Solvable model of quantum phase transitions and the symbolic-manipulation-based study of its multiply degenerate exceptional points and of their unfolding

•New elementary non-Hermitian quantum Hamiltonians with real spectra proposed.•Exceptional points found and studied.•Non-equivalent stability-loss patterns of phase transition identified.•Hermitization matrices of metrics ΘΘ constructed via symbolic manipulations and extrapolations at all NN and kk.

It is known that the practical use of non-Hermitian (i.e., typically, PTPT-symmetric) phenomenological quantum Hamiltonians H≠H†H≠H† requires an efficient reconstruction of an ad hoc   Hilbert-space metric Θ=Θ(H)Θ=Θ(H) which would render the time-evolution unitary. Once one considers just the NN-dimensional matrix toy models H=H(N)H=H(N), the matrix elements of Θ(H)Θ(H) may be defined via a coupled set of N2N2 polynomial equations. Their solution is a typical task for computer-assisted symbolic manipulations. The feasibility of such a model-completion construction is illustrated here via a discrete square well model H=p2+VH=p2+V endowed with a kk-parametric close-to-the-boundary interaction VV. The model is shown to possess (possibly, multiply degenerate) exceptional points marking the phase transitions which are attributable, due to the exact solvability of the model at any N<∞N<∞, to the loss of the regularity of the metric. In the parameter-dependence of the energy spectrum near these singularities one encounters a broad variety of alternative, topologically non-equivalent scenarios.