Why does finite size scaling shift the critical temperature downward? The question is in the title.
Say we have a system where:
$$\xi \propto |T-T_c|^{-\nu}$$
In a finite system, $\xi$ cannot diverge and is limited by the size of the system. Thus:
$$
L =|T_\text{finite size transition}-T_c(\infty)|^{-\nu}
$$
You can easily enough solve this and find the shifted transition temperature. I don't find this argument compelling, though. $\xi$ diverges on both sides of the transition temperature, so there seems to be just as good of an argument that the temperature should shift up, not down, since that is where the correlation length first reaches the sample size.
Does anyone with a better understand of this than I care to enlighten me?
Thanks!
 A: You are right, the scaling relation only implies that the $T_\mathrm{max}(L) \ne T_c$.
Your question is answered by the calculations made in the seminal paper by Ferdinand and Fisher (1969), who also provided (page 2) a physical argument:

"[$T_\mathrm{max}(L) > T_c$] might be interpreted as an indication of increased cooperation between nearby spins as a result of extra "communication" via paths that encircle the torus. (By contrast, if boundaries in the shape of "free" edges are present we expect $T_\mathrm{max} < T_c$.)"

And they go on to mention that even for periodic boundary conditions (torus) one gets $T_\mathrm{max} < T_c$ for too high aspect ratios (about 3.139), a finite-size behavior that, at least at the time, was not yet understood.
Meta-comments:


*

*The question is old, but high among Google results, so worth answering.

*My knowledge in the field is recent and very limited: improvements and corrections are welcome.

*I was pointed in the right direction by the answer to this question: Critical temperature and lattice size with the Wolff algorithm for 2d Ising model
