Incoherent optical conductivity and breakdown of the generalized Drude formula in quasi-one-dimensional bad metallic systems

The high-energy (HE) intraband optical processes in a wide class of strongly correlated quasi-one-dimensional systems are explained in a way that is consistent with the Fermi golden rule, by using the force-force correlation function (FFCF) theory. It turns out that these processes must be distinguished from the related low-energy (LE) optical processes of the generalized Drude formula by making use of two different summations of Feynman diagrams, which include, respectively, the large-momentum and $\mathbf{q}\approx 0$ electron-hole excitations. Using the FFCF approach, the common expression for the single-particle optical conductivity of clean charge-density-wave (CDW) systems is rederived and the formation of the CDW gap in the optical conductivity of dirty CDW systems is briefly discussed. It is argued that the clear distinction between the HE and LE intraband optical processes is required whenever the electrical/optical conductivity of conduction electrons is predominantly incoherent, including the cases of bad metallic compounds and various heavy-electron systems.

Figure 1

(Color online) The solid lines represent the effective numbers ${\tilde{n}}_{\text{total},\alpha }^{\text{eff}}=\left(m_{zz}/m\right)V_0{n}_{\text{total},\alpha }^{\text{eff}}$, $\alpha =z,x$, as a function of the electron doping $\delta$ in the coherent conductivity regime, for the Q1D single-band model $\epsilon \left(\mathbf{k}\right)=-2t_{\parallel }\text{cos}\mathbf{k}\cdot \mathbf{c}-2t_{\perp }\text{cos}\mathbf{k}\cdot \mathbf{a}$, with $2t_{\parallel }=1\text{eV}$, $2t_{\perp }=0.1\text{eV}$, $a/c=2$, and $T=100\text{K}$. The dashed lines show the results of the completely incoherent conductivity regime.

Figure 2

(a) The auxiliary intraband current-current correlation function in the FFCF approach. (b) The leading term in the HE limit, which consists of two regular and two anomalous contributions. The full circle is the bare current vertex and the open square is the effective current vertex. The dashed line represents the related FFCF. (c) The Bethe-Salpeter equations stands for the large-momentum electron-hole propagators.

Figure 3

The Dyson equation for ${\mathcal{G}}_{cc}\left(\mathbf{k},i{\omega }_n\right)$ in the pure CDW case in which the commensurability effects are neglected, shown in the two-band representation [Eq. (B1) in Appendix ]. Similar for ${\mathcal{G}}_{\underset{̱}{c}\underset{̱}{c}}\left(\mathbf{k},i{\omega }_n\right)$, ${\mathcal{G}}_{\underset{̱}{c}c}\left(\mathbf{k},i{\omega }_n\right)$, and ${\mathcal{G}}_{c\underset{̱}{c}}\left(\mathbf{k},i{\omega }_n\right)$. The dashed lines represent the CDW potential.

Figure 4

(Color online) The optical conductivity of the common CDW semiconductors along the highly conducting direction $\mathbf{c}$ at temperature $T=126\text{K}$, for $2t_{\parallel }=1\text{eV}$, $2t_{\perp }=1\text{meV}$, and $\Delta \left(126\text{K}\right)=20\text{meV}$. $\hslash {\Gamma }_c^{\text{intra}}=\hslash {\Gamma }_c^{\text{inter}}=1$, 5 and 20 meV, for the dashed, dot-dashed and dot-dot-dashed line, respectively. The solid line is the optical conductivity at $T>T_{\text{MF}}$ and $\hslash {\Gamma }_c^{\text{intra}}=20\text{meV}$.

Figure 5

(Color online) The direct $\left(1\to 2\right)$ and the indirect ($1\to 3\to 4$ and $1\to 5\to 6$) HE optical excitations in the dirty CDW systems, shown in the extended zone representation. The parameters are the same as in Fig. 4. The solid and the dashed lines correspond to the poles in the electron Green’s functions associated with the residues ${\text{cos}}^2\left[\phi \left(\mathbf{k}\right)/2\right]$ and ${\text{sin}}^2\left[\phi \left(\mathbf{k}\right)/2\right]$, respectively [$\phi \left(\mathbf{k}\right)$ is defined by Eq. (B3)].