Singular Field Response and Singular Screening of Vacancies in Antiferromagnets

For isolated vacancies in ordered local-moment antiferromagnets we show that the magnetic-field linear-response limit is generically singular: The magnetic moment associated with a vacancy in zero field is different from that in a finite field $h$ in the limit $h\to 0^{+}$. The origin is a universal and singular screening cloud, which moreover leads to perfect screening as $h\to 0^{+}$ for magnets which display spin-flop bulk states in the weak-field limit.

Figure 1

Illustration of singular screening for the square-lattice AF: At zero field, the vacancy induces a moment ${\overrightarrow{m}}_0$ with $m_0=S$, which is modified into $\overrightarrow{m}(h)$ in the presence of a uniform field $\overrightarrow{h}$. The size of the screening cloud, $l_h$, diverges as $h\to 0^{+}$, and $m\to 0$ in this limit. For details see text.

Figure 2

Vacancy contribution to the uniform magnetization, $m(h)$, for the square-lattice AF. (a) Crosses: classical result $m(h)/S$, obtained on lattices with $L\le 768$ and extrapolated to $L\to \infty$; the circle shows the linear-response value $|m_0/S|=1$. Open dots: $1/S$ correction $m^{(0)}(h)$, here $L\le 50$. Vertical dashed line: saturation field $h_{\mathrm{sat}}=8SJ$ [23]. Top arrows: evolution of the bulk spin state with increasing $h$. Inset: small-$h$ behavior of $m(h)/S$, together with a fit $m\propto h\ln h$ (see text). (b) Finite-size data $m(h)/S$ as function of $1/L$. $m$ saturates for $L>l_h$ with $l_h\propto SJ/h$.

Figure 3

Vacancy-induced magnetization as in Fig. 2, but for the classical triangular-lattice AF with biquadratic exchange of strength (a) $k=-0.07$, leading to coplanar states in a field, and (b) $k=0.1$, leading to umbrella states. Here $h_{\mathrm{sat}}=9SJ(1+2k)$, the shaded area in (a) indicates the magnetization plateau. Insets: small-$h$ behavior. Note that $m_0$ takes a $k$-dependent fractional value [4].

Figure 4

Field-induced screening cloud for [(a), (c)] the square lattice and [(b), (d)] the triangular lattice with $k=-0.07$, for different values of $h$. (a), (b) Vacancy-induced rotation angles $\delta \Theta (\overrightarrow{r})$ as a function of the distance $r$ to the vacancy. In (b) the zero-field angles are shown as well. Insets: scaling of $\delta \Theta /h$ vs $rh/(SJ)\propto r/l_h$, with the short-range piece projected out in (b). (c), (d) Transverse magnetization ${\overrightarrow{m}}_{\perp }(r)$ as a function of the integration radius $r$ in Eq. (2). Insets: ${\overrightarrow{m}}_{\perp }$ vs rescaled distance $rh/(SJ)$.