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Supersymmetry
Published in Harish Parthasarathy, Supersymmetry and Superstring Theory with Engineering Applications, 2023
Also, Γ0Γn,Γ0Γμ,ΓnΓ0,ΓμΓ0 are Hermitian matrices. We observe that if ψ is a Majorana Fermion, ψ¯=(ψ*)TΓ0=ψTΓ5ϵΓ0
Introducing Topological Materials: Mind the Time Reversal
Published in Grigory Tkachov, Topological Quantum Materials, 2015
Similar to band insulators, superconductors can be divided into ordinary and topological [59]. We can think of a topological superconductor (TS) as an electronic material with an energy gap in the bulk and a gapless mode at the boundary, as illustrated in Fig. 1.14. Now, the bulk energy gap and the boundary mode both come from the pairing interaction between the electrons in the material. In particular, the bulk gap characterizes the energy cost for breaking a Cooper pair into two single-particle excitations, a particle above the Fermi level and a hole below it. Besides, the pairing interaction brings about an exact symmetry between the particle and hole excitations, which has an interesting implication for the gapless boundary mode: It behaves as both a particle and a hole at the same time. An instructive analogy is the Majorana fermion. In relativistic field theory, it embodies both a particle and an antiparticle [61]. The possibility of Majorana-like modes in TSs has caused a surge of interest in their properties and potential for topological quantum computation [62].
Some Aspects of Superstring Theory
Published in Harish Parthasarathy, Advanced Probability and Statistics: Applications to Physics and Engineering, 2023
The super gravity field equations are determined by a Lagrangian density LSUGR=e[Rμvmnemμenv+χ¯aΓabcDbχc where Γabc is obtained by completely antisymmetrizing the product γaγbγc of the Dirac matrices. xa are Majorana Fermion gravitino fields. Let ωmnµ denote the spin connection of the gravitational field. Then it defines a spinor covariant derivative by Dμ=∂μ+ωμmnΓmn,Da=eaμDμ where Γmn=[Γm,Γn]
Quantum Berezinskii–Kosterltz–Thouless transition for topological insulator
Published in Phase Transitions, 2020
Ranjith Kumar R, Rahul S, Surya Narayan Sahoo, Sujit Sarkar
Here we present the quantum BKT equations of the Hamiltonian and observe there is no topological phase,Following the same procedure of Appendix 1, we obtain the RG equations for the Hamiltonian as,We do the transformation, and obtain another set of RG equations in the standard form of BKT equation,The structure of the second BKT equations (Equation (26)) is same as that of the first one (Equation (15)). Therefore, the analysis of this equation is the same as that of the first one. The only difference being, there is no evidence Majorana fermion mode in this phase. In Figure 2, the system shows only the Ising-ferromagnetic phase (region III). In practical reality, for this regime of parameter space the quantum spin Hall insulator will be in the non-topological phase. From this study, we obtain two types of BKT equations, but there is no Majorana-Ising transition. We observe Majorana-Ising transition, if we consider total RG equations of Hamiltonian H (Equation (9)) [39] in which case the quantum BKT behavior will be absent.