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Physical review. B, Condensed matter

Limits of the exchange-correlation local fields in the magnetic response of a spin-polarized electron gas

Marco Polini, Mario P. Tosi

Published Jan 9, 2001• 17 citations
Publisher

Abstract

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 $D=2.$ 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 $D=3$ by D. C. Marinescu and J. J. Quinn [Phys. Rev. B $56,$ 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.

DOI
10.1103/physrevb.63.045118

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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\ensuremathσ\ifmmode±\else\textpm\fi(q,\ensuremathω).{G}_{\ensuremath{\sigma}}^{\ifmmode\pm\else\textpm\fi{}}(\mathbf{q},\ensuremath{\omega}).G\ensuremathσ\ifmmode±\else\textpm\fi​(q,\ensuremathω). For an arbitrary degree of polarization, we determine the exact analytic expressions of G\ensuremathσ\ifmmode±\else\textpm\fi(q,\ensuremathω){G}_{\ensuremath{\sigma}}^{\ifmmode\pm\else\textpm\fi{}}(\mathbf{q},\ensuremath{\omega})G\ensuremathσ\ifmmode±\else\textpm\fi​(q,\ensuremathω) in two limiting cases: (i) the limit of large wave number q at finite frequency \ensuremathω,\ensuremath{\omega},\ensuremathω, 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 extend Niklasson's method in order to study the spin response of an electron liquid in a weak sinusoidal magnetic field. The many-body correction $G\ensuremath{-}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}},\ensuremath{\omega})$ caused by exchange and correlation is introduced to describe the correct wave-vector- and frequency-dependent spin susceptibility. The exact behavior of $G\ensuremath{-}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}},\ensuremath{\omega})$ in the large-qqq limit is related to the pair distribution function $g(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ at r=0r=0r=0. ${G}_{\ensuremath{-}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}},\ensuremath{\omega})\ensuremath{\rightarrow}\frac{[4g(0)\ensuremath{-}1]}{3}$, as $q\ensuremath{\rightarrow}\ensuremath{\infty}$. At metallic densities this value is negative, opposite in sign to the limit at small wave vector. Thus the spin susceptibility for large wave vectors is suppressed, rather than enhanced, by many-body effects.
We analyze the effective electron-electron interaction in a two dimensional polarized paramagnetic system. The spin degree of freedom, s, is manifestly present in the expressions of spin dependent local field factors that describe the short range exchange (x) and correlation (c) effects. Starting from the exact asymptotic values of the local field correction functions for large and small momentum at zero frequency we obtain self-consistent expressions across the whole spectrum of momenta. Then, the effective interaction between two electrons with spins s and s' is calculated. We find that the four effective interactions, up-up, up-down, down-up and down-down, are different. We also obtain their qualitative dependence on the electronic density and polarization and note that these results are independent of the approximation used for the local field correction functions at intermediate momenta.
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 have derived the exact limiting behavior of the many-body local fields ${G}_{\ifmmode\pm\else\textpm\fi{}}$(q,\ensuremath{\omega}) in a two-dimensional electron gas. At large wave vectors and finite frequencies these functions can be expressed in terms of g(0), the value at the origin of the pair correlation function of the electron gas. We find in particular that in this limit G+{G}_{+}G+​(\ensuremath{\infty},\ensuremath{\omega})=1-g(0), and ${G}_{\mathrm{\ensuremath{-}}}$(\ensuremath{\infty},\ensuremath{\omega})=g(0). Our results provide new insight into the problem of the many-body vertex corrections in this interacting system.
The effects of short-range correlations on the properties of the electron gas are studied using an approach phrased in the polarization potential language of Pines and Nozi\`eres. It was argued by Lowy and Brown that two-body terms make up the dominant contribution to the effective interaction for large momentum transfers. In the region of small-momentum transfers it is known that the random-phase approximation provides an adequate picture of the electron gas. With these two regions of large- and small-momentum transfers in mind, a polarization propagator is constructed to interpolate between them. A number of sum rules, among them the fff-sum rule, involving the imaginary part of the polarization propagator, were found to be satisfied. The failure of our theory to satisfy the compressibility sum rule is discussed in detail. We present here the results for the local ttt-matrix, the dynamic and static form factors, and the pair correlation function. That our pair-correlation function remains positive and is essentially the same as in the paper of Lowy and Brown indicates that the physics and not the formalism is responsible for the results we obtain.
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.