Many low-dimensional materials are well described by integrable
one-dimensional models such as the Hubbard model of electrons or the Heisenberg
model of spins. However, the small perturbations to these models required to
describe real materials are expected to have singular effects on transport
quantities: integrable models often support dissipationless transport, while
weak non-integrable terms lead to finite conductivities. We use
matrix-product-state methods to obtain quantitative values of spin/electrical
and thermal conductivities in an almost integrable gapless chain (an XXZ spin
chain with staggered fields, or equivalently a spinless fermion chain with
staggered on-site potentials). The results at low temperatures validate a
scaling theory based on bosonization
