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Journal of Physics Condensed Matter

The moments method and damped systems

C Benoit

Published Apr 25, 1994• 9 citations
Publisher

Abstract

The moments method was shown to be very useful for determining the linear response of very large harmonic systems. In the presence of anharmonicity or electron-phonon coupling, the phonon self-energy appears in the one-phonon propagator. The phonon frequency is then renormalized and the system is damped. In this paper, we show that it is possible to extend the moments method to compute the response of these damped systems. The method involves introducing a two-dimensional density functional that can be calculated by the moments techniques and which allows the computation of the response function. We illustrate these results via applications to several models: systems with constant damping, systems of coupled relaxation modes and oscillators, and systems with electron-phonon coupling.

DOI
10.1088/0953-8984/6/17/006

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The moments method was shown to be very useful for determining the linear response of very large harmonic systems. In the presence of anharmonicity or electron-phonon coupling, the phonon self-energy appears in the one-phonon propagator. The phonon frequency is then renormalized and the system is damped. In this paper, we show that it is possible to extend the moments method to compute the response of these damped systems. The method involves introducing a two-dimensional density functional that can be calculated by the moments techniques and which allows the computation of the response function. We illustrate these results via applications to several models: systems with constant damping, systems of coupled relaxation modes and oscillators, and systems with electron-phonon coupling.
The method of moments is employed to determine the ground state of the phonon-polaron system. The trial function employed in this calculation is a bare-electron wave function. It appears that the method is convergent for this problem. Although the energy is in general a complicated function of the parameter and must be determined numerically, it is possible to formally construct a power series expansion for the energy about the unperturbed energy in powers of the coupling constant. The first term in this series has been determined and agrees with the result of conventional perturbation theory. The second term has not yet been evaluated to all orders but it appears that it will have the opposite sign from the corresponding term in perturbation theory. The second and higher terms in some orders of approximation are singular for small values of the cutoff.
Coherent-state functional-integral methods are used to develop approximation schemes for the description of coupled electron-phonon systems far from equilibrium. We consider an interacting-electron system which is coupled to a system of phonons and subjected to an external---not necessarily weak---perturbation. For systems where the electron-phonon interaction is linear in the phonon variables, the phonon degrees of freedom can be accounted for exactly. From a formally exact representation of a suitable generating function for the expectation value of electronic observables, approximation schemes based on mean-field solutions can be derived. Within the lowest order of approximation, the coupled electron-phonon system may be treated as a self-consistent system of indepenent electrons. Higher-order corrections are shown to give a random-phase approximation based on self-consistent mean fields. This approach is applied to electronic two-level systems coupled to phonons under a large external bias. We find that the coupling to acoustical-phonon modes leads to a damping of the quantum beats and drives the system into a new equilibrium state. The coupling to LO phonons is shown to account correctly for resonance effects which arise when the level splitting is close to the LO-phonon energy.
Electron-phonon interactions underpin a variety of phenomena, ranging from transport and superconductivity to polarons and ultrafast carrier dynamics. Despite being one of the most intensely studied subjects in condensed matter physics, research on electron-phonon physics mostly focused on linear, first-order couplings. Second- and higher-order nonlinear couplings are commonly ignored because their calculations are too demanding and we lack computational frameworks that can provide both diagonal and off-diagonal coupling matrix elements. In this work, we report a theory and computational method for computing nonlinear electron-phonon interactions of any order in real materials. Our approach combines the advantages of unit-cell calculations of electron wavefunctions and supercell calculations of phonon perturbations, is systematically improvable, can be used with either semilocal or nonlocal exchange-correlation functionals, and is amenable to Wannier-Fourier interpolation. As a first proof of concept, we illustrate this method by computing second-order electron-phonon coupling matrix elements in diamond, lithium fluoride, and graphite as representative nonpolar semiconductors, polar semiconductors, and metals, respectively. Furthermore, we generalize the ab initio polaron equations to second-order electron-phonon couplings, and we show that second-order couplings are essential to achieve quantitative accuracy in polaron formation energies and hopping barriers. The present methodology will find application in the study of all properties and phenomena that are currently being investigated within the linear electron-phonon coupling approximation, from phonon-mediated superconductivity to excited-states dynamics, both in harmonic and anharmonic systems.
Linearizing the appropriate kinetic equation we derive general response functions including self-consistent mean fields or density functionals and collisional dissipative contributions. The latter ones are considered in relaxation time approximation conserving successively different balance equations. The effect of collisions is represented by correlation functions which are possible to calculate with the help of the finite temperature Lindhard random-phase approximation expression. The presented results are applicable to the finite temperature response of interacting quantum systems if the quasiparticle or mean-field energy is parametrized within Skyrme-type functionals including density, current, and energy dependencies which can be considered alternatively as density functionals. In this way we allow to share correlations between density functional and collisional dissipative contributions appropriate for special treatment. We present results for collective modes such as the plasmon in plasma systems and the giant resonance in nuclei. The collisions lead in general to an enhanced damping of collective modes. If the collision frequency is close to the frequency of the collective mode, resonance occurs and the collective mode is enhanced showing a collisional narrowing.
We study the application of the density-matrix renormalization group to systems with one-dimensional acoustic phonons. We show how the use of a local oscillator basis circumvents the difficulties with the long-range interactions generated in real space using the normal phonon basis. When applied to a harmonic atomic chain, we find excellent agreement with the exact solution even when using a modest number of oscillator and block states (a few times ten). We discuss the use of this algorithm in more complex cases and point out its value when other techniques are deficient.
We study the symmetric Anderson-Holstein (AH) model at zero temperature with Wilson's numerical renormalization-group (NRG) technique to investigate the interplay between the electron-electron and electron-phonon interactions. An improved method for calculating the phonon propagator using the NRG technique is presented. It is more accurate and reliable than the previous methods in that it is written in the form of the Dyson equation which calculates the phonon renormalization explicitly, and satisfies the boson sum rule better. The method is applied to calculate the renormalized phonon propagators along with the electron propagators as the on-site Coulomb repulsion U and electron-phonon coupling constant g are varied. As g is increased from 0, the phonon mode gets softened (hardened) for $U\ensuremath{\gtrsim}(\ensuremath{\lesssim}{)U}_{s}(\ensuremath{\Delta})$ depending on the parameters of the AH model, U, and the hybridization $\ensuremath{\Delta}.$ And, for $g\ensuremath{\gtrsim}{g}_{\mathrm{co}},$ one crosses over to the regime where the phonon mode splits into two components, one of which developing into a soft mode and the other approaching back to the bare frequency as g is further increased. Correlated with the emergence of the soft mode, the central peak of the electron spectral function is severely suppressed. These NRG calculations are compared with the standard Green's-function results for the weak-coupling regime to understand the phonon renormalization and soft mode.