Absence of long-range order with long-range potentials

A particular form of Mermin's inequality is analyzed for repulsive inverse power potentials V(r)=e ² r ^–m/m] in ad-dimensional space. For long-range potentials (m d) the system is put into a stabilizing background. Long-range order is shown to be excluded ford (m + 2)/2 whenm d, while for short-range potentials (m > d) we recover Mermin's result (d 2). For Coulomb systems (m=d – 2) and the experimentally studied electron surface layer (d = 2,m=1), long-range order cannot be excluded by the present argument.