Dzyaloshinskii-Moriya anisotropy and nonmagnetic impurities in the $s=\frac{1}{2}$ kagome system ${\text{ZnCu}}_3{\left(\text{OH}\right)}_6{\text{Cl}}_2$

Motivated by recent nuclear magnetic resonance experiments on ${\text{ZnCu}}_3{\left(\text{OH}\right)}_6{\text{Cl}}_2$, we present an exact-diagonalization study of the combined effects of nonmagnetic impurities and Dzyaloshinskii-Moriya (DM) interactions in the $s=1/2$ kagome antiferromagnet. The local response to an applied field and correlation-matrix data reveal that the dimer freezing which occurs around each impurity for $D=0$ persists at least up to $D/J\simeq 0.06$, where $J$ and $D$ denote, respectively, the exchange and DM interaction energies. The phase transition to the $\left(Q=0\right)$ semiclassical $120°$ state favored at large $D$ takes place at $D/J\simeq 0.1$. However, the dimers next to the impurity sites remain strong up to values $D\sim J$, far above this critical point, and thus do not participate fully in the ordered state. We discuss the implications of our results for experiments on ${\text{ZnCu}}_3{\left(\text{OH}\right)}_6{\text{Cl}}_2$.

Figure 1

(Color online) The kagome lattice of ${\text{Cu}}^{2+}$ ions $\left(s=1/2\right)$, containing a single nonmagnetic ${\text{Zn}}^{2+}$ impurity. Bond arrows define site labels $i$ and $j$ in each interaction term $\mathbf{D}\cdot {\mathbf{s}}_i\times {\mathbf{s}}_j$, while $\mathbf{D}=D{\mathbf{e}}_z$ with $D>0$ for all bonds. Curving arrows denote the ${\mathsf{C}}_6$ symmetry about the hexagon centers.

Figure 2

(Color online) In-plane magnetizations and bond spin correlation functions for the 26-site kagome cluster with $B=J/20$ and $\theta =30°$. The lengths of the arrows are proportional to ${\left({⟨s_i^x⟩}^2+{⟨s_i^y⟩}^2\right)}^{1/2}$ and the thicknesses of the bonds to $⟨{\mathbf{s}}_i\cdot {\mathbf{s}}_j⟩$.

Figure 3

(Color online) $D$ dependence of (a) $⟨s_i^x⟩$, (b) $⟨s_i^y⟩$, (c) $⟨{\mathbf{s}}_i\cdot {\mathbf{s}}_j⟩$, and (d) ${⟨{\mathbf{s}}_i\times {\mathbf{s}}_j⟩}^z$ for the 26-site kagome cluster with $B=J/20$ and $\theta =30°$. $i$ and $j$ in panels (c) and (d) are nearest-neighbor sites only. The curves which correspond to the sites or bonds closest to the impurity are designated by (red) filled symbols.

Figure 4

(Color online) In-plane magnetization pattern (red arrows) and bond strengths (bond thickness in blue) on the isolated six-site cluster discussed in Sec. 3, for $B=J/20$ and $\theta =30°$.

Figure 5

(Color online) Eigenvalues ${\lambda }_i$ of the correlation matrix (divided by $N$) as a function of $D/J$ for $N=14$ (circles), 17 (down triangles), 20 (squares), 23 (up triangles), and 26 (diamonds). Here $B=J/20$ and $\theta =30°$, but the results for $B=0$ are identical.

Figure 6

(Color online) Scaling with system size of the largest eigenvalue ${\lambda }_m$ (divided by $N$) of the correlation matrix. The solid lines correspond to the expected $1/\sqrt{N}$ scaling of the leading corrections to the thermodynamic limit in the $120°$ ordered phase.

Figure 7

(Color online) Magnetization profile for the 26-site kagome cluster in zero field, corresponding to the eigenvector ${\mathit{v}}_m$ of $\mathbf{C}$ with the largest eigenvalue ${\lambda }_m$. The in-plane moments are given by the real and imaginary parts of ${\mathit{v}}_m$. Note that this mode is unique up to a global U(1) rotation which is related to the arbitrary phase of ${\mathit{v}}_m$.

Figure 8

(Color online) In-plane spin magnitudes and orientations [represented respectively by $\sqrt{{\lambda }_m}\left|{\mathit{v}}_m\left(i\right)\right|$ and by the phase ${\theta }_i$ of ${\mathit{v}}_m\left(i\right)$] as a function of $D/J$, for clusters with $N=14$ (circles), 20 (squares), and 26 (triangles). The data corresponding to the four sites closest to the impurity are highlighted by filled symbols.

Figure 9

(Color online) Typical in-plane magnetization pattern (red arrows) and spin correlation functions (represented in blue by the bond thickness) for the minimal four-site cluster in the small-$D$ regime. Here $D=B=J/20$, $\theta =30°$, and the orientations of the DM interactions are the same as for sites 1–4 in Fig. 1.