Figure 1: A two-dimensional representation of a ten-dimensional hypercube. A RIKEN researcher has shown that quantum Ising models lack local conserved quantities even in such high-dimensional systems. © Tom Ruen
By extending a proof of a physically important behavior in one-dimensional quantum spin systems to higher dimensions, a RIKEN physicist has shown that the model lacks exact solutions1.
Theoretical physicists develop mathematical models to describe material systems, which they can then use to make predictions about how materials will behave.
One of the most important models is the Ising model, which was first developed about a century ago to model magnetic materials such as iron and nickel.
Conceptually, the classical Ising model is very simple, consisting of a grid of points that have either spin up or spin down. Spins interact with the spins closest to them, and the system tries to minimize the overall energy by aligning spins.
Despite its simplicity, the two-dimensional Ising model can predict phase transitions such as when a magnetic material loses its magnetism above a certain temperature.
The quantum version of the Ising model incorporates quantum mechanics, which allows for effects such as superposition and quantum fluctuations. It can be used to model quantum materials and could be useful for developing quantum computers.
"Quantum Ising models are quantum extensions of the classical Ising model, one of the simplest theoretical models of magnetism," explains Yuuya Chiba of the Nonequilibrium Quantum Statistical Mechanics RIKEN Hakubi Research Team. "They're among the most fundamental and widely studied models of quantum many-body spin systems."
In previous studies, one-dimensional quantum Ising models have been proved not to have 'local conserved quantities'-quantities like energy that can be defined locally in space and remain constant in time when considering the system as a whole.
"For instance, suppose you heat the right half of a solid and cool the left half, and then isolate it," says Chiba. "Energy will flow from the hot to the cold side, but the total energy of the system stays constant. This kind of spatially distributed but conserved quantity is what we mean by a local conserved quantity."
Now, Chiba has rigorously established for the first time that quantum Ising models in two and higher dimensions lack such local conserved quantities other than energy.
This proof implies that quantum Ising models do not have exact solutions, and so physicists will have to resort to computational methods to analyze them. It also means that they tend to display complex phenomena. "Such systems typically exhibit thermalization and quantum chaos," notes Chiba.
Surprisingly, the math involved was remarkably simple. "The basic strategy was quite straightforward, and the actual computation corresponded to solving linear equations-no advanced mathematical tools were needed," says Chiba. "It's actually surprising that such a result hadn't been obtained earlier, given the simplicity of the underlying method."
Yuuya Chiba has proved that quantum Ising models of two and higher dimensions lack local conserved quantities. © 2025 RIKEN
