Extinction of the Berezinskii-Kosterlitz-Thouless phase transition by nonmagnetic disorder in planar symmetry spin models

Abstract

The Berezinskii-Kosterlitz-Thouless (BKT) phase transition in two-dimensional planar rotator and $XY$ models on a square lattice, diluted by randomly placed vacancies, is studied here using hybrid Monte Carlo simulations that combine single spin flip, cluster, and over-relaxation techniques. The transition temperature $T_c$ is determined as a function of vacancy density ${\rho }_{\mathrm{vac}}$ by finite-size scaling of the helicity modulus and the in-plane magnetic susceptibility. The results for $T_c$ are consistent with those from the much less precise fourth-order cumulant of Binder. $T_c$ is found to decrease monotonically with increasing ${\rho }_{\mathrm{vac}}$, and falls to zero close to the square lattice percolation limit, ${\rho }_{\mathrm{vac}}\approx 0.41$. The result is physically reasonable: the quasi-long-range orientational order of the low-temperature phase cannot be maintained in the absence of sufficient spin interactions across the lattice.