Dzyaloshinsky-Moriya-induced order in the spin-liquid phase of the $S=1∕2$ pyrochlore antiferromagnet

We show that the $S=1∕2$ pyrochlore lattice with both Heisenberg and antisymmetric Dzyaloshinsky-Moriya (DM) interactions can order antiferromagnetically into a state with chiral symmetry, dictated by the distribution of the DM interactions. The chiral antiferromagnetic state is characterized by a small staggered magnetic moment induced by the DM interaction. An external magnetic field can also lead to characteristic field-induced ordering patterns, strongly dependent on the field direction, and generally separated by a quantum phase transition from the chiral ordered phase. The phase diagram at finite temperature is also discussed.

Figure 1

(a) Pyrochlore lattice. (b) Distribution of DM vectors on a single tetrahedron (four of the six shown, see text).

Figure 2

(Color online) (a) A fcc lattice of tetrahedra $(\text{tetrahedron}=\text{dot})$ with interactions $J^{\prime }$ between them. (b) Antiferromagnetic chiral order on the blue (dark gray) tetrahedra, with magnetic moment $\mid ⟨\mathbf{S}⟩\mid \sim \tilde{D}$, induced by the DM interactions. On the gray tetrahedra (labeled as 1,2,3) the order has the same symmetry, but is much weaker $\mid ⟨\mathbf{S}⟩\mid \sim {\tilde{D}}^3⪡\tilde{D}$, Eq. (9), and is not shown.

Figure 3

(Color online) (a), (b), and (c) Evolution of magnetic order on the blue (dark gray) tetrahedra of Fig. 2b in an external magnetic field in the (1, 1, 0) direction. Upper row: field-induced order on the rest of the tetrahedra. The tetrahedra are labeled 0,1,2,3 as in Figs. 2a, 2b. Blue (dark gray) arrows in (a) and (b) correspond to moments $\mid ⟨\mathbf{S}⟩\mid \sim \tilde{D}$, while the red (light gray) arrows on the upper row and (b) and (c) are the field- induced moments $\mid ⟨\mathbf{S}⟩\mid \sim \tilde{D}\tilde{H}$.

Figure 4

(Color online) Same as Fig. 3, but for an external magnetic field in the (0, 0, 1) direction. Blue (dark gray) arrows in (a) correspond to the DM-induced order with $\mid ⟨\mathbf{S}⟩\mid \sim \tilde{D}$, red (light gray) arrows on the upper row and (c) correspond to the field-induced component $\mid ⟨\mathbf{S}⟩\mid \sim \tilde{D}\tilde{H}$, and black arrows in (b) are a mixture of the two.

Figure 5

Schematic phase diagram at nonzero temperature in the presence of small magnetic field $(H⪡J)$ and DM interactions.