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Journal of Physics Condensed Matter

Quantum dynamics of the prototype polaron model

U. Herfort, M. Wagner

Published Mar 22, 2001• 11 citations

Abstract

The ab initio description of evolutionary processes in extended electron-phonon systems (polaronic transport, excitonic transfer, etc) up to the present is beyond numerical accessibility, since it requires the simultaneous knowledge of all eigenfunctions and eigenvalues. Therefore, usually rough approximations are made, such as a semiclassical treatment. However, as we have shown in a recent paper, the full quantum-mechanical treatment drastically deviates from the semiclassical approximation (even in a qualitative manner). In the concept discussed here unitary product transformations are introduced, the constituents of which account for the two antagonistic tendencies inherent in every coupled electron-phonon Hamiltonian. We apply our procedure to the concrete case of the dimer-oscillator model by choosing for each of the antagonistic tendencies respectively a one parameter unitary operator, such that full analytical diagonalization is reached in the opposing limits of the Hamiltonian constituents. In the intermediate regime the two parameters of the transformation are suitably optimized. In this manner the generation of the full spectrum of eigensolutions involves two analytically fixed parameters only. The evolutionary behaviour resulting from our procedure is contrasted with the exact numerical result as well as with the one from the semiclassical approach and also with a more simple (`displacive') unitary transformation frequently used in the literature. It is shown that our calculation approaches the exact result in a satisfactory manner in all intrinsic physical parameter regimes (coupling and transfer) and overcomes the drastic shortcomings of previous calculations.

DOI: 10.1088/0953-8984/13/14/306

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The ab initio description of evolutionary processes in extended electron-phonon systems (polaronic transport, excitonic transfer, etc) up to the present is beyond numerical accessibility, since it requires the simultaneous knowledge of all eigenfunctions and eigenvalues. Therefore, usually rough approximations are made, such as a semiclassical treatment. However, as we have shown in a recent paper, the full quantum-mechanical treatment drastically deviates from the semiclassical approximation (even in a qualitative manner). In the concept discussed here unitary product transformations are introduced, the constituents of which account for the two antagonistic tendencies inherent in every coupled electron-phonon Hamiltonian. We apply our procedure to the concrete case of the dimer-oscillator model by choosing for each of the antagonistic tendencies respectively a one parameter unitary operator, such that full analytical diagonalization is reached in the opposing limits of the Hamiltonian constituents. In the intermediate regime the two parameters of the transformation are suitably optimized. In this manner the generation of the full spectrum of eigensolutions involves two analytically fixed parameters only. The evolutionary behaviour resulting from our procedure is contrasted with the exact numerical result as well as with the one from the semiclassical approach and also with a more simple (`displacive') unitary transformation frequently used in the literature. It is shown that our calculation approaches the exact result in a satisfactory manner in all intrinsic physical parameter regimes (coupling and transfer) and overcomes the drastic shortcomings of previous calculations.

We describe the formation and properties of Holstein polarons in the entire parameter regime. Our presentation focuses on the polaron mass and radius, which we obtain with an improved numerical technique. It is based on the combination of variational exact diagonalization with an improved construction of phonon states, providing results even for the strong coupling adiabatic regime. In particular we can describe the formation of large and heavy adiabatic polarons. A comparison of the polaron mass for the one- and three-dimensional situation explains how the different properties in the static oscillator limit determine the behavior in the adiabatic regime. The transport properties of large and small polarons are characterized by the

f

-sum rule and the optical conductivity. Our calculations are approximation free and have negligible numerical error. This allows us to give a conclusive and impartial description of polaron formation. We finally discuss the implications of our results for situations beyond the Holstein model.

We use a one-dimensional tight binding model with an impurity site characterized by electron-vibration coupling, to describe electron transfer and localization at zero temperature, aiming to examine the process of polaron formation in this system. In particular we focus on comparing a semiclassical approach that describes nuclear motion in this many vibronic-states system on the Ehrenfest dynamics level to a numerically exact fully quantum calculation based on the Bonca-Trugman method [J. Bonča and S. A. Trugman, Phys. Rev. Lett. 75, 2566 (1995)]. In both approaches, thermal relaxation in the nuclear subspace is implemented in equivalent approximate ways: In the Ehrenfest calculation the uncoupled (to the electronic subsystem) motion of the classical (harmonic) oscillator is simply damped as would be implied by coupling to a Markovian zero temperature bath. In the quantum calculation, thermal relaxation is implemented by augmenting the Liouville equation for the oscillator density matrix with kinetic terms that account for the same relaxation. In both cases we calculate the probability to trap the electron by forming a polaron and the probability that it escapes to infinity. Comparing these calculations, we find that while both result in similar long time yields for these processes, the Ehrenfest-dynamics based calculation fails to account for the correct time scale for the polaron formation. This failure results, as usual, from the fact that at the early stage of polaron formation the classical nuclear dynamics takes place on an unphysical average potential surface that reflects the distributed electronic population in the system, while the quantum calculation accounts fully for correlations between the electronic and vibrational subsystems.

Recent development of theoretical method for dissipative quantum transport has achieved notable progresses in the weak or strong electron-phonon coupling regime. However, a generalized theory for dissipative quantum transport in a wide parameter space had not been established. In this work, a variational polaron theory for dissipative quantum transport in a wide range of electron-phonon coupling is developed. The optimal polaron transformation is determined by the optimization of the Feynman-Bogoliubov upper bound of free energy. The free energy minimization ends up with an optimal mean-field Hamiltonian and a minimal interaction Hamiltonian. Hence, second-order perturbation can be applied to the transformed system, resulting in an accurate and efficient method for the treatment of dissipative quantum transport with different electron-phonon coupling strength. Numerical benchmark calculation on a single site model coupled to one phonon mode is presented.

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.

Starting from the Hamiltonian for a dimer which includes all the electronic and electron-phonon terms consistent with a non-degenerate orbital, by a sequence of displacement and squeezing transformation we obtain an effective polaronic Hamiltonian. The renormalized electronic interactions differ from the results of semiclassical or perturbative treatments. The properties of the variationally determined ground state of two particles in the orbital are discussed for variable dimer length in the adiabatic limit