Flat Bands in Crystalline or Moiré Materials: From Topology to Electron-Electron Interactions

Abstract

Engineering exotic phases of quantum matter, where the non-trivial topology of electron wave functions encounters strong electron-electron interactions, is a central goal of condensed matter physics. This thesis explores the rich interplay between topology and electron interactions in flat band systems, focusing on crystalline materials and moiré structures like twisted bilayer and trilayer graphene. We begin by presenting a general framework for constructing perfectly flat bands in bipartite crystalline lattices, applicable to systems in all symmetry space groups, with any orbital content, and with or without spin-orbit coupling. This approach enables us to construct a comprehensive topological classification of crystalline flat bands using Topological Quantum Chemistry, providing crucial insights into their potential for hosting exotic quantum phases. In twisted bilayer graphene (TBG), we analytically compute the scanning tunneling microscopy (STM) signatures of its various correlated ground state candidates. Our study uncovers complex valley coupling effects and reveals the absence of expected Kekulé distortions in certain intervalley-coherent states. This finding highlights how the STM signal can distinguish between different many-body states, serving as a powerful tool for probing the nature of correlated phases in TBG. Expanding our investigation to twisted symmetric trilayer graphene (TSTG), we derive its many-body Hamiltonian, which incorporates Coulomb interactions and uncovers hidden symmetries, offering new perspectives on the many-body physics of TSTG. Finally, we explore the thermoelectric transport properties of TBG through the topological heavy-fermion (THF) model. The THF model elucidates the dual nature of electrons in TBG, comprising both localized, correlated f-electrons and itinerant, dispersive c-electrons. We show that the THF model provides a microscopic basis for the unconventional, negative, sign-preserving Seebeck coefficient observed at positive fillings, which also exhibits characteristic sawtooth oscillations. These findings emphasize the critical role of electron correlations in shaping the transport properties of TBG and establish the coexistence of the two carrier types in this system.