Nonequilibrium Dynamical Quantum Phase Transitions in the Extended Agassi Model

In the vast, invisible world of quantum physics, particles do not always behave like the solid objects we see around us. Instead, they exist in a state of constant potential, capable of shifting between different arrangements of energy and order. Sometimes, a group of these particles can undergo a sudden, dramatic reorganization known as a phase transition. We see this in everyday life when water freezes into ice or boils into steam; the substance changes its fundamental nature because of a shift in temperature. In the quantum realm, these changes can happen even without a change in temperature, driven instead by the sudden alteration of the forces holding the particles together. When these shifts occur in a system that has been jolted out of its calm state, they are called dynamical quantum phase transitions. Scientists study these events because they reveal how complex systems of many particles respond to sudden shocks, offering a window into the rules that govern everything from the nuclei of atoms to the behavior of exotic materials.

A team of researchers has recently explored these sudden shifts using a specific mathematical model designed to describe groups of fermions, a type of particle that makes up much of the visible universe. They focused on a system called the extended Agassi model, which acts like a simplified laboratory for watching how particles interact when they are paired up or pushed apart by different forces. The researchers wanted to see what happens when they abruptly change the strength of three specific types of interactions within this model: the pairing of particles on the same energy level, the interaction between groups of particles acting like magnetic monopoles, and the pairing of particles across different energy levels. By simulating these sudden changes, known as quenches, they could watch how the system's memory of its original state fades and how new patterns of order emerge.

The study began by mapping out the stable states of this system before any changes were made. The researchers found that depending on the strength of the forces, the system settles into one of five distinct phases, much like how water can be a solid, liquid, or gas. Some of these transitions between phases happen smoothly, where the system gradually changes its shape. Others happen abruptly, where the system snaps from one state to another. The team discovered that the transition driven by the monopole-like force was smooth and continuous, similar to how a magnet slowly aligns its internal structure. In contrast, the transitions driven by the same-level pairing and the cross-level pairing were abrupt, resembling a sudden snap rather than a gradual slide. This distinction between smooth and sudden changes in the stable state provided a baseline for understanding what would happen when the system was shaken.

To investigate the non-equilibrium behavior, the researchers simulated sudden changes to the strength of each of the three forces, one at a time. They started the system in a calm, balanced state and then instantly altered the force, watching how the system reacted over time. They measured this reaction by tracking a quantity that represents how much the system remembers its starting point. As time passed, this memory faded, but at specific moments, the fading process developed sharp, jagged features. These sharp points, which the researchers call cusps, signaled that a dynamical quantum phase transition had occurred. Crucially, the researchers found that these dramatic moments did not happen at the same time or under the same conditions as the smooth or abrupt changes seen in the stable states. The system's reaction to the sudden shock was governed by its own internal rhythm and the specific way the particles overlapped with their new environment, rather than simply mirroring the rules of its stable state.

The most surprising discovery was that the nature of these sudden transitions depended entirely on which force was changed, even when the stable states suggested they should behave similarly. When the researchers changed the same-level pairing force, the system developed a sharp, jagged signature that indicated a first-order transition, matching the abrupt nature of the stable state change. Similarly, when they changed the cross-level pairing force, the system also showed a first-order signature in its stable state and dynamical reaction. However, despite both being first-order transitions, their specific mathematical scaling behaviors were distinct: the same-level pairing followed a pattern close to three-halves, while the cross-level pairing followed a pattern close to one-third. Most notably, when they changed the monopole-like force, the system exhibited a smooth, second-order transition in its stable state, and the dynamical reaction followed a unique pattern that was distinct from the other two. This revealed that the "personality" of a quantum phase transition in a moving, changing system is not fixed solely by the type of transition it has when it is calm. Instead, the specific path the system takes and the way its particles are arranged during the shock create a unique dynamical identity, even among transitions of the same order.

To understand the depth of these changes, the researchers analyzed how the sharpness of these transitions grew as they increased the size of the simulated system. They found that as the system grew larger, the jagged features became more pronounced, eventually becoming true mathematical breaks in the smooth flow of time. By measuring how quickly these features sharpened, they extracted specific numbers that describe the scaling of the transition. For the monopole-like force, the number was close to two-thirds. For the same-level pairing, it was close to three-halves. For the cross-level pairing, it was close to one-third. These different numbers confirmed that each force drives the system into a different class of behavior, even when the starting and ending points look similar. The researchers also looked at the invisible zeros of a complex mathematical function that describes the system's evolution, finding that these zeros clustered together in the complex time plane exactly where the sharp transitions occurred, providing a microscopic explanation for the jagged features seen in the data.

The work demonstrates that the rules governing how a quantum system reacts to a sudden shock are more complex and varied than previously thought. The study shows that the order of a transition in a calm system does not dictate the order of the transition when the system is in motion. Instead, the specific channel through which the energy is injected creates a unique dynamical landscape. This finding suggests that the universe of quantum critical phenomena is richer than a simple map of stable states would imply. By using a model that can be solved exactly, the researchers provided a clear, detailed view of how different forces compete to shape the future of a quantum system. Their results offer a new way to think about how matter behaves when pushed to its limits, showing that the path a system takes through a phase transition is just as important as the destination it reaches.