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Physical review. B, Condensed matter

Coherent-state functional-integral approach to high-field transport in coupled electron-phonon systems

W. Pötz, J. Zhang

Published May 15, 1992• 23 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.45.11496

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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.
In this work, we lay down the ab initio many-body quantum theory of electrons and phonons in equilibrium as well as in steady-state or time-varying settings.Our focus is on the harmonic approximation, but the developed tools can readily incorporate anharmonic effects.We begin by showing the necessity of determining the ab initio Hamiltonian in a self-consistent manner to ensure the existence of an equilibrium state.We then identify the correct partitioning into a "noninteracting" and an "interacting" part to carry out diagrammatic expansions in terms of dressed propagators and screened interactions.The final outcome is the finite-temperature nonequilibrium extension of the Hedin equations, showcasing the emergence of the coupling between electrons and coherent phonons through the time-local Ehrenfest diagram.The Hedin equations have limited practical utility for real-time simulations of systems driven out of equilibrium by external fields.To overcome this limitation, we leverage the versatility of the diagrammatic approach to generate a closed system of differential equations for the dressed propagators and nuclear displacements.These are the Kadanoff-Baym equations for electrons and phonons.The formalism naturally merges with the theory of conserving approximations, which guarantee the satisfaction of the continuity equation and the conservation of total energy during time evolution.As an example, we show that the popular Born-Oppenheimer approximation is not conserving whereas its dynamical extension is conserving, provided that the electrons are treated in the Fan-Migdal approximation with a dynamically screened electronphonon coupling.We also derive the formal solution of the Kadanoff-Baym equations for nonequilibrium steady states, which is useful for studies in photovoltaics and optoelectronics.Interestingly, the expansion of the phononic Green's function around the quasiphonon energies points to a correlation-induced splitting of the phonon dispersion in materials with no time-reversal invariance.
A semiclassical method is presented for tunneling of self-trapped states in many-body systems with electrons and phonons. An overcomplete set of Slater determinants, lattice coordinates, and lattice momenta is used to represent a functional integral. Stationary phase equations are solved numerically without any constraint on the dynamics of electrons and phonons, i.e., without the use of the adiabatic approximation. To evaluate transition amplitudes, we integrate over small fluctuations in both electronic and phonon degrees of freedom, keeping their time order correctly. This method can be applied to general electron-phonon systems is and useful when self-trapped states have complex structures in charge or spin densities and lattice displacements. The effective hopping strength is calculated for a self-trapped kink in the commensurate charge-density-wave state in one dimension. At strong coupling in the Holstein and attractive Hubbard models, where tunneling involves effectively a single tightly bound bipolaron, this method reproduces previous analytic results. New results are obtained at intermediate coupling where the kink is extended over a couple of lattice sites and for models with both electron-electron and electron-phonon interactions.
The ground state of a one-dimensional electron-phonon system is studied by means of a small-crystal approach: an exact solution of a two-site cluster with two electrons and periodic boundary conditions. A single electron band is considered in the tight-binding approximation, together with two-electron interactions between electrons in the same site (Hubbard model); the electrons are coupled to longitudinal-acoustic phonons through a bilinear interaction. The problem is isomorphic with that of the homopolar diatomic molecule with a vibronic degree of freedom and coupling between the electronic and vibronic modes. In the adiabatic (large-ionic-mass, small-vibrational-frequency) limit there is an analytical solution which indicates that the transition between the nondistorted (weak-coupling) and the distorted (strong-coupling) phases can be either continuous or discontinuous, depending on whether the value of electron-electron interaction is smaller or larger than a critical value. In the extremely-high-frequency limit (ionic mass M\ensuremath{\rightarrow}0) it is also possible to find an analytical solution (for M=0) that shows that the system remains undistorted for any value of the electron-phonon coupling and the electron-electron interaction. In the intermediate case (finite, nonvanishing M), numerical solutions exhibit a continuous transition from the nondistorted to the distorted phase.
The ground state of a one-dimensional electron-phonon system is studied by means of a small-crystal approach: an exact solution of a two-site cluster with two electrans and periodic boundary conditions.A single electron band is considered in the tight-binding approximation, together with two-electron interactions between electrons in the same site (Hubbard model); the electrons are coupled to longitudinal acoustic phonons through a bilinear interaction.The problem is isomorphic with that of the homopolar diatomic molecule with a vibronic degree of freedom and coupling between the electronic and vibronic modes.In the adiabatic (large ionic mass, small vibrational frequency) limit there is an analytical solution which indicates that the transition between the non-distorted (weak coupling) and the distorted (strong coupling) phases can be either continuous or discontinuous, depending on whether the value of the electron-electron interaction is larger or smaller than a critical value.In the extremely high-frequency limit (ionic mass M ~ 0 ) it is also possible to find an analytical solution (for M = 0) that shows that the system remains undistorted for any value of the electron-phonon coupling and the electron-electron interaction.In the intermediate case (finite, nonvanishing M) numerical solutions exhibit a continuous transition from the non-distorted to the distorted phase.
A general iteration scheme for the many-mode degrees of freedom in locally coupled electron-phonon systems is presented. A new spectrum of phonon modes is derived and arranged into a linear chain where only the first model is vibronically coupled to the impurity, while the rest interact successively with the nearest neighbours in the chain. By dealing with a sequence of a finite number of modes along the chain an approximation scheme is developed by retaining the lower states at each step, in the spirit of quantum renormalisation-group calculations. Ground-state energies and densities of vibronic states are calculated for two-level systems for a valence-trapped mixed-valence molecular to examine the feasibility of the scheme in practice. Good convergence is achieved for both these properties. The value of the scheme for the calculation and interpretation of multi-mode effects in other electron-phonon systems is emphasised.
We develop a Fermi-Bose bootstrap embedding framework for the ground state of interacting electrons coupled to a phonon mean field. The method combines bootstrap embedding for correlated electrons with a self-consistent coherent-state mean-field treatment for phonons. This method models the interacting electron-phonon problem as a system of correlated electrons traveling in a self-consistently specified potential landscape, allowing for efficient treatment of large lattice systems. Convergence of the methods for fragment size and total system size is demonstrated for the one-dimensional Hubbard-Holstein model for up to 350 sites. Finite-size scaling is performed to extrapolate to the infinite system size. Benchmarking against the density matrix renormalization group for a small 8-site system at half- and quarter-filling shows an orders-of-magnitude runtime advantage. The comparison further reveals that the method performs best in regimes dominated by localization, such as the Mott insulating phase and the strong-coupling tiny polaron regime, where the local embedding ansatz is still valid. However, due to the mean-field treatment for phonons, we find limitations of our methods in the weakly coupled delocalized region and at the Peierls transition, where quantum phonon fluctuations and long-range kinetic correlations become substantial.