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arXiv:1008.3247 · Physical Review B

Emergence of states in the phonon spectral function of the Holstein polaron below and above the one-phonon continuum

Lev Vidmar, J. Bonča, S. A. Trugman

Published Sep 27, 2010• 21 citations
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Abstract

We investigate the low-energy properties of the Holstein polaron through calculation of the $q$-dependent phonon spectral function using an improved exact-diagonalization technique, defined over a variational Hilbert space. We perform a comprehensive study of the low-energy excitations of the polaron. Beside the energy range, where the additional phonon excitation is unbound, we observe separate coherent peaks which correspond to bound and antibound states of a polaron and additional phonon quanta. These novel states can be observed for intermediate and strong electron-phonon coupling strengths, as well as below and above the unbound one-phonon excitation spectrum. A detailed investigation of their properties is presented. We find good agreement between the phonon spectral function obtained from the first-order strong-coupling perturbation theory and numerical results.

DOI
10.1103/physrevb.82.104304

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We investigate the low-energy properties of the Holstein polaron through calculation of the qqq-dependent phonon spectral function using an improved exact-diagonalization technique, defined over a variational Hilbert space. We perform a comprehensive study of the low-energy excitations of the polaron. Beside the energy range, where the additional phonon excitation is unbound, we observe separate coherent peaks which correspond to bound and antibound states of a polaron and additional phonon quanta. These novel states can be observed for intermediate and strong electron-phonon coupling strengths, as well as below and above the unbound one-phonon excitation spectrum. A detailed investigation of their properties is presented. We find good agreement between the phonon spectral function obtained from the first-order strong-coupling perturbation theory and numerical results.
We compute the Holstein polaron spectral function on a one-dimensional ring using the finite-temperature TTT Lanczos method. With increasing TTT additional features in the spectral function emerge even at temperatures below the phonon frequency. We observe a substantial spread of the spectral weight towards lower frequencies and the broadening of the quasiparticle (QP) peak. In the weak-coupling regime the QP peak merges with the continuum in the high-TTT limit. In the strong-coupling regime the main features of the low-TTT spectral function remain detectable up to the highest TTT used in our calculations. The effective polaron mass shows a nonmonotonic behavior as a function of TTT at small phonon frequency but increases with TTT at larger frequencies. The self-energy remains kkk independent even at elevated TTT in the frequency range corresponding to the polaron band, while at higher frequencies it develops a distinguishable kkk dependence. Analytical expressions for the first few frequency moments are derived, and they agree well with those extracted from numerical calculations in a wide-TTT regime.
The phonon spectral function of the one-dimensional Holstein model is obtained within weak and strong-coupling approximations based on analytical self-energy calculations.The characteristic excitations found in the limit of small charge-carrier density are related to the known (electronic) spectral properties of Holstein polarons such as the polaron band dispersion.Particular emphasis is laid on the different physics occurring in the adiabatic and antiadiabatic regimes, respectively.Comparison is made with a cluster approach exploiting exact numerical results on small systems to yield an approximation for the thermodynamic limit.This method, similar to cluster perturbation theory, confirms the analytical findings, and yields accurate results also in the intermediate-coupling regime.
The phonon spectral function of the one-dimensional Holstein model is obtained within weak and strong-coupling approximations based on analytical self-energy calculations. The characteristic excitations found in the limit of small charge-carrier density are related to the known (electronic) spectral properties of Holstein polarons such as the polaron band dispersion. Particular emphasis is laid on the different physics occurring in the adiabatic and anti-adiabatic regimes, respectively. Comparison is made with a cluster approach exploiting exact numerical results on small systems to yield an approximation for the thermodynamic limit. This method, similar to cluster perturbation theory, confirms the analytical findings, and yields accurate results also in the intermediate-coupling regime.
A recently proposed variational approach based on an expansion in coherent states is used to investigate the single polaron features in the one-dimensional Holstein model. The method provides highly accurate results, both for static and dynamic properties, over a wide range of the polaron parameter space, from weak to strong coupling regimes, with a little numerical effort when compared with standard numerical techniques. The results for the spectral weight function and the real part of the optical conductivity are reported in the moderate adiabatic regime. We show that the spectral function exhibits a strong dependence on the polaron wave number kkk that is not present in the standard analytical approaches. The optical conductivity, in the strong coupling regime, displays a high-energy peak due to the photoionization of the small polaron, while, at lower energies, there is evidence of electronic transitions between localized states. These results show that the coherent-state basis is a very powerful basis to study interacting electron-phonon systems.
An exact-diagonalization technique is used to investigate the low-lying excited polaron states in the Holstein model for the infinite one-dimensional lattice. For moderate values of the adiabatic ratio, a comprehensive picture, involving three excited (coherent) polaron bands below the phonon threshold, is obtained. The coherent contribution of the excited states to both the single-electron spectral density and the optical conductivity is evaluated and, due to the invariance of the Hamiltonian under the space inversion, the two are shown to contain complementary information about the single-electron system at zero temperature. The chosen method reveals the connection between the excited bands and the renormalized local phonon excitations of the adiabatic theory, as well as the regime of parameters for which the electron self-energy has notable nonlocal contributions. Finally, it is shown that the hybridization of two polaron states allows a simple description of the ground and first excited states in the crossover regime.
Using an efficient variational exact diagonalization method, we computed the electron removal spectral function within the framework of the Holstein-Hubbard model containing two electrons with opposite spins coupled to dispersive quantum optical phonons. We focus on how the interplay between the electron-phonon coupling and on-site Coulomb repulsion shapes the spectral function, a quantity directly relevant to angle-resolved photoemission spectroscopy (ARPES). Tuning the strengths of the electron-phonon coupling and the Hubbard interaction allows us to examine the evolution of the spectral properties of the system as it crosses over from a bound bipolaron to separate polarons. We find that phonon dispersion plays an important role: It substantially redistributes the spectral weight across the low- and the high-energy features and controls the emergence and the structure of distinct multiphonon continua. In regimes of strong electron-phonon coupling and low Coulomb repulsion, the spectral function exhibits a broad distribution reflecting bound bipolaronic states. Increasing the Coulomb repulsion weakens bipolaron binding, shifts the primary quasiparticle peak closer to polaronic behavior and confines the spectral weight near the Brillouin-zone center. Our results provide a framework for interpreting ARPES measurements in systems where phonon–mediated attraction competes with direct Coulomb repulsion.