The two-dimensional electron self-energy: Long-range Coulomb interaction

Abstract

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 $Re{\Sigma }^{(R)}(\epsilon ,T)$ when energy $\epsilon$ and temperature $k_{B}T$ 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 $r_s\ll 1$, for temperature $r_s^{3/2}\ll k_{B}T/E_F\ll r_s$ and energy $r_s^{3/2}\ll |\epsilon |/E_F\ll r_s$. We provide the exact high-density analytical expressions for the real and imaginary parts of the electron self-energy with arbitrary value of $\epsilon /k_{B}T$, to the leading order in the dimensionless Coulomb coupling constant $r_s$, and to several higher than leading orders in $k_{B}T/r_s{E}_F$ and $\epsilon /r_s{E}_F$. We also obtain the asymptotic behavior of the self-energy in the regimes $|\epsilon |\ll k_{B}T$ and $|\epsilon |\gg k_{B}T$. The higher-order terms have subtle and highly non-trivial compound logarithmic contributions from both $\epsilon$ and $T$, explaining why they have never before been calculated in spite of the importance of the subject matter.

Published version. 16 pages, 4 figures

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• Presence versus absence of Two-Dimensional Fermi Surface Anomalies
• Odd relaxation in three-dimensional Fermi liquids