Absence of phase transition in the XY-model on Menger sponge

We have performed a Monte Carlo study of the classical XY-model on a Menger sponge with the Wolff cluster algorithm (U. Wolff, 1989). The Menger sponge is a fractal object with infinite order of ramification and fractal dimension D = log ( 20 ) / log ( 3 ) = 2.7268 . From the dependence of the helicity modulus on system size and on boundary conditions, we conclude that there is no phase transition in the system at any finite temperature.