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

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.

Figure 1

(Color online) Application of the fourth order cumulant (6) for estimating $T_c$, for the PRM at 4% vacancy concentration. The data were obtained using the Monte Carlo approach described in Sec. 2B. The inset expands the view near the estimated critical temperature $[k_B{T}_c∕J{S}^2\approx 0.815\left(10\right)]$.

Figure 2

Log-log plot of susceptibility versus system edge $L$, for the PRM at 4% vacancy concentration. The curves correspond to different values of the dimensionless temperature $\tau =k_{B}T∕J{S}^2$. Lines are guides to the eye; errors are smaller than the symbols. Least squares fits were used to determine the slopes, $(2-\eta )$, producing $\eta \left(T\right)$ as seen in Fig. 3.

Figure 3

Application of the correlation exponent $\eta$ for estimating $T_c$, for the PRM at 4% vacancy concentration, derived from using systems of sizes $L=16$, 32, 64, and 96. The inset shows how the critical temperature was estimated as $k_B{T}_c∕J{S}^2\approx 0.815$.

Figure 4

(Color online) Typical application of the helicity modulus for estimating $T_c$, for the PRM at 4% vacancy concentration. The dashed line is Eq. (12). The inset shows how the crossing points occur slightly above the critical temperature $(k_B{T}_c∕J{S}^2\approx 0.815)$. Error bars are smaller than the symbols.

Figure 5

(Color online) The helicity modulus for the PRM at 38% vacancy concentration for system sizes indicated. The dashed line is Eq. (12).

Figure 6

(Color online) Binder’s fourth cumulant for the PRM at 38% vacancy concentration for system sizes indicated. $k_B{T}_c∕J{S}^2\approx 0.05$ as estimated from the point where the data for different system sizes separate. The lines are guides to the eye.

Figure 7

(Color online) The helicity modulus for the PRM at vacancy concentrations ${\rho }_{\mathrm{vac}}$ indicated in the legend. The dashed line is Eq. (12). Part (a) shows the overall trend; error bars are smaller than the symbols. Part (b) displays the behavior as the transition is extinguished at the critical vacancy concentration.

Figure 8

(Color online) Application of the correlation exponent $\eta$ for estimating $T_c$, for the PRM at the vacancy concentrations indicated in the legend, derived from scaling of $\chi$ for systems of sizes $L=16$, 32, 64, and 96. Part (a) gives the rough overall trend and part (b) shows how $\eta$ does not fall to the value $1∕4$ at vacancy concentrations greater than 41%.

Figure 9

Results of finite-size scaling for the PRM at 4% vacancy concentration, showing the fitting parameters $c_0$ and $A$ defined in (13, 14). $T_c$ is estimated from the point where $A=1$ or where $c_0=0$. These fits were derived from MC runs for systems of sizes $L=20$, 32, 48, 64, and 96.

Figure 10

Results of finite-size scaling for the PRM at 38% vacancy concentration, as explained in Fig. 9.

Figure 11

Fitting parameter $A\left(T\right)$ for the FSS of ${\rm Y}$, Eq. (14), with vacancy concentrations ${\rho }_{\mathrm{vac}}$ in percent indicated next to the curves. Lines are guides to the eye.

Figure 12

The critical temperatures versus vacancy concentration for the PRM, extracted from fits of $\eta$ together with Eq. (8) and from FSS of ${\rm Y}\left(T\right)$. The inset shows $T_c$ as ${\rho }_{\mathrm{vac}}$ approaches the critical region.

Figure 13

(Color online) The helicity modulus for the $XY$ model at 40% vacancy concentration for system sizes indicated. The dashed line is Eq. (12).

Figure 14

(Color online) Application of the correlation exponent $\eta$ for estimating $T_c$, for the $XY$ model at the vacancy concentrations indicated in the legend, derived from scaling of $\chi$ for systems of sizes $L=16$, 32, 64, and 96. Part (a) gives the rough overall trend and part (b) shows how $\eta$ does not fall to the value $1∕4$ at vacancy concentrations greater than 41%.

Figure 15

The critical temperatures versus vacancy concentration for the $XY$ model, extracted from fits of $\eta$ together with Eq. (8) and from the crossing of ${\rm Y}\left(T\right)$ with Eq. (12) for $L=96$ systems. The inset shows $T_c$ as ${\rho }_{\mathrm{vac}}$ approaches the critical region.

Figure 16

(Color online) Application of the correlation exponent $\eta$ for estimating $T_c$, for the $XY$ model at 16% vacancy concentration, derived from using systems of sizes $L=16$, 32, 64, and 96. The inset shows how the critical temperature was estimated as $k_B{T}_c∕J{S}^2\approx 0.453$.

Figure 17

(Color online) Application of the finite-size scaling of in-plane susceptibility $\chi$ to estimate $k_B{T}_c∕J{S}^2\approx 0.453$ (common crossing point of the data) at 16% vacancy density in the $XY$ model, with fully randomly placed vacancies, using exponent $\eta =1∕4$.