Quantized thermal Hall conductance and the topological phase diagram of a superconducting bismuth bilayer

Long-range pairing in monolayer NbSe2 facilitates the emergence of topological superconducting states

“…CN = ±6, ±5, ±4, ±3, ±2, ±1) appearing and the TSC phase regions with CNs = 3 and −2 start to shrink when p A1 is greater than 0.14. The CN displays a sensitive dependence on V z and µ and distributes irregularly between CN = 3 and CN = −2 phases due to long-range pairing, which has also been observed in previous studies [19,20,22,24,[26][27][28][29]. It is notably that the sensitive non-zero CNs appear even under V z = 0 when p A1 is greater than or equal to 0.16 (see figures 2(c)-(f), 3 and 4), suggesting that there may be more intrinsic superconductors (e.g.…”

“…However, there are non-zero energy edge states inherited from the topological insulator in TSCs, affecting CNs and resulting in the mismatch [30]. Therefore, CN can predict the number of TESs crossing the E F but cannot predict the number of zero-energy Majorana edge states [19]. The closure and opening of bulk band-gap can lead to a topological phase transition [29], while the narrowing and widening of the bulk band-gap can also affect whether the TESs remain clean.…”

“…Two-dimensional TSCs with nontrivial properties exhibit topological edge states (TESs) along their edges, as dictated by the well-established bulk-boundary correspondence (BBC) principle, which associates the topological invariant with the number of TESs [2,18]. However, recent research progress on topological materials has revealed the sensitivity and richness of the CN phase diagram to variations in the chemical potential (µ), magnetic field (V z ), and superconducting order parameter amplitude [19][20][21][22][23]. Furthermore, there is evidence of a mismatch between the CN and the number of TESs, indicating a deviation from the BBC principle [19,[21][22][23][24][25][26][27][28][29][30].…”

“…However, recent research progress on topological materials has revealed the sensitivity and richness of the CN phase diagram to variations in the chemical potential (µ), magnetic field (V z ), and superconducting order parameter amplitude [19][20][21][22][23]. Furthermore, there is evidence of a mismatch between the CN and the number of TESs, indicating a deviation from the BBC principle [19,[21][22][23][24][25][26][27][28][29][30]. For example, 2 H b -stacked bilayer transition metal dichalcogenides and bilayer bismuth lattice considering on-site s-wave pairing with Kane-Mele type SOC strength 0.0075 eV or without SOC show the TSC phases with rich and high CNs up to 4 under various µ and superconducting order parameters, which do not strictly always contain the same number of TESs as the CNs [19,25,29].…”

“…Furthermore, there is evidence of a mismatch between the CN and the number of TESs, indicating a deviation from the BBC principle [19,[21][22][23][24][25][26][27][28][29][30]. For example, 2 H b -stacked bilayer transition metal dichalcogenides and bilayer bismuth lattice considering on-site s-wave pairing with Kane-Mele type SOC strength 0.0075 eV or without SOC show the TSC phases with rich and high CNs up to 4 under various µ and superconducting order parameters, which do not strictly always contain the same number of TESs as the CNs [19,25,29]. 2D square lattice of tetragonal D 4h symmetry with a mixture of on-site and off-site singlet pairing manifests non-trivial high CN, massive edge states, and zero energy modes out of high symmetry points, and the number of zero-energy modes is higher than the CN in certain cases under SOC (the strength of SOC equal to pairing) and various V z [23].…”