Thermodyamic Bounds on Drude Weights in Terms of Almost-conserved Quantities

We consider one-dimensional translationally invariant quantum spin (or fermionic) lattices and prove a Mazur-type inequality bounding the time-averaged thermodynamic limit of a finite-temperature expectation of a spatio-temporal autocorrelation function of a local observable in terms of quasi-local conservation laws with open boundary conditions. Namely, the commutator between the Hamiltonian and the conservation law of a finite chain may result in boundary terms only. No reference to techniques used in Suzuki's proof of Mazur bound is made (which strictly applies only to finite-size systems with exact conservation laws), but Lieb-Robinson bounds and exponential clustering theorems of quasi-local C* quantum spin algebras are invoked instead. Our result has an important application in the transport theory of quantum spin chains, in particular it provides rigorous non-trivial examples of positive finite-temperature spin Drude weight in the anisotropic Heisenberg XXZ spin 1/2 chain (Prosen, in Phys Rev Lett 106:217206, 2011).