Exactly solvable $\mathcal{P}\mathcal{T}$-symmetric models in two dimensions

Non-hermitian, $\mathcal{P}\mathcal{T}$-symmetric Hamiltonians, experimentally realized in optical systems, accurately model the properties of open, bosonic systems with balanced, spatially separated gain and loss. We present a family of exactly solvable, two-dimensional, $\mathcal{P}\mathcal{T}$ potentials for a non-relativistic particle confined in a circular geometry. We show that the $\mathcal{P}\mathcal{T}$ symmetry threshold can be tuned by introducing a second gain-loss potential or its hermitian counterpart. Our results explicitly demonstrate that $\mathcal{P}\mathcal{T}$ breaking in two dimensions has a rich phase diagram, with multiple re-entrant $\mathcal{P}\mathcal{T}$ symmetric phases.