17

Topological materials form a broad family including insulators, semimetals, and superconductors, of which perhaps the best known are topological insulators. For concreteness, I will focus on topological insulators as these are the ones specifically mentioned in the question. Topological insulator. A topological insulator is an insulator whose Hamiltonian ...


16

The Berry curvature is defined as: $$ \Omega_{\mu\nu}(\mathbf{k})=\partial_{\mu}A_{\nu}(\mathbf{k})-\partial_{\nu}A_{\mu}(\mathbf{k}), \tag{1} $$ where $A_{\mu}(\mathbf{k})=\langle u_{\mathbf{k}}|i\partial_{\mu}u_{\mathbf{k}}\rangle$ is the Berry connection, $|u_{\mathbf{k}}\rangle$ is a Bloch state, and $\partial_\mu\equiv \frac{\partial}{\partial k_\mu}$, ...


12

Resolution for the time reversal symmetry: I need to demonstrate: $\Omega(-\mathbf{k})=-\Omega(\mathbf{k})$ (Berry's curvature is a odd function under time reversal symmetry) Berry's curvature: $$\Omega_{\mu\nu}(\mathbf{k})=\partial_{\mu}A_{\nu}(\mathbf{k})-\partial_{\nu}A_{\mu}(\mathbf{k})\tag{1}$$ If the system is time-reversally invariant: $$T|u_k\rangle=...


9

What is the quantum anomalous Hall effect? Figure from C-X. Liu, S-C. Zhang, and X-L. Qi. "The Quantum Anomalous Hall Effect: Theory and Experiment," Annual Review of Condensed Matter Physics 7, 301-321 (2016) (arXiv link). In short, the quantum anomalous Hall effect (QAHE) is the quantized version of the regular anomalous Hall effect (AHE). That ...


9

The exponential comes from solving a linear differential equation: \begin{align} \frac{\textrm{d}|\psi(t)\rangle}{\textrm{d}t} &= -\frac{\textrm{i}}{\hbar}H|\psi(t)\rangle\tag{1}\\ |\psi(t)\rangle &=e^{-\frac{\rm{i}}{\hbar}Ht}|\psi(t=0)\rangle\tag{2}\label{eq:matrixDynamics}. \end{align} Now if you diagonalize $H$ then instead of $H$ you have a ...


7

You can just exchange the $\mu,\nu$ indices to verify the antisymmetry: $$ \Omega_{n,\mu\nu}(\mathbf{k})=\partial_{\mu}A_{n\nu}(\mathbf{k})-\partial_{\nu}A_{n\mu}(\mathbf{k})\\ \Rightarrow \Omega_{n,\nu\mu}(\mathbf{k})=\partial_{\nu}A_{n\mu}(\mathbf{k})-\partial_{\mu}A_{n\nu}(\mathbf{k}) = - \left( \partial_{\mu}A_{n\nu}(\mathbf{k})-\partial_{\nu}A_{n\mu}(\...


7

Here are some basic comments: I don't think you should call Eq(1) as the Schrödinger equation. It's just defining the energy eigen states/values, or in other words, establishing the relation of those with the Hamiltonian. The Schrödinger equation is $i\hbar\frac{d}{d t} |n(\lambda)\rangle = \hat{H}(\lambda)|n(\lambda)\rangle$, telling you how the quantum ...


7

Question 1. eq. 3.102 defines a matrix in "band-space", e.g. it's NxN where N = number of bands being considered. The right hand side is a regular vector inner product for fixed m,n. The vectors on the right hand side are in any basis you want, so long as they span the same Kohn-Sham/band subspace. Question 2. I'm not sure where you're lost. You ...


6

The figure you are showing can be made in Adobe Illustrator or Gimp I've recreated a crude rendition (very quickly) using adobe illustrator The steps to get this far are: create a sphere cut the sphere in half with pathfinder divider tool Revolve the half sphere with the 3d effect in the revolve menu select Wireframe surface in the revolve menu select map ...


4

The Bernevig-Hughes-Zhang (BHZ) model for topological insulators comes to mind, see e.g. this pedagogical introduction or the review by Qi and Zhang (arXiv link here). It's often justified to set the block-off-diagonal terms zero, as was done in the paper introducing the model (arXiv version here), which simplifies analytical calculations considerably (it ...


3

The TD-Schrodinger equation is $$H(\lambda)|\psi(t)\rangle=i\hbar\frac{\partial}{\partial t}|\psi(t)\rangle$$ So, we just need to plug in \eqref{4} and remove all terms that aren't first-order in $\dot\lambda$. Let's start with the left-hand side, since its a little simpler. $$\begin{align}H(\lambda)|\psi(t)\rangle&=e^{i\phi(\lambda(t))}e^{-i\gamma(t)}\...


3

I don't think I can use this Hamiltonian to recover the band structure. The band structure should use a tight-binding Hamiltonian with several unit cells. A colleague told me that this Hamiltonian is just for their discussion in Figure 4.


3

I think I figured this out. For most practical purposes, I think it is fine to just choose $-\pi\leq k_x\leq \pi$ and $0\leq k_y\leq \pi$ (half-BZ, with exact ranges depending on the model). I think I over-complicated the authors' work. It is unlikely that the vortices/singularities will occur near the boundary of the chosen region. So, I can just take the ...


3

As a good starting, you may take a look at this classical paper and its supplementary material: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.117.097601 Here I just show some general steps: Calculate the polarization (I assume that you can use the VASP package): Calculate the $A, B, C, D$ of Landau-Ginzburg expansion: $$E= \sum_i \dfrac{A}{2}(...


2

I found the following slides helpful: https://physics.princeton.edu//~haldane/talks/dirac.pdf Here are my answers: Yes. From what I understood, the above is not necessarily a 3D effect. The wormholes appear only at gapless points, so we can expect only one Dirac string in my example. Partly correct. We have only one wormhole from the gapless point. However, ...


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