Spin-orbit interaction driven dimerization in one dimensional frustrated magnets

Spin nematic ordering has been proposed to emerge near the saturation of field of a class of frustrated magnets. The experimental observation of this novel phase is challenging for the traditional experimental probes. Nematic spin ordering is expected to induce a local quadrupolar electric moment via the spin-orbit coupling. However, a finite spin-orbit interaction explicitly breaks the U(1) symmetry of global spin rotations down to $Z_2$, which renders the traditional nematic order no longer well-defined. In this work we investigate the relevant effect of spin-orbit interaction on the 1D frustrated $J_1-J_2$ model. The real and the imaginary parts of the nematic order parameter belong to different representations of the discrete symmetry group of the new Hamiltonian. We demonstrate that spin-orbit coupling stabilizes the real component and simultaneously induces bond dimerization in most of the phase diagram. Such a bond dimerization can be observed with X-rays or nuclear magnetic resonance. In addition, an incommensurate bond-density wave (ICBDW) appears for smaller values of $J_2/|J_1|$. The experimental fingerprint of the ICBDW is a double-horn shape of the the NMR line. These conclusions can shed light on the experimental search of this novel phase.