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2021 | OriginalPaper | Chapter

2. Method

Author : Dr. Shuntaro Sumita

Published in: Modern Classification Theory of Superconducting Gap Nodes

Publisher: Springer Singapore

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Abstract

In this chapter, we introduce modern gap classification theory using group theory (representation theory) and topological argument. First, in Sect. 2.1, we make a remark about some terminologies and notations of finite-group representation theory, which are used throughout the thesis, for the avoidance of confusion. In Sect. 2.2, we introduce the group-theoretical analysis of the superconducting gap on high-symmetry points in the BZ [1‐11]. Next, we explain the topological classification of nodes on the high-symmetry points by using the Wigner criteria and the orthogonality test in Sect. 2.3 [12‐15]. Also, an intuitive understanding of the classification methods is given by showing simple examples (Sect. 2.4).

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previous chapter Introduction
next chapter Superconducting Gap Classification on High-Symmetry Planes
Appendix
Available only for authorised users
Footnotes
1
Indeed, our topological classification theory (Sect. 2.3) can also be applied to other cases, e.g., noncentrosymmetric superconductors.
 
2
Strictly speaking, \(\alpha \) is a double-valued IR of the finite group \(\bar{\mathcal {M}}^{\varvec{k}} = \mathcal {M}^{\varvec{k}} / \mathbb {T}\) while \(j_z\) is a basis of the continuous group. Therefore there is no one-to-one correspondence between \(\alpha \) and \(j_z\) in some cases. In the \(C_{2v}\) symmetry, for example, the IR \(\alpha = 1 / 2\) includes all normal Bloch states with half-integer total angular momentum \(j_z = \pm 1 / 2, \pm 3 / 2, \pm 5 / 2, \dotsc \). In this thesis, however, we represent \(j_z\) with minimum absolute value, which satisfies \(j_z \downarrow \bar{\mathcal {M}}^{\varvec{k}} = \alpha \), as the angular-momentum counterpart of the IR \(\alpha \).
 
3
The factor system also satisfies the 2-cocycle condition:
$$\begin{aligned} \omega _{\text {ns}}^{l_1 l_2 l_3\varvec{k}}(l_1, l_2) \omega _{\text {ns}}^{l_1 l_2 l_3\varvec{k}}(l_1 l_2, l_3) = \omega _{\text {ns}}^{l_1 l_2 l_3\varvec{k}}(l_1, l_2 l_3) \omega _{\text {ns}}^{l_2 l_3\varvec{k}}(l_2, l_3)^{\phi (l_1)}. \end{aligned}$$
 
4
Here,
$$\begin{aligned} (\mathcal {I} \bar{g} \mathcal {I}) c_{\alpha j}^\dagger (\varvec{k}) (\mathcal {I} \bar{g} \mathcal {I})^{- 1}&= \sum _{j_1} (\mathcal {I} \bar{g}) c_{\alpha j_1}^\dagger (\mathcal {I} \varvec{k}) (\mathcal {I} \bar{g})^{- 1} [\bar{u}_\alpha ^{\varvec{k}}(\mathcal {I})]_{j_1 j} \\&= \sum _{j_1, j_2} \mathcal {I} c_{\alpha j_2}^\dagger (\bar{g} \mathcal {I} \varvec{k}) \mathcal {I}^{- 1} [\bar{u}_\alpha ^{\mathcal {I} \varvec{k}}(\bar{g})]_{j_2 j_1} [\bar{u}_\alpha ^{\varvec{k}}(\mathcal {I})]_{j_1 j} \\&= \sum _{j_1, j_2, j_3} c_{\alpha j_3}^\dagger (\mathcal {I} \bar{g} \mathcal {I} \varvec{k}) [\bar{u}_\alpha ^{\bar{g} \mathcal {I} \varvec{k}}(\mathcal {I})]_{j_3 j_2} [\bar{u}_\alpha ^{\mathcal {I} \varvec{k}}(\bar{g})]_{j_2 j_1} [\bar{u}_\alpha ^{\varvec{k}}(\mathcal {I})]_{j_1 j} \\&= \omega ^{\mathcal {I} \bar{g} \mathcal {I} \varvec{k}}(\mathcal {I}, \bar{g}) \omega ^{\mathcal {I} \bar{g} \mathcal {I} \varvec{k}}(\mathcal {I} \bar{g}, \mathcal {I}) \sum _{j_3} c_{\alpha j_3}^\dagger (\mathcal {I} \bar{g} \mathcal {I} \varvec{k}) [\bar{u}_\alpha ^{\varvec{k}}(\overline{\mathcal {I} g \mathcal {I}})]_{j_3 j} \\&= \frac{\omega ^{\varvec{k}}(\mathcal {I}, \bar{g})}{\omega ^{\varvec{k}}(\bar{g}, \mathcal {I})} \sum _{j'} c_{\alpha j'}^\dagger (\varvec{k}) [\bar{\lambda }_\alpha ^{\varvec{k}}(\bar{g})]_{j' j}, \end{aligned}$$
where we use \(\mathcal {I} \bar{g} \mathcal {I} \in \mathcal {G}^{\varvec{k}}\) and \(\overline{\mathcal {I} g \mathcal {I}} = \bar{g}\) in the last equal.
 
5
Taking into account the factor system in Eqs. (2.32) and (2.40), we can easily derive it by the following commutation relation:
$$\begin{aligned}{}[u(\Gamma ), u(\mathcal {M}_z)]&= \omega _{\text {in}}(\mathfrak {T}, \mathfrak {C})^{- 1} \{ u(\mathfrak {T}) u(\mathfrak {C})^* u(\mathcal {M}_z) - u(\mathcal {M}_z) u(\mathfrak {T}) u(\mathfrak {C})^* \} \\&= \omega _{\text {in}}(\mathfrak {T}, \mathfrak {C})^{- 1} \{ u(\mathfrak {T}) u(\mathfrak {C})^* u(\mathcal {M}_z) - u(\mathfrak {T}) u(\mathcal {M}_z)^* u(\mathfrak {C})^* \} \\&= \omega _{\text {in}}(\mathfrak {T}, \mathfrak {C})^{- 1} \{ u(\mathfrak {T}) u(\mathfrak {C})^* u(\mathcal {M}_z) - u(\mathfrak {T}) u(\mathfrak {C})^* u(\mathcal {M}_z) \} = 0, \end{aligned}$$
where we use \(\Gamma = \mathcal {T C} = \mathfrak {T C}\).
 
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Metadata
Title
Method
Author
Dr. Shuntaro Sumita
Copyright Year
2021
Publisher
Springer Singapore
Book
Modern Classification Theory of Superconducting Gap Nodes

Print ISBN: 978-981-334-263-7

Electronic ISBN: 978-981-334-264-4

Copyright Year: 2021

DOI
https://doi.org/10.1007/978-981-33-4264-4_2

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