September 28, 2026

feature by Tejasri Gururaj, Phys.org Author edited by Gaby Clark, reviewed by Robert Egan Scientific Editor Senior Editor This article has been reviewed according to Science X's editorial process and policies. Editors have highlighted the following attributes while ensuring the content's credibility: fact-checked peer-reviewed publication trusted source proofread In a new study published in Physical Review Letters, researchers have shown that magnetic order can survive weak quantum fluctuations in disordered magnets that lack an energy gap. The work establishes robust ferromagnetism in the two-dimensional random-bond quantum Ising model, confirming a longstanding conjecture in quantum statistical mechanics. Many magnets owe their order to spontaneous symmetry breaking (SSB). Physicists have long sought to prove that this order is stable against perturbations such as quantum fluctuations. Existing proofs, however, typically require the system to have an energy gap. Disordered magnets such as the random-bond Ising model are gapless, placing them outside the reach of these proofs. The researchers developed a proof technique that does not rely on an energy gap. They adapted an argument from statistical mechanics, known as the Peierls argument, to quantum systems. Phys.org spoke to co-author Andrew Lucas of the University of Colorado Boulder about the work. "[Co-author] Chao and I have been interested in the stability of matter for multiple years now, ever since he was my Ph.D. student, so it was natural for us to think about problems in this space. More concretely, though, 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," Lucas explained. The Ising model is a simplified description of a magnet. Each spin sits on a lattice, points either up or down, and interacts with its nearest neighbors. "An individual spin does not care whether it is up or down; it only wants to align with its neighbor. The lowest-energy configurations have all spins pointing up, or all pointing down. Classically, the system would have to pick-this would spontaneously break a symmetry of the system (the energy doesn't change if we flip all spins together). In quantum mechanics, the lowest-energy (ground) state is (roughly) a macroscopic superposition of the up state and the down state," Lucas explained. Regions of aligned spins are called domains. At domain boundaries, neighboring spins differ, resulting in what are called broken bonds. The boundary formed by these broken bonds is known as a domain wall. Each broken bond costs energy, so the total energy cost grows with the wall's length. Thermal fluctuations can create domain walls. Whether these can destroy the magnet's order comes down to a competition, which physicist Rudolf Peierls used in 1936 to show that the 2D Ising model stays ordered at low temperatures. On one hand, a longer domain wall costs more energy, making it less likely to appear. On the other hand, a long domain wall can take a huge number of different shapes. Each specific shape is unlikely, but, like lottery tickets, the more shapes there are, the greater the chance that at least one of them appears. At low temperatures, the energy cost wins. The probability of a domain wall spanning the whole system becomes vanishingly small, and the magnet remains ordered. The requirement that large domain walls cost more energy than their number of possible shapes can compensate for is known as the Peierls condition. The question the researchers set out to answer was whether a similar argument could work for quantum fluctuations in a magnet with no energy gap. Proving that a phase is stable amounts to showing that the system can be changed slightly without leaving that phase. Previous proofs have depended on the energy gap, which is the minimum energy needed to excite a system above its ground state. Discover the latest in science, tech, and space with over 100,000 subscribers who rely on Phys.org for daily insights. Sign up for our free newsletter and get updates on breakthroughs, innovations, and research that matter-daily or weekly. "Most phases we are familiar with are not gapped. [For example], you can excite a fluid or solid with an arbitrary low-energy sound wave (phonon). But some phases such as an Ising ferromagnet are gapped, and this mathematically significantly simplifies the classification of phases," Lucas explained. "Intuitively, this is because you can imagine 'tuning a knob' to convert the system between two states in the same phase, and due to the gap, the knob tuning does not excite the gapless modes (e.g., sound), preserving the character of the state." The random-bond Ising model breaks this picture. Its interactions vary randomly: Most favor alignment, but some favor anti-alignment. Clusters of spins with roughly equal numbers of each can be flipped at almost no energy cost, making the magnet gapless. "So the open challenge was basically to show that these clusters cannot globally destroy magnetism, even though they do break the previous argument that you can tune a knob and 'drag' the ground state from the unperturbed to the perturbed magnet without creating any large excitation," Lucas said. To address this, the researchers started with a classical Ising magnet and added a weak quantum perturbation, such as a transverse magnetic field. The perturbation can flip spins but does not favor up over down. They then required the Peierls condition to hold only for domain walls spanning the whole system, allowing small clusters to flip cheaply while the magnet remains gapless. From this, they formulated a quantum Peierls condition: Any quantum state containing a system-spanning domain wall must be high in energy. Low-energy states, such as the ground state, therefore cannot afford such walls. They showed that this condition holds when the quantum perturbation is weak and respects the symmetry locally. Next, they sorted the magnet's quantum state by how many broken bonds lie along a given long loop. Because the perturbation flips only a few spins at a time, it can build a wall only gradually, with each step costing energy. The chance of finding a complete wall therefore shrinks exponentially with its length. Summed over all possible walls, this suppression still outpaces their growing number, so system-spanning walls essentially never appear. In the 2D random-bond Ising model, the researchers proved that as long as the random interactions are sufficiently biased toward alignment, the magnet's symmetry breaking survives weak quantum fluctuations. Despite the cluster flips, its low-energy states remain superpositions of a mostly-up and a mostly-down configuration, each magnetically ordered. The researchers describe the work as a first step toward a rigorous classification of gapless quantum phases. According to Lucas, such a classification could connect with longstanding conjectures from statistical and quantum field theory, where physicists have an intuitive but not rigorous picture of how to classify phases based on patterns of SSB. On the other hand, if the existing classification turns out to be too coarse, "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," Lucas said. However, the method relies on the gapless clusters in the magnet being spatially localized. Excitations that travel across the whole system, such as sound waves or photons, remain beyond its reach. Lucas concluded, "The most interesting gapless phases are fluids or gauge theories which do have gapless particles. 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." Written for you by our author Tejasri Gururaj, edited by Gaby Clark, and fact-checked and reviewed by Robert Egan-this article is the result of careful human work. We rely on readers like you to keep independent science journalism alive. If this reporting matters to you, please consider a donation (especially monthly). You'll get an ad-free account as a thank-you. Chao Yin et al, Robust Symmetry Breaking in Gapless Quantum Magnets, Physical Review Letters (2026). DOI: 10.1103/5ntb-ggcz. On arXiv: arxiv.org/abs/2603.13212 Journal information: Physical Review Letters

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