Newton's third law states that for every action there is an equal reaction in the opposite direction. This law has been relieving us for over 400 years. It explains why we don’t fall through the floors (the floor pushes on us too) and why paddling makes a boat glide through water. If a system is in equilibrium, there is no energy going in or out. This is called reciprocity. These systems can be mathematically described using statistical mechanics. This branch of physics explains the behavior of collections of objects. This allows researchers to model the conditions that lead to phase transitions in matter. These are when one state transforms into another.
Many systems exist, and many of them are far from being in equilibrium. The most obvious example of this is life. Our metabolism, which transforms matter into energy, keeps us out of equilibrium. A dead body is one that has retreated into equilibrium.

Newton's third law is irrelevant in such systems. The concept of equal-and-opposite is no longer valid. Vincenzo Vitelli, a condensed-matter theorist at Chicago, said that you can imagine two particles where A interacts differently with B than A. These nonreciprocal relationships are found in systems such as neuron networks, particles in fluids, and, on a much larger scale, social groups. Predators, on the other hand, eat their prey.

These unruly systems are not well represented by statistical mechanics. Nonreciprocity prevails out of equilibrium. The law can be broken easily by flocking birds: Individuals change their flight patterns when they are unable to see behind them. Bird A and bird B don't interact in the same manner. It's not reciprocal. Similar non-reciprocal interactions exist between cars whizzing along a highway and stuck in traffic. Engineers and physicists working with metamaterials, which derive their properties from the structure of a substance, have used nonreciprocal elements in order to create acoustic and quantum devices.

Because each component has its own power source, many of these systems can be kept out of equilibrium. ATP for cells and gas for cars are two examples. All these energy sources and mismatched reactions create a dynamical system that is beyond statistical mechanics. How do we analyze the phases of such dynamic systems that are constantly changing?

Vitelli and his coworkers see the answer in mathematical objects known as exceptional points. A singularity is an area where two or more of a system's characteristic properties are indistinguishable, and mathematically become one. An exceptional point is a place where the mathematical behavior of a system is different from that at other points. These exceptional points are often used to describe strange phenomena such as lasers, in which energy is constantly gained and lost.

The team discovered that these exceptional points can also control phase transitions within nonreciprocal systems. These exceptional points are not new. Mathematicians and physicists have been studying them for decades in many settings. They've never been so commonly associated with this type phase transition. Cynthia Reichhardt, a physicist at Los Alamos National Laboratory in New Mexico, stated that this is something no one has ever thought of before. "So, you can bring all of the machinery that is already available about exceptional points to study those systems."

This new work draws connections between a variety of phenomena and areas that have, for many years, not seemed to have much to share. Robert Kohn, of New York University's Courant Institute of Mathematical Sciences, said that "I believe their research represents rich territory for mathematical advancement."

When Symmetry breaks

It was not birds or neurons that the work started, but quantum weirdness. Two years ago, Ryo Hanai (a postdoctoral researcher at University of Chicago) and Peter Littlewood (Hanai's advisor) were studying a type of quasiparticle known as a Polariton. (Littlewood sits on the scientific advisory board at the Flatiron Institute, a division of Simons Foundation that also funds this independent publication.

A quasiparticle isn't a particle per se. It is a collection quantum behaviors that look like they should be connected to particles en masse. Photons, which are the particles responsible for light, combine with excitons to form a polariton. These quasiparticles are also known as photons. Because of their low mass, polaritons can move extremely fast and can form a state called Bose-Einstein condensation (BEC). This is where separate atoms are all dissolved into one quantum state at higher temperatures than other particles.

It is however difficult to use polaritons to make a BEC. It is leaky. It is leaking because some photons continue to escape. To make up the difference, light must be continuously pumped into the system. This means that it is out of equilibrium. Hanai said, "From the theory aspect, that's what we found interesting."