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For the spin models with continuous symmetry on regular lattices and finiterange of interactions the lower critical dimension is d=2. In two dimensionsthe classical XY-model displays Berezinskii-Kosterlitz-Thouless transitionassociated with unbinding of topological defects (vortices and antivortices).We perform a Monte Carlo study of the classical XY-model on Sierpinski pyramidswhose fractal dimension is D=log4/log2=2 and the average coordination numberper site is about 7. The specific heat does not depend on the system size whichindicates the absence of long range order. From the dependence of the helicitymodulus on the cluster size and on boundary conditions we draw a conclusionthat in the thermodynamic limit there is no Berezinskii-Kosterlitz-Thoulesstransition at any finite temperature. This conclusion is also supported by ourresults for linear magnetic susceptibility. The lack of finite temperaturephase transition is presumably caused by the finite order of ramification ofSierpinski pyramid.

The absence of phase transition for the classical XY-model on Sierpinski pyramid with fractal dimension D=2