Efficient Time-Domain Approach for Linear Response Functions

We derive the general Kubo formula in a form that solely utilizes the time evolution of displacement operators. The derivation is based on the decomposition of the linear response function into its time-symmetric and time-antisymmetric parts. We relate this form to the well-known fluctuation-dissipation formula and discuss theoretical and numerical aspects of it. The approach is illustrated with an analytical example for magnetic resonance as well as a numerical example where we analyze the electrical conductivity tensor and the Chern insulating state of the disordered Haldane model. We introduce a highly efficient time-domain approach that describes the quantum dynamics of the resistivity of this model with an at least 1000-fold better performance in comparison to existing time-evolution schemes.

Figure 1

Geometric interpretation of the time evolution of the change of the response function $f_{AB}(t)-f_{AB}(0)$. Only the small displacement vectors after operator action with $\mathrm{\Delta }\widehat{A}(t)$ [and $\mathrm{\Delta }\widehat{B}(t)$ equivalently] need to be calculated instead of the large vectors related to the operators $\widehat{A}(t)$ or $\widehat{A}(0)$.

Figure 2

(a) Longitudinal and transversal resistivity components of the Haldane model for graphene calculated from the displacement form of the Kubo conductivity. The model parameters are set to ${\mathrm{\Delta }}_T=t_1$, ${\mathrm{\Delta }}_{AB}=0$ and $V=0.1t_1$. (b) Same as in (a) but with reduced topological gap ${\mathrm{\Delta }}_T=0.1t_1$. (c) and (d): Time-resolved Hall conductivity (c) and Hall resistivity (d) at the Dirac point ($E=0$) for both topological gap sizes.