Memory-driven Topological Defects and Unconventional Long-Range Order

In the study of how matter organizes itself, physicists have long relied on a set of rules that describe how systems settle down when left alone. When a collection of particles reaches a state of balance, known as equilibrium, their behavior is governed by well-understood principles. One of the most famous of these is a rule stating that in a thin, flat world, thermal jiggling from heat should always prevent a system from maintaining a perfect, long-range order. Imagine trying to keep a line of people standing perfectly still in a crowded room; the constant bumps and shoves from the crowd would eventually knock them out of alignment. This principle, known as the Mermin-Wagner theorem, suggests that in one or two dimensions, true order is impossible at any temperature above absolute zero. For decades, this seemed like a hard limit on what nature could achieve in low-dimensional spaces.

However, the natural world is not always static. Many systems, from living cells to complex machines, are constantly being pushed by forces that depend on their own past actions. This is the realm of feedback, where a system's history influences its future. While scientists have traditionally viewed the time it takes for a signal to travel through a feedback loop as a nuisance—a delay that causes errors or confusion—recent thinking suggests this delay might be a feature rather than a bug. If a system remembers its past states with a specific time lag, it might be able to create entirely new kinds of order that are impossible in a static, balanced world. This raises a compelling question: can the memory of a system's own history be used to build structures that defy the usual laws of equilibrium, creating stability where none should exist?

A team of researchers at Shanghai Jiao Tong University has taken up this challenge by designing a theoretical model of interacting rotors that possess a built-in memory. Instead of just reacting to their neighbors, these rotors also react to their own past positions, but with a deliberate time delay. The researchers found that this simple addition of a "memory force" fundamentally changes the physics of the system. In a two-dimensional arrangement, the delay causes the rotors to spontaneously start spinning, breaking the symmetry of time itself. More surprisingly, this spinning state gives rise to a unique type of topological defect—a flaw in the order of the system—that does not exist in nature under normal conditions. These defects appear as vortices, or swirling patterns, that travel along the boundaries between different regions of the material. What makes them extraordinary is that vortices spinning in opposite directions move in opposite directions along these boundaries, a behavior the researchers describe as helical. It is a structured flow where the direction of motion is locked to the direction of the spin, creating a one-way street for these swirling defects that is robust against the noise of heat.

The study goes further by looking at a one-dimensional version of this system, where the rotors are arranged in a single line and interact with their own history over long periods. In standard physics, a one-dimensional line of particles cannot maintain a true, long-range order because thermal noise would inevitably scramble any alignment. The Mermin-Wagner theorem predicts that such a system should remain disordered at any temperature. Yet, the simulations performed by the researchers show that when the memory of the system is strong and long-range, it can stabilize a state of true order even at finite temperatures. The memory effect effectively acts as a glue, holding the system together against the jiggling of heat. This results in a phase transition—a sudden change from disorder to order—at a specific temperature, a phenomenon that was previously thought impossible for a one-dimensional system without external driving forces. The researchers calculated that this transition follows a specific set of scaling laws, with a dynamic exponent of four, which is distinct from the values seen in conventional magnetic systems.

To understand how these findings were reached, the team constructed a mathematical model where each rotor is connected to a heat bath, simulating the random kicks of thermal energy, and also coupled to its neighbors. The crucial ingredient is the memory term, which calculates a force based on the difference between the rotor's current angle and its angle at a previous time. In the two-dimensional model, the memory kernel is short-range in time, meaning the rotor remembers its state from a fixed moment ago. When the researchers simulated this system, they observed that after a rapid cooling process, the system settled into a metastable state. In this state, the rotors formed distinct regions with opposite spinning directions, separated by domain walls. Trapped along these walls were vortices that oscillated in number and moved in a coordinated, helical fashion. The researchers tracked the total number of these vortices and found that it oscillated at a frequency determined by the delay time, providing direct evidence of the system's non-equilibrium nature.

In the one-dimensional model, the memory kernel was designed to decay slowly over time, representing a long-range temporal interaction. The researchers found that for a specific rate of decay, the system could maintain a correlation between distant points that did not fade away, even as the distance increased. This is the hallmark of true long-range order. By analyzing the data from their simulations, they identified a critical temperature where the system transitions from a disordered state to this ordered state. At this critical point, the correlations decay according to a power law, a signature of a phase transition. The researchers confirmed that the dynamic exponent governing the speed of fluctuations in this system is four, a value that emerges directly from the interplay between the long-range memory and the thermal noise. These results suggest that the system is not merely fluctuating but has entered a new, stable phase of matter driven by its own history.

The researchers also proposed a way to bring these theoretical ideas into the physical world using active mechatronic metamaterials. They envision a setup consisting of an array of robotic nodes, each equipped with a motor, a sensor to measure its angle, and a controller. The controller would sample the position of each node at a high frequency, store the data in a buffer, and then calculate the forces based on both the current positions of neighbors and the past positions of the node itself. By injecting a controlled amount of random noise to simulate temperature, the system could be tuned to explore the predicted phases. This experimental platform would allow scientists to observe the helical vortices and the one-dimensional phase transition in real time, verifying whether the engineered memory can indeed create the exotic states predicted by the theory.

The implications of this work extend beyond the specific models studied. The research demonstrates that time-delayed feedback is a powerful tool for engineering non-equilibrium physics, offering a route to create states of matter that have no equilibrium counterpart. By introducing a memory effect, the researchers have shown that it is possible to circumvent fundamental theorems that limit order in low-dimensional systems. The discovery of helical vortices, where opposite spins move in opposite directions, adds a new chapter to our understanding of topological defects, showing that they can exhibit directed transport in ways that are not possible in static systems. While the current results are based on simulations and theoretical models, the proposed experimental realization provides a clear path forward. If successful, this work could open the door to designing materials and machines that use their own history to maintain stability and perform complex functions, turning the delay of feedback from a problem to be solved into a resource to be harnessed.