When gold is deposited on Si(553), the surface self-assembles to form a periodic array of steps with nearly perfect structural order. In scanning tunneling microscopy these steps resemble quasi-one-dimensional atomic chains. At temperatures below ~50 K the chains develop tripled periodicity. We recently predicted, on the basis of density-functional theory calculations at T=0, that this tripled periodicity arises from the complete polarization of the electron spin on every third silicon atom along the step; in the ground state these linear chains of silicon spins are antiferromagnetically ordered. Here we explore, using ab-initio molecular dynamics and kinetic Monte Carlo simulations, the behavior of silicon spin chains on Si(553)-Au at finite temperature. Thermodynamic phase transitions at T>0 in one-dimensional systems are prohibited by the Mermin-Wagner theorem. Nevertheless we find that a surprisingly sharp onset occurs upon cooling---at about 30 K for perfect surfaces and at higher temperature for surfaces with defects---to a well-ordered phase with tripled periodicity, in good agreement with experiment.
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