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In this paper we study unitary braid group representations associated with Majorana Fermions. Majorana Fermions are represented by Majorana operators, elements of a Clifford algebra. The paper recalls and proves a general result about braid group representations associated with Clifford algebras, and compares this result with the Ivanov braiding associated with Majorana operators. The paper generalizes observations of Kauffman and Lomonaco and of Mo-Lin Ge to show that certain strings of Majorana operators give rise to extraspecial 2-groups and to braiding representations of the Ivanov type.

We use the fidelity susceptibility to study the topological properties of multi-layer nanowire system, with intrinsic spin–orbit coupling and proximity induced superconductivity. We introduce a Hamiltonian of three layer lattice chain to model the system, and consider the effects of applied Zeeman field. In pure system, we find a peak in fidelity susceptibility, which signals the topological quantum phase transition (TQPT) in the system. We then study the disorder effects in our model and find more peaks in fidelity susceptibility. We reveal that these peaks provide information on the zero energy Majorana states of the system, which appear after the TQPT.

Electron properties of semiconducting zigzag carbon nanotubes (CNTs) can be described by two uncoupled Dirac equations of dimension (1+1) for the particle with nonzero mass. The solutions of these equations are two charge-neutral Majorana fields. An analogous equation is obtained for the carbon chains. We use the approach, wherein wavefunction of charged particle is represented as the production of a rapidly oscillating exponent and the slowly varying function amplitude depending on the longitudinal coordinate.

In this paper, we study unitary braid group representations associated with Majorana fermions. Majorana fermions are represented by Majorana operators, elements of a Clifford algebra. The paper proves a general result about braid group representations associated with Clifford algebras and compares this result with the Ivanov braiding associated with Majorana operators and with other braiding representations associated with Majorana fermions such as the Fibonacci model for universal topological quantum computing.

In this paper we study unitary braid group representations associated with Majorana Fermions. Majorana Fermions are represented by Majorana operators, elements of a Clifford algebra. The paper recalls and proves a general result about braid group representations associated with Clifford algebras, and compares this result with the Ivanov braiding associated with Majorana operators. The paper generalizes observations of Kauffman and Lomonaco and of Mo-Lin Ge to show that certain strings of Majorana operators give rise to extraspecial 2-groups and to braiding representations of the Ivanov type.

A topological insulator is a material with a bulk excitation gap generated by the spin orbit interaction that is topologically distinct from an ordinary insulator. This distinction, characterized by a Z₂ topological invariant, necessitates the existence of gapless electronic states on the sample boundary, which have important implications for electronic transport. In two dimensions, the topological insulator is a quantum spin Hall insulator, which is a close cousin of the integer quantum Hall state. Here we will outline the theory of this phase and describe two recent experiments in which its signatures have been observed. (1) Transport experiments on HgTe/HgCdTe quantum wells have demonstrated the existence of the edge states predicted for the quantum spin Hall insulator. (2) Photoemission experiments on the semiconducting alloy Bi_1-xSb_x have observed the gapless surface states predicted for a three dimensional topological insulator. We will close by arguing that the proximity effect between an ordinary superconductor and a topological insulator leads to a non-trivial interface state that supports Majorana fermions and may provide a new venue for realizing proposals for topological quantum computation.

Beginning with an elementary, oscillatory discrete dynamical system associated with the square root of minus one, we study both the foundations of mathematics and physics. Position and momentum do not commute in our discrete physics. Their commutator is related to the diffusion constant for a Brownian process and to the Heisenberg commutator in quantum mechanics. We take John Wheeler's idea of It from Bit as an essential clue and we rework the structure of that bit to a logical particle that is its own anti-particle, a logical Marjorana particle. This is our key example of the amphibian nature of mathematics and the external world. We show how the dynamical system for the square root of minus one is essentially the dynamics of a distinction whose self-reference leads to both the fusion algebra and the operator algebra for the Majorana Fermion. In the course of this, we develop an iterant algebra that supports all of matrix algebra and we end the essay with a discussion of the Dirac equation based on these principles.