A quantum phase transition is a ground state phenomenon in an infinite lattice model with various defining key properties. We focus on two of them and point out their signatures in finite lattice models: Nonanalyticity of the ground state energy and strong variation of the entropy of inferred states. We study the numerical range W associated with the two energy terms of a oneparameter Hamiltonian (two arbitrary hermitian matrices). We show that parameter values with a C^2 smooth nonanalytic ground state energy correspond to C^2 smooth nonanalytic points of the boundary of W (viewed as a manifold). At these boundary points, the maximumentropy inference map, constrained on the expectation values of the energy operators, is discontinuous. We conclude that a C^2 smooth nonanalytic boundary point of W is a geometric signature of a quantum phase transition. On the way, we obtain a classification of the maximal order of differentiability of all boundary points of W.This is joint work with Ilya M. Spitkovsky (NYU Abu Dhabi, UAE). Reference: arXiv:1703.00201 [mathph]
