- Activity Report 2016 -

Tsunetsugu Group

Spin Singlet Orders in Breathing Pyrochlores

Fig. 1. Nearest-neighbor spin correlations in the breathing-pyrochlore Heisenberg model with* S* = 3/2. Four squares depict small tetrahedron units in the cubic unit cell projected onto the *xy* plane. In each unit, antiferromagnetic correlations are shown by red bonds and their width schematically shows |*S*_{i}･*S*_{j}|, while blue bonds show ferromagnetic correlations. Dashed lines show weak antiferromagnetic correlations between neighboring units.

Geometrically frustrated magnets are a good playground in the quest for new quantum phases of matter, and kagome and pyrochlore magnets are their representatives. A few years ago, Okamoto *et al.* [1] discovered an unidentified phase transition in the spinel variety Li(Ga,In)Cr_{4}O_{8}, in which the magnetic Cr ions form a breathing pyrochlore lattice, and this has motivated a theoretical investigation of its origin.

The breathing pyrochlore lattice is a staggered network of corner-sharing tetrahedrons with two sizes, and increasing In concentration enhances the size difference. The antiferromagnetic Heisenberg Hamiltonian is a minimal model to study magnetic properties in this compound, and this requires two values of nearest-neighbor exchange coupling *J*’≪*J* corresponding to different tetrahedron units. This model was theoretically studied for the case of spin *S* = 1/2 [2] and it was predicted that the spin gap is finite and the ground state exhibits a complicated spatial modulation without breaking spin rotation symmetry. Among the four sublattices of small tetrahedra, three of them show dimer-pair orders of different pairing combination, and the remaining sublattice shows either a dimer-pair order or a tetramer order. Considering Cr^{3+} ions have a spin *S* = 3/2, the important issue is if this larger spin changes an order in the ground state.

To study this problem, I have developed a systematic scheme of degenerate perturbation theory for the breathing pyrochlore Heisenberg model with general spin *S*, and derived an effective Hamiltonian for describing dynamics in the spin singlet subspace. The effective Hamiltonian is in the order of (*J*’/*J*)^{3} and represented in terms of spin-pair operators τ, and we have studied its ground state by a mean field approximation [3]. The operators are defined in the local singlet space with dimension 2*S* + 1 at each small tetrahedron, and we have solved the challenge of calculating their matrix elements for general *S*. It turns out that an essential difference from the *S* = 1/2 case is the presence of Z_{3} anisotropy in the internal τ space, and this stabilizes a different order. The anisotropy grows for larger *S* and inherits the cubic symmetry of the lattice structure. Two sublattices of small tetrahedrons now show an identical tetramer order, and the other two sublattices show another tetramer order with a small distortion. We have analyzed the spin correlations in this new ordered state in detail for *S* = 3/2 and 1, and calculated the equal-time spin structure factor *S*(** q**). The amplitudes of the components breaking the cubic lattice symmetry in

**References**

- [1] Y. Okamoto, G. Nilsen, T. Nakazono,and Z. Hiroi, J. Phys. Soc. Jpn.
**84**, 043707 (2015). - [2] H. Tsunetsugu, J. Phys. Soc. Jpn.
**70**, 640 (2001); Phys. Rev. B**65**, 024415 (2001). - [3] H. Tsunetsugu, Prog. Theor. Exp. Phys.
**2017**, 033101 (2017).

**Auther**

- H. Tsunetsugu