Optical absorption and activated transport in polaronic systems

We present exact results for the optical response in the one-dimensional Holstein model. In particular, by means of a refined kernel polynomial method, we calculate the ac and dc electrical conductivities at finite temperatures for a wide parameter range of electron-phonon interaction. We analyze the deviations from the results of standard small polaron theory in the intermediate coupling regime and discuss nonadiabaticity effects in detail.

Figure 1

(Color online) Renormalized current operator density $\tilde{j}(x,y)$ used in the 2D KPM. Data obtained for the 1D Holstein model with $\lambda =0.2$ (left panel) and $\lambda =2.0$ (right panel) at ${\omega }_0∕t=0.4$ ($L=6$, $M=50$; PBC).

Figure 2

(Color online) Schematic setup for the calculation of the finite-temperature optical conductivity (left panel). Lowest eigenvalues of the Holstein Hamiltonian for $L=6$, $M=25$, and PBC (right panel). The shaded area marks the six lowest eigenvalues to be separated from the rest of the spectrum.

Figure 3

(Color online) Optical conductivity in the 1D Holstein model at $T=0$ (in units of $\pi e^2{t}^2$) compared to the analytical small polaron result Eq. (8) (dashed blue lines). Exact diagonalization data (ED) are obtained for a system with $L=6$ and $M=45$; ${\sigma }_0$ is determined to give the same integrated spectral weight of the $\omega >0$ (regular part) of $\mathrm{Re}\sigma$.

Figure 4

(Color online) Optical absorption by Holstein polarons at finite temperatures in the adiabatic regime ($L=6$, $M=45$). Thin lines with symbols give the analytical results for the small polaron transport (Refs. 18, 26) at temperatures $\beta t=2$ (triangles), 0.5 (squares), 0.2 (circles). The deviations observed for high excitation energies at very large temperatures are caused by the necessary truncation of the phonon Hilbert space in ED.

Figure 5

(Color online) Optical absorption by Holstein polarons at finite temperatures in the nonadiabatic regime ($L=6$, $M=30$). Note that now the abscissa is scaled with respect to the phonon frequency.

Figure 6

(Color online) Optical absorption in the adiabatic intermediate EP coupling regime (the notation is the same as in Fig. 4; again we use $L=6$, $M=40$). The inset illustrates the finite-size dependence of $\mathrm{Re}\sigma \left(\omega \right)$ for $T\sim 0$ (the vertical dotted line gives the phonon absorption threshold). It demonstrates that the gap observed at low frequencies and temperatures is clearly a finite-size effect, i.e., at weak to intermediate couplings, the discrete electronic levels of our finite system show up in the conductivity spectra. These effects, of course, are of minor importance at larger EP couplings, where the polaronic bandwidth is strongly reduced, as well as for high temperatures.

Figure 7

(Color online) Conformance of the $f$-sum rule (9) in the weak (upper panel) and strong (lower panel) EP coupling regimes of the 1D Holstein model. Results for $S_{\mathrm{tot}}$, partial $S_{\mathrm{tot}}$’s and $E_{\mathrm{kin}}$ are given by solid, dot-dashed, and dashed lines, respectively. For further explanation see text.

Figure 8

(Color online) Top row: DC conductivity as a function of temperature at weak (left panel) and strong (right panel) EP coupling. The bottom panel shows, for strong coupling, a comparison with recent DMFT results (Ref. 7) and standard polaron theory (Refs. 18, 26) (for $T\gtrsim {\omega }_0∕2$). ED data do not account for the low-energy tail because of the finiteness of the system.