Chapter 4, Linear response of independent electrons Video Solutions, Quantum Theory of the Electron Liquid

Problem 1

Alternative derivation of the Lindhard function. Derive the expression (4.10) for the Lindhard function by applying the general formula (3.47) to the density-density response function of non-interacting electrons.

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Problem 2

A useful integral. Calculate $\psi_2(z)$ from the double integral (4.20). An intermediate result that is useful to arrive at the result listed in Table 4.1 is

$$
\int_0^{2 \pi} \frac{d \varphi}{z-\cos \varphi}=\frac{z+z^*}{\left|z+z^*\right|} \frac{2 \pi}{\sqrt{z^2-1}}
$$

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Problem 3

Lindhard function for an ellipsoidal band. Calculate the Lindhard function for free electrons in an ellipsoidal band with dispersion

$$
\varepsilon_{\vec{k}}=\frac{\hbar^2}{2}\left(\frac{k_x^2}{m_x}+\frac{k_y^2}{m_y}+\frac{k_z^2}{m_z}\right)
$$

[Hint: make the scale transformation $k_i \rightarrow k_i \sqrt{\frac{m_i}{m_d}}(i=x, y$, or $z)$ to reduce the calculation to that of the isotropic Lindhard function. The density of states mass $m_d= \left(m_x m_y m_z\right)^{1 / 3}$ is so chosen to ensure that the transformation preserve the volume element in momentum space.]

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Problem 4

Density-density response function for a non-interacting inhomogeneous Fermi system. Write the expression for the density-density response function $\chi_{n n}^{(0)}\left(\vec{r}, \vec{r}^{\prime}, \omega\right)$ of a non-interacting inhomogeneous electron gas in terms of one-electron orbitals $\varphi_\alpha(\vec{r})$ and their eigenvalues.

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Problem 5

Static limit of the Lindhard function at finite temperature. Equation (4.28) is valid only at $T=0$. Make use of the compressibility sum rule to show that at any temperature one has

$$
\lim _{q \rightarrow 0} \chi_0(\vec{q}, 0)=-\frac{\partial n(\mu)}{\partial \mu}=-n^2 K_0
$$

where $n(\mu)$ is the electronic density of the non-interacting electron gas as a function of the chemical potential, and $K_0$ is the compressibility.

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Problem 6

High-frequency limit of the Lindhard function. Starting from the general formulas for the Lindhard function verify the high-frequency limiting form (4.38), and show that it describes the small- $q$ behavior of the Lindhard function for any $\omega \neq 0$.

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02:21

Problem 7

Lindhard function at finite temperature. Show that the Fermi- Dirac distribution $n_F(E, T)=\frac{1}{e^{E / k_B} T+1}$ at energy $E$, temperature $T$ and chemical potential 0 satisfies the identity

$$
n_F(E, T)=\int_{-\infty}^{\infty} d E^{\prime} \frac{n_F\left(E^{\prime}, 0\right)}{4 k_B T \cosh ^2 \frac{E^{\prime}-E}{2 k_B T}}
$$

Make use of this identity to prove Eq. (4.42) for the Lindhard function at finite temperature.

Dominador Tan

Numerade Educator

Problem 8

Longitudinal current-current response function of the non-interacting electron gas. Show by direct calculation that the longitudinal current-current response function of a non-interacting electron gas, defined by means of the general formulas (3.174) and (3.172), satisfies Eq. (3.175). This amounts to proving that the Lindhard function is a number-conserving response function.

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Problem 9

Transverse spin-spin response functions of the non-interacting electron gas. Explicitly evaluate Eq. (4.14) for the transverse spin-spin response function of the non-interacting electron gas. The result is

$$
\begin{aligned}
\chi_{S_{+} S_{-}}^{(0)}(q, \omega)= & 2 N_{\uparrow}(0) \frac{k_{F \uparrow}}{q} \Psi_d\left(\frac{\omega-\Delta+i \eta}{q v_{F \uparrow}}-\frac{q}{2 k_{F \uparrow}}\right) \\
& -2 N_{\downarrow}(0) \frac{k_{F \downarrow}}{q} \Psi_d\left(\frac{\omega-\Delta+i \eta}{q v_{F \downarrow}}+\frac{q}{2 k_{F \downarrow}}\right),
\end{aligned}
$$

where $\Delta=g \mu_B B$ is the energy splitting between the up and down spin bands. Notice that in the paramagnetic state this reduces to twice the longitudinal spin-spin response function.

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26:30

Problem 10

Orbital magnetic susceptibility in two and three dimensions. In the presence of a small magnetic field a three-dimensional electron gas can be described as a stack of two-dimensional gases obtained by slicing the three-dimensional Fermi sphere with planes perpendicular to the direction of the field. Because the two-dimensional density of states does not depend on Fermi wave vector all the two-dimensional slices give equal contributions to the total density of states and to the orbital magnetization. Use this observation to explain why the Landau result (4.50) for the orbital magnetic susceptibility is valid in both two and three dimensions.

Dr. Rajveer Singh

Numerade Educator

Problem 11

Evolution of a density packet in the deformable jellium model. Obtain Eq. (4.52) for the evolution of a density packet in the deformable jellium model with random impurities from the solution of the diffusion equation (4.51). [Hint: make good use of Laplace and Fourier transforms.]

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Problem 12

Diffusion pole and diffusion equation. Show that the form (4.76) for the densitydensity response function of the electron liquid implies that neutral density fluctuations evolve according to the diffusion equation (4.51).

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Problem 13

Evolution of a density packet in the rigid jellium model. Calculate the time evolution of the density packet considered in Exercise 11, but now for a rigid jellium model with random impurities. [Hint: combine the continuity equation with the Poisson equation for the electric field produced by the excess charge density and use the Drude formula to connect the current density to the electric field. Notice that the diffusion equation (4.51) does not apply to this situation.]

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Problem 14

Evolution of a neutral electron-hole packet. Same as the previous exercise for the case of a charge-neutral packet of electrons and holes, with the electrons and the holes having different conductivities and diffusion constants.

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09:39

Problem 15

Small $q$ and $\omega$ form of Mermin's response function in 2D. Derive Eq. (4.68) for the density-density response function of a disordered 2D electron gas at small $q$ and $\omega$.

Ameer Said

Numerade Educator

Problem 16

Diffusion pole in Mermin's density-density response function. By explicitly evaluating the small $q$ and $\omega$ expression for $\chi_0\left(q, \omega+\frac{i}{\tau}\right)$, show that Mermin's density-density response function (4.67) has a diffusion pole in the form of Eq. (4.76).

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