Past exam of the phase transitions and collective phenomena course of the University of Cambridge / 2012 / Paper 1 / 1 / e

The susceptibility is the integrated connected correlation function, up to a nonsingular field-normalization factor:Above the transition at zero field the subtraction vanishes. In an ordered thermodynamic phase it is essential; integrating the disconnected constant would incorrectly give a volume divergence unrelated to the desired response.

With a microscopic cutoff $a$ and $r=|\mathbf{x}|$, radial integration and $u=r/\xi$ giveIf $g(0)$ is finite, its large-distance tail decays sufficiently rapidly and $\eta <2$, the dimensionless integral approaches a finite nonzero constant. ThusShort-distance contributions give a regular background. This is the Fisher scaling relation; it concerns the fluctuating connected part of the source correlation ansatz.