Drude and Superconducting Weights and Mott Transitions in Variation Theory

Drude weight ($D$) is a useful measure to distinguish a metal from an insulator. However, $D$ has not been justifiably estimated by the variation theory for long, since Millis and Coppersmith [Phys. Rev. B 43 (1991) 13770] pointed out that a variational wave function $\Psi_Q$, which includes the key ingredient (doublon-holon binding effect) for a Mott transition, yields a positive $D$ (namely metallic) even in the Mott-insulating regime. We argue that, to obtain a correct $D$, an imaginary part must exist in the wave function. By introducing a configuration-dependent phase factor ${\cal P}_\theta$ to $\Psi_Q$, Mott transitions are successfully represented by $D$ ($D=0$ for $U>U_{\rm c}$) for a normal and $d$-wave pairing states; thereby, the problem of Millis and Coppersmith is settled. Generally, ${\cal P}_\theta$ plays a pivotal role in describing current-carrying states in regimes of Mott physics. On the other hand, we show using a perturbation theory, the one-body (mean-field) part of the wave function should be complex for band insulators such as antiferromagnetic states in hypercubic lattices.