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

TL;DR

This study investigates the classical XY-model on Sierpinski pyramids with fractal dimension 2, finding no phase transition at finite temperature due to the fractal's finite ramification, contrasting with regular 2D lattices.

Contribution

It provides the first Monte Carlo analysis of the XY-model on Sierpinski pyramids, demonstrating the absence of Berezinskii-Kosterlitz-Thouless transition in this fractal structure.

Findings

No dependence of specific heat on system size.

Helicity modulus indicates no phase transition at finite temperature.

Results suggest finite ramification prevents long-range order.

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 transition associated with unbinding of topological defects (vortices and antivortices). We perform a Monte Carlo study of the classical XY-model on Sierpinski pyramids whose fractal dimension is D=log4/log2=2 and the average coordination number per site is about 7. The specific heat does not depend on the system size which indicates the absence of 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 Berezinskii-Kosterlitz-Thouless 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 Sierpinski pyramid.