30
votes
Accepted
What is band inversion and how to recognize it in band structure?
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 ...
- 11.2k
16
votes
Accepted
Berry's curvature and magnetic moment in TMDCs
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\...
- 11.2k
12
votes
Berry's curvature and magnetic moment in TMDCs
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:
$$\...
- 1,763
10
votes
Accepted
Anomalous Quantum Hall Effect
What is the quantum anomalous Hall effect?
Figure from C-X. Liu, S-C. Zhang, and X-L. Qi. "The Quantum Anomalous Hall
Effect: Theory and Experiment," Annual Review of Condensed Matter ...
- 4,761
9
votes
Accepted
Dynamic phase in an adiabatic system
The exponential comes from solving a linear differential equation:
\begin{align}
\frac{\textrm{d}|\psi(t)\rangle}{\textrm{d}t} &= -\frac{\textrm{i}}{\hbar}H|\psi(t)\rangle\tag{1}\\
|\psi(t)\rangle ...
7
votes
Accepted
Property related with Berry curvature: $\Omega_{n,\mu\nu}=-\Omega_{n,\nu\mu}$
You can just exchange the $\mu,\nu$ indices to verify the antisymmetry:
$$
\Omega_{n,\mu\nu}(\mathbf{k})=\partial_{\mu}A_{n\nu}(\mathbf{k})-\partial_{\nu}A_{n\mu}(\mathbf{k})\\
\Rightarrow \Omega_{n,\...
- 4,761
7
votes
Dynamic phase in an adiabatic system
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 ...
- 681
7
votes
Accepted
Help with Definitions in Numerical Calculation of Multiband Berry Phase
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 ...
- 188
6
votes
Accepted
Create an image from scratch with the vectors that represent the Berry phase for parallel transport (classic system)
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 ...
- 8,015
5
votes
Berry Curvature with VASP+WANNIER90
Yes, one can do the similar calculations with VASP.
TL;DR. Check out my example 🔗here.
There are two steps involved here, one is obtaining well localized Wannier functions and the other is using ...
- 878
5
votes
How can I prove that a material is ferroelectric using DFT?
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:...
- 15.3k
5
votes
Help with understanding topologically-protected edge states in domain wall systems
I think my understanding on this problem has improved somewhat. Let me try to clarify my confusions. If anyone sees issues with this reasoning, please let me know.
The idea of phases in this context ...
5
votes
polarization jump in berry phase method using VASP
This is quite a peculiar scenario and I think more information is needed in order to find the root casue of it.
With the information at hand, I think this might be caused by different k-samplings used ...
- 878
4
votes
Accepted
How to transform the two-dimensional Berry curvature integral into a loop integral along Fermi circle?
If the distribution function has a step discontinuity at the Fermi surface, its derivatives are Dirac delta functions which reduce the integral to one over the Fermi surface. I will illustrate this ...
- 4,761
4
votes
Accepted
Example of a standard/archetypal/simple 4-band un-gapped condensed matter model with analytic results?
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 ...
- 4,761
4
votes
How to choose half-Brillouin-zone (HBZ) in Fukui & Hatsugai's numerical scheme for the Z2 invariant?
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 ...
3
votes
Accepted
Adiabatic equation related to the Berry phase for lambda with first order terms
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 $\...
- 15.7k
3
votes
Accepted
Help with definitions in k-space twisted bilayer graphene model
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 ...
2
votes
How to calculate berry curvature directly from the quantum espresso output data
Welcome!
If you want Berry phases from plane-waves, it is already implemented in pw.x. See the input option lberry. This doesn't ...
- 1,668
2
votes
polarization jump in berry phase method using VASP
The problem is a bug in old version VASP that can occur in that unit cell because the non-polar phase polarization is not symmetric around 0. The jump happens where ionic contribution to dipole moment ...
- 101
2
votes
Confusions in interpreting Dirac strings (wormholes) in Haldane's Hilbert space picture ft. two tori joined by strings at gapless points
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 ...
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