Table 2 SC orders in the mean-field analysis

From: Monolayer Kagome metals AV3Sb5

R

Pairing in 1 × 1 unit cell

Pairing in 2 × 2 unit cell \({\hat{\Gamma }}_{R}({{{{{{{\boldsymbol{k}}}}}}}})\)

Label

Basis function

Ag

\(\frac{1}{\sqrt{3}}{\lambda }_{0},\frac{1}{\sqrt{2}}\left[\frac{\sqrt{3}}{2}{\lambda }_{7}+\frac{1}{2}{\lambda }_{8}\right],\)

\(\frac{1}{\sqrt{3}}{L}_{0}^{0},\frac{1}{\sqrt{2}}\left[\frac{\sqrt{3}}{2}{L}_{7}^{0}+\frac{1}{2}{L}_{8}^{0}\right],\)

\({\Gamma }_{{A}_{g}}^{(1)},{\Gamma }_{{A}_{g}}^{(2)},\)

x2, y2, z2

 

\(\frac{1}{\sqrt{2}}\left[{c}_{1}{\lambda }_{1}+{c}_{2}{\lambda }_{2}\right]\),

\(\frac{1}{\sqrt{2}}\left[{c}_{1}{L}_{1}^{+}+{c}_{2}{L}_{2}^{+}-({s}_{1}{L}_{4}^{-}+{s}_{2}{L}_{5}^{-})\right],\)

\({\Gamma }_{{A}_{g}}^{(3)},\)

 
 

c3λ3

\(\left[{c}_{3}{L}_{3}^{+}-{s}_{3}{L}_{6}^{-}\right]\)

\({\Gamma }_{{A}_{g}}^{(4)}\)

 

B1g

\(\frac{1}{\sqrt{2}}\left[-\frac{1}{2}{\lambda }_{7}+\frac{\sqrt{3}}{2}{\lambda }_{8}\right],\)

\(\frac{1}{\sqrt{2}}\left[-\frac{1}{2}{L}_{7}^{0}+\frac{\sqrt{3}}{2}{L}_{8}^{0}\right],\)

\({\Gamma }_{{B}_{1g}}^{(1)},\)

xy

 

\(\frac{1}{\sqrt{2}}\left[{c}_{1}{\lambda }_{1}-{c}_{2}{\lambda }_{2}\right]\)

\(\frac{1}{\sqrt{2}}\left[{c}_{1}{L}_{1}^{+}-{c}_{2}{L}_{2}^{+}-({s}_{1}{L}_{4}^{-}-{s}_{2}{L}_{5}^{-})\right]\)

\({\Gamma }_{{B}_{1g}}^{(2)}\)

 

B2u

\(\frac{1}{\sqrt{2}}\left[{s}_{1}{\lambda }_{4}-{s}_{2}{\lambda }_{5}\right]\)

\(\frac{1}{\sqrt{2}}\left[{s}_{1}{L}_{4}^{+}-{s}_{2}{L}_{5}^{+}-({c}_{1}{L}_{1}^{-}-{c}_{2}{L}_{2}^{-})\right]\)

\({\Gamma }_{{B}_{2u}}^{(1)}\)

y

B3u

\(\frac{1}{\sqrt{2}}\left[{s}_{1}{\lambda }_{4}+{s}_{2}{\lambda }_{5}\right],\)

\(\frac{1}{\sqrt{2}}\left[{s}_{1}{L}_{4}^{+}+{s}_{2}{L}_{5}^{+}+({c}_{1}{L}_{1}^{-}+{c}_{2}{L}_{2}^{-})\right],\)

\({\Gamma }_{{B}_{3u}}^{(1)},\)

x

 

s3λ6

\(\left[{s}_{3}{L}_{6}^{+}-{c}_{3}{L}_{3}^{-}\right]\)

\({\Gamma }_{{B}_{3u}}^{(2)}\)

 
  1. Spin-singlet superconducting pairings are classified by irreducible representations (R) of the point group D2h.
  2. The condensation of \({\hat{\Gamma }}_{R}\) channels can lower the interaction energy, which preserves the \({{{{{{{{\mathscr{T}}}}}}}}}_{1\times 1}\)-translational symmetry. The momentum dependence of pairing gap functions is abbreviated by \(({c}_{i},{s}_{i})\equiv (\cos {{{{{{{\boldsymbol{k}}}}}}}}\cdot {{{{{{{{\boldsymbol{R}}}}}}}}}_{i},\sin {{{{{{{\boldsymbol{k}}}}}}}}\cdot {{{{{{{{\boldsymbol{R}}}}}}}}}_{i})\). To express pairing functions in both 1 × 1 and 2 × 2 unit cells, the 3 × 3 Gell Mann matrices λa and the 12 × 12 matrices \({L}_{a}^{\pm,0}={\lambda }_{a}\otimes {M}_{a}^{\pm,0}\) are introduced with 4 × 4 matrices \({M}_{a}^{\pm,0}\) (see Supplementary Note 7). The fifth column shows the lowest-order basis functions for the corresponding irreducible representation.
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