My encounters with Alex Muller and the perovskites

The Great Crystal Identity Crisis

Imagine a world made of tiny, invisible Lego bricks. In the world of physics, these bricks are atoms, and when they snap together in a perfect, repeating pattern, they form a crystal. Sometimes, as the temperature drops, these crystals decide to change their shape, like a dancer shifting from a perfect square formation into a slanted triangle or a stretched rectangle. This is called a "phase transition." Scientists love studying these moments because they are like the crystal's "aha!" moment, where the rules of the game suddenly change.

To understand these changes, physicists use a set of mathematical tools called "critical exponents." Think of these as the crystal's personality traits—how fast it grows, how it reacts to stress, or how it wiggles as it gets ready to change shape. For a long time, scientists believed that if two crystals started with the same perfect, cube-shaped symmetry, they would behave exactly the same way when they changed shape, no matter what shape they turned into. It was like assuming that two identical twins would always have the same favorite color, even if one became a painter and the other a chef. But for decades, the data was messy. Some crystals seemed to follow the rules, while others broke them, leaving physicists scratching their heads.

A Friendship Built on Hikes and Puzzles

This paper is a tribute to a long friendship between the author, Amnon Aharony, and the late Professor K. Alex Müller. Their story began in 1974 when they met at a university in the US. They bonded over a shared obsession with "perovskites," a family of crystals that includes materials like strontium titanate (SrTiO3) and lanthanum aluminate (LaAlO3). These materials are special because they can switch from a perfect cube shape to either a stretched rectangle (tetragonal) or a slanted triangle (trigonal) as they cool down.

For over 20 years, Aharony and Müller would meet, often while hiking in the mountains near Zurich or driving to hockey games, to solve a giant puzzle: Why did these two types of shape-shifting crystals seem to act so differently, even though they started as identical cubes? They used a powerful mathematical tool called the "renormalization group" (think of it as a microscope that zooms in and out to see how the crystal's behavior changes at different scales) to try to predict what should happen.

The Old Theory vs. The New Clue

For a long time, the best theory suggested that both types of transitions (cube-to-rectangle and cube-to-triangle) belonged to the same "universality class." In plain English, this meant they were supposed to be the same species of behavior, just wearing different hats. The theory predicted they should both show "isotropic" behavior, meaning they would act the same in every direction.

However, experiments kept showing weird results. When scientists applied stress to these crystals, some seemed to turn into a "bicritical" point (a special meeting place where two different phases of matter could coexist), while others hinted at a "tetracritical" point (a more complex meeting place with four phases). The biggest confusion was about strontium titanate (SrTiO3). The math said it should eventually turn into a "triple point" (where three different phases meet), but the experiments only ever showed a "bicritical" point. It was like a map saying a city has three roads meeting, but every time you drove there, you only saw two.

The 2022 Breakthrough: Slow Motion and Hidden Paths

In 2022, Aharony, along with collaborators Ora Entin-Wohlman and Andrey Kudlis, finally cracked the code. They realized the old math was missing a crucial detail: time and speed.

They discovered that the path a crystal takes to change its behavior is not a straight line; it's a slow, winding road.

• The "Slow" Crystal (SrTiO3): This crystal starts its journey on a path that should lead to a "triple point" (where three phases meet). However, the journey is incredibly slow. For a long time, it gets stuck near the "isotropic" neighborhood (the place where the old theory said everything should look the same).
• The "Slow" Crystal (LaAlO3): This crystal starts its journey on a path that leads to a "cubic" destination (the cubic fixed point). It also moves slowly, but it eventually gets there. Its behavior is governed by the "cubic" rules.

Here is the twist: Because the journey for SrTiO3 is so slow, if you look at it at "intermediate" temperatures (not too close to the freezing point, not too far away), it looks like it's behaving like the other crystal. It's like watching a snail that is actually heading to the moon; if you only watch it for a few minutes, it looks like it's just walking in the garden.

The Final Verdict

The paper concludes that the two transitions are not the same. They belong to two different universality classes:

1. Cube-to-Triangle (Trigonal, like LaAlO3): This one belongs to the "cubic" class. Under stress, it creates a "tetracritical" map (four roads meeting).
2. Cube-to-Rectangle (Tetragonal, like SrTiO3): This one belongs to a different class. Asymptotically (if you wait long enough or get close enough to the critical temperature), it should turn into a "triple point" with three roads meeting.

Why did we get fooled?
The "triple point" for the cube-to-rectangle transition is hidden. The crystal moves so slowly away from the "isotropic" zone that in most experiments, it never gets far enough to show the triple point. Instead, it looks like a "bicritical" point (two roads meeting) with changing "effective exponents" (personality traits that shift as you get closer to the transition).

The author suggests that to see the real "triple point," scientists need to look at crystals with very specific starting conditions or measure them with extreme precision as they get very close to the critical temperature. They also propose that mixing different crystals (like mixing strontium and calcium) might change the starting point of the journey, potentially making the "triple point" easier to spot.

So, the mystery is solved: The crystals aren't breaking the rules; they are just taking a very long, slow detour that makes them look like their neighbors until you get close enough to see the truth. The author is confident in this explanation based on their new mathematical analysis, suggesting that future experiments should focus on measuring these shifting "effective exponents" to confirm the hidden triple point.