Absence of spontaneous magnetization

Let us now come back to free boundary conditions and compute the mean magnetization of a given site . By definition we have[1]:

We now use the fact that , so that in this case:
where we have used (like we have done previously) the evenness of , and now also the oddness of , i.e. . This way:
where etc. are the spin variables appropriately rearranged[2]. Let us now consider the two different contributions to . As of the first:
Considering now the second one:
we have that for every fixed this term vanishes; for example, if we consider the contribution relative to a fixed value of and sum (for example) over we have:
Therefore, also the second term vanishes and in the end:
This result perfectly agrees with what we have already seen in Absence of phase transitions for finite systems, but now it has been deduced from a direct computation.

  1. Note that the sum on nearest neighbour is done without counting the same terms twice (as we have already stressed). In fact, in our case every spin interacts with its nearest neighbours and , but the sum in the trace involves every two-spin interaction only once.
  2. The form of this last term can be understood more easily doing an explicit computation with a simple example.