Fractional Chern insulators with a non-Landau level continuum limit

Abstract

Recent developments in fractional quantum Hall (FQH) physics highlight the importance of studying FQH phases of particles partially occupying energy bands that are not Landau levels. FQH phases in the regime of strong lattice effects, called fractional Chern insulators, provide one setting for such studies. As the strength of lattice effects vanishes, the bands of generic lattice models asymptotically approach Landau levels. In this article, we construct lattice models for single-particle bands that are distinct from Landau levels even in this continuum limit. We describe how the distinction between such bands and Landau levels is quantified by band geometry over the magnetic Brillouin zone and reflected in the electromagnetic response. We analyze the localization-delocalization transition in one such model and compute a localization length exponent of 2.57(3). Moreover, we study interactions projected to these bands and find signatures of bosonic and fermionic Laughlin states. Most pertinently, our models allow us to isolate conditions for optimal band geometry and gain further insight into the stability of FQH phases on lattices.

Interplay of Band Geometry and Topology in Ideal Chern Insulators in Presence of External Electromagnetic Fields

“…We provided deformations of the Berry curvature and quantum metric upon applying a small magnetic field and went on investigating the properties of this new quantum geometry in the insulating and metallic regime. We point out that our work is to be distinguished from recent advances implementing strong magnetic fields which push the system into the Hofstadter phase [13,14]. In the insulating regime, we showed that the Hall conductivity is not affected by small magnetic fields, in agreement with previous work.…”

“…Our work dealt exclusively with properties of ideal bands and it is an interesting future endeavor to study the consequences of our work once interactions are implemented on the band. This is similar in spirit, yet significantly different from [14], which presents as well a study of fractional Chern insulators in magnetic fields. Our work differs from theirs, since they employed strong magnetic fields B allowing them to use magnetic Brillouin zones and work entirely with its quantum geometry.…”

“…In contrast our work is viable for non-rational flux as well, albeit for weak magnetic flux. It will be interesting to combine their work with ours, similar to [64], which employed a strong magnetic field B like [14] to reach the Hofstadter phase and perturbed additionally with a small magnetic field δB like ours and of non-rational flux. Taken together the magnetic field B + δB should give rise to a new quantum geometry governing the properties of the interacting system subject to large non-rational magnetic flux.…”

“…The non-uniformity of the quantum geometry is mirrored in the emergent curvatures (14). Both have two poles as depicted in figure 5 indicating that neither U xy nor V xy are gauge invariant, which is traced back to the fact that tr(g α ) and Ω α vary in the FBZ, respectively.…”

“…Its imaginary part is the well established Berry curvature characterizing the topological phase of the band, while its real part is the quantum metric. The study of the quantum metric, and thus also the quantum geometric tensor, has attracted attention on many fronts in the recent past including superconducting systems [6][7][8][9], quantum phase transitions [10,11], magnetic signatures [12][13][14][15], quantum topology and geometry [16][17][18][19][20], flat band systems [21], in nonadiabatic evolution [22], as marker distinguishing insulators from metals [23], non-hermitian systems [24,25], in twisted bilayer graphene [26][27][28][29][30][31][32][33] and in dimensions higher than two [34][35][36][37][38][39][40][41]. Theoretical measurements have been proposed and measurements of the quantum geometric tensor have been performed in [42][43][44][45][46][47][48][49].…”