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+---
+title: "Lindhard function"
+date: 2022-01-24 # Originally 2021-10-12, major rewrite
+categories:
+- Physics
+- Quantum mechanics
+layout: "concept"
+---
+
+The **Lindhard function** describes the response of
+[jellium](/know/concept/jellium) (i.e. a free electron gas)
+to an external perturbation, and is a quantum-mechanical
+alternative to the [Drude model](/know/concept/drude-model/).
+
+We start from the [Kubo formula](/know/concept/kubo-formula/)
+for the electron density operator $\hat{n}$,
+which describes the change in $\Expval{\hat{n}}$
+due to a time-dependent perturbation $\hat{H}_1$:
+
+$$\begin{aligned}
+ \delta\!\Expval{ {\hat{n}}}\!(\vb{r}, t)
+ = -\frac{i}{\hbar} \int_{-\infty}^\infty \Theta(t - t') \Expval{\Comm{\hat{n}_I(\vb{r}, t)}{\hat{H}_{1,I}(t')}}_0 \dd{t'}
+\end{aligned}$$
+
+Where the subscript $I$ refers to the [interaction picture](/know/concept/interaction-picture/),
+and the expectation $\Expval{}_0$ is for
+a thermal equilibrium before the perturbation was applied.
+Now consider a harmonic $\hat{H}_1$:
+
+$$\begin{aligned}
+ \hat{H}_{1,S}(t)
+ = e^{i (\omega + i \eta) t} \int_{-\infty}^\infty U(\vb{r}) \: \hat{n}_S(\vb{r}) \dd{\vb{r}}
+\end{aligned}$$
+
+Where $S$ is the Schrödinger picture,
+$\eta$ is a positive infinitesimal to ensure convergence later,
+and $U(\vb{r})$ is an arbitrary potential function.
+The Kubo formula becomes:
+
+$$\begin{aligned}
+ \delta\!\Expval{ {\hat{n}}}\!(\vb{r}, t)
+ = \iint_{-\infty}^\infty \chi(\vb{r}, \vb{r}'; t, t') \: U(\vb{r}') \: e^{i (\omega + i \eta) t'} \dd{t'} \dd{\vb{r}'}
+\end{aligned}$$
+
+Here, $\chi$ is the density-density correlation function,
+i.e. a two-particle [Green's function](/know/concept/greens-functions/):
+
+$$\begin{aligned}
+ \chi(\vb{r}, \vb{r}'; t, t')
+ \equiv - \frac{i}{\hbar} \Theta(t - t') \Expval{\Comm{\hat{n}_I(\vb{r}, t)}{\hat{n}_I(\vb{r}', t')}}_0
+\end{aligned}$$
+
+Let us assume that the unperturbed system (i.e. without $U$) is spatially uniform,
+so that $\chi$ only depends on the difference $\vb{r} - \vb{r}'$.
+We then take its [Fourier transform](/know/concept/fourier-transform/)
+$\vb{r}\!-\!\vb{r}' \to \vb{q}$:
+
+$$\begin{aligned}
+ \chi(\vb{q}; t, t')
+ &= \int_{-\infty}^\infty \chi(\vb{r} - \vb{r}'; t, t') \: e^{- i \vb{q} \cdot (\vb{r} - \vb{r}')} \dd{\vb{r}}
+ \\
+ &= -\frac{i}{\hbar} \frac{\Theta(t \!-\! t')}{(2 \pi)^{2D}} \iiint
+ \Expval{\Comm{\hat{n}_I(\vb{q}_1, t)}{\hat{n}_I(\vb{q}_2, t')}}_0
+ \: e^{i \vb{q}_1 \cdot \vb{r}} e^{i \vb{q}_2 \cdot \vb{r}'} e^{- i \vb{q} \cdot (\vb{r} - \vb{r}')} \dd{\vb{q}_1} \dd{\vb{q}_2} \dd{\vb{r}}
+\end{aligned}$$
+
+Where both $\hat{n}_I$ have been written as inverse Fourier transforms,
+giving a factor $(2 \pi)^{-2 D}$, with $D$ being the number of spatial dimensions.
+We rearrange to get a [Dirac delta function](/know/concept/dirac-delta-function/) $\delta$:
+
+$$\begin{aligned}
+ \chi(\vb{q}; t, t')
+ &= -\frac{i}{\hbar} \frac{\Theta(t \!-\! t')}{(2 \pi)^{2D}} \iiint
+ \Expval{\Comm{\hat{n}_I(\vb{q}_1, t)}{\hat{n}_I(\vb{q}_2, t')}}_0
+ \: e^{i (\vb{q}_1 - \vb{q}) \cdot \vb{r}} e^{i (\vb{q}_2 + \vb{q}) \cdot \vb{r}'} \dd{\vb{q}_1} \dd{\vb{q}_2} \dd{\vb{r}}
+ \\
+ &= -\frac{i}{\hbar} \frac{\Theta(t \!-\! t')}{(2 \pi)^D} \iint
+ \Expval{\Comm{\hat{n}_I(\vb{q}_1, t)}{\hat{n}_I(\vb{q}_2, t')}}_0
+ \: \delta(\vb{q}_1 \!-\! \vb{q}) \: e^{i (\vb{q}_2 + \vb{q}) \cdot \vb{r}'} \dd{\vb{q}_1} \dd{\vb{q}_2}
+ \\
+ &= -\frac{i}{\hbar} \frac{\Theta(t \!-\! t')}{(2 \pi)^D} \int
+ \Expval{\Comm{\hat{n}_I(\vb{q}, t)}{\hat{n}_I(\vb{q}_2, t')}}_0
+ \: e^{i (\vb{q}_2 + \vb{q}) \cdot \vb{r}'} \dd{\vb{q}_2}
+\end{aligned}$$
+
+On the left, $\vb{r}'$ does not appear, so it must also disappear on the right.
+If we choose an arbitrary (hyper)cube of volume $V$ in real space,
+then clearly $\int_V \dd{\vb{r}'} = V$. Therefore:
+
+$$\begin{aligned}
+ \chi(\vb{q}; t, t')
+ &= -\frac{i}{\hbar} \frac{\Theta(t \!-\! t')}{(2 \pi)^D} \frac{1}{V} \int_V \int_{-\infty}^\infty
+ \Expval{\Comm{\hat{n}_I(\vb{q}, t)}{\hat{n}_I(\vb{q}_2, t')}}_0
+ \: e^{i (\vb{q}_2 + \vb{q}) \cdot \vb{r}'} \dd{\vb{q}_2} \dd{\vb{r}'}
+\end{aligned}$$
+
+For $V \to \infty$ we get a Dirac delta function,
+but in fact the conclusion holds for finite $V$ too:
+
+$$\begin{aligned}
+ \chi(\vb{q}; t, t')
+ &= -\frac{i}{\hbar} \Theta(t \!-\! t') \frac{1}{V} \int_{-\infty}^\infty
+ \Expval{\Comm{\hat{n}_I(\vb{q}, t)}{\hat{n}_I(\vb{q}_2, t')}}_0 \: \delta(\vb{q}_2 \!+\! \vb{q}) \dd{\vb{q}_2}
+ \\
+ &= -\frac{i}{\hbar} \Theta(t \!-\! t') \frac{1}{V} \Expval{\Comm{\hat{n}_I(\vb{q}, t)}{\hat{n}_I(-\vb{q}, t')}}_0
+\end{aligned}$$
+
+Similarly, if the unperturbed Hamiltonian $\hat{H}_0$ is time-independent,
+$\chi$ only depends on the time difference $t - t'$.
+Note that $\delta{\Expval{\hat{n}}}$ already has the form of a Fourier transform,
+which gives us an opportunity to rewrite $\chi$
+in the [Lehmann representation](/know/concept/lehmann-representation/):
+
+$$\begin{aligned}
+ \chi(\vb{q}, \omega)
+ = \frac{1}{Z V} \sum_{\nu \nu'}
+ \frac{\matrixel{\nu}{\hat{n}_S(\vb{q})}{\nu'} \matrixel{\nu'}{\hat{n}_S(-\vb{q})}{\nu}}{\hbar (\omega + i \eta) + E_\nu - E_{\nu'}}
+ \Big( e^{-\beta E_\nu} - e^{- \beta E_{\nu'}} \Big)
+\end{aligned}$$
+
+Where $\Ket{\nu}$ and $\Ket{\nu'}$ are many-electron eigenstates of $\hat{H}_0$,
+and $Z$ is the [grand partition function](/know/concept/grand-canonical-ensemble/).
+According to the [convolution theorem](/know/concept/convolution-theorem/)
+$\delta{\Expval{\hat{n}}}(\vb{q}, \omega) = \chi(\vb{q}, \omega) \: U(\vb{q})$.
+In anticipation, we swap $\nu$ and $\nu''$ in the second term,
+so the general response function is written as:
+
+$$\begin{aligned}
+ \chi(\vb{q}, \omega)
+ = \frac{1}{Z V} \sum_{\nu \nu'} \bigg(
+ \frac{\matrixel{\nu}{\hat{n}(\vb{q})}{\nu'} \matrixel{\nu'}{\hat{n}(-\vb{q})}{\nu}}
+ {\hbar (\omega + i \eta) + E_\nu - E_{\nu'}}
+ - \frac{\matrixel{\nu}{\hat{n}(-\vb{q})}{\nu'} \matrixel{\nu'}{\hat{n}(\vb{q})}{\nu}}
+ {\hbar (\omega + i \eta) + E_{\nu'} - E_\nu} \bigg) e^{-\beta E_\nu}
+\end{aligned}$$
+
+All operators are in the Schrödinger picture from now on, hence we dropped the subscript $S$.
+
+To proceed, we need to rewrite $\hat{n}(\vb{q})$ somehow.
+If we neglect electron-electron interactions,
+the single-particle states are simply plane waves, in which case:
+
+$$\begin{aligned}
+ \hat{n}(\vb{q})
+ = \sum_{\sigma \vb{k}} \hat{c}_{\sigma,\vb{k}}^\dagger \hat{c}_{\sigma,\vb{k} + \vb{q}}
+ \qquad \qquad
+ \hat{n}(-\vb{q})
+ = \hat{n}^\dagger(\vb{q})
+\end{aligned}$$
+
+
+
+
+
+
+Starting from the general definition of $\hat{n}$,
+we write out the field operators $\hat{\Psi}(\vb{r})$,
+and insert the known non-interacting single-electron orbitals
+$\psi_\vb{k}(\vb{r}) = e^{i \vb{k} \cdot \vb{r}} / \sqrt{V}$:
+
+$$\begin{aligned}
+ \hat{n}(\vb{r})
+ \equiv \hat{\Psi}{}^\dagger(\vb{r}) \hat{\Psi}(\vb{r})
+ = \sum_{\vb{k} \vb{k}'} \psi_{\vb{k}}^*(\vb{r}) \: \psi_{\vb{k}'}(\vb{r})\: \hat{c}_{\vb{k}}^\dagger \hat{c}_{\vb{k}'}
+ = \frac{1}{V} \sum_{\vb{k} \vb{k}'} e^{i (\vb{k}' - \vb{k}) \cdot \vb{r}} \hat{c}_{\vb{k}}^\dagger \hat{c}_{\vb{k}'}
+\end{aligned}$$
+
+Taking the Fourier transfom yields a Dirac delta function $\delta$:
+
+$$\begin{aligned}
+ \hat{n}(\vb{q})
+ = \frac{1}{V} \int_{-\infty}^\infty
+ \sum_{\vb{k} \vb{k}'} \hat{c}_{\vb{k}}^\dagger \hat{c}_{\vb{k}'} \: e^{i (\vb{k}' - \vb{k} - \vb{q})\cdot \vb{r}} \dd{\vb{r}}
+ = \frac{(2 \pi)^D}{V} \sum_{\vb{k} \vb{k}'} \hat{c}_{\vb{k}}^\dagger \hat{c}_{\vb{k}'} \: \delta(\vb{k}' \!-\! \vb{k} \!-\! \vb{q})
+\end{aligned}$$
+
+If we impose periodic boundary conditions
+on our $D$-dimensional hypercube of volume $V$,
+then $\vb{k}$ becomes discrete,
+with per-value spacing $2 \pi / V^{1/D}$ along each axis.
+
+Consequently, each orbital $\psi_\vb{k}$ uniquely occupies
+a volume $(2 \pi)^D / V$ in $\vb{k}$-space, so we make the approximation
+$\sum_{\vb{k}} \approx V / (2 \pi)^D \int_{-\infty}^\infty \dd{\vb{k}}$.
+This becomes exact for $V \to \infty$,
+in which case $\vb{k}$ also becomes continuous again,
+which is what we want for jellium.
+
+We apply this standard trick from condensed matter physics to $\hat{n}$,
+and $V$ cancels out:
+
+$$\begin{aligned}
+ \hat{n}(\vb{q})
+ &= \frac{(2 \pi)^D}{V} \frac{V}{(2 \pi)^D} \sum_{\vb{k}} \int_{-\infty}^\infty
+ \hat{c}_{\vb{k}}^\dagger \hat{c}_{\vb{k}'} \: \delta(\vb{k}' \!-\! \vb{k} \!-\! \vb{q}) \dd{\vb{k}'}
+ = \sum_{\vb{k}} \hat{c}_{\vb{k}}^\dagger \hat{c}_{\vb{k} + \vb{q}}
+\end{aligned}$$
+
+For negated arguments, we simply define $\vb{k}' \equiv \vb{k} - \vb{q}$
+to show that $\hat{n}(-\vb{q}) = \hat{n}{}^\dagger(\vb{q})$,
+which can also be understood as a consequence of $\hat{n}(\vb{r})$ being real:
+
+$$\begin{aligned}
+ \hat{n}(-\vb{q})
+ = \sum_{\vb{k}} \hat{c}_{\vb{k}}^\dagger \hat{c}_{\vb{k} - \vb{q}}
+ = \sum_{\vb{k}'} \hat{c}_{\vb{k}' + \vb{q}}^\dagger \hat{c}_{\vb{k}'}
+ = \hat{n}^\dagger(\vb{q})
+\end{aligned}$$
+
+The summation variable $\vb{k}$ has an associated spin $\sigma$,
+and $\hat{n}$ does not carry any spin.
+
+
+
+When neglecting interactions, it is tradition to rename $\chi$ to $\chi_0$.
+We insert $\hat{n}$, suppressing spin:
+
+$$\begin{aligned}
+ \chi_0
+ &= \frac{1}{Z V} \sum_{\vb{k} \vb{k}'} \sum_{\nu \nu'} \bigg(
+ \frac{\matrixel{\nu}{\hat{c}_{\vb{k}}^\dagger \hat{c}_{\vb{k} + \vb{q}}}{\nu'}
+ \matrixel{\nu'}{\hat{c}_{\vb{k}' + \vb{q}}^\dagger \hat{c}_{\vb{k}'}}{\nu}}
+ {\hbar (\omega + i \eta) + E_\nu - E_{\nu'}}
+ - \frac{\matrixel{\nu}{\hat{c}_{\vb{k} + \vb{q}}^\dagger \hat{c}_{\vb{k}}}{\nu'}
+ \matrixel{\nu'}{\hat{c}_{\vb{k}'}^\dagger \hat{c}_{\vb{k}' + \vb{q}}}{\nu}}
+ {\hbar (\omega + i \eta) + E_{\nu'} - E_\nu} \bigg) e^{-\beta E_\nu}
+\end{aligned}$$
+
+Here, $\matrixel{\nu}{\hat{c}_{\vb{k}}^\dagger \hat{c}_{\vb{k} + \vb{q}}}{\nu'}$
+is only nonzero if $\Ket{\nu'}$ is contructed from $\Ket{\nu}$
+by moving an electron from $\vb{k}$ to $\vb{k} \!+\! \vb{q}$,
+and analogously for the other inner products.
+As a result, $\vb{k} = \vb{k}'$ (and $\sigma = \sigma'$).
+
+For the same reason, the energy difference $E_\nu \!-\! E_{\nu'}$
+can simply be replaced by the cost of the single-particle excitation
+$\xi_{\vb{k}} \!-\! \xi_{\vb{k} + \vb{q}}$,
+where $\xi_{\vb{k}}$ is the energy of a $\vb{k}$-orbital.
+Therefore:
+
+$$\begin{aligned}
+ \chi_0
+ &= \frac{1}{Z V} \sum_{\vb{k}} \sum_{\nu \nu'} \bigg(
+ \frac{\matrixel{\nu}{\hat{c}_{\vb{k}}^\dagger \hat{c}_{\vb{k} + \vb{q}}}{\nu'}
+ \matrixel{\nu'}{\hat{c}_{\vb{k} + \vb{q}}^\dagger \hat{c}_{\vb{k}}}{\nu}}
+ {\hbar (\omega + i \eta) + \xi_{\vb{k}} - \xi_{\vb{k} + \vb{q}}}
+ - \frac{\matrixel{\nu}{\hat{c}_{\vb{k} + \vb{q}}^\dagger \hat{c}_{\vb{k}}}{\nu'}
+ \matrixel{\nu'}{\hat{c}_{\vb{k}}^\dagger \hat{c}_{\vb{k} + \vb{q}}}{\nu}}
+ {\hbar (\omega + i \eta) + \xi_{\vb{k}} - \xi_{\vb{k} + \vb{q}}} \bigg) e^{-\beta E_\nu}
+\end{aligned}$$
+
+Notice that we have eliminated all dependence on $\Ket{\nu'}$,
+so we remove it by $\sum_{\nu} \Ket{\nu} \Bra{\nu} = 1$:
+
+$$\begin{aligned}
+ \chi_0
+ &= \frac{1}{Z V} \sum_{\vb{k}} \sum_{\nu} \bigg(
+ \frac{\matrixel{\nu}{\hat{c}_{\vb{k}}^\dagger \hat{c}_{\vb{k} + \vb{q}} \hat{c}_{\vb{k} + \vb{q}}^\dagger \hat{c}_{\vb{k}}}{\nu}}
+ {\hbar (\omega + i \eta) + \xi_{\vb{k}} - \xi_{\vb{k} + \vb{q}}}
+ - \frac{\matrixel{\nu}{\hat{c}_{\vb{k} + \vb{q}}^\dagger \hat{c}_{\vb{k}} \hat{c}_{\vb{k}}^\dagger \hat{c}_{\vb{k} + \vb{q}}}{\nu}}
+ {\hbar (\omega + i \eta) + \xi_{\vb{k}} - \xi_{\vb{k} + \vb{q}}} \bigg) e^{-\beta E_\nu}
+ \\
+ &= \frac{1}{Z V} \sum_{\vb{k}} \sum_{\nu}
+ \frac{\matrixel{\nu}{\comm{\hat{c}_{\vb{k}}^\dagger \hat{c}_{\vb{k} + \vb{q}}}
+ {\hat{c}_{\vb{k} + \vb{q}}^\dagger \hat{c}_{\vb{k}}} \: e^{- \beta \hat{H}_0}}{\nu}}
+ {\hbar (\omega + i \eta) + \xi_{\vb{k}} - \xi_{\vb{k} + \vb{q}}}
+\end{aligned}$$
+
+Where we recognized the commutator,
+and eliminated $E_\nu$ using $\hat{H}_0 \Ket{n} = E_\nu \Ket{\nu}$.
+The resulting expression has the form of a matrix trace $\Tr$
+and a thermal expectation $\Expval{}_0$:
+
+$$\begin{aligned}
+ \chi_0
+ &= \frac{1}{Z V} \sum_{\vb{k}} \frac{\Tr\!\big(\comm{\hat{c}_{\vb{k}}^\dagger \hat{c}_{\vb{k} + \vb{q}}}
+ {\hat{c}_{\vb{k} + \vb{q}}^\dagger \hat{c}_{\vb{k}}} \: e^{- \beta \hat{H}_0} \big)}
+ {\hbar (\omega + i \eta) + \xi_{\vb{k}} - \xi_{\vb{k} + \vb{q}}}
+ = \frac{1}{V} \sum_{\vb{k}}
+ \frac{\expval{\comm{\hat{c}_{\vb{k}}^\dagger \hat{c}_{\vb{k} + \vb{q}}}{\hat{c}_{\vb{k} + \vb{q}}^\dagger \hat{c}_{\vb{k}}}}_0}
+ {\hbar (\omega + i \eta) + \xi_{\vb{k}} - \xi_{\vb{k} + \vb{q}}}
+\end{aligned}$$
+
+This commutator can be evaluated,
+and in this particular case it turns out to be:
+
+$$\begin{aligned}
+ \comm{\hat{c}_{\vb{k}}^\dagger \hat{c}_{\vb{k} + \vb{q}}}{\hat{c}_{\vb{k} + \vb{q}}^\dagger \hat{c}_{\vb{k}}}
+ = \hat{c}_{\vb{k}}^\dagger \hat{c}_{\vb{k}} - \hat{c}_{\vb{k} + \vb{q}}^\dagger \hat{c}_{\vb{k} + \vb{q}}
+\end{aligned}$$
+
+
+
+
+
+
+In general, for any single-particle states labeled by $m$, $n$, $o$ and $p$, we have:
+$$\begin{aligned}
+ \comm{\hat{c}_m^\dagger \hat{c}_n}{\hat{c}_o^\dagger \hat{c}_p}
+ &= \hat{c}_m^\dagger \hat{c}_n \hat{c}_o^\dagger \hat{c}_p - \hat{c}_o^\dagger \hat{c}_p \hat{c}_m^\dagger \hat{c}_n
+ \\
+ &= \hat{c}_m^\dagger \big( \acomm{\hat{c}_n}{\hat{c}_o^\dagger} - \hat{c}_o^\dagger \hat{c}_n \big) \hat{c}_p
+ - \hat{c}_o^\dagger \big( \acomm{\hat{c}_p}{\hat{c}_m^\dagger} - \hat{c}_m^\dagger \hat{c}_p \big) \hat{c}_n
+\end{aligned}$$
+
+Using the standard fermion anticommutation relations, this becomes:
+
+$$\begin{aligned}
+ \comm{\hat{c}_m^\dagger \hat{c}_n}{\hat{c}_o^\dagger \hat{c}_p}
+ &= \hat{c}_m^\dagger \big( \delta_{no} - \hat{c}_o^\dagger \hat{c}_n \big) \hat{c}_p
+ - \hat{c}_o^\dagger \big( \delta_{pm} - \hat{c}_m^\dagger \hat{c}_p \big) \hat{c}_n
+ \\
+ &= \hat{c}_m^\dagger \hat{c}_p \: \delta_{no} - \hat{c}_m^\dagger \hat{c}_o^\dagger \hat{c}_n \hat{c}_p
+ - \hat{c}_o^\dagger \hat{c}_n \: \delta_{pm} + \hat{c}_o^\dagger \hat{c}_m^\dagger \hat{c}_p \hat{c}_n
+ \\
+ &= \hat{c}_m^\dagger \hat{c}_p \: \delta_{no} - \hat{c}_o^\dagger \hat{c}_n \: \delta_{pm}
+\end{aligned}$$
+
+In this case, $m = p = \vb{k}$ and $n = o = \vb{k} \!+\! \vb{q}$,
+so the Kronecker deltas are unnecessary.
+
+
+
+We substitute this result into $\chi_0$,
+and reintroduce the spin index $\sigma$ associated with $\vb{k}$:
+
+$$\begin{aligned}
+ \chi_0(\vb{q}, \omega)
+ = \frac{1}{V} \sum_{\sigma \vb{k}}
+ \frac{\expval{\hat{c}_{\sigma,\vb{k}}^\dagger \hat{c}_{\sigma,\vb{k}} - \hat{c}_{\sigma,\vb{k}+\vb{q}}^\dagger \hat{c}_{\sigma,\vb{k}+\vb{q}}}_0}
+ {\hbar (\omega + i \eta) + \xi_{\vb{k}} - \xi_{\vb{k} + \vb{q}}}
+\end{aligned}$$
+
+The operator $\hat{c}_{\sigma.\vb{k}}^\dagger \hat{c}_{\sigma.\vb{k}}$
+simply counts the number of electrons in state $(\sigma, \vb{k})$,
+which is given by the [Fermi-Dirac distribution](/know/concept/fermi-dirac-distribution/) $n_F$.
+This gives us the **Lindhard response function**:
+
+$$\begin{aligned}
+ \boxed{
+ \chi_0(\vb{q}, \omega)
+ = \frac{1}{V} \sum_{\sigma \vb{k}}
+ \frac{n_F(\xi_{\vb{k}}) - n_F(\xi_{\vb{k} + \vb{q}})}
+ {\hbar (\omega + i \eta) + \xi_{\vb{k}} - \xi_{\vb{k} + \vb{q}}}
+ }
+\end{aligned}$$
+
+From this, we would like to get the
+[dielectric function](/know/concept/dielectric-function/) $\varepsilon_r$.
+Recall its definition, where $U_\mathrm{tot}$, $U_\mathrm{ext}$, and $U_\mathrm{ind}$
+are the total, external and induced potentials, respectively:
+
+$$\begin{aligned}
+ U_\mathrm{tot}
+ = U_\mathrm{ext} + U_\mathrm{ind}
+ = \frac{U_\mathrm{ext}}{\varepsilon_r}
+\end{aligned}$$
+
+Note that these are all *energy* potentials:
+this choice is justified because all energy potentials
+are caused by electric fields in this case.
+The *electric* potential is recoverable as
+$\Phi_\mathrm{tot} = q_e U_\mathrm{tot}$,
+where $q_e < 0$ is the charge of an electron.
+
+From the Lindhard response function $\chi_0$,
+we get the induced particle density offset $\delta{\Expval{\hat{n}}}$
+caused by a potential $U$.
+The density $\delta{\Expval{\hat{n}}}$ should be self-consistent,
+implying $U = U_\mathrm{tot}$.
+In other words, we have a linear relation
+$\delta{\Expval{\hat{n}}} = \chi_0 U_\mathrm{tot}$,
+so the standard formula for $\varepsilon_r$ gives:
+
+$$\begin{aligned}
+ \boxed{
+ \varepsilon_r(\vb{q}, \omega)
+ = 1 - \frac{U_{ee}(\vb{q})}{V}
+ \sum_{\sigma \vb{k}} \frac{n_F(\xi_{\vb{k}}) - n_F(\xi_{\vb{k} + \vb{q}})}{\hbar (\omega + i \eta) + \xi_{\vb{k}} - \xi_{\vb{k} + \vb{q}}}
+ }
+\end{aligned}$$
+
+Where $U_{ee}(\vb{q}) = q_e^2 / (\varepsilon_0 |\vb{q}|^2)$
+is Coulomb repulsion.
+This is the **Lindhard dielectric function** of a free
+non-interacting electron gas,
+at any temperature and for any dimensionality.
+
+
+
+## References
+1. K.S. Thygesen,
+ *Advanced solid state physics: linear response theory*,
+ 2013, unpublished.
+2. H. Bruus, K. Flensberg,
+ *Many-body quantum theory in condensed matter physics*,
+ 2016, Oxford.
+3. G. Grosso, G.P. Parravicini,
+ *Solid state physics*,
+ 2nd edition, Elsevier.
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