  We show that the Coulomb interaction between two circuits separated by an
insulating layer leads to unconventional thermoelectric effects, such as the
cooling by thermal current effect, the transverse thermoelectric effect and
Maxwell's demon effect. The first refers to cooling in one circuit induced by
the thermal current in the other circuit. The middle represents electric power
generation in one circuit by the temperature gradient in the other circuit. The
physical picture of Coulomb drag between the two circuits is first demonstrated
for the case with one quantum dot in each circuits and then elaborated for the
case with two quantum dots in each circuits. In the latter case, the heat
exchange between the two circuits can vanish. Last, we also show that the
Maxwell's demon effect can be realized in the four-terminal quantum dot
thermoelectric system, in which the quantum system absorbs the heat from the
high-temperature heat bath and releases the same heat to the low-temperature
heat bath without any energy exchange with the two heat baths. Our study
reveals the role of Coulomb interaction in non-local four-terminal
thermoelectric transport.
