[Paper Review] Time-periodic corner states from Floquet higher-order topology

Weiwei Zhu, Haoran Xue+3 more|arXiv (Cornell University)|Dec 16, 2020

Topological Materials and Phenomena4 citations

TL;DR

This paper proposes and experimentally demonstrates a two-dimensional Floquet higher-order topological insulator (HOTI) in a 3D acoustic lattice, where time-periodic modulation along the z-axis induces topologically protected corner states with time-periodic dynamics. The key result is the observation of corner states exhibiting period-doubled oscillations—longer than the driving period—alongside coexisting chiral edge states, enabled by topological protection of both zero and π modes.

ABSTRACT

The recent discoveries of higher-order topological insulators (HOTIs) have shifted the paradigm of topological materials, which was previously limited to topological states at boundaries of materials, to those at boundaries of boundaries, such as corners . So far, all HOTI realisations have assumed static equilibrium described by time-invariant Hamiltonians, without considering time-variant or nonequilibrium properties. On the other hand, there is growing interest in nonequilibrium systems in which time-periodic driving, known as Floquet engineering, can induce unconventional phenomena including Floquet topological phases and time crystals. Recent theories have attemped to combine Floquet engineering and HOTIs, but there has thus far been no experimental realisation. Here we report on the experimental demonstration of a two-dimensional (2D) Floquet HOTI in a three-dimensional (3D) acoustic lattice, with modulation along z axis serving as an effective time-dependent drive. Direct acoustic measurements reveal Floquet corner states that have time-periodic evolution, whose period can be even longer than the underlying drive, a feature previously predicted for time crystals. The Floquet corner states can exist alongside chiral edge states under topological protection, unlike previous static HOTIs. These results demonstrate the unique space-time dynamic features of Floquet higher-order topology.

Motivation & Objective

• To extend higher-order topology to time-periodic (Floquet) systems, moving beyond static equilibrium Hamiltonians.
• To demonstrate that Floquet engineering can generate topologically protected corner states with dynamic, time-periodic behavior.
• To explore the coexistence of corner states and chiral edge states in a single topological framework, a feature absent in static HOTIs.
• To realize and measure period-doubled dynamics in corner states, a hallmark of discrete time crystals.
• To establish a platform for studying topological phenomena with tunable, independent control over zero and π bandgaps.

Proposed method

• Design a 2D tight-binding model with four-step periodic driving protocol, where coupling strengths γ and θ are modulated over time to realize time-periodic Hamiltonians.
• Use a 3D acoustic lattice with waveguides along the z-axis to emulate time evolution, where z acts as the effective time dimension.
• Engineer coupling strengths γ = 0.841π and θ = 0.841π for period-doubling corner states, and γ = 0.705π, θ = 0.283π for coexistence of corner and edge states.
• Perform direct acoustic measurements of intensity distributions at different z-evolution distances to probe state localization and dynamics.
• Simulate quasienergy spectra and eigenmode profiles to identify topological bandgaps and confirm the presence of zero and π modes.
• Apply the evolution operator U_L over one period to analyze time dynamics, showing that superpositions of |0⟩ and |π⟩ states return to initial state only after two periods.

Experimental results

Research questions

• RQ1 Can higher-order topology be extended to time-periodic systems via Floquet engineering?
• RQ2 Do Floquet HOTIs support corner states with time-periodic dynamics distinct from the driving period?
• RQ3 Can both corner states and chiral edge states coexist in a single topological system with topological protection?
• RQ4 What is the origin of period-doubled oscillations in corner states, and how is it related to the coexistence of zero and π modes?
• RQ5 Can the topological properties of zero and π bandgaps be independently tuned to realize distinct topological phases?

Key findings

• Experimental observation of time-periodic corner states in a 3D acoustic lattice with a period longer than the driving period, confirming period-doubling dynamics.
• The corner state oscillation period was found to be twice the driving period, arising from the superposition of zero-mode (|0⟩) and π-mode (|π⟩) states.
• Coexistence of topologically protected corner states and chiral edge states was directly observed when γ = 0.705π and θ = 0.283π, with distinct eigenmode profiles confirmed by simulation.
• Acoustic intensity measurements showed corner-localized excitation after 3.5 driving periods, confirming robust localization of corner states.
• Chiral edge states were observed to propagate unidirectionally along the edge for 3.5 periods, moving approximately two lattice constants upward.
• The quasienergy spectrum revealed that the zero bandgap hosts higher-order topological corner modes, while the π bandgap hosts first-order topological chiral edge states, enabling independent control of topological phases.