Quantum scars make their mark in graphene

Imagine a ball bouncing off the sides of a stadium-shaped billiard table. Regardless of where the ball starts, its trajectory will eventually cover the whole table, and it will do so in a seemingly irregular manner — in sharp contrast to the way in which a planet orbits a star, for example. This unpredictability is a hallmark of chaos, but if the ball is actually a quantum-mechanical particle, its chaotic behaviour manifests in the appearance of curious states that were predicted four decades ago¹. These ‘quantum scars’ have now been visualized directly in a solid-state system, which Ge et al.² report on in Nature.

In the classical case, if the billiard ball is aimed precisely, it can follow certain periodic trajectories across the stadium-shaped table: it can bounce back and forth between the table’s parallel sides, for instance, or trace out a bow-tie shape. But these trajectories are unstable — even a tiny deviation from its periodic path will result in the ball eventually visiting every point on the table, with negligible chance that it will return to its original trajectory. By contrast, the probability of finding a quantum particle is enhanced along orbits that correspond to the classical periodic trajectories, and these higher-probability states form the quantum scars.

Because quantum objects show both particle- and wave-like behaviour, quantum scars can also arise in classical systems. For example, quantum scars have been observed in fluid systems in which waves are confined by configurations that give rise to chaotic behaviours (such as those on a stadium-shaped table)^3,4. Researchers have also searched for signatures of quantum scars in condensed-matter systems such as quantum dots, which are semiconductor devices that can confine charge carriers. Finding such scars would have implications for the performance of electronics that are based on these tiny devices, because the enhanced probability of finding charge carriers along specific trajectories could boost their electrical conductance.

However, observing scars in quantum dots has proved to be a challenging endeavour: previous experiments^5–7 were limited by either insufficient spatial resolution or by sample imperfections. Ge et al. addressed these issues by combining a creative technique for quantum-dot fabrication with a method known as scanning tunnelling microscopy, which enabled them to create the dots and also image the scars (Fig. 1).

The quantum dots were made from graphene (a single layer of carbon atoms) on top of the material hexagonal boron nitride (hBN). To confine charge carriers, the quantum dots had to take the form of wells with sufficiently sharp edges, which the authors achieved by using a technique developed by members of the same team⁸. By tweaking this method to make their quantum dots stadium-shaped, instead of circular, Ge et al. could induce the charge carriers to undergo chaotic motion, rather than the regular dynamics that they reported previously.

The authors then imaged this motion with a scanning tunnelling microscope (STM), which is a device equipped with a sharp metal tip that scans across a sample. When a voltage is applied between the tip and the sample, the microscope can measure a property that is related to the probability of finding an electron at a given location as a function of energy, and this energy can be tuned by changing the voltage. The STM is therefore an ideal tool to visualize quantum scars, affording the authors extremely high energy resolution with which to image the patterns of charge carriers in their exquisitely crafted quantum dots.

Versatile neutral atoms take on quantum circuits

Ge et al. observed two kinds of quantum scar, which manifest as concentration patterns that resemble a bow tie and a streak. In line with theoretical predictions¹, each type of scar appears at several different energies. Crucially, the electrons producing these patterns behave as though they are quasi-relativistic, meaning that their energy is directly proportional to their momentum, as is true for particles moving at the speed of light. This was evident in the energies that the team measured. The energies of different scars corresponding to the same unstable orbit are expected to be equally spaced when the charge carriers are relativistic⁹, and Ge and colleagues’ experiments directly verified this prediction for at least two periodic orbits.

The study opens the door to many directions of research. The authors’ ability to confine electrons in precisely crafted quantum dots could enable researchers to investigate quantum scars in more-complex well geometries¹⁰. Another intriguing possibility is that creating extra wells inside the quantum dots could simulate the introduction of impurities, which could in turn be used to modify or even control the properties of the scars¹¹. And finally, Ge et al. conjecture that the creation and manipulation of relativistic scars in graphene could be useful for enhancing electron transport in the emerging field of electron optics, the study of electron beams that mimic the behaviour of light.

More broadly, being able to directly visualize scars in an electronic system could allow researchers to study how interactions (between charge carriers, for example) could affect these scars. The presence of interactions gives rise to states known as quantum many-body scars, which have attracted much attention in the past few years^12,13. But many-body scars are widely considered to occur in systems in which the ‘bodies’ are discrete — and interacting — objects, which makes it difficult to study them using classical waves. Perhaps Ge and colleagues’ success with graphene can align well-established research on quantum scars with efforts to probe their exciting many-body counterparts.

Competing Interests

The authors declare no competing interests.