Quantum transport in one dimension: from integrability to many-body localization and topology

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Recent advances in analytical theory and numerical methods
enable some long-standing questions about transport in one dimension to be
answered; these questions are closely related to transport experiments in
quasi-1D compounds.  The spinless fermion chain with nearest-neighbor
interactions at half-filling, or equivalently the XXZ model in zero magnetic
field, is an example of an integrable system in which no conventional conserved
quantity forces dissipationless transport (Drude weight); we show that there is
nevertheless a Drude weight and that at some points its contribution is from a
new type of conserved quantity recently constructed by Prosen.  Adding an
integrability-breaking perturbation leads to a scaling theory of conductivity
at low temperature.  Adding disorder, we study the question of how
Anderson localization is modified by interactions when the system remains fully
quantum coherent ("many-body localization").  We find that even
weak interactions are a singular perturbation on some quantities: entanglement
grows slowly but without limit, suggesting that dynamics in the possible
many-body localized phase are glass-like.  If time permits, some results
on the fractional Luttinger's theorem and the 1D limit of quantum Hall states
will be presented.