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REVIEW OF FERMION HAMILTONIAN IN BOZON SPACE

K. Baktybaev, N. O. Koilyk, K.E. Ramankulov

Published Aug 1, 2008• 0 citations
OpenAlex

Abstract

Fermion theory was represented into boson’s space by Daison, Belev-Zelevinski and Seniority methods. It was shown that FDSM and its boson representation gave a good explanation of experimental data. So it was shown that from the Hamiltonian of FDSM could be constructed boson type IBM-Hamiltonian by representation. So fermion theory gives microscopically base of phenomenological approaches.

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Fermion theory was represented into boson’s space by Daison, Belev-Zelevinski and Seniority methods. It was shown that FDSM and its boson representation gave a good explanation of experimental data. So it was shown that from the Hamiltonian of FDSM could be constructed boson type IBM-Hamiltonian by representation. So fermion theory gives microscopically base of phenomenological approaches.
Based on Girardeau's composite-particle theory (1971) the PPP (extended Hubbard) Hamiltonian is represented in terms of ideal-boson operators. A major unsolved problem was to find (approximate) eigenstate of the Hamiltonian which, as the same time, satisfy certain subsidiary conditions. The later are necessary to ensure that the mapping of the fermion space into the space of boson states is unique. In the present paper a partial solution to the above mentioned problem is presented by using some arguments from the representation theory of the symmetric group. The results are of immediate practical relevance for all calculation dealing with the eigenvalue problem of the Hamiltonian in ideal-boson space.
We study fermionic topological phases using the technique of fermion condensation. We give a prescription for performing fermion condensation in bosonic topological phases that contain a fermion. Our approach to fermion condensation can roughly be understood as coupling the parent bosonic topological phase to a phase of physical fermions and condensing pairs of physical and emergent fermions. There are two distinct types of objects in the resulting fermionic fusion categories, which we call “m-type” and “q-type” objects. The endomorphism algebras of q-type objects are complex Clifford algebras, and they have no analogs in bosonic theories. We construct a fermionic generalization of the tube category, which allows us to compute the quasiparticle excitations arising from the condensed theories. We prove a series of results relating data in fermionic theories to data in their parent bosonic theories; for example, if C is a modular tensor category containing a fermion, then the tube category constructed from the condensed theory satisfies Tube(C/ψ)≅C×(C/ψ). We also study how modular transformations, fusion rules, and coherence relations are modified in the fermionic setting, prove a fermionic version of the Verlinde dimension formula, construct a commuting projector lattice Hamiltonian for fermionic theories, and write down a fermionic version of the Turaev-Viro-Barrett-Westbury state sum. A large portion of this work is devoted to three detailed examples of performing fermion condensation to produce fermionic topological phases: we condense fermions in the Ising theory, the SO(3)6 theory, and the 12E6 theory and compute the quasiparticle excitation spectrum in each of the condensed theories.