1. link
   • official site
   • documentation
   • tutorial
   • gitlab
   • kwant a software for quantum transport NJP2014
2. install conda install -c conda-forge kwant
3. $t=\frac{\hbar^2}{2ma^2}$
4. 有趣的vector potential link
   • vector potential $A_x(x,y)=\Phi\delta(x)\Theta(-y)$
   • magnetic field $B_z(x,y)=\Phi\delta(x)\delta(y)$
5. normal metal, superconductor
   • N-S interface conductance $G=\frac{e^2}{h}(N-R_{ee}+R_{he})$
   • the number of electron channel $N$
   • the total probability of reflection from electrons to electrons in the normal lead $R_{ee}$
   • the total probability of reflection from electrons to holes in the normal lead $R_{he}$
   • normal metal: electron-hole conservation law, particle-hole symmetry
   • conductance that is proportional to the square of the tunneling probability within the gap, and proportional to the tunneling probability above the gap. At the gap edge, we observe a resonant Andreev reflection
6. discrete symmetry: time-reversal, particle-hole and chiral
7. local density $\rho_a=\psi^\dagger M_a \psi_i$
8. local current $J_{ab}=i(\psi_b^\dagger H_{ab}^\dagger M\psi_a - \psi_a^\dagger MH_{ab}\psi_b)$
9. the discretization of a continuum model is an approximation that is only valid in the low-energy limit
10. site: .family, .tag, .pos
11. lattice: Monatomic, Polyatomic

conda create -y -n kwant
conda install -y -n kwant -c conda-forge cudatoolkit=11.3
conda install -y -n kwant -c pytorch pytorch torchvision torchaudio
conda install -y -n kwant -c conda-forge cython ipython pytest matplotlib h5py pandas pylint jupyterlab pillow protobuf scipy requests tqdm lxml opt_einsum cupy kwant holoviews

two-dimensional electron gas

$$H = -\frac{\hbar^2}{2m}(\partial_x^2+\partial_y^2) + V(y)$$

Rashba SOI, Zeeman splitting, PRL2003 nature2010

$$H = -\frac{\hbar^2}{2m}(\partial_x^2+\partial_y^2) - i\alpha(\partial_x\sigma_y-\partial_y\sigma_x) + V(y)$$

Bogoliubov-de Gennes (BdG) Hamiltonian, time reversal operator $\mathcal{T}$

$$ H=\begin{pmatrix} H_0-\mu & \Delta \\ \Delta^\dagger & \mu-\mathcal{T}H_0\mathcal{T}^{-1} \end{pmatrix} $$

spinful Hamiltonian

$$ H=-t\sum_{\langle ij \rangle,\alpha} {|i\alpha\rangle\langle j\alpha|} + J\sum_{i,\alpha,\beta}{\vec{m}_i\cdot \vec{\sigma}_{\alpha\beta} |i\alpha\rangle\langle i\beta|} $$

magnetic texture

$$ \vec{m}_i=\left( \frac{x_i}{x_i^2+y_i^2}\sin\theta_i, \frac{y_i}{x_i^2+y_i^2}\sin\theta_i, \cos\theta_i \right) $$

$$ \theta_i=\frac{\pi}{2} (\tanh((r_i-r_0)/\delta)-1) $$

1. spectral density $\rho_A(E)=\rho(E)A(E)$
   • energy $E$
   • Hilbert space operator $A$
   • expectation value of $A$ for all the eigenstates of the Hamiltonian $H$ with energy $E$
   • density of state $\rho(E)=Tr(\delta(E-H))=\sum_k{\delta(E-E_k)}$
2. pros and cons
   • pros: suited for large systems: not interested in individual eigenvalues, but rather in obtaining an approximate spectral density
   • accuracy: controlled by the number of the moments, the lowest accuracy is at the center of the spectrum, more accurate at the edges of the spectrum
   • Jackson kernel $\sigma=\pi a/N$
   • random vectors will explore the range of the spectrum. The bigger the system is, the number of random vectors required reduces
   • NO noise for local vectors
3. optimal set of energy is not evenly distributed
4. boundary of the spectrum using scipy.sparse.linalg.eigsh
5. sesquilinear map
6. Kubo conductivity must be normalized with the area covered by the vectos