Metals in one spatial dimension are described at the lowest energy scales by
the Luttinger liquid theory. It is well understood that this free theory, and
even interacting integrable models, can support ballistic transport of
conserved quantities including energy. In contrast, realistic Luttinger-liquid
metals, even without disorder, contain integrability-breaking interactions that
are expected to lead to thermalization and conventional diffusive linear
response. We show that the expansion of energy when such a non-integrable
Luttinger liquid is locally heated above its ground state shows superdiffusive
behavior (i.e., spreading of energy that is intermediate between diffusion and
ballistic propagation), by combining an analytical anomalous diffusion model
with numerical matrix product state calculations.Comment: 5 pages, 3 figure
