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In a calculation of rotated matrix elements with angular momentum projection,the generalized Wick's theorem may encounter a practical problem ofcombinatorial complexity when the configurations have more than fourquasi-particles (qps). The problem can be solved by employing the Pfaffianalgorithm generally applicable to calculations of matrix elements forHartree-Fock-Bogoliubov states with any number of qps. This breakthrough inmany-body techniques enables studies of high-spin states in a shell-modelframework. As the first application of the Pfaffian algorithm, theconfiguration space of the Projected Shell Model is expanded to include 6-qpstates for both positive and negative parities. Taking $^{166}$Hf as anexample, we show that 6-qp states become the main configuration of the yrastband beyond spin $I \approx 34\hbar$, which explains the observed thirdback-bending in moment of inertia. Structures of multi-qp high-$K$ isomers in$^{176}$Hf are analyzed as another example.

Toward extremes of angular momentum: Application of the Pfaffian algorithm in realistic calculations