Each wall costs energy proportional to its length. A long wall costs more, so it should be rarer. But a long wall can take many shapes, and each shape has its own slim chance of appearing. Peierls showed that, at low temperatures, the energy cost outruns the sheer number of shapes. Large walls effectively never form, and the magnet keeps its order.

How did the team adapt this to quantum magnets?

Co-author Andrew Lucas explained that he and Yin developed the underlying mathematical technique in 2024.

"Chao and I have been interested in the stability of matter for multiple years now," Lucas said. "We simply happened to develop a very useful mathematical technique in 2024 that allows us to get very strong constraints on where many-body quantum states are supported. This was a highly unusual idea, and it allowed us to look at the problem in the current paper from a fresh perspective."

The team started with a classical Ising magnet and added a weak quantum nudge, such as a transverse magnetic field. The nudge can flip individual spins without favoring up over down.

Their key move: require the Peierls condition to hold only for walls spanning the whole system. Small clusters may flip freely; only full-length walls get suppressed. They call this the quantum Peierls condition: any quantum state containing a system-wide domain wall must sit high in energy.

Why the energy gap tripped up earlier proofs

Existing proofs of SSB work by slowly tuning the system from a known ordered state to a new one while staying inside the same phase. An energy gap makes this safe, because low-energy excitations cannot sneak in during the tuning. A gapless system leaks energy at every step, and the tuning fails.

In a fluid, for example, an arbitrarily soft sound wave can appear at any moment. The same issue arises in disordered Ising magnets, where cheap cluster flips mimic gapless modes.

What did the proof actually show?

For the two-dimensional random-bond Ising model, the authors proved:

• If random interactions are sufficiently biased toward alignment, the magnet's symmetry breaking survives weak quantum fluctuations.
• Low-energy states remain superpositions of mostly-up and mostly-down configurations.
• Each configuration retains long-range magnetic order, despite the gapless clusters.

What can't the new method handle?

The proof only suppresses excitations that stay in clusters. Waves that travel across the entire material, such as sound waves in a solid or photons in a vacuum, remain out of reach.

Lucas called that the most exciting frontier. "The most interesting gapless phases are fluids or gauge theories which do have gapless particles," he said. "If we could say something directly about the stability of these systems, it would verify some longstanding conjectures about quantum field theory going back over 50 years."

What's next?

The authors frame the result as a first step toward a rigorous taxonomy of gapless phases. Whether such a taxonomy will match physicists' existing intuitions remains open. If it does not, Lucas noted, "it would be interesting to know what it's missed, and whether there are meaningful experimental ways to distinguish between two phases we used to think were the same."

The paper appeared as Chao Yin et al, "Robust Symmetry Breaking in Gapless Quantum Magnets," Physical Review Letters (2026). DOI: 10.1103/5ntb-ggcz. A preprint sits at arXiv:2603.13212.