Real forms of the lowest - 3 negative parity harmonics

For point group invariant systems, the intrinsic group is G = Point group. The construction of the basis for these systems is a standard group theoretical problem. For the groups D4h, D6h and 0 it was done by Hamermesh many years ago [13]. I report here only the case of G D4i,. For positive parity one has Table 3. For negative parity one has Table 4. 

The representations here are labelled by the group theoretical notation [13] A/, A2, By, B2, E. The first four are one dimensional, while the representation E is two dimensional. For 3, some representations are contained twice and the situation is slightly more complicated. In condensed matter physics, it has become customary to label the representations with the letter T [14]. When both positive and negative parity states are considered also the parity label is added, r and V. The conversion between the two notations is Aj — Ti, A2 — T2, Bi — r3, B2 — T4, and E — Ts. 

An immediate consequence of symmetry is the nature of the quasiparticle spectrum. The density of states p(co) can be calculated from the knowledge of the quasi-particle energies. For singlet states Ek,+ = Ek, with 

As an example of unconventional pairing consider the polar state in p-wave pairing. The density of states is given here by 

In general, for rotational invariant systems, the gap function can be expanded into polynomial harmonics 

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Table IV 1 2. Real forms of the lowest ( < 3) negative parity harmonics...

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