Stacking and twisting atomically thin crystals has turned two-dimensional materials into a remarkably tunable laboratory for quantum matter. In these moiré systems, electrons can be confined to nearly flat bands where interactions dominate. There, the quantum geometry of the electronic wavefunctions, not just their energies, determines which phases emerge. One of the most dramatic consequences of this geometry is the discovery of fractional quantum Hall states at zero magnetic field in twisted MoTe₂, with superconductivity appearing nearby upon doping. Doping these states introduces anyons, fractionally charged particles that, unlike in conventional quantum Hall systems, acquire genuine kinetic energy from the lattice. I will present a controlled microscopic theory of itinerant anyons. In this theory, the space of anyon configurations is characterized by a many-body generalization of quantum geometry that enables a systematic study of anyon energetics from 1-body, to few-body, to many-body. A single anyon disperses purely through interactions and non-uniform quantum geometry, with bandwidths of a few meV in twisted MoTe₂. Two or more anyons can bind into “anyon molecules” despite purely repulsive interactions, a binding driven by Berry phases rather than attraction. Near such a binding threshold, a dilute anyon gas realizes a universal, exactly solvable regime with superconducting order visible directly in the electrons.

In the last part of the talk, I will introduce a new approach to many-body numerics called Hamiltonian bootstrap. Rather than constructing a wavefunction, it constrains correlation functions through positivity and symmetry. This yields rigorous bounds that hold directly in the thermodynamic limit, on lattices and in the continuum alike. Because it works with band-projected operators, the bootstrap can incorporate the quantum geometry of the band from the outset. I will illustrate this by demonstrating rigorous lower bounds on the superfluid stiffness of flat-band superconductors (a quantity controlled by quantum geometry) and a proof that the Laughlin state is gapped in the thermodynamic limit. The latter demonstrates the power of the method to turn numerical results into rigorous proofs.

This lecture was made possible by the William C. Ferguson Fund.