Table 2 The minimal double SIs of spinful band topology in all 1,651 double SSGs.

From: Magnetic topological quantum chemistry

Minimal Double SIs of Spinful Band Topology the 1,651 Magnetic and Nonmagnetic Double SSGs

SI

Minimal Double SSG(s)

Bulk Topology

 

SI

Minimal Double SSG(s)

Bulk Topology

η4I

2.4 \(P\bar{1}\)

WSM/QAH/AXI

 

\({z}_{4m,\pi }^{\pm }\)

83.43 P4/m

weak TI/weak TCI

z2I,i

2.4 \(P\bar{1}\)

QAH

 

\({z}_{4m,0}^{+}\)

84.51 P42/m

QAH/weak TI/weak TCI

\({\eta }_{2I}^{\prime}\)

2.4 \(P\bar{1}\)

AXI

 

z8

83.44 \(P4/m1^{\prime}\), 123.339 P4/mmm

AXI/TCI/HOTI

z2R

3.1 P2, 41.215 \(Ab^{\prime} a^{\prime} 2\)

QAH

 

z3R

147.13 \(P\bar{3}\)

QAH

δ2m

10.42 P2/m

QAH/AXI/TCI

 

z6R

168.109 P6

QAH

\({z}_{2m,\pi }^{\pm }\)

10.42 P2/m

QAH/weak TI/weak TCI

 

δ3m

174.133 \(P\bar{6}\)

QAH/AXI/TCI

z4

2.5 \(P\bar{1}1^{\prime}\), 47.249 Pmmm,

AXI/TCI/HOTI

 

\({z}_{3m,\pi }^{\pm }\)

174.133 \(P\bar{6}\)

weak TI/weak TCI

 

83.45 \(P4^{\prime} /m\)

     

\({z}_{4}^{\prime}\)

135.487 \(P{4}_{2}^{\prime}/mbc^{\prime}\)

AXI/TCI

 

δ6m

175.137 P6/m

QAH/AXI/TCI

z2w,i

2.5 \(P\bar{1}1^{\prime}\), 47.249 Pmmm,

weak TI/weak TCI

 

\({z}_{6m,\pi }^{\pm }\)

175.137 P6/m

weak TI/weak TCI

 

83.45 \(P4^{\prime} /m\)

     

z4R

75.1 P4

QAH

 

\({z}_{6m,0}^{+}\)

176.143 P63/m

QAH/weak TI/weak TCI

\({z}_{2R}^{\prime}\),

27.81 \(Pc^{\prime} c^{\prime} 2\), 54.342 \(Pc^{\prime} c^{\prime} a\),

QAH

 

z12

175.138 \(P6/m1^{\prime}\), 191.233 P6/mmm

AXI/TCI/HOTI

\({z}_{2R}^{^{\prime\prime} }\)

56.369 \(Pc^{\prime} c^{\prime} n\), 60.424 \(Pb^{\prime} cn^{\prime}\),

     
 

77.13 P42, 110.249 \(I{4}_{1}c^{\prime} d^{\prime}\)

     

z4S

81.33 \(P\bar{4}\)

QAH

 

\({z}_{12}^{\prime}\)

176.144 \(P{6}_{3}/m1^{\prime}\)

AXI/TCI/HOTI

δ2S

81.33 \(P\bar{4}\)

WSM

 

\({z}_{4R}^{\prime}\)

103.199 \(P4c^{\prime} c^{\prime}\)

QAH

z2

81.33 \(P\bar{4}\)

AXI

 

\({z}_{6R}^{\prime}\)

184.195 \(P6c^{\prime} c^{\prime}\)

QAH

δ4m

83.43 P4/m

QAH/AXI

    
  1. In order, this table contains the symbol of each double SI, the minimal double SSG(s) [i.e. the lowest-symmetry SSG(s) in which the double SI predicts nontrivial band topology, see SN 30 and 39], and the bulk topological phase(s) associated to nontrivial values of the double SI. All symmetry-indicated spinful SISM (specifically symmetry-indicated WSM), quantum anomalous Hall (QAH), TI, and TCI phases in magnetic and nonmagnetic crystalline solids necessarily exhibit nontrivial values of at least one of the double SIs listed in this table. We note that, in this table, the symbol AXI refers to both magnetic AXIs and \({{{{{{{\mathcal{T}}}}}}}}\)-symmetric 3D TIs, because AXI and 3D TI phases are both defined by the nontrivial bulk axion angle θ = π [Fig. 5b and refs. 21,55,63]. Additionally, the symbols TCI and HOTI respectively indicate helical (i.e. non-axionic) mirror Chern insulators24 and HOTIs26,28,29,31,32, which include the magnetic HOTIs in Fig. 5(c–e) introduced in this work, as well as the nonmagnetic helical HOTI phases previously identified in bismuth33 and MoTe234. Specific details of our SI calculations – including explicit SI formulas, TI and TCI layer constructions, tight-binding models, and the minimal double SSG associated to each double SSG – are provided in SN 26 and 39.
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