# Polarization Approximation

In this note, we discuss serveral approximations of the polarization

## Linearized Dispersion

The original polarization is given by,

\begin{aligned} \Pi_0(\Omega_n, \vec{q}) &= -S \int\frac{d^d \vec k}{{(2\pi)}^d} \frac{n(\epsilon_{\vec{k}+\vec{q}})-n(\epsilon_{\vec{k}})}{i\omega_n-\epsilon_{\vec{k}+\vec{q}}+\epsilon_{\vec{k}}} \\ &=-S\int \frac{d^d \vec k}{{(2\pi)}^d} n(\epsilon_{\vec k}) \left[ \frac{1}{i\Omega+\epsilon_{\vec k}-\epsilon_{\vec k+\vec q}}-\frac{1}{i\Omega+\epsilon_{\vec k-\vec q}-\epsilon_{\vec k}}\right]. \end{aligned}

One possible approximation is to replace the kinetic energy with a dispersion linearized near the Fermi surface,

$$$\xi_{\mathbf{p}+\mathbf{q}}-\xi_{\mathbf{p}}=(1 / m) \mathbf{p} \cdot \mathbf{q}+\mathcal{O}\left(q^{2}\right)$$$

so that,

$$$n_{\mathrm{F}}\left(\epsilon_{\mathbf{p}+\mathbf{q}}\right)-n_{\mathrm{F}}\left(\epsilon_{\mathbf{p}}\right) \simeq \partial_{\epsilon_p} n_{\mathrm{F}}\left(\epsilon_{p}\right)(1 / m) \mathbf{p} \cdot \mathbf{q} \simeq-\delta\left(\epsilon_{p}-\mu\right)(1 / m) \mathbf{p} \cdot \mathbf{q}$$$

where, in the zero-temperature limit, the last equality becomes exact. Converting the momentum sum into an integral, we thus obtain

$$$\Pi_0(\mathbf{q}, \omega_{m})=-S \int \frac{d^{3} p}{(2 \pi)^{3}} \delta\left(\epsilon_{p}-\mu\right) \frac{\frac{1}{m} \mathbf{p} \cdot \mathbf{q}}{i \omega_{m}+\frac{1}{m} \mathbf{p} \cdot \mathbf{q}}.$$$

Evaluate the integral gives

\begin{aligned} \Pi_0(\mathbf{q}, \omega_{m}) &=-\frac{S}{(2 \pi)^{3}} \int d p p^{2} \int d \delta\left(\epsilon_{p}-\mu\right) \frac{v_{\mathrm{F}} \mathbf{n} \cdot \mathbf{q}}{i \omega_{m}+v_{\mathrm{F}} \mathbf{n} \cdot \mathbf{q}} \\ &=-\underbrace{\frac{S}{(2 \pi)^{3}} \int d p p^{2} \int d \delta\left(\epsilon_{p}-\mu\right)}_{N_F} \frac{1}{\int d\Omega} \int d\Omega \frac{v_{\mathrm{F}} \mathbf{n} \cdot \mathbf{q}}{i \omega_{m}+v_{\mathrm{F}} \mathbf{n} \cdot \mathbf{q}} \\ &=-\frac{N_F}{2} \int_{-1}^{1} d x \frac{v_{\mathrm{F}} x q}{i \omega_{m}+v_{\mathrm{F}} x q}=-N_F\left[1-\frac{i \omega_{m}}{2 v_{\mathrm{F}} q} \ln \left(\frac{i \omega_{m}+v_{\mathrm{F}} q}{i \omega_{m}-v_{\mathrm{F}} q}\right)\right] . \end{aligned}

The above derivation is adapted from the A. Altland and B. Simons' book "Condensed Matter Field Theory" Chapter 5.2, Eq. (5.30).

### Two limits:

For the exact free-electron polarization, we expect In the limit $q ≫ ω_m$,

$$$Π_0(q, iω_m) \rightarrow -N_F \left(1-\frac{π}{2}\frac{|ω_m|}{v_{\mathrm{F}} q}\right)$$$

where we use the Taylor expansion for $\text{Log}\left[\frac{1+i x}{-1+i x}\right]$ where $x=\omega_m/(v_{\mathrm{F}} q)$,

$$$\begin{array}{cc} \{ & \begin{array}{cc} -i \pi +2 i x-\frac{2 i x^3}{3}+O\left(x^4\right) & x \ge 0 \\ i \pi +2 i x-\frac{2 i x^3}{3}+O\left(x^4\right) & x<0 \\ \end{array} \\ \end{array}$$$

and in the limit $q ≪ ω_m$,

$$$Π_0(q, iω_m) \rightarrow -\frac{N_F}{3}\left(\frac{v_{\mathrm{F}} q}{ω_m}\right)^2 = -N_F \left(\frac{q}{q_{\mathrm{TF}}}\frac{\omega_p}{\omega_m}\right)^2$$$

where the plasma-frequency and the Thomas-Fermi screening momentum is related by $ω_p=v_F q_{\mathrm{TF}}/\sqrt{3}$.

## Plasma Approximation

It is sometimes convenient to approximate the polarization with the plasma poles,

$$$Π_0(q, iω_m) \approx -N_F \frac{(q/q_{\mathrm{TF}})^2}{(q/q_{\mathrm{TF}})^2+(\omega_m/\omega_p)^2}$$$