The origin of trimerized phase in S=1 kagome magnet Na2Ti3Cl8

The kagome magnet Na2Ti3Cl8 with S=1 spins undergoes a structural transition involving trimerization or breathing distortion of the underlying kagome lattice as the temperature lowers. We investigate the magnetic ground state of this compound by constructing a model spin Hamiltonian based on first-principles density functional theory and analyzing it using exact diagonalization and density matrix renormalization group methods. Along with the nearest neighbor Heisenberg and biquadratic exchanges, we propose a new exchange term fourth order in spins - ring exchange that does not originate from spin-orbit coupling and involves simultaneous hopping of electrons from one site to its two neighboring sites. The parameters that our first principles calculations give for the high temperature crystal structure place this compound to the ferroquadrapolar region of the phase diagram predicted by exact diagonalization and density matrix renormalization group calculations. We thus suggest that a spin - lattice coupling favors the trimerized phase over the quadrapolar phase, even though density functional theory alone does not predict a lattice instability.