Exact Landau Level Description of Geometry and Interaction in a Flatband

Fractional Chern insulators with a non-Landau level continuum limit

“…FCI phases show a wide range of stability dependent on several factors, including the band flatness and strength of interactions, which were identified early on [5], as well as band geometry, which corresponds to quantities such as the Berry curvature and Fubini-Study (FS) metric [6]. Under a set of conditions that can be imposed on the band geometry, a Chern band can resemble the lowest Landau level, which is believed to yield "ideally" stable FCI phases [10][11][12][13][14]. The relevance of these conditions has been investigated in numerical simulations [15][16][17][18][19], but not so far in models where the role of the conditions can be isolated from one another.…”

Hierarchy of Ideal Flatbands in Chiral Twisted Multilayer Graphene Models

“…Recently fractional Chern insulators (FCIs) [4][5][6] are theoretically predicted and experimentally observed in TBG flatbands [8][9][10][11]. One important factor to the stability of FCIs in this system is due to the ideal geometry of the flatbands in the fixed point chiral limit [12,13], where each flatband's Berry curvature Ω k is non-vanishing and strictly proportional to its Fubini-Study metric g ab k by a constant determinant-one matrix ω ab [14][15][16]:…”

Information geometry of quantum critical submanifolds: relevant, marginal and irrelevant operators

“…The quantum metric [6]-a Riemannian metric over the space of quantum states based on state distinguishability-was shown to be a quantity able to probe zero temperature quantum phase transitions among these phases as one varies some parameter in the system [7][8][9]. In the context of Bloch bands, the quantum metric in momentum space, in the particular setting of flat bands, has received a lot of attention recently, as it gives geometrical contributions to the characterization of a range of different phenomena, such as exotic superconductivity and superfluidity [10][11][12][13], the stability of fractional Chern insulating phases [14][15][16][17][18][19] and light-matter coupling [20]. The quantum metric is also central in determining maximally-localized Wannier functions [21,22], and it can be used as a practical indicator for exotic momentum-space monopoles [23,24].…”

“…More recently, relations between the quantum metric and the Berry curvature, which determines topological invariants of the system and gives rise to Berry-phase effects, have been established and understood based on the Kähler structure of the space of quantum states [27,28]. These relations have come to play an important role in recent studies concerning the so-called ideal Chern bands, which are presumed to be ideal candidates for hosting fractional Chern insulating phases, the associated band structure engineering and transport [19,[29][30][31][32][33][34]. More general relations between the quantum metric and topology of quantum states were derived in the context of Dirac Hamiltonians [35].…”