Antisymmetric thermopower without magnetism in three-phase composites

A thermopower that turns a temperature gradient into a perpendicular electric field in every in-plane direction has relied on magnetism. Reciprocity permits it without magnetism, yet under a three-fold or higher axis it vanishes both in reciprocal elastic band transport and in composites of two isotropic phases with ideal interfaces. Here we show that composites of three isotropic phases carry it, and that under such an axis three is the minimum. A third, linearly independent transport matrix escapes the exact two-phase reduction; at weak contrast the first allowed axial term is cubic, proportional to a determinant of the three matrices times material-weighted, handed three-point correlations. Treating the phases as colours identifies the arrangements that allow the response, and crystal nets from reticular chemistry supply printable three-dimensional ones. Certified two-sided bounds give axial Seebeck coefficients of 49.5-50.2 $μ$V K$^{-1}$ in a planar cell and 76-84 $μ$V K$^{-1}$ in a printable woodpile; reflections in planes containing the axis reverse the response, and recolouring 24 nets to restore their parent symmetry switches it off. Around a line heat source the axial driving field has the $1/r$ azimuthal form of the magnetic field of a straight wire and drives circulating currents.