Many Body

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Thus the action of annihilation operators and creation operators could be

Suppose is single particle state.

For many particle state , Thus

Thus we can write

This is how we introduce first-quantized wave function from second-quantized operators.

Consider free Fermion system

Problem. Suppose is a Fermionic operator. Construct

Prove that

1. For all , , , and , thus is a Bosonic operator.

Proof. Suppose, . Use the Equation 7,

We denote

Calculate first, the first term is

We have Equation 14, thus

Thus

The first term is

The second term is

The third term is

Since , naturally commutes, hence

The last term is

However, as , Hence the last term is zero.

Thus

We know that which implies

Hence, . By conjugation is proved

Now we come to . The first term is

Thus

The first term is

The remaining terms are similar to the proof of , thus completing the whole proof. ⁠ 

Problem. Jellium model is a 3-dimensional electron gas with a uniform background of positive charge.

The Hamiltonian is given by

where we take the column interaction as the Yukawa form

in order to avoid the divergence of the interaction at the origin. When , it becomes the Coulomb interaction.

Assuming that the positive charge density is uniform, and have the same total charge as the electrons, we have

The Yukawa interaction of background positive charge can be integrated out, and we have

Similarly,

where

The term is better written in the second quantization, since it contains the kinetic energy term.

The fourier transform of Yukawa potential is well-known as

The most annoying part is when , the potential diverges. However it is shown to be cancelled by the background energy and the interaction energy. Since

By the anti-commutation relation, we have

Thus Equation 38 is reduced to

We can see the physics by making momentum dimensionless. The typical length is the Bohr radius (Gaussian unit) . Define , as the average distance between electrons. Typically, for metals, is around 2-6.

Let and , then

is the Rydberg energy as the energy scale of system.

The non-interacting ground state is a Fermi sea .

By perturbation theory, the ground state energy is

can be decomposed by Wick's theorem.

Then we have

In our context, Hartree term is cancelled. Only Fock term survives and contributes a negative energy shift.

This two terms can also be drawn in the Feynman diagrams.

We can calculate the Fock term

Problem.

With the analytical form

Write down all terms including the divergent part, hopefully they cancel out.

Hence we have

where

Thus

Which means the energy of 3d jellium electron gas is

Note that with the assumption , the perturbation is valid.

Since I'm more familiar with Mott insulator, it seems that the Wigner crystal has much similarity with Mott insulator. They both form when the potential energy dominates the kinetic energy and have strong localization of electrons. However, their context differs. While Wigner crystal forms in continuous space, Mott insulator lives on a lattice system. The Mott physics only have short range interaction, while the Wigner crystal has long range interaction (in Hartree-Fock level, we didn't cut off the long range interaction).

Problem. What is the difference between the Wigner crystal and the Mott insulator?

I cannot currently have a good picture of the Wigner crystal. In Mott insulator with short range interaction, if we perturb one electron (e.g. slightly shift its position), only the nearest neighbor electrons feel that and thus the perturbation is local and screened.

With some kind help from zhihu, now I would think Wigner crystal as isolated oscillating electrons, while Mott insulator is a system electrons still having strong correlations. Part of the reason is Mott insulator may still have the spin degree of freedom, while we would not say a Wigner crystal is "ferromagnetic" or something.

However, with long rang interaction, I would possibly expect many local minima around the Wigner crystal. This sense comes from the experience of thinking the Thompson Problem.

Problem. Is Wigner crystal stable?

If Wigner crystal has many local minima, it is unlikely to survive disorder and quantum fluctuation. Also, the discussion of transportation is hard, since the configuration will vary with time.

A friend has told me since it's quite easy to compute the Wigner crystal by numerics, it's unlikely to have many local minima. Maybe return to this problem if some day I have the chance to calculate it.

However, if the Wigner crystal is stable, it is a good candidate for the metal-insulator transition.

If the electron density can be written as

Then the system can form a ground state different from the magnetic etc. ones.

CDW breaks the translational symmetry, while SDW breaks the spin rotational symmetry and translational symmetry.

Experimentally, CDW can be observed by X-ray diffraction, while SDW can be observed by neutron diffraction.

Revisit Equation 42, we can decouple the 4 operator terms into 2 terms. Possible decomposition is

Equation 59 is actually the electron pairing term. This mean field approximation is called the Hartree-Fock-Bogoliubov Approximation. Equation 61 and Equation 60 are the Hartree term and the Fock term we have discussed before.

Problem. Why the Hartree-Fock-Bogoliubov Approximation can't be achieved by the perturbation theory?

That's because we use the Fermi sea as the reference state. The Fermi sea without instability can't excite the electron pairing term.

By Fock term, the interaction is now quadratic, and we can solve it exactly.

Thus the Hamiltonian of electron gas is now

Thus the single electron energy is

where . This is consistent with the result from perturbation theory Equation 55.

The velocity is

But diverges, leading to the un-physical result of infinite Fermi velocity.

Thomas-Fermi Approximation is a semi-classical one. It considers high density ( ) limit, thus the electron density is classical . The key difficulty in solving the jellium model is the kinetic energy term. It is because of the existing of kinetic energy term that we need to do all things in momentum space. If we can treat the kinetic energy term as a functional of , we can solve the problem in real space.

The Thomas-Fermi Approximation states, we can regard the kinetic energy term as the strong degenerate non-interacting electron gas, where

Thus

with the constraint of the total number of electrons, add a Lagrange multiplier.

Take divergence on both sides, we have

Thus we have the Thomas-Fermi equation

Thomas Fermi Approximation tells us that the interacting system at high density can be treated as a non-interacting system with an effective potential. This effective potential will affect the density of electron gas at first order. Generally, we can iteratively apply the perturbation until the density converges. There's also another approach from linear response theory to treat it.

Decompose the density into summation of occupation in space.

where Define

The density is now

The "induced" density (or say perturbated) is

Remember the linear response theory in real space,

Do the Fourier transform,

Generally we have

Fourier transform the Possion equation,

Thus we have

This can be directly achieved by the linear response theory.

Back to Equation 77, we have

which leads to a non-trivial permittivity.

This non-trivial permittivity leads to the screening of long range Coulomb interaction.

This is the Yukawa potential, which screens the long range Coulomb interaction. Because Thomas-Fermi wave vector is approximately same order of . So the typical screening length is the same order of distance between electrons.

We can treat the as an perturbation

The wave function after first order perturbation is

where The induced density is

I will try to follow AGD in this chapter, see how far I can go!

If we can divide the Hamiltonian as

where is some Hamiltonian that is relevantly easier to deal with (e.g. the free part), contains all the interaction part.

Under interaction picture we have

The schrodinger equation under interaction picture Equation 90 can be solved perturbatively. From now on we emit the subscript for simplicity.

We write out as a series:

Suppose we know some at , in zeroth order where , we have

The first order perturbation is

Similarly, the second order perturbation is

Rewrite Equation 92 as

where

Consider the case that time order doesn't hold. We insert an operator which is the operation to keep the time order. Thus the upper bound of each integral can be changed to with a price of duplicate terms.

That the time evolution operator is the time-ordered exponential of ,

Note the Time-ordering operator behaves differently in Fermionic and Bosonic systems.

Ref. 空扬笔记

This Hamiltonian doesn't commute with the translation operator , while it commutes with a magnetic translation operator

where is defined as

Check that!

The two terms cancel.

The BdG (Bogoliubov-de Genes) Hamiltonian can be derived from the negative U real space Hubbard model

Using the mean field approximation introduced in Equation 58, we have the BdG Hamiltonian,

If the system is uniform where , the Hamiltonian will be diagonalized in Fourier space

The kinetic energy can be rewritten as

Thus

The constant part can be dropped.

Write the matrix in the form of Pauli matrix

where

The notation is used to distinguish Pauli matrix in charge space from in spin space.

Thus

where

acts as a Weiss field

The energy gap is describe by the Weiss field, where

The gap is This gap is caused by non-zero density of Cooper pair at the ground state.

The Weiss field can be decomposed into magnitude and direction parts.

The direction can be described by the angle,

In the ground state the isospin is parallel to the field, which give the minimum energy of .

Here we choose the phase of , letting Thus we can represent all variables, with and ,

and tells that Thus the consistent equation tells that

In uniform case, Thus we have the self-consistent equation

Ref. Bogoliubov-de Gennes Method and Its Applications

The BdG approach relies on the assumption that there exist well-defined quasi-particles in SC.

Note that in SC state the Hatree-Fock channel mean field can be absorbed into the chemical potential, we will only consider the particle-particle pairing channel which terms Bogoliubov-Hatree-Fock mean field approxiamation. In case of s-wave superconductor, we only consider singlet-singlet pairing.

This gives the effective Hamiltonian

where

Suppose the Hamiltonian is diagonalized on the basis,

Thus we have,

These are the equations of motion (EOM) for , compared with the EOM of field operators , we have equations for and .

Problem. The here is a little bit weird.

The 4 equations can be block diagonalized to 2 sets of 2 equations, since only and are coupled, so are and .

Ref. 大黄猫笔记

Ref. 大黄猫笔记

When system is non-uniform E.g. disorder and boundary condition kicks in.

Introduce Nambu Spinor with components

Thus can be written as

Where

is diagonal matrix for order parameter, and is the tight-binding matrix.