Threetangle in the XY-model class with a non-integrable field background

The square root of the threetangle is calculated for the transverse XY-model with an integrability-breaking in-plane field component. To be in a regime of quasi-solvability of the convex roof, here we concentrate here on a 4-site model Hamiltonian. In general, the field and hence a mixing of the odd/even sectors, has a detrimental effect on the threetangle, as expected. Only in a particular spot of models with no or weak inhomogeneity $\gamma$ does a finite value of the tangle prevail in a broad maximum region of the field strength $h\approx 0.3\pm 0.1$. There, the threetangle is basically independent of the non-zero angle $\alpha$. This system could be experimentally used as a quasi-pure source of threetangled states or as an entanglement triggered switch depending on the experimental error in the field orientation.