Universality in the tripartite information after global quenches: (generalised) quantum XY models

We consider the R\'enyi-$\alpha$ tripartite information $I_3^{(\alpha)}$ of three adjacent subsystems in the stationary state emerging after global quenches in noninteracting spin chains from both homogeneous and bipartite states. We identify settings in which $I_3^{(\alpha)}$ remains nonzero also in the limit of infinite lengths and develop an effective quantum field theory description of free fermionic fields on a ladder. We map the calculation into a Riemann-Hilbert problem with a piecewise constant matrix for a doubly connected domain. We find an explicit solution for $\alpha=2$ and an implicit one for $\alpha>2$. In the latter case, we develop a rapidly convergent perturbation theory that we use to derive analytic formulae approximating $I_3^{(\alpha)}$ with outstanding accuracy.