File:Ferromagnetic behavior around critical temperature.pdf

English: Equal-time correlation functions, ${\displaystyle C(r,\tau =0)}$, as a function of radius for a ferromagnetic spin system above, at, and below at its critical temperature, ${\displaystyle T_{C}}$. Above ${\displaystyle T_{C}}$, ${\displaystyle C(r,\tau =0)}$ exhibits a combined exponential and power-law dependence on distance: ${\displaystyle C(r,\tau =0)\propto r^{-(d-2+\eta )}e^{-r/\xi (T)}}$. The power-law dependence dominates at distances short relative to the correlation length, ${\displaystyle \xi }$, while the exponential dependence dominates at distances large relative to ${\displaystyle \xi }$. At the ${\displaystyle T_{C}}$, the correlation length diverges, ${\displaystyle \xi (T_{C})=\infty }$, resulting in solely power-law behavior: ${\displaystyle C(r,\tau =0)\propto r^{-(d-2+\eta )}}$. ${\displaystyle T_{C}}$ is distinguished by the extreme non-locality of the spatial correlations between microscopic values of the relevant order parameter without long-range order. Below ${\displaystyle T_{C}}$, the spins exhibit spontaneous ordering and thus infinite correlations. Continuous order-disorder transitions can be understood as the process of the correlation length, ${\displaystyle \xi }$, transitioning from being infinite in the low-temperature, ordered state, to finite in a high-temperature, disordered state.