🔬 condensed matter

How Quasicrystals Remember: Hierarchical Memory Under Cyclic Shear

This study demonstrates that two-dimensional dodecagonal quasicrystals exhibit hierarchical memory under cyclic shear through reversible, bistable phason-like tile rearrangements, a mechanism absent in their periodic approximants and distinct from the behavior of amorphous solids.

Original authors: Edwin A. Bedolla-Montiel, Marjolein Dijkstra

Published 2026-07-15

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Edwin A. Bedolla-Montiel, Marjolein Dijkstra

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a material that is the perfect "Goldilocks" zone between a rigid, perfectly ordered crystal and a messy, disordered glass. This is a quasicrystal, a structure so unique it looks like a tiled floor that never quite repeats its pattern, no matter how far you walk. In this study, researchers at Utrecht University asked a simple but tricky question: Can these weird, non-repeating tiles "remember" where they've been after being squished and stretched over and over again?

The answer, found through computer simulations, is a resounding yes.

The Memory Game: Hysteresis Loops

Think of the quasicrystal as a giant, intricate puzzle made of squares and triangles. The researchers subjected this puzzle to a "cyclic shear," which is like grabbing the top of the puzzle and sliding it back and forth, then letting it go, then sliding it the other way.

When they pushed the puzzle gently (below a certain "yielding" point), something magical happened. The material didn't just snap back to where it started; it remembered the exact path it took. If they pushed it a little, stopped, and then pushed it a bit harder, the material would follow a specific path. But if they then went back to the smaller push, the material would return exactly to the spot it was at before the bigger push.

This behavior is called Return Point Memory (RPM). It's like a hiker who, after taking a detour up a hill, can retrace their steps perfectly to return to the exact spot they were at before the detour, no matter how many times they wander off. The researchers saw this as "nested hysteresis loops" on their graphs—tiny loops inside bigger loops, all fitting together perfectly like Russian nesting dolls.

The "Tile-Switch" Secret

How does a pile of atoms remember? The researchers zoomed in to find the microscopic heroes of the story. They discovered that the memory is stored in tiny, localized groups of tiles that act like bistable switches.

Imagine a small cluster of five tiles that can exist in two distinct shapes: State A and State B.

When the material is squeezed just right, these tiles snap from Shape A to Shape B.
When the squeeze is released, they snap right back to Shape A.

The researchers call these "tile-switch hysterons." They are the fundamental memory units. What's fascinating is that these switches are incredibly sharp and precise. They don't wobble; they just flip. Furthermore, when they flip, they create a ripple effect in the surrounding material that looks exactly like a classic physics prediction called an Eshelby field—a specific pattern of elastic stress that spreads out in a four-lobed shape, like a cloverleaf.

The "Yielding" Line

However, this memory has a limit. The researchers found a specific threshold, a strain amplitude of about 0.065.

Below 0.065: The material is a perfect memory-keeper. The tiles flip back and forth, and the system returns to its previous state.
Above 0.065: The material "yields." The gentle, reversible flips turn into permanent, messy rearrangements. The tiles get stuck in new positions, forming "shear bands" (like cracks or permanent wrinkles). Once you cross this line, the memory is lost, and the material has changed forever.

The Role of Temperature

The team tested two different "preparation temperatures" for their simulated materials: kBT/ϵ = 0.05 (cooler) and kBT/ϵ = 0.15 (warmer).

Both versions remembered perfectly.
However, the warmer version was more "active." It had more of these tile-switches firing up, and they formed longer, string-like chains of movement. The cooler version had fewer, more isolated switches.
Crucially, the ability to remember didn't change with temperature; only the amount of activity did.

What It Is NOT

It is important to note what this study ruled out. The researchers compared their quasicrystal to a periodic crystal (a standard, repeating crystal). When they tried the same memory test on the regular crystal, it failed completely. The regular crystal just bounced back elastically with no hysteresis loops and no memory.

This proves that the memory isn't just about having a dodecagonal (12-sided) shape. It specifically requires the disorder and the unique "phason" degrees of freedom found in quasicrystals. Without that specific type of structural disorder, the "tile-switch" mechanism cannot exist.

How Sure Are We?

The findings are based on athermal quasistatic shear simulations. This means the researchers used powerful computers to model the atoms at zero temperature (no thermal jiggling) and moved them very slowly. They observed these behaviors clearly in simulations with 4,096 and 16,384 particles.

While the results are robust within the simulation, the paper notes that identifying these specific "tile-switch" units is much harder, and often impossible, in real, messy amorphous solids (like glass) because they lack the underlying square-and-triangle framework that makes the quasicrystal's switches so easy to spot. The study suggests that by studying these quasicrystals, we might finally understand how memory works in more chaotic materials, but for now, this is a discovery made in the digital world of computer models.

In short: Quasicrystals are like a puzzle that can remember its history, provided you don't push it too hard. The memory lives in tiny, snapping clusters of tiles that flip back and forth, a mechanism that only works because the material is beautifully disordered.

Technical Summary: How Quasicrystals Remember: Hierarchical Memory Under Cyclic Shear

Problem Statement
Periodically driven materials can store information about their driving history, a phenomenon known as return point memory (RPM). While RPM is well-established in disordered systems like amorphous solids and ferromagnets—where it is attributed to the collective behavior of bistable switching elements (hysterons)—its existence in quasicrystals remains an open question. Quasicrystals occupy a unique structural middle ground between periodic crystals and disordered glasses, possessing long-range orientational order but lacking translational periodicity. It is unclear whether their specific structural disorder and phason degrees of freedom can support the localized, reversible rearrangements necessary for mechanical memory, or if their order precludes the rugged energy landscapes required for bistable switching.

Methodology
The authors investigate the zero-temperature mechanical behavior of two-dimensional dodecagonal quasicrystals (DDQCs) formed by square-triangle tilings.

Model System: DDQC configurations are generated using a core-softened interaction potential at a number density $\rho\sigma^2 = 0.94$ . Samples are prepared via molecular dynamics at two distinct parent temperatures ( $k_BT/\epsilon = 0.05$ and $0.15$) to test sensitivity to preparation history, followed by energy minimization to reach athermal states.
Simulation Protocol: The systems undergo athermal quasistatic shear (AQS). This involves applying an incremental affine shear strain ( $\Delta\gamma = 10^{-4}$ ), followed by energy minimization using the FIRE algorithm to return the system to a local energy minimum.
Memory Protocol: To probe memory, the authors employ a hierarchical cyclic shear protocol ( $\gamma_3 \to \gamma_2 \to \gamma_1 \to \gamma_2 \to \gamma_3$ ) with varying amplitudes below the yielding transition. They monitor the stress-strain response, energy per particle, and mean squared displacement (MSD) to identify limit cycles and hysteresis.
Microscopic Analysis: To identify the elementary units of memory, the authors utilize non-affine squared displacement ( $D^2_{min}$ ) to localize switching cores. They perform bond-network reconstruction to map particle positions to square-triangle tilings before and after switching events, allowing for the identification of specific tile rearrangements.

Key Results

Yielding Transition: The DDQC exhibits a distinct yielding transition at a critical strain amplitude $\gamma_y \approx 0.065$ . Below this threshold, the system behaves elastically with reversible dynamics. Above $\gamma_y$ , the response becomes irreversible, characterized by persistent shear bands and locally transformed regions with different symmetry, leading to a loss of quasicrystalline order.
Loop-Return Point Memory (Loop-RPM): Below the yielding transition, the system exhibits robust Loop-RPM. Hierarchical cyclic driving produces nested hysteresis loops in the stress-strain response. The system returns exactly to its pre-excursion limit cycle (within numerical precision) after a cycle of varying amplitudes. This behavior is observed in samples prepared at both low and high parent temperatures, though higher-temperature samples exhibit broader reversible excursions and larger dissipative responses.
Microscopic Mechanism: The fundamental switching units responsible for memory are identified as tile-switch hysterons. These are localized, phason-like rearrangements where a specific cluster of particles (a compact core of five particles in the representative case) undergoes a reversible transformation between two distinct tiling configurations (e.g., converting two triangles into one square and four triangles).
- These hysterons are sharply bistable in a local distance-overlap coordinate.
- They possess an Eshelby-compatible elastic far field, exhibiting a quadrupolar displacement pattern consistent with a localized shear transformation in an elastic medium.
- Unlike amorphous solids where identifying such units is difficult due to the lack of a structural framework, the tiling nature of the DDQC allows for unambiguous identification of the switching particles.
Hierarchical Organization: The spatial organization of memory is hierarchical. At higher preparation temperatures, reversible activity forms elongated, string-like segments. As the driving amplitude increases, additional particles are recruited along these same pathways, creating a nested hierarchy of activity (core $\subset$ mid $\subset$ outer) that mirrors the nested hysteresis loops.
Role of Disorder: A defect-free periodic approximant of the DDQC, when subjected to the same protocol, exhibits purely elastic, hysteresis-free behavior with no nested loops. This confirms that Loop-RPM in DDQCs arises from the structural disorder enabling reversible tile rearrangements, rather than the dodecagonal symmetry alone.

Significance and Claims
The paper demonstrates that dodecagonal quasicrystals are capable of encoding and recovering mechanical memory through a hierarchical mechanism of nested hysteresis loops, analogous to RPM in amorphous solids. The primary contribution is the microscopic identification of tile-switch hysterons as the elementary bistable units in a quasicrystal. This finding links the phason degrees of freedom inherent to quasicrystals directly to the hysteron framework of mechanical memory.

The authors emphasize that this work provides a unique platform to study mechanical memory because the underlying tiling structure allows for a precise microscopic definition of switching events—a task often impossible in amorphous solids. The study establishes that while the macroscopic memory criterion (loop closure) is robust across preparation temperatures, the temperature influences the spatial extent and organization of the cooperative reversible motion. The results suggest that quasicrystals serve as an intermediate system between ordered crystals (which lack memory) and amorphous glasses (which exhibit memory), offering a pathway to understand how mechanical memory mechanisms evolve as structural order is progressively lost.