Dirac’s Hamiltonian and Bogoliubov’s Hamiltonian as representation of the braid group

Abstract

In this paper, it is shown that Dirac’s Hamiltonian and Bogoliubov’s Hamiltonian both can be braid group matrix representations which are new type of four-dimensional matrix representation of the braid group in comparison with the well-known type (Ge et al. in Int J Mod Phys A 6:3735, 1991; Ge et al. in J Phys A 24:2679, 1991; Ge and Xue in Phys Lett A 152:266, 1991; Ge et al. J Phys A 25:L807 1992) related to the usual spin models. The Dirac’s Hamiltonian is for a free electron with mass m while the Bogoliubov’s Hamiltonian is for quasiparticles in $^{3}He$-B with the same free energy and mass being $\frac{m}{2}$ which depends on the momentum p. And this type is known that the braid matrices are related to the anyon description for FQHE with $\nu =\frac{1}{2}$ (Nayak et al. in Rev Mod Phys 80, 2008; Slingerland and Bais in Nucl Phys B 612:229, 2001), this may mean that Dirac particle could be decomposed into anyons based on the braid group relation. We also get the Temperley-Lieb matrix representations corresponding to the braid group matrix representations and investigate the entanglement and Berry phase of the corresponding Dirac system.