  We study the response to an applied flux of an interacting system of
Dirac-Weyl fermions confined in a one-dimensional (1D) ring. Combining
analytical calculations with density-matrix renormalization group results, we
show that tuning of interactions leads to a unique many-body system that
displays either a suppression or an enhancement of the Drude weight - the
zero-frequency peak in the ac conductivity - with respect to the
non-interacting value. An asymmetry in the interaction strength between same-
and different-pseudospin Dirac-Weyl fermions leads to Drude weight enhancement.
Viceversa, symmetric interactions lead to Drude weight suppression. Our
predictions can be tested in mixtures of ultracold fermions in 1D ring traps.
