Exploring Dynamical Quantum Phase Transitions via the Dynamical Phase Accumulated at an Equilibrium Critical Point

In the quantum world, matter does not always settle into a single, unchanging state. Instead, it can shift abruptly between different phases, much like water freezing into ice or boiling into steam. These shifts, known as phase transitions, usually happen when a system is cooled or heated very slowly, allowing it to adjust smoothly. However, physicists have discovered a more dramatic kind of transition that occurs not in temperature, but in time. When a quantum system is suddenly jolted out of its equilibrium by a rapid change in its environment, it can undergo a dynamical phase transition. This event is marked by a sudden, sharp change in how the system behaves as it evolves, revealing a hidden structure in its motion that is invisible when things are calm. For years, scientists have studied these transitions by watching how a system reacts to a single, sharp push. But a new study suggests that the story is far more complex if we pause the action at the exact moment of instability.

Researchers have now explored what happens when a quantum system is not just pushed, but paused at the critical point where it is most unstable. Using a theoretical model of a chain of magnetic atoms, the team simulated a process where the system is driven toward a tipping point, held there for a specific duration, and then driven away. This waiting period allows the system to accumulate a subtle, invisible shift in its internal rhythm, known as a dynamical phase. The study, conducted by a team at Southwest Petroleum University, Sichuan Agricultural University, and Sichuan University, reveals that this brief pause fundamentally rewrites the rules of the transition. By tuning the length of this wait, the researchers found they could force the system to exhibit a single, sharp transition or, surprisingly, a whole series of them.

The experiment began with a chain of atoms that behave like tiny magnets, initially aligned in one direction. The researchers simulated a process where the magnetic field acting on these atoms was slowly changed to push the system toward a critical point, a specific setting where the atoms are on the verge of flipping their alignment. Instead of passing through this point immediately, the simulation held the system there for a finite amount of time. During this pause, the atoms did not sit still; they continued to evolve, accumulating a phase shift that depended entirely on how long the system waited. When the field was finally changed again to complete the transition, the system's final state carried the memory of that pause. The researchers observed that this accumulated phase acted like a tuning knob, creating a pattern of interference in the system's behavior.

In the standard scenario, where the system is driven through the critical point without stopping, the transition is characterized by a single, dominant mode of instability. This is a straightforward event where the system's behavior changes sharply at one specific moment in time. However, the new findings show that introducing a wait at the critical point changes this picture entirely. As the waiting time increases, the accumulated phase causes the system to oscillate. These oscillations create multiple points where the system becomes unstable, effectively splitting the single transition into several distinct events. The researchers identified two distinct regimes: one where the system behaves with a single critical mode, and another where it exhibits multiple critical modes. The boundary between these two states is determined by the precise length of the wait and the speed of the initial push.

The study provides a detailed map of how these regimes interact. When the system is pushed very quickly, the transition from a single mode to multiple modes happens at a specific, fixed waiting time. When the push is slow, the required waiting time grows in a predictable way, following a scaling law that relates the duration of the wait to the speed of the push. Crucially, the researchers found that at the exact boundary where the system switches from having one critical mode to many, the nature of the transition itself changes. In the single-mode regime, the system's response curve has a sharp, linear corner, similar to the point where a road makes a sudden turn. But at the boundary, this corner smooths out into a different shape, one where the slope becomes infinitely steep. This mathematical shift means that the system's reaction becomes far more sensitive and extreme at that specific moment.

To ensure these results were not unique to their specific model, the team repeated the analysis using a different type of quantum system known as an extended Su-Schrieffer-Heeger model, which describes electrons moving through a chain of atoms with varying spacing. The results were identical: the waiting time at the critical point controlled the number of transitions, and the boundary between the single and multiple modes exhibited the same change in behavior. This consistency suggests that the phenomenon is a fundamental feature of quantum systems driven through critical points, rather than an artifact of a specific setup. The researchers also noted that if the system were held at a point away from the critical instability, the effect would be much weaker or disappear entirely, confirming that the critical point itself is essential for this accumulation of phase.

The implications of this work extend beyond just understanding a specific chain of atoms. By showing that a simple pause can restructure the very nature of a phase transition, the study offers a new way to control quantum systems. It suggests that by carefully timing how long a system waits at a critical moment, scientists could potentially engineer systems to have multiple, distinct transition points or to suppress them entirely. The research also clarifies the relationship between the speed of a change and the resulting behavior, bridging the gap between sudden, jarring shifts and slow, gradual ones. While the findings are currently based on theoretical simulations, they provide a clear roadmap for future experiments. As quantum simulators using trapped ions and superconducting circuits become more advanced, they will be able to test these predictions directly, potentially opening the door to new ways of manipulating the quantum world through the simple act of waiting.