Lower bounds for the conductivities of correlated quantum systems

We show how one can obtain a lower bound for the electrical, spin, or heat conductivity of correlated quantum systems described by Hamiltonians of the form H=H₀+gH₁. Here, H₀ is an interacting Hamiltonian characterized by conservation laws which lead to an infinite conductivity for g=0. The small perturbation gH₁, however, renders the conductivity finite at finite temperatures. For example, H₀ could be a continuum field theory, where momentum is conserved, or an integrable one-dimensional model, while H₁ might describe the effects of weak disorder. In the limit g→0, we derive lower bounds for the relevant conductivities and show how they can be improved systematically using the memory matrix formalism. Furthermore, we discuss various applications and investigate under what conditions our lower bound may become exact.