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

## TL;DR

This paper investigates how anisotropy in spin molecular-orbital coupling influences effective spin models in layered trinuclear organometallic complexes, revealing phase transitions and anisotropic magnetic behaviors.

## Contribution

It introduces a detailed effective spin model incorporating anisotropic SMOC effects and predicts phase transitions in these complex materials.

## Key findings

- Effective XXZ + 120° honeycomb quantum compass model derived
- Strong anisotropies lead to a transition from Haldane to D-phase
- External magnetic fields induce a transition from Haldane to magnetic order

## Abstract

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.