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I'm having a hard time understanding Peierl's argument for the non-existence of spontaneous magnetization in a 1D Ising model at $T>0$. Let the Hamiltonian be $$ H = -J\sum_{i} S_i S_{i+1}.$$ The argument is apparently simple: start with a chain of spins pointing all up; the energy difference to create an island of $L$ spins down around some position $i$ is 4J, and there are $L$ ways to place this $i$-th spin inside this region of size $L$ (i.e. $L$ microstates corresponding to a macrostate with an island of spins down around the $i$-th position). So the entropy difference is $\Delta S = k_B \ln L$ and free-energy difference $\Delta F = 4J - k_B T \ln L <0$ for L sufficiently large. This means the system is unstable under these fluctuations.

So far, all we actually proved is that it is favourable to create a sufficiently large island of spins down in an "all spins up" chain, but this is not the conclusion we aim at. We want to show that the favourable state will be completely disordered.

Now suppose one of these fluctuations took place, and the system now has "all spins up except for L spins down". This is clearly still not fully disordered. And if we try to apply the argument again, for a second fluctuation, the system is no longer in the "all spins up" state we originally assumed, so the argument no longer holds.

One could say: "But we are considering an infinite chain, so $L$ spins down is irrelevant compared to infinite spins up, and the argument still holds for any other fluctuation sufficiently far away from the first island".

But then how can we conclude from this argument that the resulting state after many fluctuations is completely disordered? What we assume in the proof contradicts what we want to achieve. We assume a sufficiently large region of "all spins up" (so large that any island of spins down is insignificant in comparison), but we want to prove that eventually the islands of spin down will be comparable in size to those of spin up (only then can we say that the total magnetization vanishes).

The reply in this question is possibly/probably related, but I didn' understand the argument there either...

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  • $\begingroup$ Terminology: Peierls' argument is really about the claim that there is spontaneous symmetry breaking at low enough temperatures in dimensions $2$ and above; this can be turned into an actual proof. The argument you present is not a proof: it is at best indicative that things might behave differently in one dimension. There are several related questions on this site: here, here, here and here for instance. $\endgroup$
    – Yvan Velenik
    Jul 12 at 8:33

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