Bootstrapping the Quantum Hall Problem

The bootstrap method aims to solve problems by imposing constraints on the space of physical observables, which often follow from assumptions such as positivity and symmetry. We employ this approach to study interacting electrons in the lowest Landau level by minimizing the energy as a function of the structure factor subject to constraints, bypassing the need to construct the many-body wavefunction. This approach rigorously lower bounds the ground state energy, complementing conventional variational upper bounds. The lower bound we obtain is relatively tight, within at most 5% of the ground state energy computed with exact diagonalization (ED) at small system sizes, and generally gets tighter as we include more constraints. In addition to energetics, our results reproduce the correct power law dependence of the pair correlation function at short distances and the existence of a large entanglement gap in the two-particle entanglement spectra for the Laughlin state at nu = 1/3. We further identify signatures of the composite Fermi liquid state close to half-filling. This shows that the bootstrap method is capable, in principle, of describing non-trivial gapped topologically ordered and gapless phases. At the end, we discuss possible extensions and limitations of this approach. Our work establishes numerical bootstrap as a promising method to study many-body phases in topological bands, paving the way to its application in moire platforms where the competition between fractional quantum anomalous Hall, symmetry broken, and gapless states remains poorly understood.