Julien Barrier

This semester I’ve been teaching foundation year students in experimental physics. The goal for them is to find a laser’s wavelength through diffraction experiments. Because of the covid-19 pandemic, restrictions imposed by the University don’t allow the students in labs, so I have to carry out the experiment, filmed with different cameras and broadcasted to the students at home. The far-from-ideal situation led to poor accuracy in the measurement readings, hence a substantial errors on each of their measure. I’ve noticed that this was a source of issues, as many sent me e-mails or came to the drop-in session, to understand why their measurements did not match the expected value. This actually leads to some interesting discussions about experimental errors. I’d like to share some of this.

Why we estimate errors

The foundation year students come to the class with the (justified) feeling that the job is done once they obtain a numerical value for the quantity we ask them to measure or calculate. This is different at university, as we are also concerned with the accuracy of the measurement. The accuracy is expressed with an experimental error on the measurement. As we, scientists, are interested in measurements for the sake of comparing them with different experiments, theories, or use them to predict new behaviours, the value of the error becomes crucial to the interpretation of the result. Let me develop this.

In my lab, the students are asked to find the wavelength of a laser. We give them the laser specification (including the wavelength), and most of the time, they get something different from the specification. The laser’s wavelength is supposed to be 532nm (green light beam). When they measure 528nm, how can we get an evidence for a discrepancy? Can we conclude directly that the laser’s specification is inacurate? There could be 3 possibilities:

1. if the experimental error is $\pm 5$nm then this result looks satisfactory, and in agreement with the expected value: 528±5 is consistent with 532
2. if the experimental error is $\pm 0.5$nm then the measurement loos inconsistent with the expected value. We should worry that either the experimental result or the error estimate is wrong. Another possibility is that the specification for the laser is different from its real value: 528.0±0.5 is inconsistent with 532
3. if the experimental error is $\pm50$nm (that may happen if we take only one measurement) then the result is consistent with the expectation. However, the accuracy is so low that we will be unable to detect significant differences. The range of confidence of this result would be 478 to 578nm, that is from blue to yellow: a rough estimation would do better.

For a given measurement the interpretation (our measurement is accurate, the laser specifications are inaccurate or we should find how to do a more precise experiment) depends only on the uncertainty of this measurement. If we don’t know it, we are completely unable to judge the significance of the result.

Definition of the experimental error

The measurement error (or observational error), $\delta x$, is the difference between the measured value $x$ for a determined quantity, and its true value $X$. Note that in this context, an error is not a mistake. We can write $X = x + \delta x$

This is a notion that is essential in experimental physics. There are two kinds of errors: random error and systematic error:

Random error

When we measure a physical quantity in the lab, we find that repeating the experiment results in slightly different measures. In my lab that can be due to inaccuracy of reading distances from a ruler that is filmed, for example. It comes from the inability of any measuring device (and to certain extent, to the scientist) to give infinitely accurate measures. This phenomenon is called random error and will be detected through a statistical analysis. Usually, the result of the measure is characterised by a probability distribution, centred around the true value for pure random error (see panel b in the figure below).

Systematic errors

Systematic errors are errors that do not vary with repeated measurements. They result in a measurement that is simply wrong or biased. In my lab, I place a diffraction grating on a ruler. If this grating is not exactly aligned at the center of the reading, statistical studies will not detect the error. Panel c in the figure below shows an example of a systematic error. Generally, systematic errors can have various origins:

• calibration error: A famous example is the experience of Millikan. He found a wrong value of the electron charge as he considered a wrong value for the air viscosity.
• omitted parameter: Suppose that we measure a spin echo. The spin relaxation time would depend on the temperature. Carrying out the measurement in Portugal or Finland without accounting for the temperature would give two different results.
• flawed procedure: Suppose we are measuring a resistance. If the impedance of the voltmeter is not large enough, the approximation of a constant current through the resistor will no longer be valid. Systematic errors are hard to detect, but once they are known, they can be accounted for easily.

As shown on the figure above when making a series of repeated measurements, the effect of random errors will produce a spread of measures scattered around the true value. Systematic errors on the contrary would cause the measurement to be offset from the correct value, even when individual measurements are consistent with one another.

Difference between error and uncertainty

I have found that both uncertainty and errors are mistaken, some students talking about calculating errors instead of uncertainty. This is sometimes true for established scientists.

The uncertainty $\Delta x$ estimates the importance of the random error. Without a systematic error, the uncertainty defines an interval around the measured value, that includes the true value with a determined confidence level (usually 2/3) We can estimate this two ways:

statistically (type A)

In this method we characterise the probability distribution of the $x$ values. With a good evaluation of the mean and the standard deviation of the distribution. This is done through statistical analysis over an ensemble of measures of $x$. Without systematic error, the mean value is the best estimate of the true value $X$, while the uncertainty $\Delta x$ is directly linked to the standard deviation of the distribution, and defines an interval in which the true value $X$ can be found with a known confidence level.

other means (type B)

When we have no time or resources to make additional measurements, we estimate $\Delta x$ from the specifications of the measurement instruments. For example, in the experiment I carry out in my lab, we can estimate the uncertainty in the number of lines per millimeter in the diffraction grating to be around 5%. For 300lines per millimeter, that would result in an uncertainty of $300\pm15$l/mm. When reading the position of that diffraction grating on the ruler, we can estimate the uncertainty on the reading of about half to one graduation. When reading numeric values, we typically find a precision $\Delta$ on the measure that is $\pm 2$ times the last digit and $\pm 0.1\%$ of the measured value. This, of course, depends on the quality of the instrument and calibration, and can be found in the specifications of the instrument. Calibration error is mostly accounted for in the 0.1% part of the value we read. Noise and random errors are typically found in the last digit. When we evaluate the uncertainty over a measure with a type B method, we usually divide the precision $\Delta$ by $\sqrt{3}$ (that supposes a uniform probability distribution of width $2\Delta$).

Estimating the uncertainty with a type A method

Each time we record a measure in which the source of error is purely random, we are likely to record a different value to the previous measurement. In that case, a set of repeated measurements of $x$ would give an estimate of $\Delta x$. It is likely that the dispersion of the measurements is larger than the individual error on each measurements. In that case, the standard dispersion will be used as the uncertainty: \[ \Delta x \approx \sqrt{\frac{N}{N-1}} s \] With $s$ the standard deviation of a set of $N$ measurements.

The mean value will be given by: \[ \left< x \right> = \frac{1}{N} \sum_{i=1}^N x_i \] where $x_i$ are the individual measurements in the set. Since the mean value of the set is statistically more likely to be close to the true value than any one of the individual measurement chosen at random, the uncertainty of the mean value, $\sigma_m$ is smaller than the uncertainty $\delta x$ of any one of the measurement. We can prove that $\sigma_m = \frac{\delta x}{\sqrt{N}}$.

What it means in undergrad labs

In the lab, the uncertainty is an essential value to judge the accuracy of a measurement, that makes us able to judge of the quality of a measure. There are good practices to mitigate the influence of errors and reduce the uncertainty of a measurement. The first is to analyse statistically, whenever it is possible, the value, and consider only the mean values, with the associated uncertainty. Second, I would also recommend to record everything that could be relevant to the problem on a neat notebook. It becomes all the more important in the remote lab to write down results manually, with an estimation of their uncertainty, as soon as they are collected: you cannot mitigate errors if you don’t record them in the first place.

references

• Data Analysis for Physical Science Students, L. Lyons, Cambridge University Press 1991
• Practical Physics, G. L. Squires, Cambridge University Press 2001

The past semester, I have been teaching in the physics third year lab. One of the experiments consists in emasuring the electrical resistivity of a graphene layer with a lock-in amplifier.

Before this lab, to measure a resistence, students are used to use a d.c. potentiometer to measure an electrical resistance, so I could notice some qualm about the new technique. The d.c. potentiometer is an easy system for such applications, but prone to measurement errors because of thermoelectric and drift effects that can be hard to eliminate, as well as issues when the measured signal falls below the error voltage of the instrument. For this reason, methods involvig the use of an a.c. current allow for more precise electrical measurements. In fact, if one applies an a.c. current and measure the differential output voltage, between successive maxima and minima (or, more conveniently, the root mean square of this value, $V_{rms}$) and interval this over a number of modulation cicles, one can get an improved estimate of the voltage difference, that eliminates the thermal drifts and low-frequency noise.

I think this method is of particular importance in modern mesoscopic physics, and justifies a few words on this blog.

The lock-in method.

The basic experimental arrangement in lock-in detection in a four-probe configuration consists in the following: a sine-wave current is generated at a frequency $f_0$ and constant amplitude and passed through a pair of contacts (A and B for example) of the measured device, and a voltage drop across another pair of contacts (C and D) is measured with the lock-in amplifier. This acts as a highly-tuned demodulator, and gives a d.c. output that is proportional to the amplitude of the a.c. voltage $V$ between C and D. The essential feature of the lock-in amplifier is that it works at the reference signal of frequency $f_0$, that has the same frequency as the sample. This methods allows to measure a current and a voltage with a precision of about 1 part in 10⁴.

Importance of phase-sensitive detection

As I just wrote, a lock-in is a phase-sensitive detector, measuring the difference voltage of interest, using a synchronous reference voltage. Detection with respect to a synchronous reference allows to use long averaging time to be able to improve the detection below background noise level. This is important in comparison to the use of amplitude demodulation with non-linear devices (like enveloppe detectors), that would make no difference between the signal and noise components. The phase-sensitive detector would measure only the frequency of interest $f_0$. Lock-in amplifiers are usually able to measure both the amplitude of the voltage, and the phase difference.

Noise reduction.

In a typical measurement, the resistance of metallic samples can vary from 10^-3Ω at room temperature, to 10^-6Ω at low temperatures. In the Kelvin-range, a few millivolts can heat a device substantially and counterbalance all the efforts made to reach low-temperatures. Under such conditions, the maximum range that can decently be used is about 10µV, so the measured voltage drop would be in the order of the nanovolt. In a typical experiment, leads from room temperature to liquid-helium temperatures would carry a thermoelectric voltage (noise) in the order of the µV to the mV. It is thus necessary to be able to measure below the noise level. Synchronous detection allows to single out the component of the signal at a specific reference frequency and noise, therefore reducing the measured noise.

There are three main types of noise: Johnson (or thermal) noise, shot noise and 1/f noise.

Johnson noise

Every resistive system would generate a noise voltage at its therminal, that is due to the thermal fluctuations of the electron density. We can characterise the noise as: \[ V_{JN} = \sqrt{4k_B T R \Delta f} \] where $k_B$ is Boltzmann’s constant, $T$ the temperature of the sample of interest, $R$ its resistance and $\Delta f$ the bandwith of the measurement. This means that the amount of noise that is captured by the lock-in is determined directly by the bandwidth of the input signal. State of the art lock-ins would have a bandwidth frequency around 500kHz, that means an effective noise at 300K, at the amplifier input that is $V_{JN} \approx 40\sqrt{R}$nV. Here, high input bandwidth allow to reduce Johnson noise.

Shot noise

Shot noise comes from the discrete nature of the charge carriers. As there is always non-uniformities in the electron flow generating current noise, the shot noise would be given by: \[ I_{SN} = \sqrt{2qI_{a.c.}\Delta f} \] where $q$ is the electron charge, $I_{a.c.}$ the current and $\Delta f$ the bandwidth. With a lock-in, this is minimal (that is $I_{SN} \approx 400\sqrt{I_{a.c.}}$10^-9).

1/f noise (or Flicker noise)

Flicker noise results fluctuations in resistance due to a current flowing through the resistor. It can be between 10nV and 1µV per volt applied across a resistor, with a 1/f spectrum. As a result, measurements at low frequencies are noisier.

References

• Betts, J.A, Signal processing, modulation and noise, London 1970
• Meade, ML, Lock-in amplifiers: principles and applications, London 2013
• Principles of lock-in detection and the state of the art, Zurich Instruments white paper, 2016.

This is a press release for my recent paper of high-mobility Brown-Zak fermions

A group of researchers lead by Sir Andre Geim at The University of Manchester (UK) have discovered and characterised a new family of quasiparticles named Brown-Zak fermions in graphene-based materials. The team achieved this by aligning the atomic lattice of a graphene layer to that of an insulating boron nitride sheet, dramatically changing the properties of the graphene sheet.

The study follows years of successive advances in graphene-boron nitride superlattices which allowed the observation of a fractal pattern known as the Hofstadter’s butterfly. Today the researchers report another highly surprising behaviour of particles in such structures upon the application of high magnetic fields. “It is well known that in zero magnetic field electrons move in straight trajectories and if you apply a magnetic field they start to bend and move in circles”, say Julien Barrier, a PhD student who carried out the experimental work and lead author, “in graphene aligned with the Boron nitride, they also start to bend, but if you set the magnetic field at specific values electrons have a behaviour that corresponds to straight line trajectories, as if there is no magnetic field anymore!”. “Such a behaviour is radically different from textbook physics” adds Dr Piranavan Kumaravadivel, who fabricated the graphene devices. “We attribute this fascinating behaviour to the formation of novel quasiparticles at high magnetic fields” says Dr Alexey Berdyugin who contributed to the discussion over the results, “Those quasiparticles have their own unique properties and exceptionnally high mobility despite the extremely high magnetic field”. As published in Nature Communications on 13th November 2020 (doi: 10.1038/s41467-020-19604-0), the work describes how electrons behave in an ultra-high-quality superlattice of graphene with a revised framework for the fractal features of the Hofstadter’s butterfly. Fundamental improvements in graphene devices fabrication and measurement technques in the past decade have made this work possible.

A new quasiparticle

“The concept of quasiparticles is arguably one of the most important in condensed matter physics and quantum many-body systems. It was introduced by the theoretical physicist Lev Landau in the 1940s to depict collective effects as a one-particle excitation” explains Julien Barrier “They are used in a number of complex systems to account for many-body effects.”

Until now, the behaviour of collective electrons in graphene superlattices were thought in terms of the Dirac fermion, a quasiparticle that has unique properties resembling photons (particles with no mass), that replicate at high magnetic fields. However, this did not account for some experimental features, like the additional degeneracy of the states, nor did it match the finite mass of the quasiparticle in this state. The authors propose Brown-Zak fermions to be the family of quasiparticles existing in superlattices under high magnetic field. This is characterised by a new quantum number that can directly be measured. Interestingly, working at lower temperatures allowed them to lift the degeneracy with exchange interactions at ultra-low temperatures.

“Under the presence of a magnetic field, electrons in graphene start rotating with quantised orbits. For Brown-Zak fermions, we managed to restore a straight trajectory of tens of micrometres under high magnetic fields up to 16T (500,00 times earth’s magnetic field). Under specific conditions, the ballistic quasiparticles feel no effective magnetic field.” explain Dr Kumaravadivel and Dr Berdyugin.

High mobility of Brown-Zak fermions

In an electronic system, the mobility is defined as the capacity for a particle to travel upon the application of an electrical current. High mobilities have long been the holy grail when fabricating 2D systems such as graphene, as such materials would present additional properties (integer and fractional quantum hall effects), and potentially allow for the creation of ultra-high frequency transistors, the components at the heart of a computer processor. “For this study we prepared graphene devices that are extra-large with a very high level of purity”. says Dr Kumaravadivel. This allowed us to achieve mobilities of several millions of cm²/Vs, which means particles would travel straight across the entire device without scattering. Importantly, this was not only the case for classical Dirac fermions in graphene, but also realised for the Brown-Zak fermions reported in the work.

These Brown-Zak fermions define new metallic states, that are generic to any superlattice system, not just graphene and offers a playground for new condensed matter physics problems in other 2D material based superlattices.

Julien Barrier added “The findings are important, of course for fundamental studies in electron transport, but we believe that understanding quasiparticles in novel super- lattice devices under high magnetic fields can lead to the development of new electronic devices.” The high mobility means that a transistor made out of such a device could operate at higher frequencies, allowing a processor made out of this material to perform more calculations per unit of time, resulting in a faster computer. Applying a magnetic field would usually scale down the mobility and make such a device unusable for certain applications. The high mobilities of Brown-Zak fermions at high magnetic fields open a new perspective for electronic devices operating under extreme conditions.

High resolution illustrations

Graphene and boron nitride (hBN) both present a hexagonal structure. hBN is 1.8% larger, so stacking the two will create a Moiré superlattice, with a size constant hundreds of times larger.

The following image represents different transport regime of quasiparticle in graphene and graphene superlattices: Left: an electron in a graphene sheet propagates in straight trajectories when a current is applied through the material (model of the Dirac fermion); Center: when a magnetic field is applied, perpendicular to the sheet, the electrons have circular orbits; Right: For some exact values of the magnetic field in graphene-hBN lattices, we observe Brown-Zak fermions, that have a straight trajectory despite the high magnetic field. This translates in a lower resistance of the device.

The following picture adds a doping vs magnetic field conductance map and illustrate the paper’s main result: The magnetic field is varied along the vertical axis. Horizontal Yellow streaks show propagation of Brown-Zak fermions, propagating along straight trajectories with high mobility (low resistance), whereas slanted indigo lines show the cyclotron motion around Brown-Zak fermions. The slope of these lines enabled us to get the degeneracy (and find an additional quantum number) of these new quasiparticles.

The following image combines the two:

Additional images:

• BrownZakfermions-01
• BrownZakfermions-03
• superlattice-01

Notes for editors

The paper, Long-range ballistic transport of Brown-Zak fermions in graphene superlattices (doi: 10.1038/s41467-020-19604-0), published in Nature Communications, is available from Friday, 13th November 2020. Before that, a preprint is available at: https://arxiv.org/abs/2006.15040. It should be cited as: J. Barrier et al., Long-range ballistic transport of Brown-Zak fermions in graphene superlattices, Nature Communications, Issue 11. (2020)

More information about graphene is available from www.graphene.manchester.ac. uk.

Over the past few months, anyons hit the news for two important discoveries. Here I try to give a short introduction to anyons and explain why this is interesting.

About anyons

In condensed matter physics, we usually define quasiparticles, that are not elementary particles, but excitations of collective electrons. This concept has been introduced by Landau and Kapitza to treat helium superfluidity, where they suggested that these excitations could be treated as particles, with a different mass and momentum than the electron itself. Since then, a lot of problems have been treated with quasiparticles, as Brown-Zak fermions we introduced recently, but there are a large range of quasiparticles, with exotic behaviours. Usually, quasiparticles fall into two possible categories: bosons or fermions.

Let’s focus on position, for example (but any quantum number associated with a given quasiparticle would work), and describe a two-particle system with a wavefunction $ \vert \psi (\mathbf{x}_1,\mathbf{x}_2) > $, where $\mathbf{x}_1$ is the position of the first quasiparticle, and $\mathbf{x}_2$ is the position of the second one. If we swap the positions of the two quasiparticles, the resulting state should be $ \vert \psi (\mathbf{x}_2,\mathbf{x}_1) > $.

In the case of bosons - that obey the Bose-Enstein statistics - switching their position doesn’t change the wavefunction: $ \vert \psi (\mathbf{x}_1,\mathbf{x}_2) > = \vert \psi (\mathbf{x}_2,\mathbf{x}_1) > $.

For the case of fermions, that obey the Fermi-Dirac statistics, the Pauli principle forces the wavefunction to be rotated by 180°: $ \vert \psi (\mathbf{x}_1,\mathbf{x}_2) > = - \vert \psi (\mathbf{x}_2,\mathbf{x}_1) > $, i.e. $ \vert \psi (\mathbf{x}_1,\mathbf{x}_2) > = e^{i\pi} \vert \psi (\mathbf{x}_2,\mathbf{x}_1) > $.

Anyons are neither fermions nor bosons. Switching anyons two different places would induce a rotation of some intermediate angle: $ \vert \psi (\mathbf{x}_1,\mathbf{x}_2) > = e^{i\alpha} \vert \psi (\mathbf{x}_2,\mathbf{x}_1) > $. The way to pick up a phase angle that is neither 0 or $\pi$ is to use the Aharonov-Bohm effect to switch particles in a 2D-electron gas. This would induce a phase shift. Importantly, anyons can only exist in 2D systems, as they would violate the standard model otherwise.

Fractional statistics and the evidence of the phase factor

In April, a team at ENS and Sorbonne Université (Paris) have realised anyons and observed their phases. The results have been reported in Science. The authors used a 2D electron gas made with a GaAs/AlGaAs heterostructure, cool it to about 10mK and study it in the presence of a magnetic and electric field. Experimental observation of anyon statistics is challenging as it is based on interferometry, for which theoretical interpretation is difficult.

The interferometer was studied in the fractional quantum hall regime, where quasiparticles have a charge that is 1/3 of the electron charge (Laughlin quasiparticles), for long suspected to be anyons. In this system, chiral conducting channels form along the edges of the device, and tunneling of Laughlin quasiparticles can occur between quantum point contacts (QPC). Ideally, a quasiparticle would be emitted from QPC1 and reach QPC2. It becomes interesting if the quasiparticle reaches a third one: QPC3. As the two anyons reach the QPC from opposite sides. In the case of fermions, they would block eachother rom tunneling through QPC3 because of the Pauli principle. In the case of bosons, correlations between drain currents would appear when the quasiparticles combine together. For abelian anyons, that is different: some correlation is expected, but specific details depend upon the anyon statistics. In a nutchell, the author basically swap particles from a scattering experiment and look at the correlations where particles arrive. This observation of expected anyonic properties provides experimental support to this quasiparticle.

Anyon braiding to build a quantum computer

Much recently, this work has led to further developments. A team at Purdue university interfered anyons in a similar experimental setup and posted it last week on arXiv.org (2006.14115). In their experiment, they create and destroy anyonic states on the bulk of the 2DEG, and produce anyons running on the edges. Between two QPCs, there are two posible paths anyons can undergo. At the end of the journey, an interference pattern would appear. This interference pattern shows the relative amount of rotation between the two paths, with discontinuities that show evidence of the creation and annihilation of anyons in the bulk of the material.

The ability to make anyons appear or disappear raises promises for the building of a topological quantum computer. In fact, pairs of anyons could encode information with the angle phase. The basic interpretation is using the number of circles they made around one another to store data. As the path or perturbations do not alter them, this offers a kind of topological protection to the computation.

References:

• L. Saminadayar et al 1997, Observation of the e/3 Fractionnally Charged Laughlin Quasiparticle, Phys. Rev. Lett.
• H. Bartolomei et al 2020, Fractional statistics in anyon collisions, Science
• J. Nakamura et al 2020, Direct observation of anyonic braiding statistics at the ν = 1/3 fractional quantum hall state

We just submitted the first first-author paper of my PhD, and uploaded on arXiv: “Long-range ballistic transport of Brown-Zak fermions in graphene superlattices”. (arXiv: 2006.15040) In this post, I highlight what we define as Brown-Zak fermions, and why their ballistic transport is important.

Brown-Zak fermions?

There is currently a huge interest in graphene superlattices and artificial stacks that can be obtained by crystallographic alignment of 2D crystals. Until now the community has mostly focused on additional gaps that appear in the Hofstadter-Wannier diagram. We believe that some attention should be directed towards the family of new metallic states that appear there. Particularly, the fermions appearing in rational fractions of the magnetic flux quantum have been eluded as “replica of Dirac fermions” (or described as “third-generation Dirac fermions”). This picture is ambiguous, as we show that these fermions are a different family of quasiparticles, that have a different spectrum as the Dirac fermions, and are described by a new quantum number.

High mobility in 2DEG

In usual semiconductor 2DEG (e.g. GaAs/AlGaAs), the increase in device quality and induced electronic mobility have resulted in the discovery of a plethora of new effects. For example, the highest mobilities achieved in the early 1980s were about several thousands of cm²/Vs, that allowed the discovery of the integer quantum Hall effect (von Klitzing), and rapid increase allowed further phenomena to be observed, like the odd denominator fractional quantum Hall effect (Tsui, Stormer) and the even-denominator fractional QHE (Willet, Stormer) with mobilities about 10⁶ cm²/Vs. In general, new high-quality 2D electronic experimental systems with high mobility always deliver new physics.

Ballistic Brown-Zak fermions

In this paper, we report graphene superlattices of high quality, such as the Brown-Zak fermions exhibit mobilities above 10⁶ cm²/Vs, despite the presence of high magnetic fields, that always force conventional quasiparticles into curved cyclotron trajectories. This is comparable to the best values achieved for Dirac fermions in graphene devices or semiconductor 2DEGs. We demonstrate this in an ambiguous way (negative ballistic transfer experiment), proving that Brown-Zak fermions are Bloch quasiparticles propagating along straight trajectories.

Additionally, the high quality of our superlattices allow us to get new information about the physics of Brown-Zak fermions. Particularly, we show that these quasiparticles are described by an additional quantum number q, that is the mini-valley degeneracy, and depends upon the attached fractions of the flux quantum. This comes over the spin and valley degrees of freedom from the main Dirac spectrum.

Finally, we show that at low temperatures, Brown-Zak fermions are strongly affected by electron-electron interactions. Hence, the classical single-particle Hofstadter model becomes obsolete, as it is too simplistic for the proper description of this family of new quasiparticles. The regime of strong interactions is also found to be full of strange effects, that we can only partially understand at the moment, such as bended fans ore staircase-like features.

In a nutshell, this work shows that Brown-Zak fermions can be seen as a new high-quality experimental platform to study unconventional quasiparticles (high-field but still Bloch-type). This should stimulate further experimental and theoretical work, with our paper showing the way.

References:

• J. Barrier et al 2020, Long-range ballistic transport of Brown-Zak fermions in graphene superlattices, Nat. Commun.
• K. von Klitzing, G. Dorda, and M. Pepper 1980, New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance, Phys. Rev. Lett.
• D. C. Tsui, H. L. Stormer, and A. C. Gossard 1982 Two-Dimensional Magnetotransport in the Extreme Quantum Limit, Phys. Rev. Lett.
• R. Willett, et al 1987 Observation of an even-denominator quantum number in the fractional quantum Hall effect, Phys. Rev. Lett.