Quasiparticle properties of electon gas
Renormalization factor
The renormalization constant $Z$ gives the strength of the quasiparticle pole, and can be obtained from the frequency dependence of the self-energy as
\[Z=\frac{1}{1-\left.\frac{1}{\hbar} \frac{\partial \operatorname{Im} \Sigma(k, i\omega_n)}{\partial \omega_n}\right|_{k=k_{F}, \omega_n=0^+}}\]
Effective mass
\[\frac{m^{*}}{m}= \frac{Z^{-1}}{1+\frac{m}{\hbar^{2} k_{F}} \left. \frac{\partial \operatorname{Re} \Sigma(k, i\omega_n)}{\partial k}\right|_{k=k_{F}, \omega_n=0^+}} \]
Benchmark
2D UEG
| $r_s$ | $Z$ (RPA) | $Z$ ($G_0W_0$ [1]) | $m^*/m$ (RPA) | $m^*/m$ ($G_0W_0$ [1]) |
|---|---|---|---|---|
| 0.5 | 0.787 | 0.786 | 0.981 | |
| 1.0 | 0.662 | 0.662 | 1.020 | |
| 2.0 | 0.519 | 0.519 | 1.078 | |
| 3.0 | 0.437 | 0.437 | 1.117 | |
| 4.0 | 0.383 | 0.383 | 1.143 | |
| 5.0 | 0.344 | 0.344 | 1.162 | |
| 8.0 | 0.271 | 0.270 | 1.196 | |
| 10.0 | 0.240 | 0.240 | 1.209 |
3D UEG
| $r_s$ | $Z$ (RPA) | $Z$ ($G_0W_0$) | $m^*/m$ (RPA) [5] | $m^*/m$ ($G_0W_0$ [2]) |
|---|---|---|---|---|
| 1.0 | 0.8601 | 0.859 [3] | 0.9716(5) | 0.970 |
| 2.0 | 0.7642 | 0.768 [3] 0.764 [4] | 0.9932(9) | 0.992 |
| 3.0 | 0.6927 | 0.700 [3] | 1.0170(13) | 1.016 |
| 4.0 | 0.6367 | 0.646 [3] 0.645 [4] | 1.0390(10) | 1.039 |
| 5.0 | 0.5913 | 0.602 [3] | 1.0587(13) | 1.059 |
| 6.0 | 0.5535 | 0.568 [3] | 1.0759(12) | 1.078 |
[References]
- H.-J. Schulze, P. Schuck, and N. Van Giai, Two-dimensional electron gas in the random-phase approximation with exchange and self-energy corrections. Phys. Rev. B 61, 8026 (2000).
- Simion, G. E. & Giuliani, G. F., Many-body local fields theory of quasiparticle properties in a three-dimensional electron liquid. Phys. Rev. B 77, 035131 (2008).
- G. D Mahan, Many-Particle Physics (Plenum, New York, 1991), Chap. 5.
- B. Holm and U. von Barth, Fully self-consistent GW self-energy of the electron gas. Phys. Rev. B 57, 2108 (1998).
- Calculated at the temperature $T=T_F/1000$