Categories: Physics, Quantum mechanics.

Random phase approximation

Recall that the self-energy \(\Sigma\) is defined as a sum of Feynman diagrams, which each have an order \(n\) equal to the number of interaction lines. We consider the self-energy in the context of jellium, so the interaction lines \(W\) represent Coulomb repulsion, and we use imaginary time.

Let us non-dimensionalize the Feynman diagrams in the self-energy, by measuring momenta in units of \(\hbar k_F\), and energies in \(\epsilon_F = \hbar^2 k_F^2 / (2 m)\). Each internal variable then gives a factor \(k_F^5\), where \(k_F^3\) comes from the 3D momentum integral, and \(k_F^2\) from the energy \(1 / \beta\):

\[\begin{aligned} \frac{1}{(2 \pi)^3} \int_{-\infty}^\infty \frac{1}{\hbar \beta} \sum_{n = -\infty}^\infty \cdots \:\dd{\vb{k}} \:\:\sim\:\: k_F^5 \end{aligned}\]

Meanwhile, every line gives a factor \(1 / k_F^2\). The Matsubara Green’s function \(G^0\) for a system with continuous translational symmetry is found from equation-of-motion theory:

\[\begin{aligned} W(\vb{k}) = \frac{e^2}{\varepsilon_0 |\vb{k}|^2} \:\:\sim\:\: \frac{1}{k_F^2} \qquad \qquad G_s^0(\vb{k}, i \omega_n^F) = \frac{1}{i \hbar \omega_n^F - \varepsilon_\vb{k}} \:\:\sim\:\: \frac{1}{k_F^2} \end{aligned}\]

An \(n\)th-order diagram in \(\Sigma\) contains \(n\) interaction lines, \(2n\!-\!1\) fermion lines, and \(n\) integrals, so in total it evolves as \(1 / k_F^{n-2}\). In jellium, we know that the electron density is proportional to \(k_F^3\), so for high densities we can rest assured that higher-order terms in \(\Sigma\) converge to zero faster than lower-order terms.

However, at a given order \(n\), not all diagrams are equally important. In a given diagram, due to momentum conservation, some interaction lines carry the same momentum variable. Because \(W(\vb{k}) \propto 1 / |\vb{k}|^2\), small \(\vb{k}\) make a large contribution, and the more interaction lines depend on the same \(\vb{k}\), the larger the contribution becomes.

In other words, each diagram is dominated by contributions from the momentum carried by the largest number of interactions. At order \(n\), there is one diagram where all \(n\) interactions carry the same momentum, and this one dominates all others at this order.

The random phase approximation consists of removing most diagrams from the defintion of the full self-energy \(\Sigma\), leaving only the single most divergent one at each order \(n\), i.e. the ones where all \(n\) interaction lines carry the same momentum and energy:

Where we have defined the screened interaction \(W^\mathrm{RPA}\), denoted by a double wavy line:

Rearranging the above sequence of diagrams quickly leads to the following Dyson equation:

In Fourier space, this equation’s linear shape means it is algebraic, so we can write it out:

\[\begin{aligned} \boxed{ W^\mathrm{RPA} = W + W \Pi_0 W^\mathrm{RPA} } \end{aligned}\]

Where we have defined the pair-bubble \(\Pi_0\) as follows, with an internal wavevector \(\vb{q}\), fermionic frequency \(i \omega_m^F\), and spin \(s\). Abbreviating \(\tilde{\vb{k}} \equiv (\vb{k}, i \omega_n^B)\) and \(\tilde{\vb{q}} \equiv (\vb{q}, i \omega_n^F)\):

We isolate the Dyson equation for \(W^\mathrm{RPA}\), which reveals its physical interpretation as a screened interaction: the “raw” interaction \(W \!=\! e^2 / (\varepsilon_0 |\vb{k}|^2)\) is weakened by a term containing \(\Pi_0\):

\[\begin{aligned} W^\mathrm{RPA}(\vb{k}, i \omega_n^B) = \frac{W(\vb{k})}{1 - W(\vb{k}) \: \Pi_0(\vb{k}, i \omega_n^B)} = \frac{e^2}{\varepsilon_0 |\vb{k}|^2 - e^2 \Pi_0(\vb{k}, i \omega_n^B)} \end{aligned}\]

Let us evaluate the pair-bubble \(\Pi_0\) more concretely. The Feynman diagram translates to:

\[\begin{aligned} -\hbar \Pi_0(\vb{k}, i \omega_n^B) &= - \sum_{s} \frac{1}{(2 \pi)^3} \int \frac{1}{\hbar \beta} \sum_{m = -\infty}^\infty \hbar G_s(\vb{k} \!+\! \vb{q}, i \omega_n^B \!+\! i \omega_m^F) \: \hbar G_s(\vb{q}, i \omega_m^F) \dd{\vb{q}} \\ &= - \frac{2 \hbar}{(2 \pi)^3} \int \frac{1}{\beta} \sum_{m = -\infty}^\infty \frac{1}{i \hbar \omega_n^B + i \hbar \omega_m^F - \varepsilon_{\vb{k}+\vb{q}}} \: \frac{1}{i \hbar \omega_m^F - \varepsilon_{\vb{q}}} \dd{\vb{q}} \end{aligned}\]

Here we recognize a Matsubara sum, and rewrite accordingly. Note that the residues of \(n_F\) are \(1 / (\hbar \beta)\) when it is a function of frequency, and \(1 / \beta\) when it is a function of energy, so:

\[\begin{aligned} \Pi_0(\vb{k}, i \omega_n^B) &= \frac{2}{(2 \pi)^3} \int \frac{n_F(\varepsilon_{\vb{k}+\vb{q}} - i \hbar \omega_n^B)}{(\varepsilon_{\vb{k}+\vb{q}} - i \hbar \omega_n^B) - \varepsilon_{\vb{q}}} + \frac{n_F(\varepsilon_{\vb{q}})}{i \hbar \omega_n^B + (\varepsilon_{\vb{q}}) - \varepsilon_{\vb{k}+\vb{q}}} \dd{\vb{q}} \\ &= \frac{2}{(2 \pi)^3} \int \frac{n_F(\varepsilon_{\vb{q}}) - n_F(\varepsilon_{\vb{k}+\vb{q}})} {i \hbar \omega_n^B + \varepsilon_{\vb{q}} - \varepsilon_{\vb{k}+\vb{q}}} \dd{\vb{q}} \end{aligned}\]

Where we have used that \(n_F(\varepsilon \!+\! i \hbar \omega_n^B) = n_F(\varepsilon)\). Analogously to extracting the retarded Green’s function \(G^R(\omega)\) from the Matsubara Green’s function \(G^0(i \omega_n^F)\), we replace \(i \omega_n^F \to \omega \!+\! i \eta\), where \(\eta \to 0^+\) is a positive infinitesimal, yielding the retarded pair-bubble \(\Pi_0^R\):

\[\begin{aligned} \boxed{ \Pi_0^R(\vb{k}, \omega) = \frac{2}{(2 \pi)^3} \int \frac{n_F(\varepsilon_{\vb{q}}) - n_F(\varepsilon_{\vb{k}+\vb{q}})} {\hbar (\omega + i \eta) + \varepsilon_{\vb{q}} - \varepsilon_{\vb{k}+\vb{q}}} \dd{\vb{q}} } \end{aligned}\]

This is as far as we can go before making simplifying assumptions. Therefore, we leave it at:

\[\begin{aligned} \boxed{ W^\mathrm{RPA}(\vb{k}, \omega) = \frac{e^2}{\varepsilon_0 |\vb{k}|^2 - e^2 \Pi_0(\vb{k}, \omega)} } \end{aligned}\]

References

  1. H. Bruus, K. Flensberg, Many-body quantum theory in condensed matter physics, 2016, Oxford.

© Marcus R.A. Newman, a.k.a. "Prefetch". Available under CC BY-SA 4.0.
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