Drude and superconducting weights and Mott transitions in variation theory

The Drude weight (D) is a useful measure for distinguishing a metal from an insulator. However, D has not been justifiably estimated using variation theory for a long time, since Millis and Coppersmith [Phys. Rev. B 43, 13770 (1991)] pointed out that the variational wave function Ψ_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 P_θ to Ψ_Q, Mott transitions are successfully represented by D (D = 0 for U > U_c) for normal and d-wave pairing states; thus, the problem of Millis and Coppersmith is solved. Generally, P_θ plays a pivotal role in describing current-carrying states in the regime of Mott physics. On the other hand, we show using perturbation theory that the one-body (mean-field) part of the wave function should be complex for band insulators such as antiferromagnetic states in hypercubic lattices.