Inspired by the very recent breakthrough realization of fractional Chern insulators (FCIs) in moiré materials, the purpose of this proposal is to explore novel intertwined topological orders that do not have any known fractional quantum Hall (FQH) counterparts. In particular, focusing on engineered moiré systems, we aim to theoretically study (a) Quantum geometry and competing orders in higher Chern number bands. A first step towards understanding these systems is to characterize the band structure and its concomitant fluctuating quantum geometry. Recent work has shown that this is a very fruitful approach for understanding symmetry breaking competing states in Landau level like bands with unit Chern number, C=1. Here we will focus mostly on C>1 to understand the competitors of FCIs in these bands that by virtue of the underlying band topology lack direct Landau level / FQH analogues.
(b) Higher Chern number fractional Chern insulators. With competing phases investigated in (a) we explore in this subproject the prospects and phenomenology of FCIs in moiré bands with higher Chern number. Some states of this type were theoretically discovered in toy models over a decade ago yet unpublished experimental indications of their possible realization in moiré materials pose new fundamental questions about their nature, and about the possibility of entirely new states awaiting theoretical (and experimental) discovery.
(c) Fractional Hall crystals. The possibility of coexisting charge order and the quantum Hall effect in “Hall crystals” was suggested by Halperin and co-workers in 1989. Very recent evidence suggests that moiré materials may finally provide a platform for their realization. Here we set out to investigate if an even more exotic version with intertwined topological and charge order in terms of fractional Hall crystals may prevail at fractional band filling. Both C=1 and C>1 bands will be considered gaining insights from (a) and (b). (d) Topological defects and genons in moiré materials. Topologically ordered states have a ground state degeneracy that depends on the topology, specifically the genus, of the underlying space it resides on. A novel possibility is that lattice defects such as dislocations alter this topology and hence the ground state degeneracy. Higher Chern number fractional Chern insulators provide an ideal platform as they possess an
(intertwined) internal layer structure that may in principle facilitate this phenomenology. Here we will investigate this scenario in the context of moiré materials based on insights from subproject (a) and (b).
All of the above subprojects are closely interrelated phenomenologically as indicated above with the common theme of intertwined topological order in moiré materials. Also the numerical as well as the analytical methods needed overlap considerably. While each subproject involves a number of well defined tasks there is plenty of room for serendipitous insight and new emerging directions.