Energy Current Rectification and Mobility Edges

We investigate how the presence of a single-particle mobility edge in a system can generate strong energy current rectification. Specifically, we study a quadratic bosonic chain subject to a quasiperiodic potential and coupled at its boundaries to spin baths of differing temperature. We find that rectification increases by orders of magnitude depending on the spatial position in the chain of localized eigenstates above the mobility edge. The largest enhancements occur when the coupling of one bath to the system is dominated by a localized eigenstate, while the other bath couples to numerous delocalized eigenstates. By tuning the parameters of the quasiperiodic potential it is thus possible to vary the amplitude, and even invert the direction, of the rectification.

Figure 1

(a) Rectification $\mathcal{R}$, (b) forward bias current ${\mathcal{J}}_f$, and (c) reverse bias current ${\mathcal{J}}_r$ versus the deformation parameter $\alpha$. In (a) different lines correspond to different values of $\lambda$, from 0.1 to 0.9 with steps of 0.1, increasing in the direction of the arrow. (b)–(c) Forward and reverse bias currents for $\lambda =0.1$ (red ⋄), 0.4 (blue ∘), and 0.9 (green □). ${\mathcal{J}}_f$ is represented by full symbols while ${\mathcal{J}}_r$ by empty symbols. Other parameters are chain length $L=1000$, phase $\varphi =\pi$, and temperatures of the baths are $T_h=1000.1$ and $T_c=0.1$.

Figure 2

(a) Inverse participation ratio $⟨I⟩$ for forward (filled symbols) and reverse (empty symbols) bias versus deformation parameter $\alpha$ for $\lambda =0.4$ (blue empty circle), 0.7 (red empty square), and 0.9 (green empty triangle). (b) Rectification $\mathcal{R}$ versus $\alpha$ for $\lambda =0.4$ (blue circle), 0.7 (red square), and 0.9 (green triangle). In panels (a)–(b) we have used $T_h=1000+T_c$. (c) Density plot of fraction of localized states $f_{\mathrm{loc}}$ as a function of $\alpha$ and $\lambda$. The white lines represent the values of $\alpha$ and $\lambda$ at which the rectification varies significantly. More precisely we consider $T_h=10.1$ (dotted line), $T_h=100.1$ (dot-dashed line), and $T_h=1000.1$ (dashed line). Common parameters are $L=1000$, $\varphi =\pi$, and $T_c=0.1$. Black dot-dashed lines highlight the value of $\alpha$ at which the inverse participation ratio increases significantly.

Figure 3

(a) Rectification versus phase $\varphi$ in the log-lin scale to highlight the regions in which the rectification is in different directions. (b) Maximum coupling magnitude of any $k$ mode to the first site (blue empty cirlce) or to the last site (red empty diamond). (c) Average inverse participation ratio $⟨I⟩$ as a function of $\varphi$ for the forward (blue empty circle) and reverse (red empty circle) bias. Parameters are $\alpha =\lambda =0.9$, $T_h=1000.1$, $T_c=0.1$.

Figure 4

(a) Rectification $\mathcal{R}$ versus temperature difference $\mathrm{\Delta }T$. (b) Inverse participation ratio $⟨I⟩$ and (c) steady state currents for forward ${\mathcal{J}}_f$ (filled blue symbols) and reverse ${\mathcal{J}}_f$ (empty red symbols) biases, versus $\mathrm{\Delta }T$. Common parameters are $\alpha =\lambda =0.9$, $L=1000$, $\varphi =\pi$, and $T_c=1$.