**Quantum Phase Transitions and Entanglement** - Junru Wu

We develop a general theory of the relation between quantum phase transitions (QPTs) characterized by nonanalyticities in the energy and entanglement measures. We notice a functional relation between the matrix elements of two-particle reduced density matrices and the eigenvalues of general two-body Hamiltonians of $d$-level systems. The ground state energy eigenvalue and its derivatives, whose non-analyticity characterizes a QPT, are directly tied to bipartite entanglement measures. We show that first-order QPTs are signalled by density matrix elements themselves and second-order QPTs by the first derivative of density matrix elements. Our general conclusions are illustrated via several quantum spin models.