Two-dimensional electron self-energy: Long-range Coulomb interaction

The electron self-energy for long-range Coulomb interactions plays a crucial role in understanding the many-body physics of interacting electron systems (e.g., in metals and semiconductors) and has been studied extensively for decades. In fact, it is among the oldest and the most-investigated many-body problems in physics. However, there is a lack of an analytical expression for the self-energy $Re{\mathrm{\ensuremath{\Sigma}}}^(R)(\ensuremath{\varepsilon},T)$ when energy $\ensuremath{\varepsilon}$ and temperature ${k}_{\mathsf{B}}T$ are arbitrary with respect to each other (while both being still small compared with the Fermi energy). We revisit this problem and calculate analytically the self-energy on the mass shell for a two-dimensional electron system with Coulomb interactions in the high density limit ${r}_s\ensuremath{\ll}1$, for temperature ${r}_s^3/2\ensuremath{\ll}{k}_{\mathsf{B}}T/{E}_{\mathsf{F}}\ensuremath{\ll}{r}_s$ and energy ${r}_s^3/2\ensuremath{\ll}|\ensuremath{\varepsilon}|/{E}_{\mathsf{F}}\ensuremath{\ll}{r}_s$. We provide the exact high-density analytical expressions for the real and imaginary parts of the electron self-energy with arbitrary value of $\ensuremath{\varepsilon}/{k}_{\mathsf{B}}T$, to the leading order in the dimensionless Coulomb coupling constant ${r}_s$, and to several higher than leading orders in ${k}_{\mathsf{B}}T/{r}_s{E}_{\mathsf{F}}$ and $\ensuremath{\varepsilon}/{r}_s{E}_{\mathsf{F}}$. We also obtain the asymptotic behavior of the self-energy in the regimes $|\ensuremath{\varepsilon}|\ensuremath{\ll}{k}_{\mathsf{B}}T$ and $|\ensuremath{\varepsilon}|\ensuremath{\gg}{k}_{\mathsf{B}}T$. The higher-order terms have subtle and highly nontrivial compound logarithmic contributions from both $\ensuremath{\varepsilon}$ and $T$, explaining why they have never before been calculated in spite of the importance of the subject matter.