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International Journal of Modern Physics B

STATICS AND DYNAMICS OF COUPLED ELECTRON–PHONON SYSTEMS

S. A. Trugman, J. Bonča, Li-Chung Ku

Published Aug 10, 2001• 17 citations
Publisher

Abstract

We review recent variational methods that provide extremely accurate solutions for the problem of one and two electrons coupled to dynamical, quantum phonons, including retardation effects. The dynamics of these systems far from equilibrium is explored, including the dynamics of polaron formation after electron injection by an STM. An examination of three-point functions suggests a qualitative modification of the traditional picture of a polaron. Long-range electron–phonon interactions are studied in some generality, including vector and frustrated Fröhlich models, and bounds are obtained on the possible mass enhancement.

DOI
10.1142/s0217979201006483

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We review recent variational methods that provide extremely accurate solutions for the problem of one and two electrons coupled to dynamical, quantum phonons, including retardation effects. The dynamics of these systems far from equilibrium is explored, including the dynamics of polaron formation after electron injection by an STM. An examination of three-point functions suggests a qualitative modification of the traditional picture of a polaron. Long-range electron–phonon interactions are studied in some generality, including vector and frustrated Fröhlich models, and bounds are obtained on the possible mass enhancement.
We explore the quasiparticle properties of lattice polarons on the basis of a quite general electron-phonon Hamiltonian with a long-range displacement-type interaction. To treat the dynamical quantum phonons without significant loss of accuracy we adapt an exact Lanczos diagonalization method and compute various static and dynamical quantities, such as the electron-lattice correlation function, the polaron band dispersion, the effective polaron mass, the kinetic energy, the single-particle spectral function, and the optical conductivity, on finite one-dimensional lattices for a wide range of model parameters. We compare the results with those obtained for the standard Holstein model with short-range electron-phonon interaction only.
Quantum entropies and state distances are analyzed in polaronic systems with short-range (Holstein model) and long-range (Fr\"ohlich model) electron-phonon coupling. These quantities are extracted by a variational wave function which describes very accurately polaron systems with arbitrary size in all the relevant parameter regimes. With the use of quantum information tools, the crossover region from weak to strong coupling regime can be characterized with high precision. Then, the linear entropy is found to be very sensitive to the range of the electron-phonon coupling and the adiabatic ratio. Finally, the entanglement entropy is studied as a function of the system size pointing out that it not bounded, but scales as the logarithm of the size either for weak electron-phonon coupling or for short-range interaction. This behavior is ascribed to the peculiar coupling induced by the single-electron itinerant dynamics on the phonon subsystem.
The polaron features due to electron-phonon interactions with different coupling ranges are investigated by adopting a variational approach. The ground-state energy, the spectral weight, the average kinetic energy, the mean number of phonons, and the electron-lattice correlation function are discussed for the system with coupling to local and nearest-neighbor lattice displacements comparing the results with the long-range case. For large values of the coupling with nearest-neighbor sites, most physical quantities show a strong resemblance with those obtained for the long-range electron-phonon interaction. Moreover, for intermediate values of interaction strength, the correlation function between electron and nearest-neighbor lattice displacements is characterized by an upturn as function of the electron-phonon coupling constant.
The formation of a polaron quasiparticle from a bare electron is studied in the framework of the Holstein model of electron-phonon coupling. Using Schrödinger's formalism, we calculate the time evolution of the distribution of the electron and phonon density, lattice deformation, and the electron-phonon (el-ph) correlation functions in real space. The quantum dynamical nature of the phonons is preserved. The polaron formation time is related to the dephasing time of the continuum of unbound phonon excited states in the spectral function, which depends on the phonon frequency and the el-ph coupling strength. As the el-ph coupling increases, qualitative changes in polaron formation occur when the one-phonon polaron bound state forms. In the adiabatic regime, we find that a potential barrier between the quasi-free and heavy polaron states exists in both 2D and 3D, which is crucial for polaron formation dynamics. We compare to recent experiments.
We calculate the real space and time formation of a polaron quasiparticle from a bare electron. The time-dependent Schr\"odinger equation for the Holstein model of electron-phonon coupling is numerically integrated in a large Hilbert space to obtain the time evolution of the electron and phonon densities and the electron-phonon (el-ph) correlation functions. The quantum dynamical nature of the phonons is preserved. As the el-ph coupling increases, qualitative changes in polaron formation occur when the one-phonon polaron bound state forms. A potential barrier between the quasifree and heavy polaron states exists in dimensions $D\ensuremath{\geqslant}2$, consistent with earlier adiabatic theory. We compare to recent experiments.