  Multiterminal Josephson junctions have recently been proposed as a route to
artificially mimic topological matter with the distinct advantage that its
properties can be controlled via the superconducting phase difference, giving
rise to Weyl points in 4-terminal geometries. A key goal is to accurately
determine when the system makes a transition from a gapped to non-gapped state
as a function of the phase differences in the system, the latter effectively
playing the role of quasiparticle momenta in conventional topological matter.
We here determine the proximity gap phase diagram of diffusive n-terminal
Josephson junctions, both numerically and analytically, by identifying a class
of solutions to the Usadel equation at zero energy in the full proximity effect
regime. We present an analytical equation which provides the phase diagram for
an arbitrary number of terminals n. After briefly demonstrating the validity of
the analytical approach in the previously studied 2- and 3-terminal cases, we
focus on the 4-terminal case and map out the regimes where the electronic
excitations in the system are gapped and non-gapped, respectively,
demonstrating also in this case full agreement between the analytical and
numerical approach.
