Extinction of BKT transition by nonmagnetic disorder in planar-symmetry spin models

The Berezinskii-Kosterlitz-Thouless (BKT) 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 $\rho_{vac}$ by calculations of the helicity modulus and the by finite-size scaling of 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 $\rho_{vac}$, and falls to zero close to the square lattice percolation limit, $\rho_{vac}\approx 0.41$ . The result is physically reasonable: the long-range orientational order of the low-temperature phase cannot be maintained in the absence of sufficient spin interactions across the lattice.