Solvable non-Hermitian skin effect in many-body unitary dynamics

We study unitary evolution of bipartite entanglement in a circuit with nearest-neighbor random gates. Deriving compact nonunitary description purity dynamics on qudits we find sudden transition the relaxation rate origin which is underlying boundary localized eigenmodes---the skin effect. provide full solution problem, being one simplest iterations two-site matrices, namely, that each sum only two projectors. This leads to rich influenced by Jordan normal form kernel and, most importantly, spectrum completely discontinuous thermodynamic limit. It provides simple example how seemingly innocuous many-body can harbor interesting mathematical effects: an effective nonsymmetric Toeplitz transfer matrix causes phantom relaxation, such correct not given spectrum, but rather its pseudospectrum.