The Magnetic Monopoles That Heat Up When Cooled

Cut a bar magnet in half, and you do not get an isolated north pole and a south pole. You get two smaller magnets, each with its own north and south. In classical electrodynamics, magnetic monopoles do not exist.

Yet inside a class of crystals known as spin ice, they do. These are not elementary particles, but quasiparticles—collective, emergent excitations that behave precisely like free magnetic charges. In 2024, a research team demonstrated that when these emergent monopoles are subjected to a specific magnetic field, they display a phenomenon known as negative heat capacity. As you cool the surrounding system, these monopoles draw in thermal energy and get hotter.

The Geometry of Spin Ice

To find a magnetic monopole, researchers look inside pyrochlore crystals like dysprosium titanate ($Dy_2Ti_2O_7$). In these materials, the magnetic moments—or spins—of the dysprosium ions sit on a lattice of corner-sharing tetrahedra.

The physics of this lattice is constrained by a rule analogous to the ice rules governing hydrogen bonds in water ice. For each tetrahedron, energy is minimized when two spins point inward toward the center and two spins point outward. This is the "two-in, two-out" ground state.

If you flip a single spin, you break this rule. One tetrahedron now has three spins pointing in and one pointing out (a local north pole). The adjacent tetrahedron has three pointing out and one pointing in (a local south pole).

Because the tetrahedra are connected, these flipped spins can propagate through the crystal independently. The north and south poles can drift far apart, interacting via a magnetic Coulomb potential. In 2008, Claudio Castelnovo, Roderich Moessner, and Sondhi showed that these excitations are, for all practical purposes, emergent magnetic monopoles.

Flipping the Thermodynamic Sign

Normally, adding energy to a system increases its temperature. If you put a pot of water on a stove, it absorbs heat and gets hotter. Conversely, extracting heat cools the system down. This positive relationship between energy and temperature defines a positive heat capacity.

Systems with negative heat capacity do the opposite. They grow hotter as they lose energy. While this behavior is known in astrophysics—self-gravitating systems like stars contract and heat up as they radiate energy away—it is exceedingly rare in condensed matter physics.

In a 2024 study published in Nature Physics, Jonathan Hallén and an international team of collaborators showed that spin ice can be manipulated to exhibit this inverted thermodynamic behavior.

The experimental setup requires placing the dysprosium titanate crystal under a magnetic field directed along a specific crystallographic direction: the $[111]$ axis.

This specific field direction splits the tetrahedra into two distinct types. One set of tetrahedra aligns with the field, while the other set remains free to flip. By tuning the strength of this $[111]$ magnetic field, the researchers can control the energy barrier required to create and move the magnetic monopoles.

The Measurement Apparatus

Measuring this thermodynamic inversion requires isolating the magnetic monopoles from the surrounding crystal lattice. The experimental challenge lies in the fact that the crystal lattice itself always has a positive heat capacity. If you measure the entire crystal at once, the lattice's standard thermal behavior drowns out the anomalous signal of the monopoles.

To bypass this, Hallén and his colleagues used a technique called AC susceptibility. Instead of a static measurement, they applied an oscillating magnetic field to the crystal at temperatures below 1 Kelvin, close to absolute zero.

By varying the frequency of the oscillation, the researchers separated the fast-responding degrees of freedom from the slow ones. The monopoles move and interact relatively slowly at these extreme temperatures. By measuring the phase lag between the oscillating magnetic field and the resulting magnetization of the sample, the team mapped the thermodynamic variables of the monopole subsystem alone.

When the magnetic field along the $[111]$ axis is tuned to a critical threshold, it creates a highly constrained state known as a kagome ice phase. Here, the monopoles are restricted to moving along two-dimensional planes.

As the temperature of the system is lowered, the monopoles are forced into fewer, lower-energy configurations. However, due to the tight geometric constraints of the lattice, localizing these monopoles requires them to shed entropy faster than they can shed energy. To satisfy the laws of thermodynamics under these conditions, the effective temperature of the monopole system must rise as the background temperature falls.

The quasiparticles absorb thermal energy from the crystal lattice, cooling their surroundings while the monopole subsystem itself grows warmer.

Measured.

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