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Physical review. B, Solid state

Diamagnetic susceptibility of an interacting electron gas

Harjinder Singh, K. N. Pathak

Published Jun 1, 1975• 6 citations
Publisher

Abstract

The effects of exchange and correlations on the diamagnetic susceptibility of an interacting electron gas are investigated using the double-time Green's-function method. The equation of motion of the appropriate Green's function is linearized using a moment-conserving decoupling approximation. The many-body corrections to the diamagnetic susceptibility at metallic densities are found to be small. The present expression reduces to the exact result in the high-density limit. Many of the results of earlier calculations also follow from our expression for the diamagnetic susceptibility.

DOI
10.1103/physrevb.11.4246

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Physical review. B, Condensed matter•Dec 15, 2000•1 citation
The effects of exchange and correlations on the diamagnetic susceptibility of an interacting electron gas are investigated using the double-time Green's-function method. The equation of motion of the appropriate Green's function is linearized using a moment-conserving decoupling approximation. The many-body corrections to the diamagnetic susceptibility at metallic densities are found to be small. The present expression reduces to the exact result in the high-density limit. Many of the results of earlier calculations also follow from our expression for the diamagnetic susceptibility.
A simple calculation of the wave-vector- and frequency-dependent paramagnetic spin susceptibility of an interacting electron gas is presented. The equation of motion of the double-time Green's function is solved using a moment-conserving decoupling approximation. For the short-range interaction, our expression for the dynamic spin susceptibility reduces to a previous result of Wolff. In the static and long-wavelength limit, our result for the Stonerenhanced spin susceptibility coincides with that obtained from consideration of the groundstate energy in the Hartree-Fock approximation.
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.
The calculation of the induced spin density of a normal fermion system, such as the electron gas, in the limit of small wave numbers, is carried beyond the random phase approximation (R.P.A.) by the method of canonical transformation so as to include the first non-R.P.A. corrections. An expression for the induced spin density in the same limit, exact to all orders of particle-particle coupling, is then derived by more general methods of many-body perturbation theory. It depends only on the knowledge, to all orders of interparticle coupling, of the effective mass and the forward scattering of quasi-particles at the Fermi surface, and also yields the results of the extended R.P.A. on appropriate expansion. In the limit of zero wave number, the resulting expression for the magnetic susceptibility is found identical with that deduced by Landau from a phenomenological basis and may be regarded as an additional confirmation of the microscopic validity of the theory of the Fermi liquid.
Results for the diagonal element of the second-order density matrix (pair function) and for the momentum distribution of an interacting electron gas are presented. In the low density limit, it is shown that, to a similar degree of approximation, the first-order density matrix and hence the momentum distribution may be obtained. In the highdensity problem, the pair function has also been calculated using the Green's function approach. From the information available in these high and low density limits, the range of usefulness of the concept of the Fermi surface in an interacting electron gas is discussed. Also, results of varia. tional calculations on the stability of Cverhauser spindensity waves in metals are reported. It is concluded that such waves reduce the energy below the Hartree-Fock value only when the interaction becomes very strong, or equivalently when the density is very low. Some preliminary work on the correlation problem in a non-uniform electron gas is also briefly discussed. (auth)
To investigate the effects of electron correlation in narrow energy bands, a general formula of static spin susceptibility for interacting electron systems is derived. It is expressed in terms of the one-particle Green's function and its derivative with respect to the average number of the opposite spin electrons, the existence of the latter being the characteristic of electron correlation. The formula is applied to the case of Hubbard's improved approximation (Hubbard III), and gross feature of both the correlation and temperature dependences of the susceptibility is investigated in the insulating and metallic regions. The correlation in the metallic region is reduced for the Hubbard III case so that there occurs no ferromagnetism in a half-filled band.
The spin susceptibility of the electron gas is studied using a functional integral formulation. A Hubbard-Stratonovich transformation is used to decouple the electron-electron interaction. A uniform Fock mean field is chosen, and effects of small fluctuations about this mean field are retained up to second order. The term that gives the usual spin random-phase approximation susceptibility is identified, with the ladder sum contained in the propagator for the field fluctuations. Computations of the static spin susceptibility outline the region of stability as a function of electron-gas coupling parameter and wave vector. These computations parallel those of Hamann and Overhauser [Phys. Rev. 143, 183 (1966)] for the Yukawa interaction. We use the analytic continuation method of Staunton et al. [Phys. Rev. Lett. 82, 3340 (1999)] to extend the calculations to nonzero frequency. The resulting spectral density shows the increase in low-energy spin fluctuations as the spin-density-wave transition is approached, and the development of a pseudogap in the spin fluctuation spectrum near the wave vector of the transition.