  In low-dimensional systems, the combination of reduced dimensionality, strong
interactions, and topology has led to a growing number of many-body quantum
phenomena. Thermal transport, which is sensitive to all energy-carrying degrees
of freedom, provides a discriminating probe of emergent excitations in quantum
materials. However, thermal transport measurements in low dimensions are
dominated by the phonon contribution of the lattice. An experimental approach
to isolate the electronic thermal conductance is needed. Here, we show how the
measurement of nonlocal voltage fluctuations in a multiterminal device can
reveal the electronic heat transported across a mesoscopic bridge made of
low-dimensional materials. By using graphene as a noise thermometer, we
demonstrate quantitative electronic thermal conductance measurements of
graphene and carbon nanotubes up to 70K, achieving a precision of ~1% of the
thermal conductance quantum at 5K. Employing linear and nonlinear thermal
transport, we observe signatures of long-range interaction-mediated energy
transport in 1D, in agreement with a theoretical model. Our versatile nonlocal
noise thermometry allows new experiments probing energy transport in emergent
states of matter in low dimensions.
