When Interactions Break Newton’s Third Law, Floquet Driving Can Still Give Them an Energy Map

A new Nature Physics paper gives Floquet engineering an unexpected new job: not only shaping quantum materials, but helping write an “energy book” for systems whose interactions do not normally come from an energy function at all.

The paper, “Hamiltonian description of non-reciprocal interactions” by Yu-Bo Shi, Roderich Moessner, Ricard Alert and Marin Bukov, was published online on June 12, 2026. Its problem sounds deceptively simple. In many systems, if A pushes B, B does not push A back with an equal and opposite response. That is common in active matter, driven colloids, sedimenting particles, robotic metamaterials, biological swarms and engineered feedback systems. It is also common in the broader physics of open, driven media: the environment, measurement channel or control field can make the “force” from one part to another directional.

Those asymmetric interactions are called non-reciprocal. They are powerful because they let materials rectify motion, amplify signals, self-organize and route energy or information in one preferred direction. They are also awkward because ordinary equilibrium statistical mechanics assumes a potential energy landscape. If no potential exists, many familiar tools—Hamiltonian dynamics, Monte Carlo sampling, conserved quantities and high-frequency Floquet expansions—are harder to use or may appear unavailable.

The new result says: even when the visible system has no conventional energy function, one can sometimes embed it in a larger constrained Hamiltonian system. Once that Hamiltonian exists, Floquet theory becomes a design language for non-reciprocal matter.

The core advance: adding auxiliary degrees of freedom

Shi and colleagues construct a Hamiltonian by adding auxiliary degrees of freedom to the original non-reciprocal system. Under a constraint, the enlarged Hamiltonian generates the same original dynamics. This is not merely a formal trick. The authors show that Monte Carlo simulations based on the constrained Hamiltonian reproduce both stationary and non-stationary behavior of the original Langevin dynamics for a test system: dissipative XY spins with “vision-cone” interactions.

The vision-cone model is easy to picture. Imagine spins on a lattice that interact strongly with neighbors lying inside the direction they are “looking,” but weakly or not at all with neighbors behind them. That makes the interaction directional, like a simplified flocking rule. A standard reciprocal XY magnet is recovered only when the cone becomes wide enough to see all directions symmetrically.

That agreement matters. If the embedding predicted a different phase transition, it would be a nice mathematical object but a poor physical proxy. Instead, the paper reports that the constrained Monte Carlo and Langevin approaches match the same transition within statistical uncertainty. In the language of practical simulation, the result gives researchers a way to bring fast Hamiltonian-based algorithms to systems that previously looked outside the Hamiltonian toolbox.

Why Floquet engineering enters the story

Floquet engineering is the art of controlling a system with periodic driving—laser pulses, microwaves, oscillating fields, modulated couplings or synthetic clocks—so that its long-time behavior resembles a new effective Hamiltonian. The site’s usual examples include driven topological bands, time crystals, polar-molecule spin models, trapped-ion simulators and light-controlled superconducting or photonic systems.

The Nature Physics paper extends that mindset to non-reciprocal systems. Once the authors have an enlarged Hamiltonian with a symplectic phase-space structure, they can apply high-frequency periodic-drive methods. In their example, a periodic drive along one lattice direction weakens the effective spin interactions along that direction while preserving interactions in the transverse direction. The result is a kind of dimensional crossover: a two-dimensional square lattice can behave more like a collection of one-dimensional chains.

The authors connect this idea to the familiar physics of Josephson-junction arrays, where an alternating magnetic field can be described by a time-dependent vector potential coupled to phase differences. But the broader message is platform independent: if a non-reciprocal material cannot have its couplings tuned directly, a periodic drive may still tune the effective couplings indirectly.

Why this matters for quantum energy

At first glance, bird flocks and active colloids may seem far from quantum batteries or beyond-Carnot devices. The bridge is controlled non-equilibrium flow. Quantum energy science increasingly studies systems where energy, heat, coherence and information move through open driven networks rather than static equilibrium machines. Non-reciprocity is one of the most useful ingredients in such networks because it can make transport directional.

Consider three recurring goals in quantum-energy research:

• Routing energy without backflow. A non-reciprocal channel can protect a load from feeding noise or energy back into a source.
• Amplifying weak signals. Driven non-reciprocal media, including photonic and polaritonic platforms, can act as directional amplifiers or sensors.
• Stabilizing useful non-equilibrium states. Energy devices often rely on states that are maintained by drive and dissipation rather than by equilibrium free energy minima.

Hamiltonians are the grammar used to compute work, heat, response and effective dynamics in many of these proposals. A Hamiltonian embedding for non-reciprocal interactions therefore gives theorists a cleaner way to ask: where is energy stored, which effective couplings are being engineered by the drive, and how does the driven system transition between ordered and disordered phases?

For energy applications, the most exciting phrase is not “flocking” or “active matter.” It is “directional control with an effective energy description.” That is exactly the combination needed for better thermal routers, quantum sensors and driven energy-transfer networks.

How it fits with recent Floquet experiments

The paper arrives amid a wave of experiments showing that Floquet control is no longer just a theoretical convenience. In 2024, del Valle-Inclan Redondo and collaborators demonstrated non-reciprocal band structures in an exciton-polariton Floquet optical lattice, using a gigahertz-detuned optical “conveyor belt” to create directional band tilts. In the same year, Miller, Carroll, Lin and colleagues used Floquet-engineered spin models with ultracold KRb polar molecules to realize tunable many-body dynamics relevant to squeezing and entanglement. In 2025, Katz, Feng, Porras and Monroe reported Floquet control of interactions and edge states in a trapped-ion programmable quantum simulator, implementing a spin-based Su-Schrieffer-Heeger model with up to 22 interacting spins.

These experiments share a theme: periodic driving turns difficult static engineering problems into timing problems. Instead of fabricating a perfect material with the exact desired coupling pattern, one applies a drive whose averaged effect produces the desired model. The new Hamiltonian-embedding work asks what happens when the desired model is not even reciprocal enough to start from a normal energy function. Its answer is that the Floquet toolbox can still be used if the non-reciprocal dynamics are lifted into the right constrained Hamiltonian space.

A careful limitation: not yet an energy device

This is a theory paper, not a working quantum heat engine. It does not claim a beyond-Carnot violation, a battery with higher stored ergotropy or a new superconducting device. Its benchmark is a classical dissipative spin model, and the experimental systems in the supplementary discussion span active and soft matter as well as engineered feedback platforms. That distinction matters.

Still, foundational tools often precede devices. Floquet master equations, Sambe-space methods, high-frequency expansions and stochastic thermodynamics all became useful in energy research because they gave a common language for driven open systems. A Hamiltonian description of non-reciprocity could play a similar role for directional energy networks, especially where control fields create effective one-way couplings.

What to watch next

The immediate follow-up questions are concrete. Can the embedding be adapted to quantum non-Hermitian or Lindblad systems with engineered gain and loss? Can Floquet drives control non-reciprocal heat currents in superconducting circuits, optomechanical devices or photonic time crystals? Can auxiliary-variable Hamiltonians improve numerical optimization for quantum batteries whose charging protocols involve feedback or asymmetric couplings?

Another important direction is thermodynamic accounting. If the effective Hamiltonian lives in an enlarged space, researchers must track what the auxiliary variables mean operationally. Are they mathematical bookkeeping, controllable laboratory modes, reservoir degrees of freedom, or hidden costs paid by the feedback controller? Practical energy claims will depend on that accounting.

For now, the paper’s value is conceptual and methodological. It tells researchers that non-reciprocal interactions need not be excluded from the Hamiltonian and Floquet playbook. In a field where the best devices may be open, driven, directional and far from equilibrium, that is a significant expansion of the design space.

Sources and further reading

Primary paper: Yu-Bo Shi, Roderich Moessner, Ricard Alert & Marin Bukov, “Hamiltonian description of non-reciprocal interactions,” Nature Physics (published June 12, 2026), DOI: 10.1038/s41567-026-03317-0.

Related Floquet experiments: Yago del Valle-Inclan Redondo et al., “Non-reciprocal band structures in an exciton–polariton Floquet optical lattice,” Nature Photonics 2024, DOI: 10.1038/s41566-024-01424-z; Calder Miller et al., “Two-axis twisting using Floquet-engineered XYZ spin models with polar molecules,” Nature 2024, DOI: 10.1038/s41586-024-07883-2; Or Katz, Lei Feng, Diego Porras & C. Monroe, “Floquet control of interactions and edge states in a programmable quantum simulator,” Nature Communications 2025, DOI: 10.1038/s41467-025-62897-2.