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

Abstract

For the spin models with continuous symmetry on regular lattices and finite range of interactions, the lower critical dimension is d = 2. In two dimensions the classical XY-model displays Berezinskii–Kosterlitz–Thouless (BKT) transition associated with unbinding of topological defects (vortices and antivortices). We perform a Monte Carlo study of the classical XY-model on Sierpiński pyramids (SPs) whose fractal dimension is D  =  log 4/log 2 = 2 and the average coordination number per site is ≈ 7. The specific heat does not depend on the system size which indicates the absence of a long-range order. From the dependence of the helicity modulus on the cluster size and on boundary conditions, we draw a conclusion that in the thermodynamic limit there is no BKT transition at any finite temperature. This conclusion is also supported by our results for linear magnetic susceptibility. The lack of finite temperature phase transition is presumably caused by the finite order of ramification of SP.

Keywords:

• XY-model
• fractals
• Sierpiński pyramid
• Monte Carlo simulations

Acknowledgements

We thank Professor S.K. Bose for many useful discussions. This work was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada. The work of M. Przedborski was also supported in part through an NSERC Undergraduate Student Research Award (USRA).