Order from disorder: Quantum spin gap in magnon spectra of ${\mathrm{LaTiO}}_3$

A theory of the anisotropic superexchange and low-energy spin excitations in a Mott insulator with $t_{2g}$ orbital degeneracy is presented. We observe that the spin-orbit coupling induces frustrating Ising-like anisotropy terms in the spin Hamiltonian, which invalidate noninteracting spin-wave theory. The frustration of classical states is resolved by an order from disorder mechanism, which selects a particular direction of the staggered moment and generates a quantum spin gap. The theory explains well the observed magnon gaps in ${\mathrm{LaTiO}}_3.$ As a test case, a specific prediction is made on the splitting of magnon branches at certain momentum directions.