Conservation of the number of fermions

Maarten Stroeks, Barbara M. Terhal

We introduce the fermionic satisfiability problem, Fermionic k -SAT: this is the problem of deciding whether there is a fermionic state in the null-space of a collection of fermionic, parity-conserving, projectors on n fermionic modes, where each fermionic projector involves at most k fermionic modes. We prove that this problem can be solved efficiently classically for k = 2 . In addition, we show that deciding whether there exists a satisfying assignment with a given fixed particle number parity can also be done efficiently classically for Fermionic 2-SAT: this problem is a quantum-fermionic extension of asking whether a classical 2-SAT problem has a solution with a given Hamming weight parity. We also prove that deciding whether there exists a satisfying assignment for particle-number-conserving Fermionic 2-SAT for some given particle number is NP-complete. Complementary to this, we show that Fermionic 9-SAT is QMA 1 -hard.

Kevin E. Schmidt, Malvin H. Kalos

Y. Tsue, Constança Providência, João da Providência +1 more

The relation between two approaches to the su(2)-algebraic many-fermion model is discussed: (1) the BCS-Bogoliubov approach in terms of the use of the quasiparticles representing all the degrees of freedom except those related to the Cooper-pairs and (2) the conventional algebraic approach in terms of the use of the minimum weight states, from which the Cooper-pairs are excluded. In order to arrive at the goal, the idea of the quasiparticles is brought up in the conservation of the fermion number. Under the c-number replacement for the three su(2)-generators, the quasiparticles suggested in this paper are reduced to those in the BCS-Bogoliubov approach. It is also shown that the two approaches are equivalent through the c-number replacement. Further, a certain modification of the BCS-Bogoliubov approach is discussed.

Chao-Te Li, V. G. Pedrocchi, Taro Tamura

A boson expansion theory based on a number-conserving quasiparticle approach is applied to Ginocchio's SO(8) fermion model. Energy spectra and E2 transition rates calculated by using this new boson mapping are presented and compared against the exact fermion values. A comparison with other boson approaches is also given.

Kevin E. Schmidt, Malvin H. Kalos