Floquet Engineering Topological Many-Body Localized Systems

We show how second-order Floquet engineering can be employed to realize systems in which many-body localization coexists with topological properties in a driven system. This allows one to implement and dynamically control a symmetry protected topologically ordered qubit even at high energies, overcoming the roadblock that the respective states cannot be prepared as ground states of nearest-neighbor Hamiltonians. Floquet engineering—the idea that a periodically driven nonequilibrium system can effectively emulate the physics of a different Hamiltonian—is exploited to approximate an effective three-body interaction among spins in one dimension, using time-dependent two-body interactions only. In the effective system, emulated topology and disorder coexist, which provides an intriguing insight into the interplay of many-body localization that defies our standard understanding of thermodynamics and into the topological phases of matter, which are of fundamental and technological importance. We demonstrate explicitly how combining Floquet engineering, topology, and many-body localization allows one to harvest the advantages (time-dependent control, topological protection, and reduction of heating, respectively) of each of these subfields while protecting them from their disadvantages (heating, static control parameters, and strong disorder).

Figure 1

We combine insights from the subfields of Floquet engineering, many-body localization, and topology. By doing so, we demonstrate how to harvest the full advantages promised by these fields—flexible control, suppression of heating, and topological protection—hile removing their respective disadvantages. These advantages are accessible with programmable quantum simulators. We explicitly illustrate this in Fig. 3 (topological protection), Fig. 4 (suppression of heating), and Fig. 5 (control), which are schematically given as insets here.

Figure 2

Representation of the nonvanishing part of the commutator $[H_1(t),H_1(t^{\prime })]$ relevant to the second-order Magnus expansion for a chain consisting of eight sites. The vertical lines represent the sites, while the light blue rectangles represent the terms acting on the sites. The red crosses mark terms that are commuting even though they are acting on the same site. The commutator of two two-body interactions effectively generates a three-body interaction.

Figure 3

Topological protection at high energies.—Time evolution of the edge spin $⟨L_z⟩$ governed by Eqs. (1) and (2) for different system sizes and a driving frequency of $\omega =1000$. The edge spin decays exponentially with $\sim \exp [-\sqrt{t/\tau }]$. As the system size increases, the lifetime $\tau \sim \exp [0.752L]$ of the spin becomes exponentially large in the system size (inset) due to the protection by topology and many-body localization even at high energies. The deviation from $⟨L_z⟩=1$ is due to the fact that, although this operator has a nonzero overlap to the operator describing the topologically protected gapless edge mode, they are not identical. We choose ${\varphi }_0=0$, disorder strengths ${\sigma }_{\alpha ,\beta ,\gamma }=1.0$, ${\sigma }_V=0.1$, ${\sigma }_h=0.05$, and average over 1000 random configurations. For $L=8$, we show data obtained using the effective time-independent Hamiltonian of Eq. (3) for comparison.

Figure 4

Suppression of heating.—Time evolution of the excess energy pumped into the system by the periodic drive. The energy saturates quickly, and the system does not approach an infinite-temperature state. This suppression of heating is due to many-body localization and detuning from single-particle resonances. We choose $L=10$, ${\varphi }_0=0$, ${\sigma }_{\alpha ,\beta ,\gamma }=1.0$, ${\sigma }_V=0.1$, ${\sigma }_h=0.05$, and average over 1000 random configurations. Note that the scaling of the linear time axis changes at $t=100$.

Figure 5

Dynamical control.—The edge spin can be flipped on a short timescale $T_1\le t\le T_2$, $T_2-T_1\sim 1$ by applying a constant magnetic field $B_x$ as well as a brief $\pi /2$ phase shift to the drive (lower panel). Before ($t<T_1$) and after ($t>T_2$), the spin is stable on exponentially long timescales (upper panel). We choose $L=10$, a driving frequency of $\omega =1000$, and disorder strengths ${\sigma }_{\alpha ,\beta ,\gamma }=1.0$, ${\sigma }_V=0.1$, and ${\sigma }_h=0.05$. Disorder averages are performed over 1265 random configurations.