The Grüneisen ratio $\mathrm{\Gamma }$, 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 $\mathrm{\Gamma }$ cannot be employed. We propose a quantum analog to $\mathrm{\Gamma }$ that computes entanglement as a function of a tuning parameter $\lambda$ and show that QPTs take place only for systems in which the ground-state energy depends on $\lambda$ 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 mass” close to any QCP/QPT is also discussed.

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• Magnetic order
• Quantum computation
• Quantum phase transitions