Current Status of Polarons and Open Problems
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At present, the basic properties of single polarons are well understood theoretically, and to a large extent they are analytically under control at all coupling. It is remarkable how the Fröhlich continuum polaron, one of the simplest examples of a Quantum Field Theoretical problem, as it basically consists of a single fermion interacting with a scalar Bose field, has resisted full analytical solution at all coupling since ∼1950, when its Hamiltonian was first written. Although a mechanism for the optical absorption of Fröhlich polarons was already proposed a long time ago [73,98], some subtle characteristics were only clarified very recently  by combining numerical DQMC studies  and improved analytical methods [66,112] (Sect. 2.4.6). Of special interest are several sum rules derived for the optical conductivity spectra of arbitrary-coupling Fröhlich polarons [118, 119]. A variety of magneto-optical and transport experiments were successfully analyzed with Fröhlich polaron theory (see, e.g., [74, 83, 106,516], and references therein).
The charge carriers in a rich variety of systems of reduced dimension and dimensionality (submicron- and nanostructures including heterojunctions, quantum wells, quantum wires, quantum dots, etc.) turn out to be Fröhlich polarons. Several scaling relations were derived , which connect polaron characteristics (the self-energy, the effective mass, the impedance, and the mobility) in different dimensions.
The Fröhlich polaron has led to many generalizations. The stability region of the Fröhlich large bipolaron is now firmly established [296,301,302] (Sect. 4.3). Here, the surprise is double (cf. [302, 310]) (a) only in a very limited sector of the phase diagram (Coulomb repulsion vs. α), the bipolaron is stable and (b) most traditional Fröhlich polaron materials (alkali halides and the like) lie completely outside (and “far” from) this bipolaron stability sector, but several high-T c superconductors lie very close and even inside this very restricted area of the stability diagram. This should be a very hopeful sign for bipolaronic quasiparticles in the high-T c superconductors.