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arXiv:cond-mat/0305195 · Physical review. B, Condensed matter

Correlation effects and the high-frequency spin susceptibility of an electron liquid: Exact limits

Gurinder S. Atwal, Neil William Ashcroft

Published Jun 27, 2003• 6 citations
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Abstract

Spin correlations in an interacting electron liquid are studied in the high-frequency limit and in both two and three dimensions. The third-moment sum rule is evaluated and is used to derive exact limiting forms (at both long and short wavelengths) for the spin-antisymmetric local-field factor, ${limop}_{\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\omega}}\ensuremath{\infty}}{G}_{\ensuremath{-}}(\mathbf{q},\ensuremath{\omega}).$ In two dimensions ${limop}_{\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\omega}}\ensuremath{\infty}}{G}_{\ensuremath{-}}(\mathbf{q},\ensuremath{\omega})$ is found to diverge as $1/q$ at long wavelengths, and the spin-antisymmetric exchange-correlation kernel of time-dependent spin density functional theory diverges as ${1/q}^{2}$ in both two and three dimensions. These signal a failure of the local-density approximation, one that can be redressed by alternative approaches.

DOI
10.1103/physrevb.67.233104

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Spin correlations in an interacting electron liquid are studied in the high-frequency limit and in both two and three dimensions. The third-moment sum rule is evaluated and is used to derive exact limiting forms (at both long and short wavelengths) for the spin-antisymmetric local-field factor, limop\ensuremathω\ensuremath→\ensuremath∞G\ensuremath−(q,\ensuremathω).{limop}_{\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\omega}}\ensuremath{\infty}}{G}_{\ensuremath{-}}(\mathbf{q},\ensuremath{\omega}).limop\ensuremathω\ensuremath→​\ensuremath∞​G\ensuremath−​(q,\ensuremathω). In two dimensions limop\ensuremathω\ensuremath→\ensuremath∞G\ensuremath−(q,\ensuremathω){limop}_{\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\omega}}\ensuremath{\infty}}{G}_{\ensuremath{-}}(\mathbf{q},\ensuremath{\omega})limop\ensuremathω\ensuremath→​\ensuremath∞​G\ensuremath−​(q,\ensuremathω) is found to diverge as 1/q1/q1/q at long wavelengths, and the spin-antisymmetric exchange-correlation kernel of time-dependent spin density functional theory diverges as 1/q2{1/q}^{2}1/q2 in both two and three dimensions. These signal a failure of the local-density approximation, one that can be redressed by alternative approaches.
The exchange-correlation kernel in the spin channel in an electron liquid has the structure ${f}_{xc,\ensuremath{-}}^{L,T}(q,\ensuremath{\omega})\stackrel{q\ensuremath{\rightarrow}0}{\ensuremath{\rightarrow}}A(\ensuremath{\omega})∕{q}^{2}+{B}^{L,T}(\ensuremath{\omega})$ in the limit of the long wavelength. Here LLL denotes the longitudinal component and TTT the transverse component relative to the direction of the wave vector q\mathbf{q}q. A collection of exact results for $A(\ensuremath{\omega})$ and ${B}^{L,T}(\ensuremath{\omega})$ is obtained at limiting low and high frequency, respectively, in two dimensions. Based on these results, we further give approximate interpolations for $A(\ensuremath{\omega})$ and ${B}^{L,T}(\ensuremath{\omega})$ at all frequencies in the paramagnetic case.
We present an analytical study of the dynamical local-field factors associated with the response of a homogeneous two-dimensional interacting electron liquid as functions of momentum, frequency, and density. We derive sum rules that constrain their asymptotic forms (in momentum and frequency) for both the spin-symmetric and spin-antisymmetric cases. Parametrized expressions for the local-field factors are proposed, based on all available sum rules and on many-body perturbation theory, and these are found to be in good agreement with quantum Monte Carlo calculations. Finally, these expressions are used to evaluate the effective electron-electron interaction in a local approximation for two-dimensional systems. It is shown that both the quantitative and qualitative behaviors of the interaction are sensitive to the inclusion of dynamical correlations.
The high-frequency limits of the singular component $A(\ensuremath{\omega})$ of the small wave-vector expansion of the longitudinal (L)(L)(L) and transverse (T)(T)(T) components of the spin-resolved exchange-correlation kernel tensor ${f}_{xc,\ensuremath{\sigma}{\ensuremath{\sigma}}^{\ensuremath{'}}}^{L,T}(q,\ensuremath{\omega})=\ensuremath{-}v(q){G}_{\ensuremath{\sigma}{\ensuremath{\sigma}}^{\ensuremath{'}}}^{L,T}(q,\ensuremath{\omega})$ in a two-dimensional isotropic electron liquid with arbitrary spin polarization are studied. Here ${G}_{\ensuremath{\sigma}{\ensuremath{\sigma}}^{\ensuremath{'}}}^{L,T}(q,\ensuremath{\omega})$ is the spin-resolved local-field factor, v(q)v(q)v(q) is the Coulomb interaction in momentum space, and $\ensuremath{\sigma}$ denotes spin. Particularly, the real part of $A(\ensuremath{\omega})$ is found to be logarithmically divergent at large $\ensuremath{\omega}$. The large wave-vector structure of the corresponding spin-resolved static structure factor is also established.
We analyze the spin and charge susceptibilities of a spin-polarized electron gas subject to a weak space- and time-dependent field coupled to the electronic spins, with the main attention given to the case of space dimensionality D=2.D=2.D=2. Exchange and correlations enter the dynamic susceptibilities through spin-dependent local-field factors ${G}_{\ensuremath{\sigma}}^{\ifmmode\pm\else\textpm\fi{}}(\mathbf{q},\ensuremath{\omega}).$ For an arbitrary degree of polarization, we determine the exact analytic expressions of ${G}_{\ensuremath{\sigma}}^{\ifmmode\pm\else\textpm\fi{}}(\mathbf{q},\ensuremath{\omega})$ in two limiting cases: (i) the limit of large wave number q at finite frequency $\ensuremath{\omega},$ already considered in D=3D=3D=3 by D. C. Marinescu and J. J. Quinn [Phys. Rev. B 56,56,56, 1114 (1997)]; and (ii) the static limit at small wave number. In the latter case, we obtain thermodynamic sum rules of general validity in both dimensionalities. Our work gives insight into many-body vertex corrections and basic information for calculations of the effects of the electron-electron interactions on physical properties.
We use the Ceperley and Alder Green's-function Monte Carlo data for the static form factor, S(q)S(q)S(q), of an electron liquid to calculate the static local-field correction, G(q)G(q)G(q), to the dielectric function which takes into account short-range electron correlations. The resulting local-field correction in the long-wavelength limit is obtained by smoothly fitting to the results of Iwamoto and Pines, which are exact in this limit. We use the G(q)G(q)G(q) thus obtained to calculate the static and dynamic form factors, the contribution to the correlation energy from different momentum transfers, plasmon dispersion, and the plasmon-dispersion coefficient. We compare these theoretical quantities with the available experimental data, and use the ${\ensuremath{\omega}}^{3}$-sum rule (which we also calculate from the Monte Carlo data) to discuss the limitations arising from the use of the static local-field correction.
The parametrized form of local-field corrections is introduced to construct the dielectric function and the nonlocal spin susceptibility that describe the dielectric properties of a paramagnetic degenerate electron liquid in two dimensions beyond the random-phase approximation (RPA). The parameters that take into account the spin-parallel and spin-antiparallel electron-electron interactions at short distances are determined in such a way that the response functions correctly give the Monte Carlo values for the compressibility and the spin susceptibility, respectively, in the long-wavelength limit. The local-field correction thus obtained is compared with that in the Hubbard approximation; the static dielectric function and the spin susceptibility are also compared with those in the RPA. It is found that the sign of the static dielectric function depends on both the dimensionless coupling constant and the wave number, which is a feature markedly different from the RPA results.