HEG/jellium

For the study of electrons in solids, a popular basis set is plane waves. Eigenstates are described by \((p, \sigma)\), where \(\sigma\) is the spin index, which is ±1 for spin up or down. The Hamiltonian then has the form:

[Mahan, 1.160] H = On site energy + Interaction b/w electionrs and atoms or ions + electron-electron interaction.

\[ \text{Electron Density Operator} = \rho(q) = \Sigma_{k\sigma} c_{k+q\sigma}^{\dagger}c_{k\sigma} \]

Aside: Particle Density Operator

The particle density operator is a fundamental quantity in quantum mechanics and solid-state physics. It describes how particles are distributed in space and is often expressed in the plane-wave representation.

In second quantization, the particle density operator in real space is given by:

\[ \hat{n}(\mathbf{r}) = \sum_{k,k'} \psi_k^*(\mathbf{r}) \psi_{k'}(\mathbf{r}) c_k^\dagger c_{k'} \]

For a plane-wave basis, where the wavefunctions are:

\[ \psi_k(\mathbf{r}) = \frac{1}{\sqrt{V}} e^{i \mathbf{k} \cdot \mathbf{r}} \]

the density operator simplifies to:

\[ \hat{n}(\mathbf{r}) = \sum_{k,k'} e^{i (\mathbf{k'} - \mathbf{k}) \cdot \mathbf{r}} c_k^\dagger c_{k'} \]

This expression shows that the density operator measures the particle distribution in space by summing over all possible transitions between states ( k ) and ( k’ ).

To study density fluctuations, we define the Fourier transform of ( () ):

\[ P_q = \int d\mathbf{r} \, e^{-i \mathbf{q} \cdot \mathbf{r}} \hat{n}(\mathbf{r}) \]

Substituting \(\hat{n}(\mathbf{r})\) gives:

\[ P_q = \sum_k c^\dagger_{k+q} c_k \]

This shows that the density operator in reciprocal space describes how particles scatter between momentum states differing by ( q ). The operator ( P_q ) is crucial in studying response functions, such as those appearing in linear response theory and dielectric screening.

Problem: Too complicated a Hamiltonian.

Solution: Use a simpler model e.g. HEG which gets rid of the middle term.

[Mahan, 1.164] H_HEG = On site energy + electron-electron interaction.

Replaces atoms with a uniform positive background charge of density \(n_o\). To preserve charge neutrality, the average particle density of the electron gas must also be \(n_o\).

Question: What is \(n_o\), the average density:

It’s the \(q=0\) value of the density operator.

\[ n_o = \frac{1}{\nu} \langle \rho (q=0) \rangle = \frac{1}{\nu} \Sigma_{p\sigma}N_p = \frac{N_e}{\nu} \]

Sources

• Mahan. Many Particle Physics.