Effects of anisotropy in spin molecular-orbital coupling on effective spin models of trinuclear organometallic complexes

We consider layered decorated honeycomb lattices at two-thirds filling, as realized in some trinuclear organometallic complexes. Localized $S=1$ moments with a single-spin anisotropy emerge from the interplay of Coulomb repulsion and spin molecular-orbit coupling (SMOC). Magnetic anisotropies with bond dependent exchange couplings occur in the honeycomb layers when the direct intracluster exchange and the spin molecular-orbital coupling are both present. We find that the effective spin exchange model within the layers is an XXZ + 120$^\circ$ honeycomb quantum compass model. The intrinsic non-spherical symmetry of the multinuclear complexes leads to very different transverse and longitudinal spin molecular-orbital couplings, which greatly enhances the single-spin and exchange coupling anisotropies. The interlayer coupling is described by a XXZ model with anisotropic biquadratic terms. As the correlation strength increases the systems becomes increasingly one-dimensional. Thus, if the ratio of SMOC to the interlayer hopping is small this stabilizes the Haldane phase. However, as the ratio increases there is a quantum phase transition to the topologically trivial `$D$-phase'. We also predict a quantum phase transition from a Haldane phase to a magnetically ordered phase at sufficiently strong external magnetic fields.Comment: 22 pages, 11 figures. Final version of paper to be published in PRB. Important corrections to appendix

Heisenberg and Dzyaloshinskii-Moriya interactions controlled by molecular packing in tri-nuclear organometallic clusters

Motivated by recent synthetic and theoretical progress we consider magnetism in crystals of multi-nuclear organometallic complexes. We calculate the Heisenberg symmetric exchange and the Dzyaloshinskii-Moriya antisymmetric exchange. We show how, in the absence of spin-orbit coupling, the interplay of electronic correlations and quantum interference leads to a quasi-one dimensional effective spin model in a typical tri-nuclear complex, Mo$_3$S$_7$(dmit)$_3$, despite its underlying three dimensional band structure. We show that both intra- and inter-molecular spin-orbit coupling can cause an effective Dzyaloshinskii-Moriya interaction. Furthermore, we show that, even for an isolated pair of molecules the relative orientation of the molecules controls the nature of the Dzyaloshinskii-Moriya coupling. We show that interference effects also play a crucial role in determining the Dzyaloshinskii-Moriya interaction. Thus, we argue, that multi-nuclear organometallic complexes represent an ideal platform to investigate the effects of Dzyaloshinskii-Moriya interactions on quantum magnets.Comment: This update incorporates the corrections described in a recently submitted erratum. Changes are confined to sections IV.A and B. The conclusions of the paper are unchanged. 12 + 4 pages, 9 figure

Spin-orbit coupling in {Mo3_3S7_7(dmit)3_3}

Spin-orbit coupling in crystals is known to lead to unusual direction dependent exchange interactions, however understanding of the consequeces of such effects in molecular crystals is incomplete. Here we perform four component relativistic density functional theory computations on the multi-nuclear molecular crystal {Mo$_3$S$_7$(dmit)$_3$} and show that both intra- and inter-molecular spin-orbit coupling are significant. We determine a long-range relativistic single electron Hamiltonian from first principles by constructing Wannier spin-orbitals. We analyse the various contributions through the lens of group theory. Intermolecular spin-orbit couplings like those found here are known to lead to quantum spin-Hall and topological insulator phases on the 2D lattice formed by the tight-binding model predicted for a single layer of {Mo$_3$S$_7$(dmit)$_3$}

Galois groups of multivariate Tutte polynomials

The multivariate Tutte polynomial $\hat Z_M$ of a matroid $M$ is a generalization of the standard two-variable version, obtained by assigning a separate variable $v_e$ to each element $e$ of the ground set $E$. It encodes the full structure of $M$. Let \bv = \{v_e\}_{e\in E}, let $K$ be an arbitrary field, and suppose $M$ is connected. We show that $\hat Z_M$ is irreducible over K(\bv), and give three self-contained proofs that the Galois group of $\hat Z_M$ over K(\bv) is the symmetric group of degree $n$, where $n$ is the rank of $M$. An immediate consequence of this result is that the Galois group of the multivariate Tutte polynomial of any matroid is a direct product of symmetric groups. Finally, we conjecture a similar result for the standard Tutte polynomial of a connected matroid.Comment: 8 pages, final version, to appear in J. Alg. Comb. Substantial revisions, including the addition of two alternative proofs of the main resul

Haldane phase in the hubbard model at 2/3-filling for the organic molecular compound Mo3 S7 (dmit)3

We report the discovery of a correlated insulator with a bulk gap at 2/3 filling in a geometrically frustrated Hubbard model that describes the low-energy physics of Mo3S7(dmit)(3). This is very different from the Mott insulator expected at half-filling. We show that the insulating phase, which persists even for very weak electron-electron interactions (U), is adiabatically connected to the Haldane phase and is consistent with experiments on Mo3S7(dmit)(3)

Low-energy effective theories of the two-thirds filled Hubbard model on the triangular necklace lattice

Motivated by Mo3S7(dmit)(3), we investigate the Hubbard model on the triangular necklace lattice at two-thirds filling. We show, using second-order perturbation theory, that in the molecular limit, the ground state and the low-energy excitations of this model are identical to those of the spin-one Heisenberg chain. The latter model is known to be in the symmetry-protected topological Haldane phase. Away from this limit we show, on the basis of symmetry arguments and density matrix renormalization group (DMRG) calculations, that the low-energy physics of the Hubbard model on the triangular necklace lattice at two-thirds filling is captured by the ferromagnetic Hubbard-Kondo lattice chain at half-filling. This is consistent with and strengthens previous claims that both the half-filled ferromagnetic Kondo lattice model and the two-thirds filled Hubbard model on the triangular necklace lattice are also in the Haldane phase. A connection between Hund's rules and Nagaoka's theorem is also discussed