When nothing adds up — until it does

Imagine a strange material being rhythmically pushed— tapped again and again by invisible hands. These are periodically driven quantum systems, or Floquet systems, where energy is no longer conserved in the usual sense. Instead, physicists speak of quasienergy — a looping spectrum with no clear start or end.

When scientists measure how such a system responds to a magnetic field, every single contribution seems to vanish — like adding an infinite list of zeros. And yet, the total stubbornly comes out finite, quantized, and very real.

The resolution is a mathematical sleight of hand called Cesàro summation — a way to assign meaning to infinite series that refuse to converge. Think of standing in a hall of mirrors: each reflection is static, but the infinite chain of images encodes a hidden, coherent structure. In the same way, the “sum of zeros” hides a deeper pattern that Cesàro summation can reveal.

That hidden pattern is topology at work. In quantum materials, topology governs the bulk–boundary correspondence: the rule that properties deep in the bulk dictate the behavior of robust edge channels. In static systems, this is elegantly captured by the celebrated Středa formula, which links magnetic responses to protected edge modes.

In this work, Lucila Peralta Gavensky and Nathan Goldman (Faculty of Science at ULB and Collège de France), in collaboration with Gonzalo Usaj (Balseiro Institute, Argentina) extend the Středa framework into the driven world of Floquet matter. They show that the Cesàro-summed response corresponds to a quantized bulk magnetization, a signature of quasienergy flowing along the edge. They also find that applying a magnetic field induces a steady energy exchange between the system and its surroundings — a kind of “energy pump” unique to driven systems.

This reframing not only resolves a conceptual paradox but also offers a roadmap for experiments. The authors propose detecting Floquet–Středa responses via particle-density measurements, even in disordered systems. The energy pump effect hints at links to cavity quantum materials, where the driving field itself is a quantum object and could experience a Středa-type back-action.

In short, what looks like nothing — a sum of zeros — becomes a profound something. A mathematical trick exposes a topological truth, opening new paths to classify exotic nonequilibrium phases and explore the frontiers of driven quantum matter.