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arXiv:1211.1945 · Physical Review B

Perturbational study of the lifetime of a Holstein polaron in the presence of weak disorder

Hadi Ebrahimnejad, Mona Berciu

Published Nov 8, 2012• 5 citations
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Abstract

Using the momentum average (MA) approximation, we find an analytical expression for the disorder-averaged Green's function of a Holstein polaron in a three-dimensional simple cubic lattice with random on-site energies. The on-site disorder is assumed to be weak compared to the kinetic energy of the polaron, and is treated perturbationally. Within this scheme, the states at the bottom of the polaron band are found to have an infinite lifetime, signaling a failure of perturbation theory at these energies. The higher-energy polaron states have a finite lifetime. We study this lifetime and the disorder-induced energy shift of these eigenstates for various strengths of disorder and electron-phonon coupling. We compare our findings to the predictions of Fermi's golden rule and the average $T$-matrix method, and find a significant quantitative discrepancy at strong electron-phonon coupling, where the polaron lifetime is much shorter than Fermi's golden rule prediction. We attribute this to the renormalization of the on-site potential by the electron-phonon coupling.

DOI
10.1103/physrevb.86.205109

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Using the momentum average (MA) approximation, we find an analytical expression for the disorder-averaged Green's function of a Holstein polaron in a three-dimensional simple cubic lattice with random on-site energies. The on-site disorder is assumed to be weak compared to the kinetic energy of the polaron, and is treated perturbationally. Within this scheme, the states at the bottom of the polaron band are found to have an infinite lifetime, signaling a failure of perturbation theory at these energies. The higher-energy polaron states have a finite lifetime. We study this lifetime and the disorder-induced energy shift of these eigenstates for various strengths of disorder and electron-phonon coupling. We compare our findings to the predictions of Fermi's golden rule and the average TTT-matrix method, and find a significant quantitative discrepancy at strong electron-phonon coupling, where the polaron lifetime is much shorter than Fermi's golden rule prediction. We attribute this to the renormalization of the on-site potential by the electron-phonon coupling.
We study the bound states of a three-dimensional Holstein polaron near various kinds of single impurities, using the momentum-average approximation. We show that the electron-phonon coupling is responsible for a strong renormalization of the impurity potential, resulting in an effective potential with significant retardation effects, which describes essential physics ignored by ``instantaneous'' approximations. The accuracy of our approximation is gauged by comparison with results from diagrammatic Monte Carlo for the case of an impurity that modifies the on-site energy of the electron. We also discuss impurities that modify the local strength of the electron-phonon coupling as well as isotope substitutions that change both the electron-phonon coupling and the phonon frequency, and contrast and highlight the difference between these cases.
The ground state and finite-temperature properties of polarons are studied considering a two-site and a four-site Holstein model by exact diagonalization of the Hamiltonian. The kinetic energy, Drude weight, correlation functions involving charge and lattice deformations, and the specific heat have been evaluated as a function of electron–phonon (e–ph) coupling strength and temperature. The effects of site diagonal disorder on the above properties have been investigated. The disorder is found to suppress the kinetic energy and the Drude weight, and reduces the spatial extension of the polaron. Increasing temperature also reduces the kinetic energy, Drude weight and the polaron size when the e–ph interaction is weak or intermediate. For strong coupling the effect of temperature is small but opposite. For sufficiently strong coupling the kinetic energy arises mainly from the incoherent hopping processes owing to the motion of electrons within the polaron and is almost independent of the disorder strength. The specific heat shows a peak in the intermediate range of coupling, and the peak is suppressed in the presence of disorder. From the coherent and incoherent contributions to the kinetic energy, the temperature above which the incoherent part dominates is determined as a function of e–ph coupling strength.
In this thesis, we investigated the spectral properties of polaronic quasiparticles resulting from the coupling of a charge carrier to the bosonic excitations of ordered environments. Holstein polarons, describing an electron locally coupled to dispersionless phonons, and spin polarons in hole-doped antiferromagnets are considered. The Green's function of the Holstein polaron in a lattice with various kinds of impurities is calculated using the momentum average approximation. The main finding is that the scenario where the mere effect of the coupling to phonons is to enhance the electron's effective mass is incomplete as the phonons are found to also renormalize the impurity potential the polaron interacts with. This formalism is applied first to the case of a single impurity and the range of parameters where the polaron's ground state is localized is identified. The lifetime of the polaron due to scattering from weak but extended disorder is then studied and it is shown that the renormalization of the disorder potential leads to deviation of the strong coupling results from Fermi's golden rule's predictions. Motivated by the hole-doped cuprate superconductors, the motion of a hole in a two-dimensional Ising antiferromagnet and its binding to an attractive impurity is studied next, based on a variational scheme that allows for configurations with a certain maximum number of spin flips. The role of the magnetic sublattices in determining the symmetry of the resulting bound states is discussed. Next, a more realistic model describing a hole in a CuO₂ layer which retains the O explicitly, is considered. By neglecting the fluctuations of the Cu spins and using a variational principle similar to that of the previous chapter, a semi-analytic solution for the Green's function of the hole in an infinite 2D lattice is constructed. The resulting quasiparticle dispersion shows the proper ground state and other features observed in experiments. The lack of importance of the background spin fluctuations is justified based on the hole's ability to move on the O sublattice without disturbing the Cu spins. Finally, the model is generalized to gauge the importance of other relevant O orbitals.
Non-local, inhomogeneous and retarded response similar to that observed in experiments is studied theoretically by introducing the Inhomogeneous Momentum Average (IMA) approximation for single-polaron problems with disorder in the on-site potential and/or spatial variations of the electron-phonon couplings and/or phonon frequencies. We show that the electron-phonon coupling gives rise to an additional inhomogeneous, strongly retarded potential. This potential describes essential physics ignored by "instanteneous" approximations. The accuracy of IMA is demonstrated by comparison with single-impurity results from the approximation-free Diagrammatic Monte Carlo (DMC) method. Its simplicity allows for easy study of many problems that were previously unaccessible.
We describe a variational method to solve the Holstein model for an electron coupled to dynamical, quantum phonons on an infinite lattice. The variational space can be systematically expanded to achieve high accuracy with modest computational resources (12-digit accuracy for the one-dimensional polaron energy at intermediate coupling). We compute ground-state and low-lying excited-state properties of the model at continuous values of the wave vector k in essentially all parameter regimes. Our results for the polaron energy band, effective mass, and correlation functions compare favorably with those of other numerical techniques, including the density-matrix renormalization-group technique, the global-local method, and the exact diagonalization technique. We find a phase transition for the first excited state between a bound and unbound system of a polaron and an additional phonon excitation. The phase transition is also treated in strong-coupling perturbation theory.
Using the perturbation method based on a variational phonon basis obtained by the modified Lang-Firsov (MLF) transformation, the two-site double-exchange Holstein model is studied in presence of a finite difference in bare site energies (ǫd=ǫ2-ǫ1). The polaronic ground-state wave function is calculated up to the fifth order of perturbation. The effect of ǫd (acts as a site-energy disorder) on the polaron crossover, polaronic kinetic energy and polaron localization are studied. Role of disorder on the properties of the double-exchange system is discussed.