[Yokoyama-Hotta, PRB

The basic picture of spin nematics is that a quantum magnet may also act like a liquid crystal,
breaking spin-rotation symmetry without breaking time-reversal symmetry.
Such ``spin-nematics'' right now became a quite realistic phase of matter that may appear in
the cubic spinel CdCr_{2}O_{4}, 3He, FeSe, and volborthite.

The electrons in solids have spin-1/2 and when it is localized per orbital in the Mott insulators, form a quantum magnet interacting by the antiferromagnetic neighboring exchange as well as more complicated higher order spin exchange interactions. The order parameter of spin nematics is a rank-2 tensor, a quadrupole, with finite S but without having any directions like dipoles. To form this kind of quadrupole, a spin-1 is required, as the definition of this tensor does not allow for the spin-1/2 as a building block. In order to have such spin nematics in solid crystals, one needs to construct a spin-1 from spin-1/2: there are two ways. One is to define this spin-1 on bonds connecting two spin-1/2's. The other is to make use of the ferromagnetic couplings of two spins on the same site belonging to different orbitals (Hund's coupling).

The electrons in solids have spin-1/2 and when it is localized per orbital in the Mott insulators, form a quantum magnet interacting by the antiferromagnetic neighboring exchange as well as more complicated higher order spin exchange interactions. The order parameter of spin nematics is a rank-2 tensor, a quadrupole, with finite S but without having any directions like dipoles. To form this kind of quadrupole, a spin-1 is required, as the definition of this tensor does not allow for the spin-1/2 as a building block. In order to have such spin nematics in solid crystals, one needs to construct a spin-1 from spin-1/2: there are two ways. One is to define this spin-1 on bonds connecting two spin-1/2's. The other is to make use of the ferromagnetic couplings of two spins on the same site belonging to different orbitals (Hund's coupling).

The studies of spin-1/2 spin nematics are done along the former direction: starting from the ferromagnetically ordered state/or fully polarized state above the saturation field, and exciting sz=-1 magnons. When there are some reasons to bind a pair of magnons, they form a nematic order where these nematic objects itenerate and contribute to the Bose Einstein condensation. [Momoi, Shannon, Tsunetsugu, Penc, Lauhili (2005)]. The "reason" is attributed to the frustration effect: if there is a kinetic frustration effect on the lattices like triangles, the single magnon propagation is prohibited by the destructive interference effect. In such a case, in order to gain kinetic energy, it is better to bind two magnons and to propagate together. The frustration is introduced by the geometry of the lattice and by the ring exchange interactions.

However, in the above studies, one needs to polarize the magnets, which needs large fields (also ferromagnets are quite elusive in nature). We thus wanted to consider a more realistic situation that can easily be realized in laboratories. For that we consider a dumbell dimer unit consisting of spin-1/2 which are interacting antiferromagnetically. In such state, usually, the starting point is a spin singlet, S=0, on a dimer. Again an important issue here is, (1) how to generate spin-1 from a singlet, (2) How to form spin nematics, which is a strongly fluctuating/resonating (locally) spin-1's.

When the interactions between the singlets are absent, the singlet product state is the ground state, which we consider as a vacuum. Introducing inter-dimer interactions, one can dope the triplons(Sz=1,0,-1) in a unit of dimers into this sea of singlets. Then the triplons hop, interact, and also shows magnetic exchange when they occupy the neighboring dimers. The key to understanding the physics here is that this ring exchange interaction transforms to the bilinear-biquadratic interactions between S=1 pairs: in toy model, biquadratic interaction is known to stabilize the nematic order.

Now, considering (2) one needs a higher order exchange term in a unit of spin-1/2, which is the ring exchange interaction. This interaction makes a cyclic permutation of spins along the loops. We consider two different types of loops, square and twisted square that connects two dimers. By writing down the phase diagram on the plane of these two different types of ring exchange, we found that it is quite reasonable to replace the singlet S=1 to the spin-nematic phase. We also find a very exotic nematic kagome BEC phase: the occupancy of triplets is 3/4 on a triangle, and there is no symmetry breaking in terms of the configuration of triplets. Still, the triplons gain magnetic energy by forming a self-organized correlation along a kagome geometry by depleting one of the sites of the triangle.