Many Body Theory
Jellium Model
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
Remark
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
Remark
In thermodynamics limit, only term survives, and it cancels the and term. This is also why we introduce the background charge density.
Thus Equation 7 is reduced to
Corollary 1
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.
Remark
When , perturbation theory is unfeasible. We can think that the electron is well-separated. Maybe a Wigner crystal is formed.
Perturbation Calculation: Fock energy
The non-interacting ground state is a Fermi sea .
By pertubation theory, the ground state energy is
can be decomposed by Wick's theorem.
Theorem 1 Wick's theorem
Then we have
Definition 2
The first term is called the direct term or Hartree term, and the second term is called the exchange term or Fock term.
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.
Remark
Different from the calculation of scattering amplitude in high energy physics, in condensed matter physics, we are more interested in the vacuum diagrams. The above two diagrams are all vacuum 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
Corollary 2.1
Note that with the assumption , the perturbation is valid.
Remark
The calculation of jellium model gives us a good approximation approach – if your system is not too far from the jellium model, you can ignore the Hartree self energy term. This concept leads to the RPA (Random Phase Approximation) method.
Wigner Crystal
Definition 1 Wigner crystal
If electron density is less than a critical value, the jellium model electron gas will crystallize into a Wigner crystal.
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.
Definition 2 Thompson Problem
The Thompson problem is a problem of finding the minimum energy configuration of electrons on a sphere.
As far as I know, few configurations are known to have determined lowest energy.
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 question 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.
CDW and SDW
If the electron density can be written as
Then the system can form a ground state different from the magnetic etc. ones.
Definition 1 Charge Density Wave
if , the spin density , the charge density is periodic with wave vector . We call this the Charge Density Wave.
Spin Density Wave
If , the spin density is periodic with wave vector , the charge density is uniform. We call this the Spin Density Wave.
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.
Hartree-Fock Approximation (Mean Field Theory)
Definition 1 Hartree-Fock Approximation (Mean Field Theory)
If the operator has non-zero expectation value, we can assume the fluctuation is small and decompose the two operators into the mean field and the fluctuation.
Revisit Equation 11, we can decouple the 4 operator terms into 2 terms. Possible decomposition is
Equation 28 is actually the electron pairing term. This mean field approximation is called the Hartree-Fock-Bogoliubov Approximation. Equation 30 and Equation 29 are the Hartree term and the Fock term we have discussed before.
Remark
Key Observation: By this Hamiltonian decomposition, we can achieve the same results from the perturbation theory
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 24.
The velocity is
But diverges, leading to the un-physical result of infinite Fermi velocity.
Remark
The Fermi velocity tells us how electrons near Fermi surface move. The infinite Fermi velocity is due to the unscreened long range Coulomb interaction.
Thomas-Fermi Approximation
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.
Definition 1 The effective potential is
Take divergence on both sides, we have
Thus we have the Thomas-Fermi equation
Definition 2 The Thomas-Fermi equation is
Screen Long Range Coulomb Interaction
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
Corollary 1
This can be directly achieved by the linear response theory.
Back to Equation 46, we have
which leads to a non-trivial permissivity.
Definition 1 Thomas-Fermi wave vector is
This non-trivial permissivity 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.
Remark
The Thomas-Fermi wave vector is proportional to the DOS at Fermi surface. If we have large DOS at Fermi surface (e.g. flat band), the screening length is short. If the DOS is small, or even zero like at the Dirac point, the screening length is long.
Lindhard Function (RPA)
We can treat the as an perturbation
The wave function after first order perturbation is
where
The induced density is
QFT at
I will try to follow AGD in this chapter, see how far I can go!
Interaction picture
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.
Definition 1 unofficial definition
Interaction picture says that the state evolves with and the operator evolves with .
Under interaction picture we have
The schrodinger equation under interaction picture Equation 59 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 61 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 ,
Definition 3
Note the Time-ordering operator behaves differently in Fermionic and Bosonic systems.
Corollary 3.1 In Bosonic system
While Fermionic system there's an extra sign
A Glimpse into Topology
Why is Landau Level State an Ideal Topological Flat Band?
Ref. 空扬笔记
Definition 1 The Hamiltonian of a free electron in a magnetic field is
where and is periodical.
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.
BCS theory Is Deep
BdG formalism
The BdG (Bogoliubov-de Genes) Hamiltonian can be derived from the negative U real space Hubbard model
Definition 1 Attractive Hubbard model
Using the mean field approxiamation introduced in Equation 27, we have the BdG Hamiltonian,
Definition 2 Define the order parameter , because Cooper pair of Fermions condense at the ground state.
Thus,
which give the BdG Hamiltonian
If the system is uniform where , the Hamiltonian will be diagonalized in Fourier space
The kinetic energy can be rewritten as
Definition 3 Nambu Spinor in space
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
Remark
The important thing is the Zeeman field here is momentum dependent.
Unlike normal metals, the Weiss field of superconductor remains finite at the Fermi energy, giving rise to a gap in the excitation spectrum.
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
Corollary 3.1 BCS gap equation at
BdG Equation in Continuum
Ref. Bogoliubov-de Gennes Method and Its Applications
The BdG approach relies on the assumption that there exist well-defined quasi-particles in SC.
Remark
By Ref. BdG is correct in the weak-coupling regime, but also yiels qualitative results in strong-coupling regime.
Definition 1 The attractive interaction Hamiltonian is
where is the single particle Hamiltonian defined as
With the presence of vector potential,
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
Definition 2 Bogoliubov canonical transformation is helpful.
where is a Fermionic operator. In lattice system the scope of is restricted by the lattice sites, while in continuum, can reach infinity as a field operator.
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 .
Remark
This is such a messy formalism! I would stop here.
Non-uniform BCS theory
Anderson Theorem
Ref. 大黄猫笔记
Non-uniform BdG
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.
Definition 1 The impurity Hamiltonian
where and are the strength of non-magnetic and ferromagnetic impurity scattering.