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Magic-angle twisted bilayer graphene (MATBG) is a material created by stacking two sheets of graphene onto each other, with a small twist angle of about 1.1°. At this "magic angle," electrons move very slowly, which can lead to the emergence of highly correlated electron states.

Due to its unique properties and characteristics, MATBG has become the focus of numerous studies rooted in physics and materials science. Some physicists discovered that when an external magnetic field is applied to MATBG, the flat energy bands in the material transform into a fractal-like energy pattern known as a Hofstadter spectrum.

Researchers at University of Washington, Florida State University and other institutes recently carried out a study aimed at further investigating the emergence of these energy patterns in ultraclean MATBG.

Their paper, in Nature Â鶹ÒùÔºics, reports the observation of two distinct strongly interacting topological phases in the material, known as Chern insulators and fractional quantum Hall states.

"Our findings were unexpected, coming as a pleasant byproduct of another project," Dr. Xiaodong Xu, senior author of the paper, told Â鶹ÒùÔº.

"Since its discovery in 2018, there has been great interest in MATBG, owing to its exotic quantum phases of matter, including unconventional superconductivity, new forms of ferromagnetism, and topology. Our original intention was to study orbital ferromagnetic states underlying the anomalous Hall effect in MATBG."

To better understand the orbital ferromagnetic states that could underpin the anomalous Hall effect previously reported in MATBG, the researchers first created a device based on a monolayer WSe₂ substrate.

Their plan was to directly control the anomalous Hall effect in MATBG by optically pumping the WSe₂ with circularly polarized light. To do this, however, they first needed to characterize the basic properties of the MATBG device they developed.

"To our surprise, our device appeared to be among the highest quality samples ever created, yielding a wealth of exciting new data to understand," said Dr. Matthew Yankowitz, a co-senior author of the work.

"Despite the many new features we observed, understanding our measurements was a challenge that took several years to crack. Our results appeared to fit within the framework of the Hofstadter butterfly—a recursive electronic structure first predicted by Douglass Hofstadter over 50 years ago and realized only in recent years with the advent of moiré materials—yet the details were too complicated to be captured by any existing theory at the time."

A breakthrough in the team's research occurred when Dr. Minhao He, lead author of the paper, attended a conference—Moore foundation's EPiQS Postdoctoral Symposium—alongside co-lead author Dr. Xiaoyu Wang. At this conference, Wang presented a new interesting theory that could explain the team's experimental observations.

The theory extended the Hofstadter butterfly framework to a regime in which strong interactions between electrons play an important role. At the conference, He and Wang initiated a close collaboration that culminated in the publication of the recent paper.

"In our experiment, we studied strongly interacting Hofstadter states by performing electrical transport measurements of our MATBG sample at millikelvin temperatures and in high magnetic fields of several Teslas," explained Xu.

"We then compared our measurements with state-of-the-art Hartree-Fock calculations, which were recently developed to specifically capture the role of interactions in modifying the electronic band structure and inducing symmetry breaking phases in MATBG under a strong magnetic field."

The experiments performed by the researchers led to two important observations. Firstly, in their ultra-pure MATBG device, the team observed cascades of topological states known as symmetry-broken Chern insulators (SBCI).

"These states are unusual forms of Chern insulators that spontaneously enlarge the area of the moiré unit cell by a rational value," said Minhao He.

"Although SBCI states have previously been observed in MATBG, they have so far only appeared at apparently random moiré filling factors. What we have uncovered is a remarkable cascade sequence of SBCI states, with a sequence of Chern numbers mimicking their parent correlated Chern insulators."

The second notable achievement of this recent study was the observation of a series of fractional quantum Hall (FQH) states that only appeared when a strong magnetic field was applied to MATBG. These states follow a so-called Jain-sequence, a pattern or hierarchy in their emergence that is aligned with predictions of the so-called composite fermion theory.

"In stark contrast to FQH states in conventional 2D electron gas systems that strengthen upon increasing magnetic field, the observed FQH states in our sample abruptly disappear above a magnetic field of ≈10 T," explained Xu.

"This phenomenon arises because the effective magnetic length can approach the moiré wavelength at relatively low magnetic fields, which is impossible to achieve in non-moiré 2D systems."

The measurements collected by the researchers and the calculations they performed suggest that the FQH states observed in their MATBG platform are of an unconventional nature. To explain their emergence, the team drew from the theory devised by Wang and Dr. Oskar Vafek (co-senior author of the work), showing that they could be understood as fractional Chern insulators forming in a magnetic field.

"Our Hartree–Fock band analysis shows that these fractional states arise out of strained magnetic sub-bands with finite bandwidth and non-uniform, non-ideal quantum geometric properties. This unusual quantum geometric structure points to a potential description of these states within the framework of magnetic field-induced FCIs," explained Vafek.

The recent work by this team of researchers further highlights the unique potential of MATBG for studying correlated quantum states and topological phases of matter. In addition, it offers a new theoretical interpretation for emerging quantum states, which could be tested and improved in future studies.

"An exciting opportunity for our next studies will be to investigate the connection and interplay between fractional Chern insulator and fractional quantum Hall effect," added Xu.

"Meanwhile, we also plan to continue the exploration of optical probes and control of correlated and topological states in moire superlattice heterostructure."

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