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May 19 |
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Why is planar geometry preferred to observe ordinary Hall effect? The Hall effect can be observed only in even spatial dimensions. Refer to the PRB by Qi, Hughes, and Zhang with the title: "Topological field theories of time reversal invariant insulators" |
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May 19 |
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Some questions about the edge states for time-reversal invariant topological superconductors? I'm not sure if this counts as a proof; but you could naïvely justify the linearity in this way: We know that two Majorana bands, which are Kramers partners of one another, cross at the TRIM. Therefore, we can say that these Majoranas satisfy the massless Dirac equation. Massless Dirac fermions (even if they belong to a specific category, i.e. Majorana in this case) have a linear dispersion. (Minor correction: I should haved used "Off the top of my head..." in my last comment. But I cannot edit it anymore!) |
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May 19 |
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Some questions about the edge states for time-reversal invariant topological superconductors? Of the top of my head I would be tempted to say that they must cross at a TRI momentum (TRIM) point without exception. In 2D your edge states would be counter-propagating Majorana Kramers pairs. Therefore, by Kramers theorem, the two nontrivial edge bands must cross at a TRIM. Also, the dispersion is not guaranteed to be "perfectly" linear. It will approximately be linear very close to the TRIM. For example see Fig 2. of arxiv.org/abs/1212.4232. Also, if you could please share your calculations, I would be happy to go over them and resolve the apparent paradox. Scanned copy is fine. |
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May 17 |
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What is the importance of the Fermi energy $E_F$ or the chem. potential $\mu$ for topological superconductors (continued) what Kramers theorem does is that it guarantees that you will have edge/surface states will remain gapless despite any time-reversal symmetric perturbations; these perturbations destroy edge states in pairs. So if you had odd edge states you'll be left with at least one. It is the violation of the Nielsen-Ninomiya theorem theorem that permits odd Kramers pairs (or Dirac cones) |
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May 17 |
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What is the importance of the Fermi energy $E_F$ or the chem. potential $\mu$ for topological superconductors Both Kramers theorem and Nielsen-Ninomiya theorem are relevant in this discussion. Please check this quote: “This (Nielsen-Ninomiya) theorem states that there is always an even number of Kramers pairs at the Fermi energy for a TR invariant, but otherwise arbitrary 1D band structure. A single pair of helical states can occur only “holograhically”, i.e. when the 1D system is the boundary of a 2D system. This fermion doubling theorem is a TR invariant generalization of the Nielsen-Ninomiya no-go theorem for chiral fermions on a lattice” from arxiv.org/abs/1008.2026 at the end of page 10/54 |
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May 6 |
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What is the importance of the Fermi energy $E_F$ or the chem. potential $\mu$ for topological superconductors deleted 5 characters in body |
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May 6 |
awarded | Critic |
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May 4 |
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What is the importance of the Fermi energy $E_F$ or the chem. potential $\mu$ for topological superconductors deleted 1 characters in body |
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May 4 |
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What is the importance of the Fermi energy $E_F$ or the chem. potential $\mu$ for topological superconductors deleted 1 characters in body |
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May 4 |
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What is the importance of the Fermi energy $E_F$ or the chem. potential $\mu$ for topological superconductors added 18 characters in body |
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May 4 |
answered | What is the importance of the Fermi energy $E_F$ or the chem. potential $\mu$ for topological superconductors |
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May 2 |
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Is it possible to have topological degeneracy in 1D ? Perhaps if you have $N$ segments of the wire which are topological superconductors ($p$-wave), all of these segments have non-topological segments between them, and the boundaries between the non-topological and topological sections are well separated then you would have a topologically protected $2^N$-fold degenerate ground state. However, it won't have topological order (i.e. long-range entanglement); I'm not 100% sure why. In a recent talk Kitaev mentioned that the 1-D $p$-wave chain does not have topological order. |
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Apr 26 |
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Would HgTe be a topological insulator? Murakami et. al (prb.aps.org/abstract/PRB/v76/i20/e205304) have considered the effect of bulk inversion asymmetry on the location of gap closing in k-space. But this is simply a mathematical exercise; the qualitative physics is still unchanged. |
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Apr 26 |
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Would HgTe be a topological insulator? annualreviews.org/na101/home/literatum/publisher/ar/journals/… Observe the bulk sub-bands for a HgTe strip (blue), which is finite in the $y$-direction, and the Dirac-like dispersion of the edge states (red). You can note that the edge dispersion only exists in $-|k_{{\rm max}}|\le k\le|k_{{\rm max}}|$. This $k_{{\rm max}}$ is typically very small. Hence the key physics away from the $\Gamma$ point is not very relevant. |
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Apr 26 |
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Would HgTe be a topological insulator? Check the figure at: |
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Apr 25 |
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Would HgTe be a topological insulator? added 28 characters in body |
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Apr 25 |
answered | Would HgTe be a topological insulator? |
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Apr 21 |
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What is the mathematical reason for topological edge states? added 4 characters in body |
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Apr 20 |
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What is the mathematical reason for topological edge states? added 86 characters in body |
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Apr 20 |
answered | What is the mathematical reason for topological edge states? |
