Spin Seebeck Effect of Triangular-lattice Spin Supersolid

Using thermal tensor-network approach, we investigate the spin Seebeck effect (SSE) of the triangular-lattice quantum antiferromagnet hosting spin supersolid phase. We focus on the low-temperature scaling behaviors of the normalized spin current across the interface. For the 1D Heisenberg chain, we find a negative spinon spin in the bulk current with algebraic temperature scaling; at low fields, boundary effects induce a second sign reversal at lower temperatures. These benchmark results are consistent with field-theoretical analysis. On the triangular lattice, spin frustration dramatically enhances the low-temperature SSE, with distinct spin-current signatures -- particularly the sign reversal and characteristic temperature dependence -- distinguishing different spin states. Remarkably, we discover a persistent, negative spin current in the spin supersolid phase, which saturates to a non-zero value in the low-temperature limit and can be ascribed to the Goldstone-mode-mediated spin supercurrents. Moreover, a universal scaling $T^{d/z}$ is found at the U(1)-symmetric polarization quantum critical points. These distinct quantum spin transport traits provide sensitive spin current probes for spin supersolid states in quantum magnets such as Na$_2$BaCo(PO$_4$)$_2$. Furthermore, our results also establish spin supersolids as a tunable quantum platform for spin caloritronics in the ultralow-temperature regime.