Abstract
The recent discoveries of higher-order topological insulators (HOTIs) have shifted the paradigm of topological materials, previously limited to topological states at boundaries of materials, to include topological states at boundaries of boundaries, such as corners. So far, all HOTI realisations have been based on static systems described by time-invariant Hamiltonians, without considering the time-variant situation. There is growing interest in Floquet systems, in which time-periodic driving can induce unconventional phenomena such as Floquet topological phases and time crystals. Recent theories have attempted to combine Floquet engineering and HOTIs, but there has been no experimental realisation so far. Here we report on the experimental demonstration of a two-dimensional (2D) Floquet HOTI in a three-dimensional (3D) acoustic lattice, with modulation along a spatial axis serving as an effective time-dependent drive. Acoustic measurements reveal Floquet corner states with double the period of the underlying drive; these oscillations are robust, like time crystal modes, except that the robustness arises from topological protection. This shows that space-time dynamics can induce anomalous higher-order topological phases unique to Floquet systems.
Highlights
The recent discoveries of higher-order topological insulators (HOTIs) have shifted the paradigm of topological materials, previously limited to topological states at boundaries of materials, to include topological states at boundaries of boundaries, such as corners
A HOTI has been implemented in a coupled-resonator lattice[24], which can be described as a Floquet system using a scattering matrix framework[25,26]; the topological phase implemented in that work was a quadrupole insulator, which belongs to the class of static HOTIs
These oscillations are reminiscent of time crystals, but have different origins[27]: whereas, a time crystal requires quantum many-body interactions to stabilise the breaking of discrete time-translation symmetry[28,29], the period-doubled Floquet corner states are topologically protected by the space-time symmetries of the lattice
Summary
The recent discoveries of higher-order topological insulators (HOTIs) have shifted the paradigm of topological materials, previously limited to topological states at boundaries of materials, to include topological states at boundaries of boundaries, such as corners. Acoustic measurements reveal Floquet corner states with double the period of the underlying drive; these oscillations are robust, like time crystal modes, except that the robustness arises from topological protection This shows that space-time dynamics can induce anomalous higher-order topological phases unique to Floquet systems. Unlike static HOTIs, the corner states in the Floquet HOTI can oscillate in time, with oscillation period either equal to, or double of, the driving period This extends the concept of anomalous Floquet band topology, previously limited to first order, to higher order. The quasienergy bands have zero quadrupole moment in addition to zero polarisation; in a static system, such features would point to a topologically trivial phase This anomalous Floquet higher-order topology allows for the coexistence of two distinct Floquet topological states, satisfying the condition to observe period-doubled oscillation. We perform a series of experiments to probe the various unusual dynamical properties that arise from the system’s Floquet higher-order topology
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