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

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.

Figure 1

Cross ($\partial {E_0}^2/\partial J\partial B$) and second $-B({\partial }^2{E}_0/\partial B^2$) derivatives of the ground-state energy $E_0$ (black line) and the von Neumann entropy $S_N$ as a function of $\lambda =J/B$. The gray region indicates the $\lambda$ range where both $E_0$ derivatives and $S_N$ vary more expressively with $\lambda$. The spin configurations depict the ferromagnetic phase for $\lambda <1$ and the quantum paramagnet close to $\lambda =1$. Inset: $S_N$ vs $\lambda$ for large values of $\lambda$, i.e., for $S_N\to 1$, the so-called Greenberger-Horne-Zeilinger-like state is achieved [30]. The white bullet depicted in $S_N$ vs $\lambda$ both in the main panel and inset indicates $S_N\to \infty$ for $\lambda =1$.

Figure 2

Schematic representation of the here-proposed optimization of quantum information processing for the 1D IMTF. (a) For $B\to 0$ $\Rightarrow$ $\lambda \to \infty$, the ferromagnetic phase state is the Greenberger-Horne-Zeilinger-like state (GHZ) [30], i.e., a superposition of $\uparrow$ and $\downarrow$ states. The von Neumann entropy $S_N$ is insensitive to $\lambda$ changes. (b) Upon applying the transverse field, the system is brought to the quantum paramagnetic phase, i.e., $\lambda \to 1$, where the system state is defined as the product of each eigenstate and $|+\rangle =1/\sqrt{2}\left(\right|\uparrow \rangle +|\downarrow \rangle )$ [35], which corresponds to the simplest entangled state. In this regime, $S_N$ dramatically changes upon varying $\lambda$. For a real system, one has to properly set the initial state [(a)] before applying a transverse $B$ to achieve the here-discussed optimal conditions for quantum computing.

Figure 3

Cross $(\partial E^2/\partial J^{\prime }\partial B)$ and second $-J^{\prime }({\partial }^{2}E/\partial J^{\prime 2})$ derivatives of Eq. (9) for Kane's quantum computer using $B=1$ T. See details in the main text.