in neutron star matter, it becomes energetically favourable to replace neutrons by $\Lambda$ hyperons via weak interactions. For a baryonic species $j$, the chemical potential $\mu_j$ is determined by the derivative of the energy density ${\cal E}$ with respect to the partial density $\rho_j$ of that species: $$\mu_j = {\partial{\cal E}\over\partial\rho_j}~.$$ Given a realistic neutron star equation-of-state, $P({\cal E})$, the Gibbs-Duhem relation for multicomponet sytems $$P({\cal E}) + {\cal E} = \rho_n\,\mu_n + \rho_p\,\mu_p + \rho_\Lambda\,\mu_\Lambda\,+ \dots~~.$$ determines the chemical potentials. In practice we use the microscopic EoS computed from a chiral nucleon-meson field theory in combination with functional renormalization group methods [@DW2017; @FW2019]. This EoS is consistent with all important nuclear physics constraints and neutron star observations. For simplicity we neglect the small (few percent) proton fraction from beta equilibrium which is part of the neutron star EoS, but its effect is marginal in the present context.

At the possible onset of $\Lambda$ hyperons, their kinetic energy vanishes as there is no $\Lambda$ Fermi sea to start with. Therefore the minimal $\Lambda$ chemical potential at that point is simply given by $$\mu_\Lambda(\rho) = M_\Lambda + U_\Lambda(\rho)~,
\label{eq:mulambda}$$

where $M_\Lambda$ is the $\Lambda$ hyperon mass and $\rho\simeq\rho_n$ is the neutron 