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Physical Review

A Collective Description of Electron Interactions: III. Coulomb Interactions in a Degenerate Electron Gas

David Böhm, David Pines

Published Nov 1, 1953• 1,913 citations
Publisher

Abstract

The behavior of the electrons in a dense electron gas is analyzed quantum-mechanically by a series of canonical transformations. The usual Hamiltonian corresponding to a system of individual electrons with Coulomb interactions is first re-expressed in such a way that the long-range part of the Coulomb interactions between the electrons is described in terms of collective fields, representing organized "plasma" oscillation of the system as a whole. The Hamiltonian then describes these collective fields plus a set of individual electrons which interact with the collective fields and with one another via short-range screened Coulomb interactions. There is, in addition, a set of subsidiary conditions on the system wave function which relate the field and particle variables. The field-particle interaction is eliminated to a high degree of approximation by a further canonical transformation to a new representation in which the Hamiltonian describes independent collective fields, with ${n}^{\ensuremath{'}}$ degrees of freedom, plus the system of electrons interacting via screened Coulomb forces with a range of the order of the inter electronic distance. The new subsidiary conditions act only on the electronic wave functions; they strongly inhibit long wavelength electronic density fluctuations and act to reduce the number of individual electronic degrees of freedom by ${n}^{\ensuremath{'}}$. The general properties of this system are discussed, and the methods and results obtained are related to the classical density fluctuation approach and Tomonaga's one-dimensional treatment of the degenerate Fermi gas.

DOI
10.1103/physrev.92.609

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Coulomb Correlations in Inhomogeneous Many-Particle Systems

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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 ${S}_{0}$. 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 $K$ is the response function of the electron density to an external potential field, and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{K}$ its conjugate; $v$ 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.

Physical Review•Oct 11, 1965•14 citations

Particles and quasi-particles

Richard M. Martin, Lucia Reining, David M. Ceperley

Art is the elimination of the unnecessary. Pablo Picasso Summary Condensed matter is constituted by a huge number of electrons and nuclei interacting with Coulomb potentials. The topic of this chapter is a way to deal with the full, coupled problem by separating it into a part that is tractable and the rest that is approximated. This naturally leads to the appearance of dynamical fields, and to the concept of “quasi-particles” that have the same quantum numbers as non-interacting electrons. The quasi-particles obey equations where the potential and bare interactions in the hamiltonian are replaced by dynamical self-energies and screened interactions that can describe many of the effects of correlation. In this chapter we discuss the intuitive concepts behind this approach in general terms to motivate the more rigorous formulations and approximations used in the Green's function methods of the following chapter and in Part III. The first equations of this book, Eqs. (1.1) and (1.2), express the fundamental theory of matter in terms of electrons and nuclei that interact with Coulomb potentials. For systems with only a few electrons, a solution of the problem can be obtained by exact diagonalization, using configuration interaction methods. For many-electron systems one has to resort to other methods. For example, QMC stochastic simulations (Chs. 22–25) are among the most accurate methods known to calculate certain expectation values, such as equilibrium thermodynamic properties, total energies, the density, and various static correlation functions like those in Ch. 5. Other properties such as excitation spectra are more difficult to access with QMC. Straightforward perturbation theory is, in general, not appropriate since interactions among the electrons are of the same order of magnitude as the independent-particle terms. Hence, we must develop other approaches. One strategy that gives access to equilibrium thermodynamic as well as dynamical excitation properties is to separate the interacting many-body problem into a part that is tractable by some means and the rest of the problem that is dealt with more approximately. The present chapter is devoted to this idea. It is a prelude to many-body Green's function methods; the aim is to provide a unified framework for the developments in Chs. 8–21, along with a qualitative description of the most relevant quantities.

Cambridge University Press eBooks•May 31, 2016•0 citations

Theory of a two-level system interacting with a degenerate electron gas. II. Scaling theory of a one-dimensional Coulomb gas

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A one-dimensional Coulomb gas is considered in which the particles represent both charges and dipoles. The values of the charges and dipole moments and the fugacities of that gas have been obtained in the preceding paper by mapping the dynamics of a two-level system (TLS) interacting with a degenerate electron gas onto the present problem. The scaling procedure of Anderson, Yuval, and Hamann is applied to eliminate the close pair which corresponds to the short-time behavior of the TLS. The main contribution arises from pairs of zero total charge, but the role of other pairs is considered as well. The scaling equations show that the electron-assisted transitions of the TLS play a very important role if their value is different from zero for the starting Hamiltonian. In that case localization cannot occur in the case of a spin-(1/2 fermionic system. Localization may occur only if the fermion spin has an artificially large value with spin degeneracy ${N}_{s}$>2. That clearly demonstrates that a particle moving in a double potential well may have different dissipative behavior depending upon whether it is coupled to a fermionic or a bosonic heat bath. The screening of the TLS by electrons is described by an arbitrarily large phase shift \ensuremath{\delta} which renormalizes towards the value \ensuremath{\delta}=\ensuremath{\pi}/2${N}_{s}$, but the scaling equations lose their validity as the fugacities corresponding to the TLS transitions become of the order of unity. The method developed is a very general one, and thus it can be applied to other similar one-dimensional gases as well.

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Phase space formulation of the quantum many-body problem

Paul H. Levine

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The Collective Treatment of a Fermi Gas: II

T. F. GASKELL

The ground state energy of the free electron gas is calculated using the Rayleigh-Schrödinger variational method with the wave function ψ= D Π i<j f ( x ij ) where D is a determinant of plane waves and f ( x ij ) a correlation function. Consideration of the wave function in terms of the collective coordinates p k' the Fourier components of the density, suggests an accurate approximation for the energy integral which is then evaluated over the coordinates of the particles so that the use of subsidiary conditions is avoided. Effects omitted in the random phase approximation are included and the final results extend continuously over plasma and particle modes and should be valid in the range of densities encountered in real metals. The results agree closely with those of Nozières and Pines, and of Hubbard obtained by more elaborate methods.

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Semiclassical Behaviour of Quasiparticle Excitations in Electron Gas

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Abstract In current research we investigate macroscopic collective quantum effects in interacting electron gas with arbitrary degree of degeneracy due to electrostatic field-density coupling. It is found that in the one dimensional excitation model deterministic quantum charge transport can take place due to the electrostatic field coupling of the interacting electron correlations with number density variations of the electron gas. A new expression for probability current is found to be valid for collective excitations in interacting electron gas. It is also revealed that the phase-space evolution of modified Wigner distribution function for linear quasiparticle excitations resembles that of the classical free particle. In other words, semiclassical linear evolution of collective excitations in interacting quantum electron gas is quite analogous to dynamics of classical system of particles in the absence of external forces, where the interparticle electrostatic forces play the analogous role of internal classical forces. It is shown that the electrostatic field of electrons when coupled to collective electron wavefunction can also lead to directly measurable macroscopic phenomena. It is remarked that the interacting system of electrons simultaneously exhibits microscopic and macroscopic effects that are linked through the dual lengthscale behavior of collective quantum electron excitations.

•Jan 8, 2024•0 citations
The behavior of the electrons in a dense electron gas is analyzed quantum-mechanically by a series of canonical transformations. The usual Hamiltonian corresponding to a system of individual electrons with Coulomb interactions is first re-expressed in such a way that the long-range part of the Coulomb interactions between the electrons is described in terms of collective fields, representing organized "plasma" oscillation of the system as a whole. The Hamiltonian then describes these collective fields plus a set of individual electrons which interact with the collective fields and with one another via short-range screened Coulomb interactions. There is, in addition, a set of subsidiary conditions on the system wave function which relate the field and particle variables. The field-particle interaction is eliminated to a high degree of approximation by a further canonical transformation to a new representation in which the Hamiltonian describes independent collective fields, with n\ensuremath′{n}^{\ensuremath{'}}n\ensuremath′ degrees of freedom, plus the system of electrons interacting via screened Coulomb forces with a range of the order of the inter electronic distance. The new subsidiary conditions act only on the electronic wave functions; they strongly inhibit long wavelength electronic density fluctuations and act to reduce the number of individual electronic degrees of freedom by n\ensuremath′{n}^{\ensuremath{'}}n\ensuremath′. The general properties of this system are discussed, and the methods and results obtained are related to the classical density fluctuation approach and Tomonaga's one-dimensional treatment of the degenerate Fermi gas.
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.
Art is the elimination of the unnecessary. Pablo Picasso Summary Condensed matter is constituted by a huge number of electrons and nuclei interacting with Coulomb potentials. The topic of this chapter is a way to deal with the full, coupled problem by separating it into a part that is tractable and the rest that is approximated. This naturally leads to the appearance of dynamical fields, and to the concept of “quasi-particles” that have the same quantum numbers as non-interacting electrons. The quasi-particles obey equations where the potential and bare interactions in the hamiltonian are replaced by dynamical self-energies and screened interactions that can describe many of the effects of correlation. In this chapter we discuss the intuitive concepts behind this approach in general terms to motivate the more rigorous formulations and approximations used in the Green's function methods of the following chapter and in Part III. The first equations of this book, Eqs. (1.1) and (1.2), express the fundamental theory of matter in terms of electrons and nuclei that interact with Coulomb potentials. For systems with only a few electrons, a solution of the problem can be obtained by exact diagonalization, using configuration interaction methods. For many-electron systems one has to resort to other methods. For example, QMC stochastic simulations (Chs. 22–25) are among the most accurate methods known to calculate certain expectation values, such as equilibrium thermodynamic properties, total energies, the density, and various static correlation functions like those in Ch. 5. Other properties such as excitation spectra are more difficult to access with QMC. Straightforward perturbation theory is, in general, not appropriate since interactions among the electrons are of the same order of magnitude as the independent-particle terms. Hence, we must develop other approaches. One strategy that gives access to equilibrium thermodynamic as well as dynamical excitation properties is to separate the interacting many-body problem into a part that is tractable by some means and the rest of the problem that is dealt with more approximately. The present chapter is devoted to this idea. It is a prelude to many-body Green's function methods; the aim is to provide a unified framework for the developments in Chs. 8–21, along with a qualitative description of the most relevant quantities.
A one-dimensional Coulomb gas is considered in which the particles represent both charges and dipoles. The values of the charges and dipole moments and the fugacities of that gas have been obtained in the preceding paper by mapping the dynamics of a two-level system (TLS) interacting with a degenerate electron gas onto the present problem. The scaling procedure of Anderson, Yuval, and Hamann is applied to eliminate the close pair which corresponds to the short-time behavior of the TLS. The main contribution arises from pairs of zero total charge, but the role of other pairs is considered as well. The scaling equations show that the electron-assisted transitions of the TLS play a very important role if their value is different from zero for the starting Hamiltonian. In that case localization cannot occur in the case of a spin-(1/2 fermionic system. Localization may occur only if the fermion spin has an artificially large value with spin degeneracy Ns{N}_{s}Ns​>2. That clearly demonstrates that a particle moving in a double potential well may have different dissipative behavior depending upon whether it is coupled to a fermionic or a bosonic heat bath. The screening of the TLS by electrons is described by an arbitrarily large phase shift \ensuremath{\delta} which renormalizes towards the value \ensuremath{\delta}=\ensuremath{\pi}/2Ns{N}_{s}Ns​, but the scaling equations lose their validity as the fugacities corresponding to the TLS transitions become of the order of unity. The method developed is a very general one, and thus it can be applied to other similar one-dimensional gases as well.
By means of a quantum mechanical phase space distribution function introduced by von Roos, the Schroedinger equation for a non-relativistic system of N identical particles with scalar interactions is transformed into a quantum mechanical generalization of the Liouville equation, thereby formulating the problem in terms of a generalized density in phase space, a quantity of primary interest in most treatments of the corresponding classical system (or plasma). This transformation permits a parallel development of the theories of classical and quantum plasmas and thus allows the quantum many-body problem to be discussed virtually completely in classical terms. In particular, a kinetic theory of quantum plasmas is obtained by deriving the quantum analogue of the BBGKY hierarchy, and applying thereto approximation techniques similar to those of Rostocker and Rosenbluth, and Bogoliubov. The point of departure from similar previous studies based on the Wigner distribution function is that the proper exchange symmetry can be tractably introduced into the formalism. Attention is first focused on the Hartree and Hartree-Fock approximations, in which case the quantum BBGKY system reduces to a simple quantum generalization of the Vlasov equation. This equation is used to study the response of spatially homogeneous systems to weak external forces, and the associated problems of plasmon and spin-wave excitations. It is also used to derive the quantum and exchange corrected equations of inviscid hydrodynamical transport which are then applied to the problem of sound propagation in the degenerate electron gas. The second part of the study is concerned with the theory of the many-electron atom in the Hartree and Hartree-Fock approximations. The relevant quantum Vlasov equations lead naturally to a statistical theory of the atom which reduces to the Thomas-Fermi-Amaldi and Thomas Fermi models (respectively) as ħ → 0. For ħ ≠ 0, the quantum and exchange corrections to these models are simultaneously generated. The quantum hydrodynamical theory developed earlier is used to determine the influence of these corrections on the boundary conditions of the model, and a theory of the compressed atom is consequently obtained. Considered in somewhat less detail are the effects of non-zero temperature, net orbital angular momentum, relativity and correlations, as well as time dependent processes. The final part deals with the problem of the degenerate electron gas with a uniform neutralizing background. Going beyond the Hartree-Fock approximation, the pair functions for particles with parallel and anti-parallel spin are obtained by neglecting three particle correlations. From these functions, a quantum-mechanical collision integral is derived which differs from that obtained by Silin and Guernsey and conjectured by Wyld and Pines in that dynamical exchange effects are included. Also obtained from the pair function is an expression for the correlation which reduces in the high density limit to the result of Gell-Mann and Brueckner. At intermediate densities an additional term appears in the energy due to the screening of the exchange interaction by the dielectric properties of the medium. It is evaluated in the high density limit and found to be -0.151 r_s ln r_s Ryd/electron in marked disagreement with the corresponding value obtained by DuBois.
The ground state energy of the free electron gas is calculated using the Rayleigh-Schrödinger variational method with the wave function ψ= D Π i<j f ( x ij ) where D is a determinant of plane waves and f ( x ij ) a correlation function. Consideration of the wave function in terms of the collective coordinates p k' the Fourier components of the density, suggests an accurate approximation for the energy integral which is then evaluated over the coordinates of the particles so that the use of subsidiary conditions is avoided. Effects omitted in the random phase approximation are included and the final results extend continuously over plasma and particle modes and should be valid in the range of densities encountered in real metals. The results agree closely with those of Nozières and Pines, and of Hubbard obtained by more elaborate methods.
Abstract In current research we investigate macroscopic collective quantum effects in interacting electron gas with arbitrary degree of degeneracy due to electrostatic field-density coupling. It is found that in the one dimensional excitation model deterministic quantum charge transport can take place due to the electrostatic field coupling of the interacting electron correlations with number density variations of the electron gas. A new expression for probability current is found to be valid for collective excitations in interacting electron gas. It is also revealed that the phase-space evolution of modified Wigner distribution function for linear quasiparticle excitations resembles that of the classical free particle. In other words, semiclassical linear evolution of collective excitations in interacting quantum electron gas is quite analogous to dynamics of classical system of particles in the absence of external forces, where the interparticle electrostatic forces play the analogous role of internal classical forces. It is shown that the electrostatic field of electrons when coupled to collective electron wavefunction can also lead to directly measurable macroscopic phenomena. It is remarked that the interacting system of electrons simultaneously exhibits microscopic and macroscopic effects that are linked through the dual lengthscale behavior of collective quantum electron excitations.