Classification of excited-state quantum phase transitions for arbitrary number of degrees of freedom

•Classical stationary points cause singularities in quantum level density & dynamics.•An analysis for general Hamiltonian forms and arbitrary dimensions is presented.•For quadratic stationary points a classification is given in terms of Morse theory.•Results are applicable in the theory of Excited-State Quantum Phase Transitions.

Classical stationary points of an analytic Hamiltonian induce singularities of the density of quantum energy levels and their flow with a control parameter in the system's infinite-size limit. We show that for a system with f degrees of freedom, a non-degenerate stationary point with index r causes a discontinuity (for r even) or divergence (r   odd) of the (f−1)(f−1) th derivative of both density and flow of the spectrum. An increase of flatness for a degenerate stationary point shifts the singularity to lower derivatives. The findings are verified in an f=3f=3 toy model.