Polarons in Advanced Materials pp 393-461 | Cite as

# Numerical Solution of the Holstein Polaron Problem

Noninteracting itinerant electrons in a solid occupy Bloch one-electron states. Phonons are collective vibrational excitations of the crystal lattice. The basic electron-phonon (EP) interaction process is the absorption or emission of a phonon by the electron with a simultaneous change of the electron state. From this it is clear that the motion of even a single electron in a deformable lattice constitutes a complex many-body problem, in that phonons are excited at various positions, with highly non-trivial dynamical correlations.

The mutual interaction between the charge carrier and the lattice deformations may lead to the formation of a new quasiparticle, an electron dressed by a phonon cloud. This composite entity is called a polaron [1, 2]. Since the induced distortion (polarisation) of the lattice will follow the electron when it is moving through the crystal, one of the most important ground-state properties of the polaron is an increased inertial mass. A polaronic quasiparticle is referred to as a ‘large polaron’ if the spatial extent of the phonon cloud is large compared to the lattice parameter. By contrast, if the lattice deformation is basically confined to a single site, the polaron is designated as ‘small’. Of course, depending on the strength, range and retardation of the electronphonon interaction, the spectral properties of a polaron will also notably differ from those of a normal band carrier. Since there is only one electron in the problem, these findings are independent of the statistics of the particle, i.e. we can think of any fermion or boson, such as an electron-, hole-, exciton- or Jahn-Teller polarons (for details see [3–5]). The paper is organised as follows: In the remaining introductory part, Sect. 2 presents the Holstein model and outlines the numerical methods we will employ for its solution. The second, main part of this paper reviews our numerical results for the ground-state and spectral properties of the Holstein polaron. The polaron's effective mass and band structure, as well as static electron-lattice correlations, will be analysed in Sect. 3. Section 4 is devoted to the investigation of the excited states of the Holstein model. The dynamics of polaron formation is studied in Sect. 5. Characteristic results for electron and phonon spectral functions will be presented in Sect. 6. The optical response is examined in Sect. 7. Here also finite-temperature properties such as activated transport will be discussed. In the third part of this paper finite-density and correlation effects will be addressed. First we investigate the possibility of bipolaron formation and discuss the many-polaron problem (Sect. 8). Second we comment on the interplay of strong electronic correlations and EP interaction in advanced materials (Sect. 9). Some open problems are listed in the concluding Sect. 10.

### Keywords

Anisotropy Manganese Uranium Coherence Perovskite## Preview

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