Notes

Here are some notes on topics that interest me. They will cover numerical methods, interesting physics problems, and reading notes on papers that strongly caught my attention.

The purpose of these notes is not for proposing anything new, but rather for providing detailed derivations, which I believe will be helpful for understanding. 

I would be grateful for any feedback and comments about the notes!

• Exact Diagonalization method for interacting fermions

The most canonical way of studying an interacting quantum many-body fermion system is simply writing the Hamiltonian as a matrix under suitable basis. 

• Variational Monte Carlo method for interacting fermions

Variational Monte Carlo (VMC) method is a useful numerical technique to evaluate and optimize complicated Ansatz wave functions, which is also very useful in the study of neural quantum states recently.

• Entanglement depth with translation symmetry

Translation symmetry and filling constraint guarantees the ground state wave functions in most common condensed matter systems are highly entangled in terms of entanglement depth, which is also interestingly related to the parity super-selection rule (P-SSR)  in many-body fermionic systems.

• Fourier transformation of double-gate Coulomb potential in 2D

An interesting definite integral problem using Bessel function and residue theorem.

• Fourier transformation of Gaussian orbitals with finite angular momentum in 2D

Another interesting definite integral problem, which has some practical usage in the Wannierization for 2D Bloch states.

• Magnetic Bloch states for lowest Landau levels

And this is why Landau levels are inherently "Chern bands".

• Why do (or do not) we choose normal ordered projected interacting Hamiltonian?

I don't think I am the only one who got confused by the "normal order" procedure in projected interacting Hamiltonian...

• Some non-perturbative conclusions in electronic linear response

One of the most "interesting" statement made by Piers Coleman in his great textbook Introduction to many-body physics is in the beginning of Chapter 10:

Surprisingly, Ohm’s law,

V= IR, 

requires quite a sophisticated understanding of quantum many-body physics.

Ohm's law is one of the things I learned first as a high school student that still puzzles me until this very day. In this note, a number of non-perturbative statements for interacting electronic systems are derived.

• Schwinger-Dyson, Bethe-Salpeter and random phase approximation (to be finished).

• Basics of quantum electrodynamics (to be finished).

• Work in progress...