Quantized thermal and spin transports of dirty planar topological superconductors

Abstract: Nontrivial bulk topological invariants of quantum materials can leave their signatures on charge, thermal and spin transports. In two dimensions, their imprints can be experimentally measured from well-developed multiterminal Hall bar arrangements. Here, we numerically compute the low temperature ($T$) thermal ($\kappa_{xy}$) and zero temperature spin ($\sigma<sup>{sp}_{xy}$) Hall conductivities, and longitudinal thermal conductance ($G<sup>{th}_{xx}$) of various prominent two-dimensional fully gapped topological superconductors, belonging to distinct Altland-Zirnbauer symmetry classes, namely $p+ip$ (class D), $d+id$ (class C) and $p \pm ip$ (class DIII) paired states, in mesoscopic six-terminal Hall bar setups from the scattering matrix formalism using Kwant. In both clean and weak disorder limits, the time-reversal symmetry breaking $p+ip$ and $d+id$ pairings show half-quantized and quantized $\kappa_{xy}$ [in units of $\kappa_0=\pi² k^2_B T/(3h)$], respectively, while the latter one in addition accommodates a quantized $\sigma<sup>{sp}_{xy}$ [in units of $\sigma^{{sp}_0=\hbar/(8} \pi)$]. By contrast, the time-reversal invariant $p \pm ip$ pairing only displays a quantized $G<sup>{th}_{xx}$ at low $T$ up to a moderate strength of disorder. In the strong disorder regime, all these topological responses ($\kappa_{xy}$, $\sigma<sup>{sp}_{xy}$, and $G<sup>{th}_{xx}$) vanish. Possible material platforms hosting such paired states and manifesting these robust topological thermal and spin responses are discussed.