Vacancy effects in an easy-plane Heisenberg model:  Reduction of $T_c$ and doubly charged vortices

Magnetic vortices in thermal equilibrium in two-dimensional magnets are studied here in the presence of a low concentration of nonmagnetic impurities (spin vacancies). A nearest-neighbor Heisenberg (XXZ) spin model with easy-plane exchange anisotropy is used to determine static thermodynamic properties and vortex densities via combined cluster and over-relaxation Monte Carlo simulations. Especially at low temperatures, a large fraction of the thermally generated vortices nucleate centered on vacancies, where they have a lower energy of formation. This fact is responsible for the reduction of the vortex-unbinding transition temperature with increasing vacancy concentration, similar to that seen in the planar rotator model. Spin vacancies also present the possibility of the appearance of vortices with double topological charges ($\pm 4\pi$ change in in-plane spin angle), stable only when centered on vacancies.

Figure 1

For the model with edge $L=64$, at the XY limit, the internal energy, absolute magnetization, and specific heat per spin for the uniform system and with 16% vacancy density.

Figure 2

Application of the $L$ dependence of the fourth-order cumulant $U_L$ on various system sizes to estimate $T_c∕J\approx 0.48$ (common crossing point of the data) at 16% vacancy density in the XY model.

Figure 3

Application of the finite-size scaling of in-plane susceptibility to estimate $T_c∕J\approx 0.478$ (common crossing point of the data) at 16% vacancy density in the XY model, using exponent $\eta =1∕4$.

Figure 4

A spin configuration from MC simulations for $L=32$, $\lambda =0$, ${\rho }_{\mathrm{vac}}=0.16$, at $T=0.85J$, with vortices indicated by ± signs. The projections of the xy spin components are shown as arrows, with line (triangular) heads indicating positive (negative) $z$ spin components. The three larger plus signs are vortices of charge $q=+2$ centered on vacancies. Many vortices have formed centered on vacancies.

Figure 5

Thermally induced vorticity density for the uniform XY model $\left[\rho \left(0\right)\right]$ and at 16% vacancy density $\left[\rho \left(0.16\right)\right]$. Also displayed are the vorticity fraction pinned on vacancies $\left[f_{\mathrm{pin}}\right]$ and the fraction with doubled charges $\left[f_{\mathrm{dbl}}\right]$, both when ${\rho }_{\mathrm{vac}}=0.16$.

Figure 6

For the model with edge $L=64$, at the vortex-in-plaquette critical anisotropy, the internal energy, absolute magnetization, and specific heat per spin for the uniform system and with 16% vacancy density.

Figure 7

For the model with edge $L=64$, at the vortex-on-vacancy critical anisotropy, the internal energy, absolute magnetization, and specific heat per spin for the uniform system and with 16% vacancy density.

Figure 8

Application of the finite-size scaling of in-plane susceptibility to estimate $T_c∕J\approx 0.404$ (common crossing point of the data) at 16% vacancy density at the vortex-on-vacancy critical anistropy.

Figure 9

Thermally induced vorticity density $\left[\rho \left(0\right)\right]$ at the vortex-on-vacancy critical anisotropy with 16% vacancy density $\left[\rho \left(0.16\right)\right]$. Also displayed are the vorticity fraction pinned on vacancies $\left[f_{\mathrm{pin}}\right]$ and the fraction due to double charges $\left[f_{\mathrm{dbl}}\right]$, at 16% vacancy density.

Figure 10

Out-of-plane susceptibilities ${\chi }^{zz}$ vs temperature for $L=128$, at the three anisotropies studied. The lower curves at low $T$ (open symbols) correspond to ${\rho }_{\mathrm{vac}}=0$, the upper curves (solid symbols) correspond to ${\rho }_{\mathrm{vac}}=0.16$. Unlike ${\chi }^{xx}$ or ${\chi }^{yy}$, there is only a very weak dependence of ${\chi }^{zz}$ on $L$, mostly in the high-temperature phase.

Figure 11

Various total system energies with a vortex present versus system radius $R$.

Figure 12

After relaxation of a $q=2$ vortex initially centered on an isolated vacancy in a circular system of radius $R=50$, the total system energy (solid curve) is shown as a function of the anisotropy constant $\lambda$. The vertical bars indicate the net out-of-plane magnetization of the relaxed configuration, on the same numerical scale.

Figure 13

Final state of relaxation of a $q=2$ vortex initially centered on an isolated vacancy in a circular system of radius $R=50$ with $\lambda =0.7$ (only the central region of the system is shown). (a) shows the projection of xy spin components on the plane, as explained in Fig. 4. In (b) the lengths of the arrows are equal to the $z$ spin components, while the directions are still given by the xy projections.

Figure 14

A spin configuration for $L=32$, $\lambda =0$, ${\rho }_{\mathrm{vac}}=0.16$, at $T\approx T_c\approx 0.48J$, where nine out of the ten vortices present have formed on vacancies.