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I am trying to understand the following statement from notes which I found: "For electron gas with long-range Coulomb interaction there is a problem with its instability". What does it mean? I know the problem of Cooper instability, but in that case the problem was related to the attractive potential between pair of electron exchanging phonon. How should I understand meaning of this sentence (I do not have any richer context - it was just one sentence given as a remark)?

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The coulomb potential is $1/r$ type potential. When you calculate the total interaction energy as,$E=\frac{1}{4\pi \epsilon_0}\sum_{<i,j>}\frac{q_iq_j}{|r_{ij}|}$ the summation will not readily converge. The described sum converge very slowly and also conditionally convergent. So we can use the Ewald summation technique to break it up into two parts. One summation is carried out in real space and another part is carried out in reciprocal space. This technique is often useful for these $1/r^n$ potentials and for dipolar interaction. I think the problem lies in the convergence in the series.
PS: The Ewald Summation Technique:
Let us assume there is N number of charged atoms placed in a box of dimension $L\times L\times L$. The position of each particles having charge $q_1,q_2,q_3,\hdots,q_n$ from a suitable chosen origin are denoted by $r_1,r_2,r_3,\hdots r_n$. So the total Coulomb interaction among the particles can be expressed as, \begin{align} E=\frac{1}{4 \pi \epsilon_0}\sum_{<i,j>} \frac{q_i q_j}{|r_{ij}|} \label{coulomb1} \end{align} and $r_{ij}$ is denoted as $r_i-r_j$. Under periodic boundary condition, we have an identical particle at a distance $r_i+n1a1+n2a2+n3a3$ as the particle situated at a distance $r_i$ from the origin. Here $a1,a2 and a3$ are the primitive vector and $n1,n2 and n3$ are any arbitrary integers. For the sake of simplicity let us choose the box cubic and $L=|a1|=|a2|=|a3|$ and vector $n$ from a simple cubic lattice $(n1,n2,n3)$. So we can modify our \cref{coulomb1} as, \begin{align} E&=\frac{1}{4 \pi \epsilon_0}\sum_{n} \sum_{<i,j>} \frac{q_i q_j}{|r_{ij}+nL|} \nonumber \\ &=\frac{1}{4 \pi \epsilon_0} \frac{1}{2} \sum_{n} \sum_{i=1}^{N} \sum_{j=1}^{N} {'} \frac{q_i q_j}{|r_{ij}+nL|}\label{coulomb2} \end{align}

The factor $\frac{1}{2}$ comes due to double counting and the symbol ($'$) denotes the fact that $i\neq j$. The described sum converge very slowly and also conditionally convergent. So we can use the Ewald summation technique to break it up into two parts. We will use simple tricks to do that. We can write the \cref{coulomb2} as, \begin{align} E=\frac{1}{2} \sum_{i=1}^{N} q_i \phi_{[i]} (r_i) \label{Energy} \end{align} where, \begin{align*} \phi_{[i]}(r_i)=\frac{1}{4 \pi \epsilon_0} \sum_{n}\sum_{j=1}^{N} {'} \frac{ q_j}{|r_{ij}+nL|} \end{align*} Or, changing variable we can identify the potential field generated by all the ions excluding the ion $i$ as, \begin{align} \phi_{[i]}(r)=\phi(r)-\phi_{i}(r)=\frac{1}{4 \pi \epsilon_0} \sum_{n}\sum_{j=1}^{N} {'} \frac{ q_j}{|r-r_j+nL|} \end{align} The sum will give the potential field at any point $r$ of the box due to the other charges. But the direct sum is not easy to determine because it is divergent. So we will use a little trick to handle this problem. The potential field created by only one charge $q_i$ at the point $r_i$ is given by; \begin{align} \varphi_{i}(r)=\frac{1}{4 \pi \epsilon_0} \sum_{n} \frac{q_i}{|r-r_i+nL|} \end{align} The Fourier transformation of it is \begin{align} \varphi_{i}(K)= \frac{1}{\epsilon_0 K^2}e^{-i|K|r_i} \label{phi} \end{align} Where $K$ is the reciprocal lattice vector. We can further write $k^2$ (where $k=|K|$) in terms of integral. \begin{align} \frac{1}{k^2}=\int_{0}^{\infty} e^{-k^2 t}dt \end{align} Putting back into the \cref{phi} we get, \begin{align} \varphi_{i}(K)= \frac{1}{\epsilon_0 }e^{-ikr_i} \int_{0}^{\infty} e^{-k^2 t}dt \label{phi1} \end{align} Here we can divide the \cref{phi1} into two parts using a suitably chosen Ewald cut off parameter ($\Gamma$) as both the integral part converge rapidly. \begin{align} \varphi_{i}(K) &= \varphi_{i}^{S}(K)+\varphi_{i}^{L}(K)\nonumber \\ &= \frac{1}{\epsilon_0 }e^{-ikr_i} \int_{0}^{\Gamma} e^{-k^2 t}dt+\frac{1}{\epsilon_0 }e^{-ikr_i} \int_{\Gamma}^{\infty} e^{-k^2 t}dt \end{align} We can easily carry out the integral and get the expression for long range interaction term. \begin{align} \varphi_{i}^{L}(K)= &\frac{1}{\epsilon_0 }e^{-ikr_i} \int_{\Gamma}^{\infty} e^{-k^2 t}dt \nonumber \\ &=\frac{1}{\epsilon_0 }e^{-ikr_i} e^{-\Gamma k^2} \label{long} \end{align} And the $\varphi_{i}^{S} (K)$ is not convergent in reciprocal space. So we pull it back to the real space to calculate the integral. So using inverse Fourier transformation we get, \begin{align} \varphi_{i}^{S} (r) &=\frac{1}{V} \sum_{k} \varphi_{i}^{S} (K) e^{ikr} \nonumber \\ &=\frac{1}{V \epsilon_0} \sum_{k} \int_{0}^{\Gamma} e^{-ik(r-r_i)} e^{-k^2t} dt \nonumber \\ &= \frac{1}{4 \pi \epsilon_0} \sum_{n} \frac{1}{|r-r_i|} erfc\Bigg(\frac{|r-r_i|}{\sqrt{2}\Gamma} \Bigg) \label{short} \end{align} Now putting \cref{short} and \cref{long} back to \cref{Energy} we can find out the Energy terms. \begin{align} E^S &=\frac{1}{2} \sum_{i=1}^{N} q_i \phi_{[i]}^{S} (r_i) \nonumber \\ &= \frac{1}{4 \pi \epsilon_0} \frac{1}{2} \sum_{n} \sum_{i=1}^{N} \sum_{j=1}^{N} {'} \frac{q_i q_j}{|r_{i}-r_{j}+nL|} erfc\Bigg(\frac{|r_i-r_j+nL|}{\sqrt{2}\Gamma} \Bigg) \end{align} where we have used a variable transformation $r=r_i+nL$. Similarly we get, \begin{align} E^L &=\frac{1}{2} \sum_{i=1}^{N} q_i \phi_{[i]}^{L} (r_i) \nonumber \\ &= \frac{1}{2} \sum_{i=1}^{N} q_i \phi^{L} (r_i)-\frac{1}{2} \sum_{i=1}^{N} q_i \phi_{i}^{L} (r_i) \nonumber \\ &= \frac{1}{2 V \epsilon_0} \sum_{k \neq 0} \sum_{i=1}^{N} \sum_{j=1}^{N} \frac{q_i q_j}{k^2} e^{-ik(r_i-r_j)} e^{-\Gamma k^2} -E^{Self} \end{align}

Combining the Energy expression can be expressed as $E_E^S+E^L-E^{Self}$ where $E^{Self}=\frac{1}{2} \sum_{i=1}^{N} q_i \phi_{i}^{L} (r_i)$ term is the self interaction energy term. The final expression of Energy becomes, \begin{align} E=\frac{1}{4 \pi \epsilon_0} \frac{1}{2} \sum_{n} \sum_{i=1}^{N} \sum_{j=1}^{N} {'} \frac{q_i q_j}{|r_{i}-r_{j}+nL|} erfc\Bigg(\frac{|r_i-r_j+nL|}{\sqrt{2}\Gamma} \Bigg) + \nonumber \\ \frac{1}{2 V \epsilon_0} \sum_{k \neq 0} \sum_{i=1}^{N} \sum_{j=1}^{N} \frac{q_i q_j}{k^2} e^{-ik(r_i-r_j)} e^{-\Gamma k^2} -\frac{1}{4 \pi \epsilon_0} \frac{1}{\sqrt{4 \pi \Gamma}} \sum_{i=1}^{N} q_{i}^2 \end{align}

Reference:
1. http://micro.stanford.edu/mediawiki/images/4/46/Ewald_notes.pdf
2. https://www.academia.edu/29151377/Ewald_Summation_and_its_Applications

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