Magnetically Hidden Order of Kramers Doublets in $d^1$ Systems: ${\mathrm{Sr}}_2{\mathrm{VO}}_4$

We formulate and study an effective Hamiltonian for low-energy Kramers doublets of $d^1$ ions on a square lattice. We find that the system exhibits a magnetically hidden order in which the expectation values of the local spin and orbital moments both vanish. The order parameter responsible for a time-reversal symmetry breaking has a composite nature and is a spin-orbital analog of a magnetic octupole. We argue that such a hidden order is realized in the layered perovskite ${\mathrm{Sr}}_2{\mathrm{VO}}_4$.

Figure 1

(a) A schematic view of the local level structure: the $t_{2g}$ level is split into three sets of Kramers doublets by a tetragonal crystal field ${\Delta }_{\mathrm{cf}}$ and a spin-orbit coupling $\lambda$. (b) Density profile of an electron in the ground state doublet. The chirality of the isospin up wave function, associated with the gradient of its phase, is shown by small arrows. The electron spin $\overrightarrow{S}$ and orbital angular momentum $\overrightarrow{L}$ are coupled antiparallel to each other.

Figure 2

Left: The dispersions of elementary excitations along the direction $X(\pi ,0)\to M(\pi ,\pi )\to \Gamma$ in the Brillouin zone. The lower (upper) curve corresponds to the isospin (orbital) waves. Right: The spectrum of in-plane, $M^x$ component, magnetic excitations. It consists of a continuum of composite, isospin-orbital excitations (shaded area), and a quasiparticle part (lower curve, the width scales with intensity). The corresponding diagrams are also shown: The solid (wavy) lines denote the isospin (orbital) excitations and the open dots stand for the three-particle vertex in Eq. (6). Energies are given in units of $J$.

Figure 3

The isospin $m$ and orbital $\sigma$ order parameters, isospin exchange $J_s$, and orbital gap $\Delta$ versus temperature. The broken lines mark corresponding jumps at the first-order transition.