Transport and optical conductivity in the Hubbard model: A high-temperature expansion perspective

Abstract

We derive analytical expressions for the spectral moments of the dynamical response functions of the Hubbard model using the high-temperature series expansion. We consider generic dimension $d$ as well as the infinite-$d$ limit, arbitrary electron density $n$, and both finite and infinite repulsion $U$. We use moment-reconstruction methods to obtain the one-electron spectral function, the self-energy, and the optical conductivity. They are all smooth functions at high temperature and, at large $U$, they are featureless with characteristic widths of the order of the lattice hopping parameter $t$. In the infinite-$d$ limit, we compare the series expansion results with accurate numerical renormalization group and interaction expansion quantum Monte Carlo results. We find excellent agreement down to surprisingly low temperatures, throughout most of the bad-metal regime, which applies for $T\ensuremath{\gtrsim}(1\ensuremath{-}n)D$, the Brinkman-Rice scale. The resistivity increases linearly in $T$ at high temperature without saturation. This results from the $1/T$ behavior of the compressibility or kinetic energy, which play the role of the effective carrier number. In contrast, the scattering time (or diffusion constant) saturates at high $T$. We find that $\ensuremath{\sigma}(n,T)\ensuremath{\approx}(1\ensuremath{-}n)\ensuremath{\sigma}(n=0,T)$ to a very good approximation for all $n$, with $\ensuremath{\sigma}(n=0,T)\ensuremath{\propto}t/T$ at high temperatures. The saturation at small $n$ occurs due to a compensation between the density dependence of the effective number of carriers and that of the scattering time. The $T$ dependence of the resistivity displays a kneelike feature which signals a crossover to the intermediate-temperature regime where the diffusion constant (or scattering time) starts increasing with decreasing $T$. At high temperatures, the thermopower obeys the Heikes formula, while the Wiedemann-Franz law is violated with the Lorenz number vanishing as $1/{T}^{2}$. The relevance of our calculations to experiments probing high-temperature transport in materials with strong electronic correlations or ultracold atomic gases in an optical lattice is briefly discussed.