Upper and lower bounds for the large polaron dispersion in 1, 2, or 3 dimensions

Abstract

Numerical results for the polaron dispersion are presented for an arbitrary number of space dimensions. Upper and lower bounds are calculated for the dispersion curves. They are rather close to each other in the cases of small electron-phonon couplings usual for real polar materials. To describe the dispersion in other materials, we suggest a simple fitting formula which can be applied at intermediate values of the Fröhlich electron-phonon coupling constant. Its validity is approved by the comparison with direct calculations and previously obtained results. This makes our results not only reliable and highly accurate but also easy reproducible.

Diagrammatic Monte Carlo study of Fröhlich polaron dispersion in two and three dimensions

“…To our knowledge, there do not exist any DMC results for the 2D Fröhlich polaron in the literature. We find that the present DMC results, both in 2D and 3D, agree with the analytically known limits, thus refuting the critique of the DMC method formulated in [19,20]. In addition, we compare the obtained dispersion relations with analytic upper and lower bounds (where available) and a fitting function [20].…”

“…Early work on the behavior of the dispersion curve [16,17] allowed one to conclude that the energy-momentum relation starts off quadratically at low k (thus allowing one to define a polaron mass) but bends over when approaching the continuum edge E c (α) = E 0 (0,α) +hω 0 . Later it was found that in 3D the dispersion hits the continuum edge whereas for 2D it approaches it asymptotically, and upper and lower bounds for the dispersion were obtained [18][19][20]. These bounds, as well as some analytically known limits, constitute good benchmarks for any theory of the polaron dispersion.…”

“…An important difference between the 3D and 2D case is that in 3D the dispersion curve crosses the continuum edge at a finite critical wave vector length k c (α). Instead, in 2D, it has been proven that this edge constitutes an asymptote and is approximated from below as k → ∞ [18][19][20]. For small α, there exist rigorous upper and lower bounds for the polaron dispersion [20] that restrict this dispersion to a narrow domain.…”

“…The DMC not only allows for the calculation of the ground state energies but as well the polaron dispersion curves, Z factors (quasiparticle weights), and phonon statistics. However, the DMC results [21,22] were criticized [19,20]: the reported results disagree with the analytically known second-order coefficient in α for the polaron ground state energy, as well as the large-α expansion coefficient.…”

Handbook of Materials Modeling

“…Both large and small polarons have been observed in several experiments, as reported in the next section, and studied by simulations and computational techniques. Historically, large polarons have been investigated mostly via effective Hamiltonians, in particular by means of variational treatments solved by Feynmann's path integral techniques, and by diagrammatic Monte Carlo [27][28][29][40][41][42][43] approaches. First principles techniques are more suitable for the description of the small polaron, but successful attempts to address the large-polaron case have been presented in the last few years [11,44,45].…”