Hall number, optical sum rule and carrier density for thet−t′−Jmodel

We revisit the relationship between three classical measures of particle number, namely, the chemical doping $x$, the Hall number ${x}_{\mathit{\text{Hall}}}$, and the particle number inferred from the optical sum rule ${x}_{\mathit{opt}}$. We study the $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}J$ model of correlations on a square lattice, as a minimal model for high ${T}_{c}$ systems, using numerical methods to evaluate the low temperature Kubo conductivities. These measures disagree significantly in this type of system, owing to Mott Hubbard correlations. The Hall constant has a complex behavior with several changes of sign as a function of filling $x$, depending on the model parameters. Thus, ${x}_{\mathit{\text{Hall}}}$ depends sensitively on ${t}^{\ensuremath{'}}$ and $J$ due to a kind of quantum interference.