In physics, substances exist in phases, such as solid, liquid, or gas. When a material crosses from one phase into another, we talk about a phase transition: think of water boiling into steam, turning from a liquid into a gas.
From jumps to smooth transitions
In our kitchens, water boils at 100°C, at which point its density changes dramatically, making a discontinuous jump from the liquid phase into the gas phase. If we turn up the pressure, the boiling point of water also increases, until at a pressure of 221 atmospheres it boils at 374°C. Here, something strange happens: the liquid and gas merge into a single phase. Beyond this “critical point”, there is no longer a phase transition at all, and so by controlling its pressure and temperature water can be steered from liquid to gas without ever encountering a ‘jump’ in its properties.
An intriguing physical question is: are there quantum analogues of these water-like phase transitions? Henrik Rønnow is a professor at the École Polytechnique Fédérale de Lausanne (EPFL), the institute that was in the lead of this project. He says: “The current directions in quantum magnetism and spintronics require highly spin-anisotropic interactions to produce the physics of topological phases and protected qubits, but these interactions also favour discontinuous quantum phase transitions.”
Whereas previous studies have focused on smooth, continuous phase transitions in quantum magnetic materials, in a joint experimental and theoretical project, the physicists have now studied a discontinuous phase transition to observe the first ever critical point in a quantum magnet appearing at a specific pressure and temperature, which is similar to the critical point of ordinary water. The work was published in Nature this week.
The scientists used a quantum antiferromagnet, known in the field as SCBO, a shorthand for its chemical composition: SrCu2(BO3)2. Quantum antiferromagnets are magnetic materials that are especially useful for understanding how the quantum aspects of a material’s structure affect its overall properties – for example, how the spins of its electrons interact to give the material its magnetic properties. SCBO is also a ‘frustrated’ magnet, meaning that its electron spins can’t stabilize in some orderly structure, and instead adopt some uniquely quantum fluctuating states.
In a complex experiment, the researchers controlled both the pressure and the magnetic field applied to milligram pieces of SCBO. “This allowed us to look all around the discontinuous quantum phase transition. In that way, we found critical-point physics in a pure spin system,” says Rønnow.
The team performed high-precision measurements of the so-called specific heat of SCBO, a quantity which shows its readiness to “suck up energy”. For example, water absorbs only small amounts of energy at -10°C, but at exactly 0°C or 100°C it can take up huge amounts of energy as every molecule is driven across the transitions from ice to liquid or from liquid to gas. Just like what happens in water, the pressure-temperature relationship of SCBO also forms a phase diagram showing a discontinuous transition line separating two quantum magnetic phases, with the line ending at a critical point.
“Now when a magnetic field is applied, the problem becomes richer than water,” says Frédéric Mila, another researcher from EPFL. “Neither magnetic phase is strongly affected by a small field, so the line becomes a wall of discontinuities in a three-dimensional phase diagram – but then one of the phases becomes unstable and the field helps push it towards a third phase.”
A theoretical explanation
This is where the UvA physicists come into the story. To explain this macroscopic quantum behaviour, the researchers teamed up with Philippe Corboz and Schelto Crone at the UvA-Institute of Physics, who have been developing powerful new computer-based techniques.
Previously it had not been possible to calculate the properties of frustrated quantum magnets in a realistic two- or three-dimensional model. “Using so-called 2D tensor network simulations, we managed to simulate SCBO,” says Corboz. “The material is effectively described by a theoretical model known as the Shastry-Sutherland model. From the numerical side, simulating this model is a very challenging problem. For a long time, it was inaccessible to numerical approaches, due to a technical difficulty known as the negative sign problem.”
However, thanks to recent advances with 2D tensor network methods, the theorists did manage to tackle the problem and it was possible to numerically confirm that the sharp peak feature in the specific heat found in experiments corresponds to a critical point at finite temperature, analogous to the critical point of water.
Henrik Rønnow concludes: “Looking forward, the next generation of functional quantum materials will be switched across discontinuous phase transitions, so a proper understanding of their thermal properties will certainly include the critical point, whose classical version has been known to science for two centuries.”
A quantum magnetic analogue to the critical point of water, J. Larrea Jiménez, S. P. G. Crone, E. Fogh, M. E. Zayed, R. Lortz, E. Pomjakushina, K. Conder, A. M. Läuchli, L. Weber, S. Wessel, A. Honecker, B. Normand, Ch. Rüegg, P. Corboz, H. M. Rønnow and F. Mila. Nature 592, (2021) 370–375.