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Journal of Physics C Solid State Physics

Dielectric response of a one-dimensional electron gas

W. I. Friesen, B. Bergersen

Published Dec 30, 1980• 147 citations
Publisher

Abstract

The dielectric response of a quasi-one-dimensional system of interacting electrons has been investigated. In the authors' model the particles move freely in one direction and are tightly bound in the transverse direction. Calculation of the Coulomb matrix elements leads to an effective one-dimensional electron-electron interaction nu (q) which is less singular for small q than the 1/q 2 potential assumed by previous authors. As a result, the excitonic instability which they find even for a rigid positive background does not arise in their calculation. The calculations are based on the approximation of Singwi, Tosi, Land and Sjolander (1968). As expected in this approach, the pair correlation function at small distances remains positive over a much larger range of densities than that for which the compressibility sum rule is satisfied. The static dielectric function is negative around q=2k F , which suggests the possibility of charge ordering when the background is allowed to be polarisable. If nu (q) is replaced by a delta -function potential, a numerical calculation of the energy for the exact solution of Yang may be made. On comparison with the result obtained with the method of Singwi et al. it is found that the approximate scheme gives good correlation energies for high densities.

DOI
10.1088/0022-3719/13/36/016

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The dielectric response of a quasi-one-dimensional system of interacting electrons has been investigated. In the authors' model the particles move freely in one direction and are tightly bound in the transverse direction. Calculation of the Coulomb matrix elements leads to an effective one-dimensional electron-electron interaction nu (q) which is less singular for small q than the 1/q 2 potential assumed by previous authors. As a result, the excitonic instability which they find even for a rigid positive background does not arise in their calculation. The calculations are based on the approximation of Singwi, Tosi, Land and Sjolander (1968). As expected in this approach, the pair correlation function at small distances remains positive over a much larger range of densities than that for which the compressibility sum rule is satisfied. The static dielectric function is negative around q=2k F , which suggests the possibility of charge ordering when the background is allowed to be polarisable. If nu (q) is replaced by a delta -function potential, a numerical calculation of the energy for the exact solution of Yang may be made. On comparison with the result obtained with the method of Singwi et al. it is found that the approximate scheme gives good correlation energies for high densities.
For nearly a decade, quasi-one-dimensional conductors have been the subject of intensive study. Theoretically, much attention has been devoted to the development of one-dimensional Fermi gas models, some which may be solved exactly, and to the calculation of their response functions. After a review of this theory, a different approach is adopted in the investigation of two models. The dielectric response theory of the three-dimensional Coulomb gas has been applied to an anisotropic system in which the particles interact with an effective one-dimensional long-range potential. Within the framework of the approximation of Singwi, Tosi, Land, and Sjolander, the dielectric properties of the model are examined in order to determine the conditions under which it is unstable with respect to formation of a charge density wave state. It is found that the positive neutralizing background must be polarizable in order for such an instability to occur. The same approximation method, when applied to a one-dimensional fermion gas with a ʃ-function interaction may be compared with the exact solution of Yang. This solution, which exists in the form of coupled integral equations, has been calculated numerically, and, as predicted by the Lieb-Mattis theorem, the ground state is found to be non-magnetic. The approximation of Singwi et al. proves to give better correlation energies than other inexact methods, particularly at higher densities.
We have evaluated the density-density response of the two-dimensional electron gas at zero temperature by solving the Dyson equation for the particle-hole Green's function, including exchange Coulomb matrix elements and short-range contributions in the ladder approximation. We study the effect of these correlations on the total energy, compressibility per particle, local field factor G ( q ), static structure factor and pair-correlation function. Results are compared with the normal random-phase approximation, local field theories and quantum Monte Carlo calculations.
We study the ground state properties of a quasi-one-dimensional electron gas interacting via an effective potential with a harmonic transversal confinement and long-range Coulomb tail. The exact correlation energy has been calculated for a wide range of electron densities by using the lattice regularized diffusion Monte Carlo method, which is a recent development of the standard projection Monte Carlo technique. In this case it is particularly useful as it allows one to sample the exact ground state of the system, even in the low-density regime when the exchange between electrons is extremely small. For different values of the width parameter bbb $(0.1\phantom{\rule{0.3em}{0ex}}{a}_{0}^{*}\ensuremath{\leqslant}b\ensuremath{\leqslant}4\phantom{\rule{0.3em}{0ex}}{a}_{0}^{*})$, we give a simple parametrization of the correlation energy, which provides an accurate local density energy functional for quasi-one-dimensional systems. Moreover, we show that static correlations are in qualitative agreement with those obtained for the Luttinger liquid model with long-range interactions.
In this paper, we study correlations in quasi-one-dimensional electron systems. To account for the short-range correlations we have used a ladder approximation (LA), and to account for the long-range correlations we have used the Singwi, Tosi, S\"olander, and Land (STLS) scheme. After solving the integral equation appearing in the LA and solving the set of self-consistent equations of the STLS scheme, we join the resulting static structure factor in the large- and small-qqq limit smoothly. We then calculate the pair-correlation functions, the ground-state energy, the screened potential, and the plasmon dispersion relation. Our pair-correlation function remains positive everywhere (for rs< 6.5{r}_{s}<~6.5rs​< 6.5) and its value at the origin, g(0),g(0),g(0), is compared with the results of other calculational schemes. The screened potential is repulsive for rs< 2{r}_{s}<~2rs​< 2 and shows a weak attractive part for rs>2.{r}_{s}>2.rs​>2. We have argued that in these systems the sss-wave intrinsic superconductivity is absent. Finally, the quantities calculated in this paper are compared with the results of other calculational schemes.
The ground-state energy of a one-dimensional fermion system interacting via a repulsive $\ensuremath{\delta}$-function potential is calculated rigorously within the "ladder" approximation as a function of the coupling constant and the results compared with those given by the exact solution of Yang. Comparison with other many-body approximate theories has also been made. The density-density response function has also been calculated in an approximation in which the particle-hole interaction in the polarization bubble is taken into account to all orders. Results of this dynamic theory are compared with those of other approximations.