  We study the quantum thermal transport of elastic excitations through a
two-dimensional elastic waveguide between two thermal reservoirs. We solve the
classical Kirchhoff-Love equation for rectangular wires and explore the
dispersion relation for both the symmetric and antisymmetric solutions. Then,
we study the phonon transport of these modes within the second quantization
framework by analyzing the mean quadratic displacement, from which the energy
density current, the temperature field, and conductance are determined. We
identify the relevant modes contributing to thermal transport and explore the
average temperature difference to reach the high-temperature limit. We expect
our results to pave the way for understanding phonon-mediated thermal transport
in two-dimensional mesoscopic quantum devices.
