Topological Charge Pumping in Excitonic Insulators

We show that in excitonic insulators with $s$-wave electron-hole pairing, an applied electric field (either pulsed or static) can induce a $p$-wave component to the order parameter, and further drive it to rotate in the $s+ip$ plane, realizing a Thouless charge pump. In one dimension, each cycle of rotation pumps exactly two electrons across the sample. Higher dimensional systems can be viewed as a stack of one-dimensional chains in momentum space in which each chain crossing the Fermi surface contributes a channel of charge pumping. Physics beyond the adiabatic limit, including in particular dissipative effects is discussed.

Figure 1

(a) The $s+ip$ plane for the excitonic order parameter, with electric field-driven evolution shown as a dashed line. (b) The quasiparticle dispersion in a one-dimensional EI (solid lines) along with bands in the metallic phase (dashed lines). (c) The band dispersion of a two-dimensional EI with an $s+ip$ order parameter and ${\mathrm{\Delta }}_s\ll {\mathrm{\Delta }}_p$.

Figure 2

(a) The surface $S$ in the $(k,{\mathrm{\Delta }}_s,{\mathrm{\Delta }}_p)$ space used to calculate the flux of the Berry curvature for a 1D EI for which the order parameter evolution completes a full cycle in the $s+ip$ plane. The left and right ends of the cylinder are identified so that $S$ is a 2-torus. Blue dots are Berry curvature monopoles and red dashed lines are “Dirac strings” with the direction shown by black arrows. (b) The surface of the torus shown in (a) parametrized by $k$ and $\theta$ with $k=\pm \pi$ and $\theta =0,2\pi$ identified. The blue contours yield the charge pumped during a full cycle. The red rectangles are used to compute the flux for a partial cycle in the BCS limit.

Figure 3

Left is the evolution of the edge state energies as a function of $\theta$. Right is the spatial profile of one of the two edge states (labeled by red line on the left) neglecting its quick oscillating detail. Being filled means the state is occupied. The blue edge state is not shown but is the mirror image of the red one.

Figure 4

Electric field pulse induced dynamics of a 2D isotropic EI. (a) The polarization as a function of time during the dynamics with $P_{\mathrm{dis}}$ being small. Inset is the free energy landscape $F({\mathrm{\Delta }}_s,{\mathrm{\Delta }}_p)$ plotted on the $s+ip$ plane. Lower energy appears bluer. The black dashed order parameter trajectory is caused by a pulse $E(t)=E_{\max }{\tanh }^{\prime }[(t-t_0)/w]$ with maximum electric field $E_{\max }=0.39E_0$. (b) The pumped charge by a single pulse as a function of $E_{\max }$. The units are $P_{2D}=2k_F/\pi$ and $E_0={\mathrm{\Delta }}^2/v_F$. Top inset is a schematic of a train of well separated pulses which can induce a “steady” current. Bottom inset is a schematic of the device with EI shown in blue and the contacts in gold. The parameters are $g_s\nu =0.3$, $g_p\nu =0.58$, $\mathrm{\Delta }=2\mathrm{\Lambda }{e}^{-1/(g_s\nu )}$, $\gamma =0.07\mathrm{\Delta }$, and $w=2\pi /(2\mathrm{\Delta })$.