Signatures of quantum phase transitions from the boundary of the numerical range

We analyze the smoothness of the ground state energy of a one-parameter Hamiltonian by studying the differential geometry of the numerical range and continuity of the maximum-entropy inference. The domain of the inference map is the numerical range, a convex compact set in the plane. We show that its boundary, viewed as a manifold, has the same order of differentiability as 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. 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).