Categories: Physics, Quantum mechanics.

Random phase approximation

Recall that the self-energy Σ\Sigma is defined as a sum of Feynman diagrams, which each have an order nn equal to the number of interaction lines. We consider the self-energy in the context of jellium, so the interaction lines WW 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 kF\hbar k_F, and energies in ϵF=2kF2/(2m)\epsilon_F = \hbar^2 k_F^2 / (2 m). Each internal variable then gives a factor kF5k_F^5, where kF3k_F^3 comes from the 3D momentum integral, and kF2k_F^2 from the energy 1/β1 / \beta:

1(2π)31βn=dkkF5\begin{aligned} \frac{1}{(2 \pi)^3} \int_{-\infty}^\infty \frac{1}{\hbar \beta} \sum_{n = -\infty}^\infty \cdots \:\dd{\vb{k}} \quad\sim\quad k_F^5 \end{aligned}

Meanwhile, every line gives a factor 1/kF21 / k_F^2. The Matsubara Green’s function G0G^0 for a system with continuous translational symmetry is found from equation-of-motion theory:

W(k)=e2ε0k21kF2Gs0(k,iωnF)=1iωnFεk1kF2\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 nnth-order diagram in Σ\Sigma contains nn interaction lines, 2n ⁣ ⁣12n\!-\!1 fermion lines, and nn integrals, so in total it evolves as 1/kFn21 / k_F^{n-2}. In jellium, we know that the electron density is proportional to kF3k_F^3, so for high densities we can rest assured that higher-order terms in Σ\Sigma converge to zero faster than lower-order terms.

However, at a given order nn, not all diagrams are equally important. In a given diagram, due to momentum conservation, some interaction lines carry the same momentum variable. Because W(k)1/k2W(\vb{k}) \propto 1 / |\vb{k}|^2, small k\vb{k} make a large contribution, and the more interaction lines depend on the same k\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 nn, there is one diagram where all nn 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 Σ\Sigma, leaving only the single most divergent one at each order nn, i.e. the ones where all nn interaction lines carry the same momentum and energy:

RPA self-energy definition

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

RPA screened interaction definition

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

Dyson equation for screened interaction

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

WRPA=W+WΠ0WRPA\begin{aligned} \boxed{ W^\mathrm{RPA} = W + W \Pi_0 W^\mathrm{RPA} } \end{aligned}

Where we have defined the pair-bubble Π0\Pi_0 as follows, with an internal wavevector q\vb{q}, fermionic frequency iωmFi \omega_m^F, and spin ss. Abbreviating k~(k,iωnB)\tilde{\vb{k}} \equiv (\vb{k}, i \omega_n^B) and q~(q,iωnF)\tilde{\vb{q}} \equiv (\vb{q}, i \omega_n^F):

Internal variables of pair-bubble diagram

We isolate the Dyson equation for WRPAW^\mathrm{RPA}, which reveals its physical interpretation as a screened interaction: the “raw” interaction W ⁣= ⁣e2/(ε0k2)W \!=\! e^2 / (\varepsilon_0 |\vb{k}|^2) is weakened by a term containing Π0\Pi_0:

WRPA(k,iωnB)=W(k)1W(k)Π0(k,iωnB)=e2ε0k2e2Π0(k,iωnB)\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 Π0\Pi_0 more concretely. The Feynman diagram translates to:

Π0(k,iωnB)=s1(2π)31βm=Gs(k ⁣+ ⁣q,iωnB ⁣+ ⁣iωmF)Gs(q,iωmF)dq=2(2π)31βm=1iωnB+iωmFεk+q1iωmFεqdq\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 nFn_F are 1/(β)1 / (\hbar \beta) when it is a function of frequency, and 1/β1 / \beta when it is a function of energy, so:

Π0(k,iωnB)=2(2π)3nF(εk+qiωnB)(εk+qiωnB)εq+nF(εq)iωnB+(εq)εk+qdq=2(2π)3nF(εq)nF(εk+q)iωnB+εqεk+qdq\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 nF(ε ⁣+ ⁣iωnB)=nF(ε)n_F(\varepsilon \!+\! i \hbar \omega_n^B) = n_F(\varepsilon). Analogously to extracting the retarded Green’s function GR(ω)G^R(\omega) from the Matsubara Green’s function G0(iωnF)G^0(i \omega_n^F), we replace iωnFω ⁣+ ⁣iηi \omega_n^F \to \omega \!+\! i \eta, where η0+\eta \to 0^+ is a positive infinitesimal, yielding the retarded pair-bubble Π0R\Pi_0^R:

Π0R(k,ω)=2(2π)3nF(εq)nF(εk+q)(ω+iη)+εqεk+qdq\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:

WRPA(k,ω)=e2ε0k2e2Π0(k,ω)\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}


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