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Physical review. B, Condensed matter

Boson approximations for localized dynamic perturbations in metals

O. Gunnarsson, Kurt Schönhammer

Published Feb 15, 1982• 19 citations
Publisher

Abstract

Recently we proposed two boson approximations to describe the creation of electronhole---pair excitations by a localized time-dependent perturbation. These boson approximations were shown to be exact in certain limiting cases. In this paper we compare the results in these approximations with exact calculations where the electrons are treated as fermions. We show that the validity of the approximations is broader than had earlier been proven.

DOI
10.1103/physrevb.25.2503

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Physical review. B, Solid state•Feb 15, 1971•182 citations
Recently we proposed two boson approximations to describe the creation of electronhole---pair excitations by a localized time-dependent perturbation. These boson approximations were shown to be exact in certain limiting cases. In this paper we compare the results in these approximations with exact calculations where the electrons are treated as fermions. We show that the validity of the approximations is broader than had earlier been proven.
A model of hybridized bosons and fermions is studied beyond the mean field approximation. The divergent boson self-energy at zero temperature makes the Cooper pairing of fermions impossible. The frequency and momentum dependence of the self- energy and the condensation temperature TcT_{c}Tc​ of initially localized bosons are calculated analytically. The value of the boson condensation temperature TcT_{c}Tc​ is below 1K1K1K which rules out the boson-fermion model with the initially localized bosons as a phenomenological explanation of high-temperature superconductivity. The intra-cell density-density fermion-boson interaction dominates in the fermion self-energy. The model represents a normal metal with strongly damped bosonic excitations. The latter play the role of normal impurities.
Abstract In recent years it has been shown how approximate bosonization can be used to justify the random phase approximation for the correlation energy of interacting fermions in a mean-field scaling limit. At the core is the interpretation of particle-hole excitations close to the Fermi surface as bosons. The main two approaches however differ in emphasizing collective degrees of freedom (particle-hole pairs delocalized over patches on the Fermi surface) or particle-hole pairs exactly localized in momentum space. Both methods lead to equal precision for the correlation energy with regular interaction potentials. This poses the question how big the influence of delocalizing particle-hole pairs really is. In the present note we show that a description with few, completely collective bosonic degrees of freedom only yields an upper bound of about 92% of the optimal value. Nevertheless it is remarkable that such a simple approach comes that close to the optimal bound.
A model of hybridized bosons and fermions is studied beyond the mean-field approximation. The divergent boson self-energy at zero temperature makes the Cooper pairing of fermions impossible. The frequency and momentum dependence of the self-energy and the condensation temperature of initially localized bosons are calculated analytically. The value of the boson condensation temperature is below 1 K and the long-wave spectrum is strongly damped which rules out the boson - fermion model with the initially localized bosons as a phenomenological explanation of high-temperature superconductivity. The intra-cell density - density fermion - boson interaction dominates in the fermion self-energy. The model represents a dirty normal metal, in which localized bosons play the role of impurities.
In this thesis, an approximate method introduced to solve time-dependent many-body problems known as time-dependent many-body perturbation theory is studied. Many-body perturbation theory for interacting electrons and phonons is reviewed. In particular, the electron propagator G and an unconventional two-component phonon propagator, which satisfy coupled integral Dyson equations, are introduced. In practice, the associated integral kernels known as the electron Σ and phonon self-energies need to be approximated. The conserving approximations known as the Hartree (-Fock) and the first and second Born approximations, which respect the continuity equation between the electron density and current, are considered in this work. Time-dependent systems of interest are studied in this thesis by using the integro-differential forms of the Dyson equations referred to as the Kadanoff-Baym Equations (KBE). The Kadanoff-Baym equations are introduced for the electron and phonon propagators unconventionally as coupled firstorder integro-differential equations. It is reviewed how these equations are solved numerically by describing an integration rule, a class of practical methods and a parallel implementation of the numerical method. In addition, documentation of how the Kadanoff-Baym equations allow to solve the Bethe-Salpeter Equation (BSE) with the kernel δΣ/δG for the density response function, is provided. In two of the enclosed publications, we benchmarked observables obtained by using the Hartree and partially and fully self-consistent Born approximations against numerically exact results for the two- site, two-electron Holstein model. In this model, the two electrons which are constrained to move between two lattice sites interact indirectly via the electron-phonon coupling. It was observed that only the fully self-consistent Born approximation could cope qualitatively correctly with the competition between the delocalizing and localizing effects of the kinetic and interaction energies. For the other two approximations, spurious bifurcative symmetry breaking with an associated unbounded density response was observed. Nevertheless, also the self-consistent Born approximation was concluded to fail in describing the bipolaronic behavior of the true system. In the third publication, we benchmarked the Hartree-Fock and second Born approximations against an exact method for the few-site Hubbard and Pariser-Parr-Pople models in which the underlying lattice is inert and the electrons interact amongst themselves directly. It was found that the second Born approximation is capable of describing the so-called correlation induced doubly-excited states. This is not possible for time-local approximations such as Hartree-Fock. In addition to the qualitative results, which highlight successes of the applied simple self-energy approximations, the approximate and exact results were also compared on a more quantitative level. It is the quantitative and qualitative results combined which are used in this thesis to assess the quality of the many-body approximations used, with the aim to better understand when these methods are predictive.
We develop a method that allows us to map models of interacting fermions onto bosonic models describing collective excitations in an arbitrary dimension. We present arguments that this mapping may become exact in the thermodynamic limit in the presence of a bath. The boson models can be written either in the form of a model of noninteracting bosons in a fluctuating auxiliary Hubbard-Stratonovich field or in the form of a superfield theory of interacting bosons. We show how one can study the latter version within the perturbation theory. Using a diagrammatic technique developed for the boson model we compare the first two orders of the perturbation theory with the corresponding results for the original fermion model and find a perfect agreement. As concerns the former representation, we suggest a scheme that may be suitable for Monte Carlo simulations. We demonstrate both analytically and numerically that the partition function of the boson model in the auxiliary Hubbard-Stratonovich field is, in contrast to the fermion model, always positive and thus the boson model can be suitable for the Monte Carlo routine. At the same time, our arguments showing the equivalence of the boson and fermion models in the thermodynamic limit are not completely rigorous and it is still to be investigated numerically how accurate the boson model can approximate the original fermion one.
The effect of a localized time-dependent potential on a Fermi gas is discussed. A simple bosonlike model for the particle-hole excitations is developed. Its use appears justified for the dissipative part of the response of the Fermi gas to localized time-dependent potentials. General expressions are obtained for the probability that the Fermi gas remains in the ground state and for the energy spectrum of excited states. Several illustrative examples are discussed. We consider the effect of an exponentially switched-on localized potential. It is shown that the state reached when the potential attains its full strength VVV is orthogonal to the ground state of the Fermi gas with the potential VVV, no matter how slow the switching. It consists of a spectrum of excited states of the full Hamiltonian with a width proportional to the switching rate. Other aspects of this failure of adiabaticity such as nonreversibility and path dependence are investigated. Some possible applications are pointed out. The potential source may have a finite mass. The effect of its recoil is calculated for the special case of a suddenly switched-on potential. The probability of finding the Fermi gas in its final ground state after the perturbation is switched on is found to be ${P}_{G}=\mathrm{exp}[\frac{{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}}^{2}\mathrm{ln}\ensuremath{\gamma}}{(1\ensuremath{-}{\ensuremath{\gamma}}^{2})}]$, where $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}=V{\ensuremath{\rho}}_{\ensuremath{\epsilon}F}$ and is a measure of the coupling of the localized perturbation to the electron gas. $\ensuremath{\gamma}$ is the mass ratio mM\frac{m}{M}Mm​. The potential source is initially at rest. The above expression is critically discussed and corrections to it are calculated. They are found to be small. The spectrum of excitations produced is also calculated. In conclusion, connections are pointed out with other particle field problems exhibiting a similar infrared singularity.