One of the central open theoretical problems in the solid state physics is the understanding of strongly correlated electrons, where the properties are dominated by strong electron-electron repulsion and Pauli exclusion principle. We discuss the quantum electronic transport in such systems , in particular electrical conductivity, spin diffusion and heat conductivity. The concept of charge stiffness is introduced which makes qualitative distinction between conductors and insulators in the quantum ground state, while at finite temperatures it leads to possibilities of usual resistors, but also of anomalous ideal conductors and ideal insulators [2,3]. It is shown that the singular transport appears in many integrable systems of interacting fermions, even when the current is not a conserved quantity. The evidence comes from the relation with level dynamics , the existence of conserved quantities , from exact results as well as from numerical studies of small correlated systems using exact diagonalization method and finite-temperature Lanczos method . Several open theoretical problems in this connection will be addressed: a) necessary ingredients for the quantum dissipationless transport, b) transport in systems close to integrability, and c) the existence of ideal insulators at finite temperatures.
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