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arXiv:cond-mat/0202186 · Europhysics Letters (EPL)

Photoemission spectroscopy and sum rules in dilute electron-phonon systems

P. E. Kornilovitch

Published Sep 1, 2002• 26 citations
arXivHTMLPDF

Abstract

A family of exact sum rules for the one-polaron spectral function in the low-density limit is derived. An algorithm to calculate energy moments of arbitrary order of the spectral function is presented. Explicit expressions are given for the first two moments of a model with general electron-phonon interaction, and for the first four moments of the Holstein polaron. The sum rules are linked to experiments on momentum-resolved photoemission spectroscopy. The bare electronic dispersion and the electron-phonon coupling constant can be extracted from the first and second moments of the spectrum. The sum rules could serve as constraints in analytical and numerical studies of electron-phonon models.

DOI
10.1209/epl/i2002-00187-x

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A family of exact sum rules for the one-polaron spectral function in the low-density limit is derived. An algorithm to calculate energy moments of arbitrary order of the spectral function is presented. Explicit expressions are given for the first two moments of a model with general electron-phonon interaction, and for the first four moments of the Holstein polaron. The sum rules are linked to experiments on momentum-resolved photoemission spectroscopy. The bare electronic dispersion and the electron-phonon coupling constant can be extracted from the first and second moments of the spectrum. The sum rules could serve as constraints in analytical and numerical studies of electron-phonon models.
Abstract We derive a general procedure for evaluating the nth derivative of a time-dependent operator in the Heisenberg representation and employ this approach to calculate the zeroth to third spectral moment sum rules of the retarded electronic Green’s function and self-energy for a system described by the Holstein–Hubbard model allowing for arbitrary spatial and time variation of all parameters (including spatially homogeneous electric fields and parameter quenches). For a translationally invariant (but time-dependent) Hamiltonian, we also provide sum rules in momentum space. The sum rules can be applied to various different phenomena like time-resolved angle-resolved photoemission spectroscopy and benchmarking the accuracy of numerical many-body calculations. This work also corrects some errors found in earlier work on simpler models.
We derive frequency moment sum rules for the retarded electronic Green's function and self-energy of the Holstein model for both equilibrium and nonequilibrium cases. We also derive sum rules for the phonon propagator in equilibrium and nonequilibrium. These sum rules allow one to benchmark nonequilibrium calculations and help with interpreting the behavior of electrons driven out of equilibrium by an applied electric field. We exactly evaluate the sum rules when the system is in the atomic limit. We also discuss the application of these sum rules to pump/probe experiments like time-resolved angle-resolved photoemission spectroscopy.
We derive frequency moment sum rules for the retarded electronic Green's function and self-energy of the Holstein model for both equilibrium and nonequilibrium cases. We also derive sum rules for the phonon propagator in equilibrium and nonequilibrium. These sum rules allow one to benchmark nonequilibrium calculations and help with interpreting the behavior of electrons driven out of equilibrium by an applied electric field. We exactly evaluate the sum rules when the system is in the atomic limit. We also discuss the application of these sum rules to pump/probe experiments like time-resolved angle-resolved photoemission spectroscopy.
This thesis first reviews and examines the Angle Resolved Photoemission Spectroscopy (ARPES) experiment. It is shown that the spectral density function, familiar from the Green's function method of studying correlated systems, can be directly measured. A model spectral function with a nontrivial self-energy is then used to test an improvement to a recently arrived method[15,17] to analyze ARPES data. This new method relies on self-consistency between the real and imaginary parts of the self-energy (as measured through the Kramers-Kronig transform) to overcome the requirement of knowing the bare electronic structure. Through this, the method extracts both the complex self-energy and the bare electronic structure from the spectral function. The method described here is an improvement on this idea as previously implemented in that a strict form for the bare band dispersion (previously considered linear or quadratic) is never assumed. Although the method here utilized a polynomial of arbitrary degree, it could be trivially expanded to use any other functional form so long as both the value and first derivative are known analytically as a function of the fitting parameters. Using Mona Berciu's first order momentum average (MA) approximation[2] as implemented by Glen Goodvin[8] the spectral function as well as momentum independent self-energy were calculated for the Holstein polaron for a wide range of parameters. It was found that self-consistent spectral function analysis was highly successful at extracting the self energy and bare electronic dispersion from the spectral function over a consistent subset of these parameters. For studies outside this range of parameters the more traditional ARPES analysis method of measuring renormalization is also examined.
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.
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.