  A geometry-based mechanism for generating steady-state internal circulation
of local thermal currents is demonstrated by harmonically coupled quantum
oscillators formulated by the Redfield quantum master equation (RQME) and the
Bose Hubbard model (BHM) of phonons formulated by the Lindblad quantum master
equation (LQME) using the simple multi-path geometry of a triangle. Driven by
two reservoirs at different temperatures, both systems can exhibit an atypical
local thermal current flowing against the total current. However, the total
thermal current behaves normally. While the RQME of harmonically coupled
quantum oscillators allows an analytical solution, the LQME of the interacting
BHM can be solved numerically. The emergence of the geometry-based circulation
in both systems demonstrates the ubiquity and robustness of the mechanism. In
the high-temperature limit, the results agree with the classical results,
confirming the generality of the geometric-based circulation across the quantum
and classical boundary. Possible experimental implications and applications are
briefly discussed.
