Spin--lattice coupling and the emergence of the trimerized phase in the $S=1$ Kagome antiferromagnet Na$_2$Ti$_3$Cl$_8$

Spin-1 antiferromagnets are abundant in nature, but few theories or results exist to understand their general properties and behavior, particularly in situations when geometric frustration is present. Here we study the $S=1$ Kagome compound Na$_2$Ti$_3$Cl$_8$ using a combination of Density Functional Theory, Exact Diagonalization, and Density Matrix Renormalization Group methods to achieve a first principles supported explanation of exotic magnetic phases in this compound. We find that the effective magnetic Hamiltonian includes essential non-Heisenberg terms that do not stem from spin-orbit coupling, and both trimerized and spin-nematic magnetic phases are relevant. The experimentally observed structural transition to a breathing Kagome phase is driven by spin--lattice coupling, which favors the trimerized magnetic phase against the quadrupolar one. We thus show that lattice effects can be necessary to understand the magnetism in frustrated magnetic compounds, and surmise that Na$_2$Ti$_3$Cl$_8$ is a compound which cannot be understood from only electronic or only lattice Hamiltonians, very much like VO$_2$.