  Non-ergodic dynamical systems display anomalous transport properties. A
prominent example are integrable quantum systems, whose exceptional property
are diverging DC conductivities. In this Letter, we explain the microscopic
origin of ideal conductivity by resorting to the thermodynamic particle content
of a system. Using group-theoretic arguments we rigorously resolve the
long-standing controversy regarding the nature of spin and charge Drude weights
in the absence of chemical potentials. In addition, by employing a hydrodynamic
description, we devise an efficient computational method to calculate exact
Drude weights from the stationary currents generated in an inhomogeneous quench
from bi-partitioned initial states. We exemplify the method on the anisotropic
Heisenberg model at finite temperatures for the entire range of anisotropies,
accessing regimes which are out of reach with other approaches. Quite
remarkably, spin Drude weight and asymptotic spin current rates reveal a
completely discontinuous (fractal) dependence on the anisotropy parameter.
