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
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
, we can use the Hubbard-Stratonovich identity
. 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
is distributed along a Gaussian.
We can therefore write:
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
Now, since the exponent in the integral that defines
is extensive (
doesn't depend on
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:
is the maximum of
, thus given by the condition:
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:
- :in this case , 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.