Quantum phase transition induced by Dzyaloshinskii-Moriya interactions in the kagome antiferromagnet

We argue that the $S=1/2$ kagome antiferromagnet undergoes a quantum phase transition when the Dzyaloshinskii-Moriya coupling is increased. For $D<D_c$ the system is in a moment-free phase, and for $D>D_c$ the system develops antiferromagnetic long-range order. The quantum critical point is found to be $D_c\simeq 0.1J$ using exact diagonalizations and finite-size scaling. This suggests that the kagome compound ${\text{ZnCu}}_3{\left(\text{OH}\right)}_6{\text{Cl}}_3$ may be in a quantum critical region controlled by this fixed point.

Figure 1

(Color online) The kagome lattice with the allowed Dzyaloshinskii-Moriya interactions. The orientations of the bonds specify the order of the operators in ${\mathbf{S}}_i\times {\mathbf{S}}_j$. $m$ is a mirror plane.

Figure 2

(Color online) Low-energy levels of the Heisenberg kagome lattice with Dzyaloshinskii-Moriya interactions for $N=36$. The inset gives the slope of the lowest-energy states versus $S_z^2$, denoted by $1/\left(2I_N\right)$, as a function of $1/N$. $(D=0.1J.)$

Figure 3

(Color online) Finite-size extrapolation of the largest eigenvalue of $S_{ab}\left(\mathbf{Q}=0\right)$ showing (i) a scaling in $N^{-1/2}$; (ii) the decrease in the sublattice moment from $m_{\text{AF}}^2=0.326$ $\left(D=1,J=0\right)$ to zero $\left(D=0.1J\right)$ and negative values $\left(D=0.06,0.08J\right)$. Inset: finite-size scaling of the ground-state energy in $N^{-3/2}$ for $D=0.1J$.

Figure 4

The Néel order parameter as a function of $D/J$ obtained from extrapolations to infinite size. The system has no magnetic moment for $D<D_c\simeq 0.1J$ and has Néel order for $D>D_c$.