Grüneisen parameter as an entanglement compass and the breakdown of the Hellmann-Feynman theorem

The Grüneisen ratio Γ, i.e., the singular part of the ratio of thermal expansion to the specific heat, has been broadly employed to explore both finite T and quantum critical points (QCPs). For a genuine quantum phase transition (QPT), thermal fluctuations are absent and thus the thermodynamic Γ cannot be employed. We propose a quantum analog to Γ that computes entanglement as a function of a tuning parameter λ and show that QPTs take place only for systems in which the ground-state energy depends on λ nonlinearly. Furthermore, we demonstrate the breakdown of the Hellmann-Feynman theorem in the thermodynamic limit at any QCP. We showcase our approach using the quantum one-dimensional Ising model with a transverse field and Kane's quantum computer. The slowing down of the dynamics and thus the creation of massclose to any QCP/QPT is also discussed.