  We define a `hyperconductor' to be a material whose electrical and thermal DC
conductivities are infinite at zero temperature and finite at any non-zero
temperature. The low-temperature behavior of a hyperconductor is controlled by
a quantum critical phase of interacting electrons that is stable to all
potentially-gap-generating interactions and potentially-localizing disorder. In
this paper, we compute the low-temperature DC and AC electrical and thermal
conductivities in a one-dimensional hyperconductor, studied previously by the
present authors, in the presence of both disorder and umklapp scattering. We
identify the conditions under which the transport coefficients are finite,
which allows us to exhibit examples of violations of the Wiedemann-Franz law.
The temperature dependence of the electrical conductivity, which is
characterized by the parameter $\Delta_X$, is a power law, $\sigma \propto
1/T^{1 - 2(2-\Delta_X)}$ when $\Delta_X \geq 2$, down to zero temperature when
the Fermi surface is commensurate with the lattice. There is a surface in
parameter space along which $\Delta_X = 2$ and $\Delta_X \approx 2$ for small
deviations from this surface. In the generic (incommensurate) case with weak
disorder, such scaling is seen at high-temperatures, followed by an exponential
increase of the conductivity $\ln \sigma \sim 1/T$ at intermediate temperatures
and, finally, $\sigma \propto 1/T^{2-2(2-{\Delta_X})}$ at the lowest
temperatures. In both cases, the thermal conductivity diverges at low
temperatures.
