Physical review. B, Condensed matter

Correlations in a two-dimensional quantum electron gas: The ladder approximation

Seido Nagano, K. S. Singwi, Shuhei Ohnishi

Published Feb 1, 1984• 72 citations

Abstract

The Bethe-Goldstone integral equation for the effective interaction between two electrons in a two-dimensional electron gas has been solved numerically. Ground-state properties of the system have been calculated within the ladder approximation and the results compared with those obtained from other many-body approximations.

DOI: 10.1103/physrevb.29.1209

Related papers

Correlations in two-dimensional electron gas: Random-phase approximation with exchange and ladder results

Francesco Pederiva, E. Lipparini, K Takayanagi

Europhysics Letters (EPL)Dec 15, 19976 citations

Pair-correlation function of the electron gas with long-range Coulomb interaction: Ladder theory

L. Calmels, Alfred Gold

Physical review. B, Condensed matterJan 15, 199819 citations

On-top pair-correlation function in the homogeneous electron liquid

Zhixin Qian

arXiv:cond-mat/0509177Physical Review BJan 9, 200610 citations

Short-range correlations in a two-dimensional electron gas

Marco Polini, Gerardo Sica, B. Davoudi +1 more

Journal of Physics Condensed MatterMar 29, 200120 citations

Low-density expansion for the two-dimensional electron gas

Francesca Sauli, Peter Kopietz

arXiv:cond-mat/0608423Physical Review BNov 17, 200612 citations

Electron correlations. II. Ground-state results at low and metallic densities

R. F. Bishop, K.H. Lührmann

Physical review. B, Condensed matterNov 15, 1982142 citations

Physical review. B, Condensed matter

Correlations in a two-dimensional quantum electron gas: The ladder approximation

Seido Nagano, K. S. Singwi, Shuhei Ohnishi

Published Feb 1, 1984• 72 citations

Publisher

Abstract

DOI: 10.1103/physrevb.29.1209

Related papers

Correlations in two-dimensional electron gas: Random-phase approximation with exchange and ladder results

Francesco Pederiva, E. Lipparini, K Takayanagi

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 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.

Europhysics Letters (EPL)Dec 15, 19976 citations

Pair-correlation function of the electron gas with long-range Coulomb interaction: Ladder theory

L. Calmels, Alfred Gold

We present analytical and numerical results for the pair-correlation function $g(r=0)$ of the electron gas with long-range Coulomb interaction obtained within the ladder theory. Analytical results obtained by perturbation calculation and a Pad\'e approximation are derived and are found to be in good agreement with our numerical solution of the ladder equation. Finite-width effects in the two-dimensional electron gas (quantum well) are also described. For quasi-one-dimensional systems (quantum wires) numerical and analytical results are derived for an oscillator confinement model.We present analytical and numerical results for the pair-correlation function $g(r=0)$ of the electron gas with long-range Coulomb interaction obtained within the ladder theory. Analytical results obtained by perturbation calculation and a Pad\'e approximation are derived and are found to be in good agreement with our numerical solution of the ladder equation. Finite-width effects in the two-dimensional electron gas (quantum well) are also described. For quasi-one-dimensional systems (quantum wires) numerical and analytical results are derived for an oscillator confinement model.

Physical review. B, Condensed matterJan 15, 199819 citations

On-top pair-correlation function in the homogeneous electron liquid

Zhixin Qian

The ladder theory, in which the Bethe-Goldstone equation for the effective potential between two scattering particles plays a central role, is well known for its satisfactory description of the short-range correlations in the homogeneous electron liquid. By solving exactly the Bethe-Goldstone equation in the limit of large transfer momentum between two scattering particles, we obtain accurate results for the on-top pair-correlation function $g(0)$ , in both three dimensions and two dimensions. Furthermore, we prove, in general, that the ladder theory satisfies the cusp condition for the pair-correlation function $g(r)$ at zero distance $r=0$ .The ladder theory, in which the Bethe-Goldstone equation for the effective potential between two scattering particles plays a central role, is well known for its satisfactory description of the short-range correlations in the homogeneous electron liquid. By solving exactly the Bethe-Goldstone equation in the limit of large transfer momentum between two scattering particles, we obtain accurate results for the on-top pair-correlation function $g(0)$, in both three dimensions and two dimensions. Furthermore, we prove, in general, that the ladder theory satisfies the cusp condition for the pair-correlation function $g(r)$ at zero distance $r=0$.

arXiv:cond-mat/0509177Physical Review BJan 9, 200610 citations

Short-range correlations in a two-dimensional electron gas

Marco Polini, Gerardo Sica, B. Davoudi +1 more

We construct a model for the value of the pair distribution function g ↑↓ (0) between antiparallel-spin electrons at contact in a two-dimensional electron gas with e 2 / r interactions, as a function of the coupling-strength parameter r s . The model involves an interpolation between the result of a low- r s expansion, including the second-order direct and exchange contributions to the energy in the paramagnetic state, and the result of a partial-wave phase-shift analysis near Wigner crystallization. The interpolation formula is in excellent agreement with many-body calculations based on the ladder approximation. We further show through an STLS self-consistent calculation that g ↑↓ (0) is essentially independent of the state of spin polarization of the electron gas.We construct a model for the value of the pair distribution function g ↑↓ (0) between antiparallel-spin electrons at contact in a two-dimensional electron gas with e 2 / r interactions, as a function of the coupling-strength parameter r s . The model involves an interpolation between the result of a low- r s expansion, including the second-order direct and exchange contributions to the energy in the paramagnetic state, and the result of a partial-wave phase-shift analysis near Wigner crystallization. The interpolation formula is in excellent agreement with many-body calculations based on the ladder approximation. We further show through an STLS self-consistent calculation that g ↑↓ (0) is essentially independent of the state of spin polarization of the electron gas.

Journal of Physics Condensed MatterMar 29, 200120 citations

Low-density expansion for the two-dimensional electron gas

Francesca Sauli, Peter Kopietz

We show that in two dimensions (2D) a systematic expansion of the self-energy and the effective interaction of a dilute electron gas in powers of the two-body $T$ matrix ${T}_{0}$ can be generated from an exact hierarchy of the functional renormalization group equations for one-particle irreducible vertices using the chemical potential as flow parameter. Due to the interference of particle-particle and particle-hole channels at order ${T}_{0}^{2}$ , in 2D the ladder approximation for the self-energy is not reliable beyond the leading order in ${T}_{0}$ . We also discuss two-body scattering in vacuum in arbitrary dimensions from the renormalization group point of view and argue that the singular interaction proposed by Anderson [Phys. Rev. Lett. 65, 2306 (1990)] cannot be ruled out on the basis of the ladder approximation.We show that in two dimensions (2D) a systematic expansion of the self-energy and the effective interaction of a dilute electron gas in powers of the two-body $T$ matrix ${T}_{0}$ can be generated from an exact hierarchy of the functional renormalization group equations for one-particle irreducible vertices using the chemical potential as flow parameter. Due to the interference of particle-particle and particle-hole channels at order ${T}_{0}^{2}$, in 2D the ladder approximation for the self-energy is not reliable beyond the leading order in ${T}_{0}$. We also discuss two-body scattering in vacuum in arbitrary dimensions from the renormalization group point of view and argue that the singular interaction proposed by Anderson [Phys. Rev. Lett. 65, 2306 (1990)] cannot be ruled out on the basis of the ladder approximation.

arXiv:cond-mat/0608423Physical Review BNov 17, 200612 citations

Electron correlations. II. Ground-state results at low and metallic densities

R. F. Bishop, K.H. Lührmann

In this second work in a series of papers the coupled-cluster or $\mathrm{exp}(S)$ formalism is applied to the problem of ground-state correlations in the one-component Fermi plasma. The main approximation employed is the so-called SUB2 approximation for the two-body subsystem correlation operator ${S}_{2}$ which provides a measure of the two-particle---two-hole component in the true ground-state wave function, and in which the coupling to larger subsystems is neglected. The SUB2 equations for Fermi systems are brought into tractable form by a "state-averaging" procedure which we develop by analogy with the comparable Bose equations, and which is shown by comparison with earlier exact results to be accurate in the metallic density regime at about the 1% level. Our final results for metallic densities include the completely integrated and self-consistent effects of the terms which by themselves generate (i) the random-phase approximation (RPA) and its associated exact long-range screening effects, (ii) the extra random-phase approximation exchange terms necessary to keep the RPA explicitly antisymmetric, (iii) the self-consistent particle-particle ladders (LAD) that describe two-particle scattering within the many-body medium and which describe the exact short-range behavior, (iv) a class of particle-hole ladder terms, and (v) the self-consistent hole-potential terms. Particular attention is paid to the important effects caused by the interference at intermediate separations of the long-range RPA and the short-range LAD effects. By comparison with recent and essentially exact stochastic simulations of the many-body Schr\"odinger equation, our results are seen to be accurate to about 1% for metallic densities, and hence to provide what is probably the currently best microscopic description available for this system. We show also that even in the low-density limit the SUB2 approximation provides a good "translationally-invariant-solid" description of both charged Fermi and Bose systems in this exact Wigner crystal regime.In this second work in a series of papers the coupled-cluster or $\mathrm{exp}(S)$ formalism is applied to the problem of ground-state correlations in the one-component Fermi plasma. The main approximation employed is the so-called SUB2 approximation for the two-body subsystem correlation operator ${S}_{2}$ which provides a measure of the two-particle---two-hole component in the true ground-state wave function, and in which the coupling to larger subsystems is neglected. The SUB2 equations for Fermi systems are brought into tractable form by a "state-averaging" procedure which we develop by analogy with the comparable Bose equations, and which is shown by comparison with earlier exact results to be accurate in the metallic density regime at about the 1% level. Our final results for metallic densities include the completely integrated and self-consistent effects of the terms which by themselves generate (i) the random-phase approximation (RPA) and its associated exact long-range screening effects, (ii) the extra random-phase approximation exchange terms necessary to keep the RPA explicitly antisymmetric, (iii) the self-consistent particle-particle ladders (LAD) that describe two-particle scattering within the many-body medium and which describe the exact short-range behavior, (iv) a class of particle-hole ladder terms, and (v) the self-consistent hole-potential terms. Particular attention is paid to the important effects caused by the interference at intermediate separations of the long-range RPA and the short-range LAD effects. By comparison with recent and essentially exact stochastic simulations of the many-body Schr\"odinger equation, our results are seen to be accurate to about 1% for metallic densities, and hence to provide what is probably the currently best microscopic description available for this system. We show also that even in the low-density limit the SUB2 approximation provides a good "translationally-invariant-solid" description of both charged Fermi and Bose systems in this exact Wigner crystal regime.

Physical review. B, Condensed matterNov 15, 1982142 citations

We present analytical and numerical results for the pair-correlation function

g(r=0)

of the electron gas with long-range Coulomb interaction obtained within the ladder theory. Analytical results obtained by perturbation calculation and a Pad\'e approximation are derived and are found to be in good agreement with our numerical solution of the ladder equation. Finite-width effects in the two-dimensional electron gas (quantum well) are also described. For quasi-one-dimensional systems (quantum wires) numerical and analytical results are derived for an oscillator confinement model.

g(0)

, in both three dimensions and two dimensions. Furthermore, we prove, in general, that the ladder theory satisfies the cusp condition for the pair-correlation function

g(r)

at zero distance

r=0

We show that in two dimensions (2D) a systematic expansion of the self-energy and the effective interaction of a dilute electron gas in powers of the two-body

T

matrix

{T}_{0}

can be generated from an exact hierarchy of the functional renormalization group equations for one-particle irreducible vertices using the chemical potential as flow parameter. Due to the interference of particle-particle and particle-hole channels at order

{T}_{0}^{2}

, in 2D the ladder approximation for the self-energy is not reliable beyond the leading order in

{T}_{0}

. We also discuss two-body scattering in vacuum in arbitrary dimensions from the renormalization group point of view and argue that the singular interaction proposed by Anderson [Phys. Rev. Lett. 65, 2306 (1990)] cannot be ruled out on the basis of the ladder approximation.

In this second work in a series of papers the coupled-cluster or

\mathrm{exp}(S)

formalism is applied to the problem of ground-state correlations in the one-component Fermi plasma. The main approximation employed is the so-called SUB2 approximation for the two-body subsystem correlation operator

{S}_{2}

which provides a measure of the two-particle---two-hole component in the true ground-state wave function, and in which the coupling to larger subsystems is neglected. The SUB2 equations for Fermi systems are brought into tractable form by a "state-averaging" procedure which we develop by analogy with the comparable Bose equations, and which is shown by comparison with earlier exact results to be accurate in the metallic density regime at about the 1% level. Our final results for metallic densities include the completely integrated and self-consistent effects of the terms which by themselves generate (i) the random-phase approximation (RPA) and its associated exact long-range screening effects, (ii) the extra random-phase approximation exchange terms necessary to keep the RPA explicitly antisymmetric, (iii) the self-consistent particle-particle ladders (LAD) that describe two-particle scattering within the many-body medium and which describe the exact short-range behavior, (iv) a class of particle-hole ladder terms, and (v) the self-consistent hole-potential terms. Particular attention is paid to the important effects caused by the interference at intermediate separations of the long-range RPA and the short-range LAD effects. By comparison with recent and essentially exact stochastic simulations of the many-body Schr\"odinger equation, our results are seen to be accurate to about 1% for metallic densities, and hence to provide what is probably the currently best microscopic description available for this system. We show also that even in the low-density limit the SUB2 approximation provides a good "translationally-invariant-solid" description of both charged Fermi and Bose systems in this exact Wigner crystal regime.