Topological energy conversion through the bulk or the boundary of driven systems

Combining physical and synthetic dimensions allows a controllable realization and manipulation of high-dimensional topological states. In our work, we introduce two quasiperiodically driven one-dimensional systems which enable tunable topological energy conversion between different driving sources. Using three drives, we realize a four-dimensional quantum Hall state which allows energy conversion between two of the drives within the bulk of the one-dimensional system. With only two drives, we achieve energy conversion between the two at the edge of the chain. Both effects are a manifestation of the effective axion electrodynamics in a three-dimensional time-reversal-invariant topological insulator. Furthermore, we explore the effects of disorder and commensurability of the driving frequencies, and show the phenomena are robust. We propose two experimental platforms, based on semiconductor heterostructures and ultracold atoms in optical lattices, in order to observe the topological energy conversion.

Figure 1

(a) Schematic representation of a quasiperiodically driven 1D system along $x$ subjected to multidrives. The phase of each can be spatially dependent, denoted as the arrow at each site. When subjected to three drives, the system can exhibit an energetic version of 4D QH effect with bulk energy conversion between two drives, for example, with frequencies ${\omega }_3$ and ${\omega }_4$. When subjected to two drives (frequencies ${\omega }_2,{\omega }_3$), the system can be mapped to a 3D topological insulator, which realizes the surface quantum Hall effect on its surfaces. This corresponds to boundary energy conversion between the two drives. (b) Bulk energy conversion and 4D quantum Hall effect. The chain is driven with three frequencies ${\omega }_2$, ${\omega }_3$, and ${\omega }_4$. The drive with ${\omega }_2$ on each site in addition has a phase $Bx$ linearly in the site coordinate, which creates an effectively magnetic field. The number of energy quanta $n_3,n_4$ in the other two drives behaves as the coordinate in synthetic dimensions. In the bulk of the system, the energy quanta with frequency ${\omega }_3$ will be converted into the energy quanta with frequency ${\omega }_4$. (c) Boundary energy conversion and surface quantum Hall effect. The chain is driven with two frequencies ${\omega }_2$ and ${\omega }_3$, whose energy quanta number behaves as the synthetic dimensions. At the left and right ends of the chain, the energy quanta with frequencies ${\omega }_2$ and ${\omega }_3$ will be converted into each other, in opposite directions.

Figure 2

Numerical simulation of the work gained by different drives (denoted by solid lines) as a function of time at different parameter regimes. The energy conversion rate predicted by Eq. (21) is indicated by the slope of black dashed lines, whose intercept is arbitrary. The common parameters for all cases are ${\omega }_2=0.02$, ${\omega }_3=\sqrt{3}{\omega }_2$, ${\omega }_4=(\sqrt{5}+1){\omega }_2$, and $m=1$. The parameters for each figure are (a) $N=30$, $t=\lambda =V_b=1/2$, $B=1$. (b) $N=30$, $J=\lambda =V_b=1/6$, $B=1$. (c) $N=30$, $J=\lambda =V_b=1/2$, $B=0.5$. (d) $N=30$, $J=\lambda =V_b=1/9$, $B=1$. (e) $N=20$, $J=\lambda =V_b=1/2$, $B=1$. (f) $N=60$, $J=\lambda =V_b=1/2$, $B=1$. The parameters are chosen to fulfill the adiabatic condition (23). The system is assumed to be half-filled.

Figure 3

Numerical simulation of the work gained by drive 2 at different regions of a 1D chain with $N=40$ sites, as a function of time at different parameter regimes. The energy conversion rate predicted by Eq. (40) is indicated by the slope of black dashed lines, whose intercept is arbitrary. The two driving frequencies are ${\omega }_2=0.1$, ${\omega }_3=(\sqrt{5}+1){\omega }_2/2$. We choose $J=\lambda =1$, and the boundary contains $M=5$ sites. The rest of the parameters for each figure are (a) $m=4$, $V_b=1$. (b) $m=4$, $V_b=-1$. (c) $m=0$, $V_b=1$. (d) $m=7$, $V_b=1$. The parameters are chosen to fulfill the adiabatic condition (42). The system is assumed to be half-filled.

Figure 4

(a) Experimental setup for the quasiperiodically driven 1D system in GaAs/AlGaAs quantum wells. The 1D system is composed of two coupled quantum wires, created by confining electrons of the 2DEG by gate voltages, which are assumed to be time dependent. In addition, in-plane oscillating magnetic fields $B_x,B_y$ along and perpendicular to the wires are applied. (b) Tight-binding description for the coupled wires. The couplings within and between the wires are denoted as red/blue and black lines, which are used to engineer various Pauli matrices.

Figure 5

Numerical simulation of the work gained by different drives (denoted by solid lines) as a function of time with different disorder strengths in the system with bulk energy conversion. The energy conversion rate predicted by Eq. (21) is indicated by the slope of black dashed lines, whose intercept is arbitrary. The common parameters for all cases are ${\omega }_2=0.02$, ${\omega }_3=\sqrt{3}{\omega }_2$, ${\omega }_4=(\sqrt{5}+1){\omega }_2$, $N=30$, $m=1$, $t=\lambda =V_b=1/2$, and $B=0.5$. The disorder strengths are (a) ${\eta }_1={\eta }_2=0.02$. (b) ${\eta }_1={\eta }_2=0.1$.

Figure 6

Numerical simulation of the work in $W_2^L$ as a function of time with different disorder strengths in the second setup (with boundary energy conversion). The energy conversion rate predicted by Eq. (40) is indicated by the slope of black dashed lines, whose intercept is arbitrary. The two driving frequencies are ${\omega }_2=0.1$, ${\omega }_3=(\sqrt{5}+1){\omega }_2/2$, $N=40$, $m=4$, $t=\lambda =V_b=1$. We consider cases with boundary region including $M=10$ or 5 sites, denoted by red and blue lines. The disorder strengths are (a) ${\eta }_1={\eta }_2=0$, (b) ${\eta }_1={\eta }_2=0.02$, (c) ${\eta }_1={\eta }_2=0.05$, (d) ${\eta }_1={\eta }_2=0.1$.

Figure 7

Numerical simulation of the work gained by drives 3 and 4 (denoted by solid lines) as a function of time at different parameter regimes. The energy conversion rate predicted by Eq. (21) is indicated by the slope of black dashed lines, whose intercept is arbitrary. The common parameters for all cases are ${\omega }_2=0.01$, ${\omega }_3=1.5{\omega }_2$, ${\omega }_4=3{\omega }_2$, and $m=1$. The parameters for each figure are (a) $N=30$, $t=\lambda =V_b=1/2$, $B=1$. (b) $N=30$, $t=\lambda =V_b=1/9$, $B=1$.

Figure 8

Numerical simulation of the work gained by drive 2 on the left and right of a 1D chain with $N=40$ sites, as a function of time at different parameter regimes. The energy conversion rate predicted by Eq. (40) is indicated by the slope of black dashed lines, whose intercept is arbitrary. The two driving frequencies are ${\omega }_2=0.1$, ${\omega }_3=1.5{\omega }_2$. We choose $t=\lambda =1$, and the boundary contains $M=5$ sites. The rest of the parameters for each figure are (a) $m=4$, $V_b=1$. (b) $m=7$, $V_b=1$.

Figure 9

Numerical simulation of the work gained by drives 3 and 4 (denoted by solid lines) as a function of time at different parameter regimes after disorder average over 20 realizations. The energy conversion rate predicted by Eq. (21) is indicated by the slope of black dashed lines, whose intercept is arbitrary. The common parameters for all cases are ${\omega }_2=0.01$, ${\omega }_3=1.5{\omega }_2$, ${\omega }_4=3{\omega }_2$, $m=1$, ${\tau }_d=5/{\omega }_2$, and ${\sigma }_d={10}^{-4}$. The parameters for each figure are (a) $N=30$, $t=\lambda =V_b=1/2$, $B=1$. (b) $N=30$, $t=\lambda =V_b=1/9$, $B=1$.

Figure 10

Numerical simulation of the work gained by drive 2 on the left and right of a 1D chain with $N=40$ sites after disorder average over 20 realizations. as a function of time at different parameter regimes. The energy conversion rate predicted by Eq. (40) is indicated by the slope of black dashed lines, whose intercept is arbitrary. The two driving frequencies are ${\omega }_2=0.1$, ${\omega }_3=1.5{\omega }_2$. We choose $t=\lambda =1$, and the boundary contains $M=5$ sites. The parameters for temporal disorder are ${\tau }_d=10/{\omega }_2$ and ${\sigma }_d=5\times {10}^{-5}$. The rest of the parameters for each figure are (a) $m=4$, $V_b=1$. (b) $m=7$, $V_b=1$.