Exact Landau Level Description of Geometry and Interaction in a Flatband

Abstract

Flatbands appear in many condensed matter systems, including the two-dimensional electron gas in a high magnetic field, correlated materials, and moiré heterostructures. They are characterized by intrinsic geometric properties such as the Berry curvature and Fubini-Study metric. The influence of the band geometry on electron-electron interaction is difficult to understand analytically because the geometry is in general nonuniform in momentum space. In this work, we study the topological flatband of Chern number C = 1 with a momentum-dependent but positive definite Berry curvature that fluctuates in sync with Fubini-Study metric. We derive an exact correspondence between such ideal flatbands and Landau levels and show that the band geometry fluctuation gives rise to a new type of interaction in the corresponding Landau levels that depends on the center of mass of two particles. We characterize such interactions by generalizing the usual Haldane pseudopotentials. This mapping gives exact zero-energy ground states for short-ranged repulsive generalized pseudopotentials in flatbands, in analogy to fractional quantum Hall systems. Driving the center-of-mass interactions beyond the repulsive regime leads to a dramatic reconstruction of the ground states towards gapless phases. The generalized pseudopotential could be a useful basis for future numerical studies.

Quantum Metric Induced Phases in Moiré Materials

“…It was argued [12,45] that the role of the weak magnetic field, similar to a smaller r, is to make the Berry curvature more uniform and thus stabilizing the zero-field FCI. Noting that in this system, the FS metric fluctuates in sync with the Berry curvature [12,39,41], the magnetic field or the ratio r also flattens the FS metric. As a consequence, we here provide a complementary interpretation of the competition between FCI and CDW based on the FS metric.…”

“…The most prominent example thereof is the quantum Hall system exhibiting exactly flat bands in the continuum limit, and a rich phenomenology of strongly correlated states [32]. Flat bands of lattice models are known to in principle exhibit an even richer phenomenology [33][34][35][36], for which the recently engineered superlattice Moiré materials provide remarkably versatile flat band structures [37][38][39][40][41][42][43] that can be controlled in experiments [44,45].…”

Interacting topological Dirac magnons

“…The almost flat topological bands give rise to a more uniform Berry curvature and quantum metric tensors that mimic Landau level physics of interacting particles [46]. The quantum geometric properties are illustrated in Fig.…”

Relating the topology of Dirac Hamiltonians to quantum geometry: When the quantum metric dictates Chern numbers and winding numbers

“…Recent advances have revealed the central role played by the Fubini-Study metric [1] in various fields of quantum sciences [2], with a direct impact on quantum technologies [3,4] and many-body quantum physics [2,5]. In condensed matter, the quantum metric generally defines a notion of distance over momentum space, and it was shown to provide essential geometric contributions to various phenomena, including exotic superconductivity [6][7][8] and superfluidity [9], orbital magnetism [10,11], the stability of fractional quantum Hall states [12][13][14][15][16][17], semiclassical wavepacket dynamics [18,19], topological phase transitions [20], and lightmatter coupling in flat-band systems [21]. Besides, the quantum metric plays a central role in the construction of maximally-localized Wannier functions in crystals [22,23], and it provides practical signatures for exotic momentum-space monopoles [24,25] and entanglement in topological superconductors [26,27].…”

“…In the context of Chern insulators, relations between the Berry curvature, the Chern number, the quantum metric and the quantum volume were recently established and understood based on the Kähler structure of the quantum-states space [41,42]. These relations are known to play an important role in the formation and stabilization of fractional Chern insulators [12][13][14][15][16][17]43].…”