Geometry of metal-insulator transitions in one-dimension

We use the geometric approach to quantum critical points to study the metal-insulator transitions driven by chemical potential, $\mu$, or repulsion, $U$, in the one-dimensional Hubbard model. The transition to the band-insulator, as $\mu\rightarrow \mu_c,$ exhibits conventional scaling of the ground-state fidelity metric tensor $G_{\mu,\nu}\equiv{\rm Re}\left [\left \langle \partial_\mu\psi | \partial_\nu\psi\right \rangle - \left \langle \partial_\mu\psi |\psi\right\rangle\left\langle\psi| \partial_\nu\psi\right \rangle \right ]$. For example, the metric diverges as $G_{U,U}\sim 1/n$, where, $n\sim\sqrt{\mu-\mu_c}$, is the band filling. At the Mott transition, the metric behavior depends on the path of approach to the critical point.