , and thus is not an acceptable solution of . On the other hand, if then (also using the evenness of ):

and this time this derivative is negative, because:

which is always true.

Therefore, if the only acceptable value for the magnetization of the system is , while if then : a phase transition has occurred, since now the system can exhibit a net spontaneous magnetization. We thus see explicitly that if we let the interactions to be long-ranged the Ising model can undergo phase transitions already in the one-dimensional case.

- ↑ It can be easily verified completing the square in the exponential: and computing the Gaussian integral.