Driven Floquet systems can realize topological phases with no static counterparts. This so-called anomalous Floquet topology breaks the bulk-boundary correspondence based on the Chern number. The number of edge modes in each band gap is instead determined by another integer index, a winding number, which is calculated from the time evolution operator of the bulk states within one driving period. While topological Chern numbers can be inferred from real-space measurements, it is not straightforward to obtain information about the anomalous Floquet topological phase from bulk measurements. Here we consider two-band models and show that the area enclosed during a Floquet period by an initially localized particle signals the presence of an anomalous phase when it approaches half the unit cell area. In general, we show that for dispersionless dynamics during the micromotion, the enclosed area is quantized and an exact proportionality relation exists between the area and the winding number. Direct detection of anomalous topology in real space could be realized in several quantum simulation platforms, and could be useful for systems with disorder or interactions. Building on the connection between area and winding number, we also show a way to realize arbitrarily high winding numbers.
Driven Floquet systems can realize topological phases with no static counterparts. This so-called anomalous Floquet topology breaks the bulk-boundary correspondence based on the Chern number. The number of edge modes in each band gap is instead determined by another integer index, a winding number, which is calculated from the time evolution operator of the bulk states within one driving period. While topological Chern numbers can be inferred from real-space measurements, it is not straightforward to obtain information about the anomalous Floquet topological phase from bulk measurements. Here we consider two-band models and show that the area enclosed during a Floquet period by an initially localized particle signals the presence of an anomalous phase when it approaches half the unit cell area. In general, we show that for dispersionless dynamics during the micromotion, the enclosed area is quantized and an exact proportionality relation exists between the area and the winding number. Direct detection of anomalous topology in real space could be realized in several quantum simulation platforms, and could be useful for systems with disorder or interactions. Building on the connection between area and winding number, we also show a way to realize arbitrarily high winding numbers.
Topology has emerged as a central concept for classifying phases of matter. The situation is especially rich in periodically driven systems, where anomalous Floquet topological phases break the usual bulk-boundary correspondence between Chern number and edges modes of two-dimensional systems. These phases were so far realized by periodic modulation of the tunneling elements at frequencies near-resonant with respect to the system's bandwidth, a regime where Floquet heating plays a significant role in interacting systems. Here we show that such anomalous Floquet topological phases can also be realized by means of an adiabatic protocol, where the system is always in the instantaneous ground state of the cyclic path in parameter space, like in a Thouless charge pump. We experimentally realize such a state using ultracold atoms in a hexagonal lattice where we adiabatically modulate the lattice geometry, including the sublattice offset. To infer the topology, we use the micromotion area in real-space, which was recently identified as a proxy for the winding number. This way of realizing anomalous phases avoids resonant Floquet heating and imperfect loading into the target state. We demonstrate the robustness of the adiabatic construction by observing the anomalous phase even in the presence of mean-field interactions of magnitude comparable to all other energy scales. These findings are promising for engineering novel topological states in a more robust way.
Floquet topological phases emerge when systems are periodically driven out-of-equilibrium. They gained attention due to their external control, which allows to simulate a wide variety of static systems by just tuning the external field in the high frequency regime. However, it was soon clear that their relevance goes beyond that, as for lower frequencies, anomalous phases without a static counterpart are present and the bulk-to-boundary correspondence can fail. In this work we discuss the important role of resonances in Floquet phases. For that, we present a method to find analytical solutions when the frequency of the drive matches the band gap, extending the well-known high frequency analysis of Floquet systems. With this formalism, we show that the topology of Floquet phases with resonances can be accurately captured in analytical terms. We also find a bulk-to-boundary correspondence between the number of edge states in finite systems and a set of topological invariants in different frames of reference, which crucially do not explicitly involve the micromotion. To illustrate our results, we periodically drive a SSH chain and a π -flux lattice, showing that our findings remain valid in various two-band systems and in different dimensions. In addition, we notice that the competition between rotating and counter-rotating terms must be carefully treated when the undriven system is a semi-metal. To conclude, we discuss the implications to experimental setups, including the direct detection of anomalous topological phases and the measurement of their invariants.
Recently, anomalous Floquet topological phases without static counterparts have been observed in different systems, where periodically driven models are realized to support a winding number of 1 and a pair of edge modes in each quasienergy gap. Here, we focus on cold atomic gases in optical lattices and propose a driving scheme that breaks rotation symmetry but maintains inversion symmetry of the instantaneous Hamiltonian, and discover a type of anomalous Floquet topological phase with a winding number larger than 1. By analyzing the condition of band touching under the constraint of symmetries, we map out the phase diagram exactly by varying the driving parameters and discuss the quasienergy spectra of typical topological phases, which can present multiple pairs of edge modes within a single gap. Finally, we suggest to characterize the topology of such phases by detecting the band inversion surfaces via quench dynamics.
The Středa formula establishes a fundamental physical connection between the topological invariants characterizing the bulk of topological matter and the presence of gapless edge modes. In this work, we extend the Středa formula to periodically driven systems, providing a rigorous framework to elucidate the unconventional bulk-boundary correspondence of Floquet systems while offering a physically grounded link between Floquet winding numbers and tractable response functions. Using the Sambe representation of periodically driven systems, we analyze the response of the unbounded Floquet density of states to a magnetic perturbation. This Floquet-Středa response is regularized through Cesàro summation, yielding a well-defined, quantized result within spectral gaps. The response features two physically distinct contributions: a quantized charge flow between edge and bulk and an anomalous energy flow between the system and the drive, offering new insight into the nature of anomalous edge states. This fundamental result rigorously connects Floquet winding numbers to the orbital magnetization density of Floquet states and holds broadly, from clean to disordered and inhomogeneous systems. This is further supported by providing a real-space formulation of the Floquet-Středa response, which introduces a local topological marker suited for periodically driven settings. In translationally invariant systems, the framework yields a remarkably simple expression for Floquet winding numbers involving geometric properties of Floquet-Bloch bands. A concrete experimental protocol is proposed to extract the Floquet-Středa response via particle-density measurements in systems coupled to engineered baths. Finally, by expressing the topological invariants through the magnetic response of the Floquet density of states, this approach opens a promising route toward the topological characterization of interacting driven phases.
The classification of topological Floquet systems with time-periodic Hamiltonians transcends that of static systems. For example, spinless fermions in periodically driven two-dimensional lattices are not completely characterized by the Chern numbers of the quasienergy bands, but rather by a set of winding numbers associated with the gaps. We propose a feasible scheme for measuring these winding numbers in a periodically driven optical lattice efficiently and directly. It is based on the construction of a one-parameter family of drives, continuously connecting the Floquet system of interest to a trivial reference system. The winding numbers are then determined by the identification and the tomography of the band-touching singularities occurring on the way. As a by-product, we also propose a method for probing spectral properties of time evolution operators via a time analog of crystallography.
Coherent control via periodic modulation, also known as Floquet engineering, has emerged as a powerful experimental method for the realization of novel quantum systems with exotic properties. In particular, it has been employed to study topological phenomena in a variety of different platforms. In driven systems, the topological properties of the quasienergy bands can often be determined by standard topological invariants, such as Chern numbers, which are commonly used in static systems. However, due to the periodic nature of the quasienergy spectrum, this topological description is incomplete and new invariants are required to fully capture the topological properties. Most prominently, there exist two-dimensional anomalous Floquet systems that exhibit robust chiral edge modes, despite all Chern numbers are equal to zero. Here, we realize such a system with bosonic atoms in a periodically-driven hexagonal lattice and infer the complete set of topological invariants from energy gap measurements and local Hall deflections.