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Středa Formula for Floquet Systems: Topological Invariants and Quantized Anomalies from Cesàro Summation

Lucila Peralta Gavensky1,2,*, Gonzalo Usaj1,3,4, and Nathan Goldman1,2,5

  • *Contact author: lucila.peralta.gavensky@ulb.be
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Phys. Rev. X 15, 031067 – Published 10 September, 2025

DOI: https://doi.org/10.1103/b3pw-my97

Abstract

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.

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Popular Summary

Topological matter is shaped by the bulk-boundary correspondence, where subtle bulk properties give rise to robust edge channels. A compelling demonstration of this principle is the Středa formula, which rigorously links quantized bulk magnetic responses to topologically protected edge modes. Here, we extend this concept to Floquet systems—materials and quantum simulators driven periodically in time—where bulk-edge connections become richer and more intricate. Our Floquet-Středa framework provides a unified approach for understanding the topology of driven matter, showing how quantized bulk responses set edge structure and offering a physical route to define Floquet topological invariants.

A mathematical twist lies at the core of our approach. Because energy is not conserved, one must consider quasienergy, which forms a looping, periodically repeating spectrum. Its magnetic response arises from summing infinitely many seemingly trivial contributions. Using Cesàro summation, we show these apparent zeros combine into a finite, quantized response. This reveals two complementary processes: a quantized bulk magnetization reflecting the spectral flow of quasienergy states, and a magnetic-field-induced energy pump transferring energy between the system and the drive. Together, these capture anomalous edge modes unique to Floquet systems and reveal a deeper anomaly intrinsic to periodically driven quantum matter.

Building on this perspective, we propose experimental protocols to detect Floquet-Středa responses via particle-density measurements, even in disordered platforms. The energy pump further suggests a connection to cavity quantum materials, where the driving field is a quantum degree of freedom and a Středa-type backaction may arise. Our work lays a foundation for classifying exotic nonequilibrium phases and motivates extensions to symmetry-protected and interacting Floquet systems.

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Phys. Rev. X 15, 031067– Published 10 September, 2025
  • Received 20 September 2024
  • Accepted 23 July 2025
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