Matrix elements of unitary group generators in many-fermion correlation problem. II: Graphical methods of spin algebras. (English) Zbl 1466.81148
Summary: [For parts I, and III see the author, ibid. 59, No. 1, 1–36 (2021; Zbl 1466.81147), and 59, No. 1, 72–118 (2021; Zbl 1466.81149) ] This second instalment continues our survey inter-relating various approaches to the evaluation of matrix elements (MEs) of the unitary group generators, their products, and spin-orbital \(U(2n)\) generators, in the basis of the electronic Gel’fand-Tsetlin (G-T) [or Gel’fand-Paldus (G-P) or P-] states spanning the carrier spaces of the two-column irreducible representations (irreps) of the unitary group approach (UGA) to the many-electron correlation problem. Exploiting an intimate relationship between the \(U(n)\), \(\mathcal{S}_N\), and SU(2) groups we rely here on the SU(2) formalism within the confines of UGA rather than on an earlier approach relying on the permutation group \(\mathcal{S}_N\). The key is a close relationship between the P-states of UGA and the Yamanouchi-Kotani (Y-K) states of SU(2) that differ only in a phase. This enables one to exploit the powerful graphical techniques of spin (or angular momentum) algebras as first developed by Jucys. We show the usefulness and power of these techniques in the problem of UGA ME evaluation which, moreover, leads automatically to their segmentation, first derived for single generators by Shavitt and for generator products by Boyle and Paldus. We also show how the same technique can be successfully employed for MEs of spin-dependent \(U(2n)\) generators in the UGA P-basis.
MSC:
| 81V70 | Many-body theory; quantum Hall effect |
| 81V74 | Fermionic systems in quantum theory |
| 81R25 | Spinor and twistor methods applied to problems in quantum theory |
| 81P16 | Quantum state spaces, operational and probabilistic concepts |
| 20F05 | Generators, relations, and presentations of groups |
| 22D10 | Unitary representations of locally compact groups |
| 22E70 | Applications of Lie groups to the sciences; explicit representations |
Keywords:
unitary group approach (UGA); graphical UGA (GUGA); many-electron correlation problem; UGA generator matrix elements (MEs); one- and two-body MEs; graphical methods of spin algebras; MEs of spin-dependent \(U(2n)\); me segmentationReferences:
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