Lifting the veil of topological censorship

Topological protection gives physical phenomena unprecedented robustness to all sorts of perturbations; but in doing so it exercises topological censorship by hiding all sorts of interesting and important microscopic information. Recent experiments have yielded microscopic information of precisely the kind that is hidden by such topological censorship.

The work of Douçot, Kovrizhin, and Moessner provides a detailed microscopic theory that goes beyond this topological censorship. They not only identify an unexpected phenomenon—the meandering edge state carrying a topologically quantized current—that contradicts common expectations, but also identify mechanisms that allow tuning between qualitatively different microscopic implementations corresponding to a single topologically protected global quantity.

The research is published in the journal Proceedings of the National Academy of Sciences.

Introduction: Topological Physics

The 2016 Nobel Prize in Physics was awarded to David J. Thouless, F. Duncan M. Haldane and J. Michael Kosterlitz “for their theoretical discoveries on topological phase transitions and topological phases of matter.”

One of the key predictions behind the Nobel Prize was that at very low temperatures, atoms and electrons can form new, exotic topological states of matter. These are distinct from the usual states of matter, which include crystals, gases and liquids.

These new states have been termed “topological” because their special properties arise from the geometric structure of their quantum wave functions, which in turn make these states extremely robust.

In other words, to destroy these states, it would be necessary to “undo” the knots of their wave function. This robustness is the basis of the phenomenal accuracy of quantization observed in the quantum Hall effect, the founding experiment of the discipline in the 1980s (Nobel Prize for Klaus von Klitzing, 1985), and which led to a redefinition of the resistance standard in metrology.

Topological protection implies topological censorship

Scientists have been particularly interested in this “topological protection” because it could be used (as proposed by Alexei Kitaev) in future quantum computers to protect quantum information from errors. This has given rise to new theoretical designs for quantum computers. These designs are currently being explored by experimental laboratories and industry.

However, topological protection also implies “topological censorship.” In other words, topology casts a veil over the local properties of these states, making it difficult to test them at a deeper level using various experimental probes.

What we usually observe in experiments are global universal properties, such as quantized resistance. Topological censorship therefore hides from the observer entire classes of interesting and potentially useful information. We can make a comparison with a black hole, whose internal properties are hidden from us by the event horizon.

Topological censorship is useful because it ensures that even oversimplified theories ultimately provide topologically correct results, at the potential expense of the microscopic accuracy of any given experiment. Such a simple theory implies the notion that all current in the quantum Hall effect is carried exclusively by states that hug the boundary (edge) of the experimental sample.

This is indeed the standard theoretical picture of the quantum Hall effect. This picture has been tested in many experiments and very often justifies their observations.

However, exciting new experiments by groups at Stanford and Cornell have made surprising observations that challenge this standard picture. They have discovered that the current in a so-called Chern insulator can be tuned from a current along the edge, as in the standard picture, to a much stronger volume current.

This directly challenges topological censorship: the quantized current can be set to flow in the volume). In the article by Proceedings of the National Academy of SciencesA collaboration of researchers from MPI-PKS (Dresden) and Paris have provided an analysis that theoretically lifts the veil of topological censorship: they reproduce the experimental results very well.

Their work has made it possible to identify the mechanisms that enable mass transport. In particular, they have identified a sinuous conduction channel capable of transporting a quantized mass current.

They note that “the existence of states that have the visual appearance of a narrow-sided channel (is not necessary). Rather, a broad, winding channel, resembling a stream flowing through its marshy floodplain rather than a barren channel, is perfectly adequate.”

The authors of the theoretical work state in their paper: “Our work answers the question: ‘Where does the charge current, famous for being quantized, flow in a Chern insulator?’

“This issue has received considerable attention in the context of the quantum Hall effect, but progress in this field has been hampered by the lack of local probes, and no consensus has emerged so far. The fundamental problem is that topological protection is excellent at hiding local information (such as the spatial distribution of the current), a phenomenon we call topological censorship.

“Two recent experiments, which used local probes to determine the spatial distribution of current in insulating Chern (Bi, Sb) heterostructures₂You₃have addressed the lack of experimental data in the case of the anomalous quantum Hall effect. These experiments have led to unexpected, although very different, conclusions. Here we provide the theory explaining one of these experiments.”

Background: Chern insulator experiments

Chern insulators were predicted in 1988 by Duncan Haldane, one of the 2016 Nobel Prize winners, and considered a mathematical curiosity until their experimental realization in 2009. Importantly, they do not require a magnetic field to realize the quantum Hall effect.

In the new experiments with Chern insulators, it was possible to detect local magnetic fields using a SQUID magnetometer and, in doing so, to map the distribution of the current flowing through the sample. According to the standard picture, the current should have flowed strictly along the edges of the sample, but what Katja Nowack et al. discovered was quite remarkable. They observed the electron current flowing everywhere, depending on the voltage applied to the system.

These experimental observations were at odds with the standard picture of the quantum Hall effect, and for a while there was no theoretical explanation for this behavior. The work of the three theorists offers such an explanation. Their theory explains the distribution of current flow measured by the Cornell group, thus confirming that current in a Chern insulator can indeed flow inside the sample.

This experimental and theoretical work begins to end the topological censorship that has reigned for almost half a century and calls for new experimental investigations into the topological states of matter.