Shekhtman, Entin-Wohlman, and Aharony reply

Quantum phase transitions in the spin-1/2 XXZ model with staggered γ interaction

The one-dimensional spin-1/2 XXZ model with the limited case of the staggered Gamma interaction which includes the staggered DM and KSEA interactions is investigated by utilizing the infinite-time evolving block decimation method in the infinite matrix product state representation. It is shown that the staggered DM interaction does not lead to any new phase but expands the XY phase. Contrasted to the DM interaction, the region of the ferromagnetic phase expands into the nematic-ferromagnetic and dimer-Haldane phases with the deconfined quantum phase transition happening in the case of KSEA interaction. We observe a pronounced effect in the entanglement entropy and Schmidt gap, the odd-bond and even-bond von Neumann entropies exhibit different values, and the value of the Schmidt gap on the even bond becomes zero in the dimer-Haldane phase. This implies that quantum entanglement and the Schmidt gap can distinguish the two degenerate ground states caused by the existence of dimerized interaction. The zero value of the Schmidt gap works as a sign of the topological phase, which is also evidenced by the string correlation. From the perspective of magnetism, the existence of chiral order and nematic order suggests the helical magnetic structure induced by Gamma interaction. By comparing the DM and KSEA interactions, it is found that DM interaction does not change ferromagnetism, whereas the KSEA interaction inhibits ferromagnetism and destroys the product state in the ferromagnetic phase. Furthermore, the behaviors of the spin structure factor for the transversal spin correlation can distinguish different phases. From an appropriate fitting exponent of the spin correlation oscillating decay, commensurate or incommensurate phases can be identified.

Dzyaloshinskii–Moriya Coupling in 3d Insulators

We present an overview of the microscopic theory of the Dzyaloshinskii–Moriya (DM) coupling in strongly correlated 3d compounds. Most attention in the paper centers around the derivation of the Dzyaloshinskii vector, its value, orientation, and sense (sign) under different types of the (super)exchange interaction and crystal field. We consider both the Moriya mechanism of the antisymmetric interaction and novel contributions, in particular, that of spin–orbital coupling on the intermediate ligand ions. We have predicted a novel magnetic phenomenon, weak ferrimagnetism in mixed weak ferromagnets with competing signs of Dzyaloshinskii vectors. We revisit a problem of the DM coupling for a single bond in cuprates specifying the local spin–orbital contributions to the Dzyaloshinskii vector focusing on the oxygen term. We predict a novel puzzling effect of the on-site staggered spin polarization to be a result of the on-site spin–orbital coupling and the cation-ligand spin density transfer. The intermediate ligand nuclear magnetic resonance (NMR) measurements are shown to be an effective tool to inspect the effects of the DM coupling in an external magnetic field. We predict the effect of a strong oxygen-weak antiferromagnetism in edge-shared CuO 2 chains due to uncompensated oxygen Dzyaloshinskii vectors. We revisit the effects of symmetric spin anisotropy directly induced by the DM coupling. A critical analysis will be given of different approaches to exchange-relativistic coupling based on the cluster and the DFT (density functional theory) based calculations. Theoretical results are applied to different classes of 3d compounds from conventional weak ferromagnets ( α -Fe 2 O 3 , FeBO 3 , FeF 3 , RFeO 3 , RCrO 3 , ...) to unconventional systems such as weak ferrimagnets (e.g., RFe 1 - x Cr x O 3 ), helimagnets (e.g., CsCuCl 3 ), and parent cuprates (La 2 CuO 4 , ...).