Abstract
The ground state energy of a finite-dimensional one-parameter Hamiltonian and the continuity of a maximum-entropy inference map are discussed in the context of quantum critical phenomena. The domain of the inference map is a convex compact set in the plane, called the numerical range. We study the differential geometry of its boundary in relation to the ground state energy. We prove that discontinuities of the inference map correspond to C¹-smooth crossings of the ground state energy with a higher energy level. Discontinuities may appear only at C¹-smooth points of the boundary of the numerical range considered as a manifold. Discontinuities exist at all C²-smooth non-analytic boundary points and are essentially stronger than at analytic points or at points which are merely C¹-smooth (non-exposed points).