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A spin-U(1)-symmetry protected momentum-dependent integer-$Z$-valuedtopological invariant is proposed to time-reversal-invariant (TRI)superconductivity (SC) whose nonzero value will lead to exactly flat surfaceband(s). The theory is applied to the weakly spin-orbit coupled quasi-1DA$_2$Cr$_3$As$_3$ (A=Na, K, Rb, Cs) superconductors family with highest $T_c$up to 8.6 K with $p_z$-wave pairing in the $S_z=0$ channel. It's found that upto the leading atomic spin-orbit-coupling (SOC), the whole (001) surfaceBrillouin zone is covered with exactly-flat surface bands, with some regimehosting three flat bands and the remaining part hosting two. Such exactly-flatsurface bands will lead to very sharp zero-bias conductance peak in thescanning tunneling microscopic spectrum. When a tiny subleading spin-flippingSOC is considered, the surface bands will only be slightly split. Theapplication of this theory can be generalized to other unconventionalsuperconductors with weak SOC, particularly to those with mirror-reflectionsymmetry.

physical review research

Intrinsic topological superconductivity with exactly flat surface bands  in the quasi-one-dimensional A$_2$Cr$_3$As$_3$ (A=Na, K, Rb, Cs)  superconductors