  Integrable models provide emblematic examples of non-ergodic phenomena. One
of their most distinguished properties are divergent zero-frequency
conductivities signalled by finite Drude weights. Singular conductivities owe
to long-lived quasiparticle excitations that propagate ballistically through
the system without any diffraction. The case of the celebrated quantum
Heisenberg chain, one of the best-studied many-body paradigms, turns out to be
particularly mysterious. About a decade ago, it was found that the spin Drude
weight in the critical phase of the model assumes an extraordinary, nowhere
continuous, dependence on the anisotropy parameter in the shape of a `popcorn
function'. This unprecedented discovery has been afterwards resolved at the
level of the underlying deformed quantum symmetry algebra which helps
explaining the erratic nature of the quasiparticle spectrum at commensurate
values of interaction anisotropy. This work is devoted to the captivating
phenomenon of discontinuous Drude weights, with the aim to give a broader
perspective on the topic by revisiting and reconciling various perspectives
from the previous studies. Moreover, it is argued that such an anomalous
non-ergodic feature is not exclusive to the integrable spin chain but can be
instead expected in a number of other integrable systems that arise from
realizations of the quantum group $\mathcal{U}_{q}(\mathfrak{sl}(2))$,
specialized to unimodular values of the quantum deformation parameter $q$. Our
discussion is framed in the context of gapless anisotropic quantum chains of
higher spin and the sine-Gordon quantum field theory in two space-time
dimensions.
