# Flat Electronic Bands and Where to Find Them

It is possible to classify all the flat bands in the quantum materials in a "periodic table".

**W. Kandinsky - Intersecting Lines (1923) **

Dispersionless electronic states ("flat bands") is a paradigm change in condensed matter physics. Occasionally, in past we were discovering different cases of flat bands, such as (nearly) flat bands in Argon ice, Landau levels (flat bad in magnetic field), and more theoretical constructions such as flat bands in artificial atomic lattices (Lieb, Kagome, etc.). However, with advent of __twisted bilayer graphene____ (TBG)__, where the controllable engineering of flat bands is possible, we have entered into the new era of the flat band research.

It has become clear shortly after that the [__direct experiment__] that the flat bands in twisted bilayer graphene are no coincidence, and not a lucky choice of material parameters; in fact, they are of the __fundamental origin__, the same way as the Landau levels. We have later realized that this construction is not limited to twisted bilayer, but one can add extra layers to enhance the Tc (__trilayer__; __multilayer and "infinite"-layer construction__), while a more careful __optimization work__ is required (in practice, multilayers are generically not stable against layer displacement, which ruins the flat bands). One can go even further with replacing the graphene with other van der Waals materials, at still have a (nearly) flat bands at certain twist angles, for example with __transition metals dichalcogenides__.

With all these diverse and *heterogeneous* examples in mind, one might ask a simple question: What is a controllable recipe for creating a new flat band system with the given properties (bandwidth, Chern numbers etc.)? To compose this cookbook, one first need to systematize and classify the flat bands within the same framework.

The systematization of the flat bands should start from the most fundamental cases of perfectly flat bands; all the other (nearly) flat bands can be derived from those classes. However, now we face a fundamental difficulty, the perfectly flat bands are featureless in momentum space; they might or might not carry auxiliary information on the band topology (e.g. compare Atomic Insulator to Landau Levels), however from the momentum-space observational point of view, they are hardly distinguishable.

After studying the flat bands for several years, I have decided to reformulate the criteria for the band flatness in the real space, through the wave functions in the coordinate representation. This has an important advantage that the wave function per se contain all the information on band flatness, topology and "quantum geometry" (a concept important in optimizing a flat band for the correlated electronic phases, such as Fractional Chern Insulators). The new band flatness criterion, to my surprise, points on the __common origin of the band flatness__, which is (self) trapping in the real space: this applies to atomic insulators, Landau Levels, artificial atomic lattices (wave function localized inside the plaquette), twisted bilayer graphene, etc. Thus, the fundamental classification of the flat bands is possible, together with understanding of the routes and obstructions towards flat electronic bands of higher Chern numbers.

My motivation for this study has been to find a perfect flat band for __realizing the Sachdev-Ye-Kitaev state__ (spoiler: such a perfect flat band should be written through meromorphic functions; the work in progress is how to approximate this in the real material). However, I am not alone who is looking in this direction, towards the classification of perfectly flat bands on the lattice; I want to highlight the recent __work by the Princeton group__ on the similar topic from the quantum chemistry point of view.

With the fundamental understanding of where the band flatness is rooting from, the further steps to be taken is to prepare a realistic system with flat bands of higher Chern numbers, with the TBG-like heterostructures being a good platform for development.

Preprint: Alexander Kruchkov - __Origin of band flatness and constraints of higher Chern numbers. __