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arXiv:cond-mat/0404325 · Physical Review B

Long-wavelength behavior of the dynamical spin-resolved local-field factor in a two-dimensional electron liquid

Zhixin Qian

Published Dec 17, 2004• 2 citations
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Abstract

The high-frequency limits of the singular component $A(\ensuremath{\omega})$ of the small wave-vector expansion of the longitudinal $(L)$ and transverse $(T)$ 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, $v(q)$ 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.

DOI
10.1103/physrevb.70.235118

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The high-frequency limits of the singular component A(\ensuremathω)A(\ensuremath{\omega})A(\ensuremathω) 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 fxc,\ensuremathσ\ensuremathσ\ensuremath′L,T(q,\ensuremathω)=\ensuremath−v(q)G\ensuremathσ\ensuremathσ\ensuremath′L,T(q,\ensuremathω){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})fxc,\ensuremathσ\ensuremathσ\ensuremath′L,T​(q,\ensuremathω)=\ensuremath−v(q)G\ensuremathσ\ensuremathσ\ensuremath′L,T​(q,\ensuremathω) in a two-dimensional isotropic electron liquid with arbitrary spin polarization are studied. Here G\ensuremathσ\ensuremathσ\ensuremath′L,T(q,\ensuremathω){G}_{\ensuremath{\sigma}{\ensuremath{\sigma}}^{\ensuremath{'}}}^{L,T}(q,\ensuremath{\omega})G\ensuremathσ\ensuremathσ\ensuremath′L,T​(q,\ensuremathω) is the spin-resolved local-field factor, v(q)v(q)v(q) is the Coulomb interaction in momentum space, and \ensuremathσ\ensuremath{\sigma}\ensuremathσ denotes spin. Particularly, the real part of A(\ensuremathω)A(\ensuremath{\omega})A(\ensuremathω) is found to be logarithmically divergent at large \ensuremathω\ensuremath{\omega}\ensuremathω. The large wave-vector structure of the corresponding spin-resolved static structure factor is also established.
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.
The components of the exchange-correlation kernel tensor of an isotropic electron liquid in the spin channel have the structure ${f}_{\mathrm{xc},\ensuremath{-}}^{L,T}(q,\ensuremath{\omega})\stackrel{\stackrel{\ensuremath{\rightarrow}}{q}0}{\ensuremath{\rightarrow}}A(\ensuremath{\omega}{)/q}^{2}{+B}^{L,T}(\ensuremath{\omega}),$ where L denotes the longitudinal component and T the transverse component relative to the direction of the wave vector q.\mathbf{q}.q. In this paper we calculate analytically the high- and low-frequency limits of $A(\ensuremath{\omega})$ and ${B}^{L,T}(\ensuremath{\omega})$ and combine these limiting forms with the Kramers-Kr\"onig dispersion relations to obtain approximations for $A(\ensuremath{\omega})$ and ${B}^{L,T}(\ensuremath{\omega})$ at all frequencies.
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/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.
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 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.
A microscopic derivation of the equations of transverse spin dynamics of a spin-polarized Fermi liquid at zero temperature is given in the leading-order approximation in the frequencies and wave vectors characterizing the spin motion. The equations are applicable for arbitrary degree of polarization and arbitrary deviations of the spin direction from the equilibrium orientation. The solutions describing a coherently precessing two-domain structure and spin waves are examined. In contrast to the assertion discussed in the literature that spin waves are damped at zero temperature, spin waves are found to be undamped in the long-wavelength limit.