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 ${\mathrm{\Psi }}_Q$, 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$\Psi_Q$,Motttransitionsaresuccessfullyrepresentedby$D$($D=0$for$U>U{\rm c}$)foranormaland$d$-wave pairing states; thereby, the problem of Millis and Coppersmith is settled. Generally, ${\mathcal{P}}_{\theta }$ 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.