Temperature Dependence of Charge and Exciton Transport in One-Dimensional Systems Subject to Static and Dynamic Disorder

Abstract

The temperature-dependence of dynamical properties (e.g., the asymptotic diffusion coefficient and the sub-diffusive exponent) are calculated for charges and excitons in one-dimensional systems subject to static and dynamic disorder. These properties are determined by three complementary methods. One approach is via the time-integration of the velocity autocorrelation function. The second is via the mean-squared-displacement of thermal wavepackets subject to stochastic collapse via Lindblad jump operators. These two methods are applicable in the high-temperature regime, where the noise is temporally uncorrelated. In this regime the noise causes particle localization and the transport is diffusive. The third approach -- applicable in the low-temperature regime -- is weak-coupling Redfield theory. Here, static disorder causes particle localization. When the dynamics is diffusive, the diffusion coefficient is a non-monotonic function of temperature, increasing with temperature in the low-temperature Environment Assisted Quantum Transport regime and decreasing with temperature in the high-temperature quantum-Zeno regime. For any temperature, static and dynamic disorder decreases the diffusion coefficient. The dynamics is non-diffusive for thermal energies deep within the manifold of local-ground-states, where the sub-diffusive exponent decreases with increasing disorder and decreasing temperature.

Figures (10)

• Figure 1: Density of states, $g(E)$ (where $\int g(E) \textrm{d}E = 1$), for LGS and for all states near to the band edge (at $E = -2J$ when $\sigma =0$). The width of the LGS density of states is $W_{\textrm{LGS}} \sim J(\sigma/J)^{4/3}$. For these results the static on-site energy disorder is $\sigma/J = 0.2$.
• Figure 2: The ensemble-averaged maximum projection of $|\Psi_{\beta}\rangle$ onto a LGS, $|\psi_{LGS}\rangle$, as a function of temperature ($k_B = 1$). $\sigma/J = 0.1$, chains of 200 sites and 100 realizations of the disorder. The inset shows the ensemble-averaged deviation of $\langle E_{\Psi} \rangle$ (Eq. \ref{['Eq:28']}) from the Boltzmann average, $\langle E_{\textrm{B}} \rangle = \sum_a P_a^{\textrm{B}} E_a$.
• Figure 3: Ensemble-averaged thermal energies, $\langle E_{\textrm{B}} \rangle$, versus temperature ($k_B = 1$) for various disorder strengths, $\sigma' = \sigma/J$. Linear chains of 500 sites and 500 realizations of the disorder.
• Figure 4: The diffusion coefficient as a function of temperature for different values of the disorder, $\sigma' = \sigma/J$. The dephasing factor, $\Gamma_0 = 1$ (where $\Gamma(T) = \Gamma_0\times T$). ($k_B = \hbar = d = 1$.) Solid curves and filled symbols: results using the velocity autocorrelation function (i.e., Eq. (\ref{['Eq:15']})) with chains of 400 sites and 200 realizations of the disorder. Dashed curves and open symbols: results from quantum trajectories and the ETH with chains of 400 sites and $10^5$ quantum jumps. The dotted curve with star symbols is the diffusion coefficient computed via the Redfield equation, as described in Sections \ref{['Se:2.3']} and \ref{['Se:3.2']}.
• Figure 5: Solid curves and filled symbols: the diffusion coefficient (via Eq. (\ref{['Eq:15']})) as a function of temperature for different values of the dephasing factor $\Gamma_0$ (where $\Gamma(T) = \Gamma_0\times T$) when Boltzmann populations are enforced at finite temperatures, i.e., $P_a =P_a^{\textrm{B}}(T)$. The onsite disorder $\sigma/J = 0.2$. ($k_B = \hbar = d = 1$.) Dashed curves and open symbols: the diffusion coefficient determined via Eq. (\ref{['Eq:15']}) in the 'high-T' limit when the eigenstates are equally populated, i.e., $P_a= 1/N$. (Note that since $D_{\infty}$ is multiplied by $\Gamma_0$ the low temperature results differ by a factor of $\Gamma_0^2$, whereas the high temperature results converge to the same value.)