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arXiv:cond-mat/0610236 · Physical Review B

Study of two-dimensional electron systems in the renormalized-ring-diagram approximation

Xin-Zhong Yan, C. S. Ting

Published Jan 31, 2007• 5 citations
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Abstract

With a superhigh-efficiency numerical algorithm, we are able to self-consistently calculate Green's function in the renormalized-ring-diagram approximation for a two-dimensional electron system with long-range Coulomb interactions. The obtained ground-state energy is found to be in excellent agreement with that of the Monte Carlo simulation. The numerical results of the self-energy, the effective mass, the distribution function, and the renormalization factor of Green's function for the coupling constants in the range $0\ensuremath{\leqslant}{r}_{s}\ensuremath{\leqslant}30$ are also presented.

DOI
10.1103/physrevb.75.035342

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With a superhigh-efficiency numerical algorithm, we are able to self-consistently calculate Green's function in the renormalized-ring-diagram approximation for a two-dimensional electron system with long-range Coulomb interactions. The obtained ground-state energy is found to be in excellent agreement with that of the Monte Carlo simulation. The numerical results of the self-energy, the effective mass, the distribution function, and the renormalization factor of Green's function for the coupling constants in the range 0\ensuremath⩽rs\ensuremath⩽300\ensuremath{\leqslant}{r}_{s}\ensuremath{\leqslant}300\ensuremath⩽rs​\ensuremath⩽30 are also presented.
The efficiency of the variational perturbation theory (You et al 2000 Phys. Rev. C 62 045503) formulated recently for many-particle systems is examined by calculating the ground state correlation energy of the 3D electron gas with the Coulomb interaction. The perturbation beyond a variational result can be carried out systematically by the modified Wick's theorem which defines a contraction rule about the renormalized perturbation. Utilizing the theorem, variational ring diagrams of the electron gas are summed. As a result, the correlation energy is found to be much closer to the result of the Green function Monte Carlo calculation than that of the conventional ring approximation.
The ground state of a two-dimensional charged-boson system is investigated over the range of densities 1\ensuremath{\lesssim}Rs{R}_{s}Rs​\ensuremath{\lesssim}10 in the self-consistent-field approximation; Rs{R}_{s}Rs​=(a02{a}_{0}^{2}a02​\ensuremath{\pi}\ensuremath{\rho}${)}^{\mathrm{\ensuremath{-}}1/2}$, where a0{a}_{0}a0​ is the Bohr radius and \ensuremath{\rho} is the number density. Starting with numerical self-consistent calculations of the static structure factor, the elementary excitation, pair-correlation functions, pressure, and ground-state energy are evaluated. These results are compared with those of the two-dimensional and three-dimensional systems obtained from other methods. The ground-state energy is given as ${\mathit{E}}_{0=\mathrm{\ensuremath{-}}1.2918{\mathit{R}}_{\mathit{s}}}^{\mathrm{\ensuremath{-}}2/3}$+0.03, which improves that result from the ring-diagram approximation.
The discrete Fourier transform is approximated by summing over part of the terms with corresponding weights. The approximation reduces significantly the requirement for computer memory storage and enhances the numerical computation efficiency with several orders without losing accuracy. As an example, we apply the algorithm to study the three-dimensional interacting electron gas under the renormalized-ring-diagram approximation where the Green's function needs to be self-consistently solved. We present the results for the chemical potential, compressibility, free energy, entropy, and specific heat of the system. The ground-state energy obtained by the present calculation is compared with the existing results of Monte Carlo simulation and random-phase approximation.
Within the infinite series of ring (or bubble) diagram approximation for the electronic self-energy as appropriate for the long-range Coulomb interaction, we calculate the density-dependent T=0T=0T=0 Fermi-liquid quasiparticle effective-mass renormalization in two-dimensional (2D) electron systems in the random-phase approximation. We find that the quasiparticle effective mass increases very strongly with increasing rs{r}_{s}rs​, exhibiting an unexpected divergence at a critical rs{r}_{s}rs​ value ($\ensuremath{\sim}16$ in the ideal 2D system). We discuss the possibility for and the consequences of such an interaction-induced effective-mass divergence as the electron density decreases and critically comment on the likely theoretical scenarios which could lead to such a low-density effective-mass divergence in electron liquids.
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 kBT{k}_{\mathsf{B}}TkB​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 rs{r}_{s}rs​, and to several higher than leading orders in kBT/rsEF{k}_{\mathsf{B}}T/{r}_{s}{E}_{\mathsf{F}}kB​T/rs​EF​ 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 TTT, explaining why they have never before been calculated in spite of the importance of the subject matter.
Motivated by a number of recent experimental studies we have revisited the problem of the microscopic calculation of the quasiparticle self-energy and many-body effective mass enhancement in an unpolarized two-dimensional electron liquid. Our systematic study is based on the many-body local field theory and takes advantage of the results of the most recent diffusion Monte Carlo calculations of the static charge and spin response of the electron liquid. We report extensive calculations of both the real and imaginary parts of the quasiparticle self-energy. We also present results for the many-body effective mass enhancement and the renormalization constant over a broad range of electron densities. In this respect we critically examine the relative merits of the on-shell approximation, commonly used in weak coupling situations versus the actual self-consistent solution of the Dyson equation. We show that already for ${r}_{s}\ensuremath{\simeq}3$ and higher, a solution of the Dyson equation proves necessary in order to obtain a well-behaved effective mass. Finally we find confirmation that the inclusion of both charge- and spin-density fluctuations beyond the random phase approximation is indeed crucial to get reasonable agreement with recent measurements.