According to the usual Seebeck effect, you can make a loop of wire with two different metals, apply a temperature difference between two opposite sites of the ring and measure a current. See picture
taken from here.
An answer to this question Seebeck effect and the need for two conductors claims that the difference in metals is not necessary, you can have a single metal you just need to break the symmetry so that you select the current to go clockwise or anti-clockwise.
(One reason to doubt this statement is that we could make the distances between the two temperature contacts asymmetric with respect to the circumference and this in principle should not generate any currents)
My question is, in a single metal loop, with a point on the loop at temperature $T_1$ and the antipode point in the loop at temperature $T_2<T_1$, would an external static magnetic field (perpendicular to the plane of the ring) produce a current? And if so does this lead to some kind of out of equilibrium Aharonov-Bohm effect (where current only is created at certain value of the magnetic field) without a battery?
