Flat Electronic Bands and Quantum Geometry

Electrons with very heavy mass reveal their secrets.

W. Kandinsky - Intersecting Lines (1923)

Flat bands, also known as dispersionless electronic states, have revolutionized condensed matter physics. Previously, we encountered various instances of flat bands, such as Landau levels in magnetic fields. But the game-changer came with twisted bilayer graphene (TBG), which allowed us to engineer and control flat bands in real materials, ushering in a new era of research.

Direct experiments quickly revealed that the flat bands in twisted bilayer graphene were not coincidental or based on lucky material choices. They stemmed from fundamental principles, much like Landau levels. We soon realized that this concept extended beyond twisted bilayer systems. By adding more layers (trilayer, multilayer, or "infinite"-layer structures), we could enhance properties like critical temperature, albeit with meticulous optimization. Unfortunately, multilayers often faced instability due to layer displacement, disrupting the flat bands. However, replacing graphene with other van der Waals materials could still yield almost flat bands at specific twist angles, like in transition metal dichalcogenides.

Amidst this fascinating array of diverse examples, an intriguing question arises: How can we unravel the secrets of creating a recipe—a recipe of immense power—for crafting a new system with flat bands that can be precisely controlled? A recipe that allows us to tailor properties such as bandwidth and Chern numbers to our exact specifications. To embark on this transformative journey, we must first embark on the systematic classification and organization of flat bands, weaving them together into a unified framework—a cookbook that holds the key to unlocking extraordinary possibilities.

After years of studying flat bands, I embarked on a journey to redefine the criteria for band flatness in real space, using wave functions in coordinate representation. This approach offers a pivotal advantage, as the wave function encapsulates essential information about band flatness, topology, and "quantum geometry" (crucial for optimizing flat bands for correlated electronic phases like Fractional Chern Insulators). To my astonishment, the new band flatness criterion highlights a common origin of band flatness—self-trapping in real space. This principle applies to atomic insulators, Landau Levels, artificial atomic lattices (with wave functions localized inside the plaquette), twisted bilayer graphene, and more. As a result, we can achieve a fundamental classification of flat bands, enhancing our understanding of the pathways and obstacles towards realizing flat electronic bands with higher Chern numbers.

My motivation has been to find a perfect flat bandit realise holographic quantum matter (such as the Sachdev-Ye-Kitaev state). In the process, I have found that the band flatness is tightly linked to the concept of quantum geometry, is a unified formalism merging geometry and topology, and is responsible for the shapes and size of electronic orbitals in solids. Armed with a fundamental understanding of the origins of band flatness, the next steps would involve understanding how quantum geometry affects the electronic transport in dispersionless quantum states.

Further reading: Alexander Kruchkov - Quantum geometry, flat Chern bands, and Wannier orbital quantization