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We subtract the constant term from (Equation presented) and choose an appropriate basis, making (Equation presented) an irreducible matrix. Also, when dim(Equation presented), our classification is applicable only in the stable regime (Equation presented) for the complex case (Equation presented) and (Equation presented) for the real case (Equation presented).
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We subtract the constant term from (Equation presented) and choose an appropriate basis, making (Equation presented) an irreducible matrix. Also, when dim(Equation presented), our classification is applicable only in the stable regime (Equation presented) for the complex case (Equation presented) and (Equation presented) for the real case (Equation presented).
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For the (Equation presented) class, the reflection symmetry satisfies (Equation presented) on the reflection plane, namely, (Equation presented) is block diagonalized by an eigenvalue of the reflection symmetry and each sector does not have PHS. The absence of PHS in the sector implies that the stability of the line node is directly determined by that of the Fermi surface intersecting with the reflection plane. In fact, the reflection sector belongs to the topologically nontrivial state of the A class in the AZ classes. Similarly, for the (Equation presented) case, the commutation relations are given by (Equation presented). Hence PHS also breaks in the reflection sector, but the TRS remains in each sector, i.e., the reflection sector belongs to the AII class of the AZ classes.
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For the (Equation presented) class, the reflection symmetry satisfies (Equation presented) on the reflection plane, namely, (Equation presented) is block diagonalized by an eigenvalue of the reflection symmetry and each sector does not have PHS. The absence of PHS in the sector implies that the stability of the line node is directly determined by that of the Fermi surface intersecting with the reflection plane. In fact, the reflection sector belongs to the topologically nontrivial state of the A class in the AZ classes. Similarly, for the (Equation presented) case, the commutation relations are given by (Equation presented). Hence PHS also breaks in the reflection sector, but the TRS remains in each sector, i.e., the reflection sector belongs to the AII class of the AZ classes.
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84905047054
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(Equation presented)-SRS is equivalent with (Equation presented)-SRS around a quantization axis in the mean-field level. We choose the (Equation presented)-SRS as a sufficient condition of SRS, since (Equation presented)-SRS is a looser condition than (Equation presented)-SRS.
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(Equation presented)-SRS is equivalent with (Equation presented)-SRS around a quantization axis in the mean-field level. We choose the (Equation presented)-SRS as a sufficient condition of SRS, since (Equation presented)-SRS is a looser condition than (Equation presented)-SRS.
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For the (Equation presented) state of (Equation presented) B phase, provided that (Equation presented), the line node is protected by the mirror reflection (Equation presented) and the SRS (Equation presented). This is proved as follows. The combined operator (Equation presented) gives a new reflection operator satisfying (Equation presented) and (Equation presented). Thus (Equation presented) belongs to the (Equation presented) class, which has a stable line node.
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For the (Equation presented) state of (Equation presented) B phase, provided that (Equation presented), the line node is protected by the mirror reflection (Equation presented) and the SRS (Equation presented). This is proved as follows. The combined operator (Equation presented) gives a new reflection operator satisfying (Equation presented) and (Equation presented). Thus (Equation presented) belongs to the (Equation presented) class, which has a stable line node.
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