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Physical review. B, Solid state

Free energy of the interacting electron gas including higher-order exchange effects

D. J. Kim

Published Apr 15, 1974• 8 citations
Publisher

Abstract

We discuss the free energy of an electron gas with the long-range Coulomb interaction by using the dielectric-response function which includes the higher-order exchange processes. Our result reproduces earlier results of Gell-Mann and Brueckner; Nozi\`eres and Pines; Hubbard; Englert and Brout; and others as special cases. The longitudinal spin-fluctuation effect is included in the present result, but its role seems to be significantly different from what is expected in the Hubbard-type model due to the large cross effect of the spin fluctuation and charge fluctuation even in the paramagnetic state.

DOI
10.1103/physrevb.9.3307

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The dielectric response of a quasi-one-dimensional system of interacting electrons has been investigated. In the authors' model the particles move freely in one direction and are tightly bound in the transverse direction. Calculation of the Coulomb matrix elements leads to an effective one-dimensional electron-electron interaction nu (q) which is less singular for small q than the 1/q 2 potential assumed by previous authors. As a result, the excitonic instability which they find even for a rigid positive background does not arise in their calculation. The calculations are based on the approximation of Singwi, Tosi, Land and Sjolander (1968). As expected in this approach, the pair correlation function at small distances remains positive over a much larger range of densities than that for which the compressibility sum rule is satisfied. The static dielectric function is negative around q=2k F , which suggests the possibility of charge ordering when the background is allowed to be polarisable. If nu (q) is replaced by a delta -function potential, a numerical calculation of the energy for the exact solution of Yang may be made. On comparison with the result obtained with the method of Singwi et al. it is found that the approximate scheme gives good correlation energies for high densities.

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We discuss the free energy of an electron gas with the long-range Coulomb interaction by using the dielectric-response function which includes the higher-order exchange processes. Our result reproduces earlier results of Gell-Mann and Brueckner; Nozi`eres and Pines; Hubbard; Englert and Brout; and others as special cases. The longitudinal spin-fluctuation effect is included in the present result, but its role seems to be significantly different from what is expected in the Hubbard-type model due to the large cross effect of the spin fluctuation and charge fluctuation even in the paramagnetic state.
A microscopic formula for the effective two-body interaction between electrons or holes in an ideal metal is derived within the linear response theory. Both the zero-range Hubbard interaction U0 and the long-range Coulomb interaction, denoted by ΔV(q), are included. The effective interaction, which is necessarily spin-dependent, is expressed in terms of the exact charge and spin density-density correlation functions and of their higher order mixtures. These correlation functions are analyzed diagrammatically. Attention is paid to clarifying the different roles played by U0 and ΔV(q). We also display the corresponding new formula for the free energy. Comparison is made with previous theories.
Abstract We prove a rigorous upper bound on the correlation energy of interacting fermions in the mean-field regime for a wide class of interaction potentials. Our result covers the Coulomb potential, and in this case we obtain the analogue of the Gell-Mann–Brueckner formula c1ρlog⁡(ρ)+c2ρc_{1}\rho \log \left( \rho \right) +c_{2}\rho c1​ρlog(ρ)+c2​ρ c 1 ρ log ρ + c 2 ρ in the high density limit. We do this by refining the analysis of our bosonization method to deal with singular potentials, and to capture the exchange contribution which is absent in the purely bosonic picture.
A simple approximate expression in real and reciprocal spaces is given for the static exchange-correlation kernel of a uniform electron gas interacting with the long-range part only of the Coulomb interaction. This expression interpolates between the exact asymptotic behaviors of this kernel at small and large wave vectors which in turn requires, among other things, information from the momentum distribution of the uniform electron gas with the same interaction that have been calculated in the G0W0{G}_{0}{W}_{0}G0​W0​ approximation. This exchange-correlation kernel as well as its complement analog associated to the short-range part of the Coulomb interaction are more local than the Coulombic exchange-correlation kernel and constitute potential ingredients in approximations for recent adiabatic connection fluctuation-dissipation and/or density functional theory approaches of the electronic correlation problem based on a separate treatment of long- and short-range interaction effects.
The fluctuation-dissipation theorem is applied to the study of the pair correlation functions of the system containing charged particles which is not invariant under spatial translation. The results are expressed in terms of the longitudinal response functions and involve correlation functions which depend only on screened short-range Coulomb interactions S0{S}_{0}S0​. In the case of the electron gas, the density-density correlation function is $S(\mathbf{r}{\mathbf{r}}^{\ensuremath{'}};\ensuremath{\omega})=[\ensuremath{\delta}(\mathbf{r}\ensuremath{-}\mathbf{R})\ensuremath{-}K(r{\mathbf{R}}^{\ensuremath{'}\ensuremath{'}};\ensuremath{\omega})v({\mathbf{R}}^{\ensuremath{'}\ensuremath{'}}\mathbf{R})]{S}_{0}(\mathbf{R}{\mathbf{R}}^{\ensuremath{'}};\ensuremath{\omega})[\ensuremath{\delta}({\mathbf{R}}^{\ensuremath{'}}\ensuremath{-}{\mathbf{r}}^{\ensuremath{'}})\ensuremath{-}v({\mathbf{R}}^{\ensuremath{'}}{\mathbf{R}}^{\ensuremath{'}\ensuremath{'}\ensuremath{'}})\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{K}({\mathbf{R}}^{\ensuremath{'}\ensuremath{'}\ensuremath{'}}{\mathbf{r}}^{\ensuremath{'}};\ensuremath{-}\ensuremath{\omega})]$where KKK is the response function of the electron density to an external potential field, and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{K}$ its conjugate; vvv represents the Coulomb interaction; and the repeated spatial coordinates are integrated over the volume. It is shown that this expression implies the principle of superposition of dressed particles (the bare particles plus their associated screening clouds); for the calculations of the correlation functions, the dressed particles may be superposed without further consideration of the long-range parts of Coulomb interactions. Similar reductions are carried out for the various correlation functions in the electron-ion system, and the superposition principle is confirned. A possible extension of the results to a nonequilibrium stationarysystem is suggested and discussed. The interrelationship among various longitudinal response functions, which is characteristic of the inhomogeneous system, is discussed and clarified. Simplifications arising from the homogeneity and discussions from diagrammatic considerations are given in Appendices.
We determine the free energy of strongly correlated electron systems in the example of the Hubbard model by calculating the contribution of spin and charge fluctuations to the Gutzwiller approximation mean-field result. We employ the slave-boson formulation of Kotliar and Ruckenstein in its spin-rotation-invariant form in the usual continuous-time approximation of the functional integral representation, corrected by “high-frequency contributions” ( SRIKR + ). In contrast to all previous attempts to calculate the fluctuation contribution to the free energy within the spin-rotation-invariant Kotliar-Ruckenstein slave-boson representation (SRIKR) scheme our result preserves the noninteracting limit. We show that the correct form of the kinetic energy renormalization is essential. The results for the ground-state energy in the paramagnetic phase are in very good agreement with state-of-the-art results obtained by methods such as density matrix embedding theory, quantum Monte Carlo, and others. The leading low-temperature behavior of the free energy allows us to extract the quasiparticle effective mass, in particular its enhancement near a continuous phase transition into an ordered state. Our work demonstrates that the SRIKR + method is competitive with the best available alternative methods and equips the slave-boson approach with an improved synoptic power to explore strongly correlated electron systems.
The dielectric response of a quasi-one-dimensional system of interacting electrons has been investigated. In the authors' model the particles move freely in one direction and are tightly bound in the transverse direction. Calculation of the Coulomb matrix elements leads to an effective one-dimensional electron-electron interaction nu (q) which is less singular for small q than the 1/q 2 potential assumed by previous authors. As a result, the excitonic instability which they find even for a rigid positive background does not arise in their calculation. The calculations are based on the approximation of Singwi, Tosi, Land and Sjolander (1968). As expected in this approach, the pair correlation function at small distances remains positive over a much larger range of densities than that for which the compressibility sum rule is satisfied. The static dielectric function is negative around q=2k F , which suggests the possibility of charge ordering when the background is allowed to be polarisable. If nu (q) is replaced by a delta -function potential, a numerical calculation of the energy for the exact solution of Yang may be made. On comparison with the result obtained with the method of Singwi et al. it is found that the approximate scheme gives good correlation energies for high densities.