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arXiv:0711.3260 · Physical Review A

Bogoliubov’s Hamiltonian as a derivative of Dirac’s Hamiltonian via a braid relation

Bao-Xing Xie, Kang Xue, Mo‐Lin Ge

Published Jun 3, 2008• 3 citations
arXivHTMLPDF

Abstract

In this paper we discuss a new type of four-dimensional representation of the braid group. The matrices of braid operations are constructed by $q$ deformation of Hamiltonians. One is Dirac's Hamiltonian for a free electron with mass $m$, the other, which we find, is related to Bogoliubov's Hamiltonian for quasiparticles in $^{3}\text{H}\text{e}\ensuremath{-}\text{B}$ with the same free energy and mass being $m/2$. In the process, we choose the free $q$-deformation parameter as a special value in order to be consistent with the anyon description for fractional quantum Hall effect with $\ensuremath{\nu}=1/2$.

DOI
10.1103/physreva.77.064101

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Physical review. B, Condensed matter•May 1, 1991•21 citations
In this paper we discuss a new type of four-dimensional representation of the braid group. The matrices of braid operations are constructed by qqq deformation of Hamiltonians. One is Dirac's Hamiltonian for a free electron with mass mmm, the other, which we find, is related to Bogoliubov's Hamiltonian for quasiparticles in 3He\ensuremath−B^{3}\text{H}\text{e}\ensuremath{-}\text{B}3He\ensuremath−B with the same free energy and mass being m/2m/2m/2. In the process, we choose the free qqq-deformation parameter as a special value in order to be consistent with the anyon description for fractional quantum Hall effect with \ensuremathν=1/2\ensuremath{\nu}=1/2\ensuremathν=1/2.
The path integral approach to representing braid group is generalized for particles with spin. Introducing the notion of {\\em charged} winding number in the super-plane, we represent the braid group generators as homotopically constrained Feynman kernels. In this framework, super Knizhnik-Zamolodchikov operators appear naturally in the Hamiltonian, suggesting the possibility of {\\em spinning nonabelian} anyons. We then apply our formulation to the study of fractional quantum Hall effect (FQHE). A systematic discussion of the ground states and their quasi-hole excitations is given. We obtain Laughlin, Halperin and Moore-Read states as {\\em exact} ground state solutions to the respective Hamiltonians associated to the braid group representations. The energy gap of the quasi-excitation is also obtainable from this approach.
A bstract Fibonacci anyons ε provide the simplest possible model of non-Abelian fusion rules: [1] × [1] = [0] ⊕ [1]. We propose a conformal field theory construction of topological quantum registers based on Fibonacci anyons realized as quasiparticle excitations in the ℤ 3 parafermion fractional quantum Hall state. To this end, the results of Ardonne and Schoutens for the correlation function of four Fibonacci fields are extended to the case of arbitrary number n of quasi-holes and N = 3 r electrons. Special attention is paid to the braiding properties of the obtained correlators. We explain in details the construction of a monodromy representation of the Artin braid group B \mathcal{B} B B n acting on n -point conformal blocks of Fibonacci anyons. The matrices of braid group generators are displayed explicitly for all n ≤ 8. A simple recursion formula makes it possible to extend without efforts the construction to any n . Finally, we construct N \mathcal{N} N N qubit computational spaces in terms of conformal blocks of 2N 2\mathcal{N} 2N 2 N + 2 Fibonacci anyons.
It is shown that composite fermions—particles which appear in fractional quantum Hall effect, and are statistically different from anyons—could be described within braid group formalism. In the proposed model some ways of exchange of bare particles of composite particles are restricted. The new group describing exchanges of particles is defined. The description of such a group is included and the unitary one-dimensional representation of this group is found. An explicit form of this group for two exemplary cases is presented.
We introduce and study a class of anyon models that are a natural generalization of Ising anyons and Majorana fermion zero modes. These models combine an Ising anyon sector with a sector associated with SO(m)2\mathit{\text{SO}}{(m)}_{2}SO(m)2​ Chern-Simons theory. We show how they can arise in a simple scenario for electron fractionalization and give a complete account of their quasiparticle types, fusion rules, and braiding. We show that the image of the braid group is finite for a collection of 2n2n2n fundamental quasiparticles and is a proper subgroup of the metaplectic representation of $Sp(2n\ensuremath{-}2,{\mathbb{F}}_{m})\ensuremath{\ltimes}H(2n\ensuremath{-}2,{\mathbb{F}}_{m})$, where $Sp(2n\ensuremath{-}2,{\mathbb{F}}_{m})$ is the symplectic group over the finite field Fm{\mathbb{F}}_{m}Fm​ and $H(2n\ensuremath{-}2,{\mathbb{F}}_{m})$ is the extra special group (also called the $(2n\ensuremath{-}1)$-dimensional Heisenberg group) over Fm{\mathbb{F}}_{m}Fm​. Moreover, the braiding of fundamental quasiparticles combined with a restricted set of measurements can be efficiently simulated classically. However, computing the result of braiding a certain type of composite quasiparticle followed by fusion into the identity is $#P$-hard. It is not universal for quantum computation because it has a finite braid group image. This is a rare example of a topological phase that is not universal for quantum computation through braiding but nevertheless has $#P$-hard link invariants. We argue that our models are closely related to recent analyses finding non-Abelian anyonic properties for defects in quantum Hall systems, generalizing Majorana zero modes in quasi-1D systems.
We present a careful construction of anyons on a torus starting with the braid-group analysis. The rules of Wen, Dagotto, and Fradkin for putting anyons on a torus are reproduced with some minor improvements. The existence of noncontractible loops leads to braid-group representations characterized not only by anyon statistics \ensuremath{\theta} but also by the magnetic fluxes ${\mathrm{\ensuremath{\Phi}}}_{\mathit{x}}$ and ${\mathrm{\ensuremath{\Phi}}}_{\mathit{y}}$ threading through the holes of the torus. The three parameters are tangled with each other. We explore the symmetries of the torus to separate the effects of ${\mathrm{\ensuremath{\Phi}}}_{\mathit{x}}$ and ${\mathrm{\ensuremath{\Phi}}}_{\mathit{y}}$ from those of \ensuremath{\theta}. It is shown that the anyon system always has a smaller period \ensuremath{\theta}/\ensuremath{\pi} in ${\mathrm{\ensuremath{\Phi}}}_{\mathit{x}}$ and ${\mathrm{\ensuremath{\Phi}}}_{\mathit{y}}$ than the natural period 1. We perform several numerical calculations to investigate the spectral flow and consistency of the method and find interesting features in the spectral flow, which are relevant in understanding the fractional quantum Hall effect.