Ferromagnetism is hard to model. One of just a few rigorous models, Nagaoka ferromagnetism has long been popular with theorists for its simplicity. But whether any material satisfies the model’s assumptions was unclear. Now Lieven Vandersypen of Delft University of Technology in the Netherlands and his colleagues have observed it in the lab by using a quantum simulation with quantum dots.

Adapted from J. P. Dehollain et al., Nature 579, 528 (2020)

Nagaoka ferromagnetism is based on the Hubbard model, a simple way to represent particles in a lattice. When applied to fermions, its Hamiltonian includes terms for site-to-site tunneling and on-site electron–electron repulsion, and that’s all. In 1966 Yosuke Nagaoka found that for certain lattices with infinitely strong interactions, he could solve the Hubbard model analytically. In a 2 × 2 array filled with three out of four possible electrons, for example, the result is two low-energy states: the total spin s = 1/2 state and the ferromagnetic s = 3/2 state.

The ferromagnetic state has the lowest energy, which can be understood by imaging the lone hole’s path as it swaps places with one of its two nearest neighbors. For the s = 1/2 state, in which the electron spins point in different directions, the resulting spin state depends on which neighbor the hole swaps places with. If the electron spins all point in the same direction, as in the ferromagnetic state, the spin state stays the same. In the ferromagnetic case, the paths interfere constructively, and that leads to a lower kinetic energy for the hole.

In their attempt to realize Nagaoka ferromagnetism, Vandersypen and his colleagues arranged gated quantum dots in a 2 × 2 array. Quantum dots are ideal for quantum simulations because they easily cool to energies below those of site-to-site hopping and on-site interactions. They also have many tunable parameters, including local disorder, boundary conditions, and external magnetic fields.

The researchers used a sensor that converts spin to an electrical signal to check the spin state of quantum dots cooled to 70 mK. They saw both predicted spin states, including the previously unobserved Nagaoka ferromagnetic state. And that state was stable; it survived the addition of disorder in the form of one dot’s energy being offset by an amount, dV, larger than the tunneling energy, as shown by the potential wells in the figure.

The result settles a decades-long debate: Nagaoka ferromagnetism is possible in practice. It also demonstrates the power of quantum simulations (see the article by Ignacio Cirac and Peter Zoller, Physics Today, March 2004, page 38). Although the Hubbard model captures some of the system’s essential physics, a complete ab initio calculation would take on the order of 10 000 hours to compute classically. (J. P. Dehollain et al., Nature 579, 528, 2020.)