Solving the Bethe-Salpeter equation with exponential convergence

The Bethe-Salpeter equation plays a crucial role in understanding the physics of correlated fermions, relating to optical excitations solids as well resonances high-energy physics. Yet, it is notoriously difficult control numerically, typically requiring an effort that scales polynomially with energy and accuracy. This puts many interesting systems out computational reach. Using intermediate representation sparse modelling for two-particle objects on Matsubara axis, we develop algorithm solves $O(L^8)$ time $O(L^4)$ memory, where $L$ grows only logarithmically inverse temperature, bandwidth, desired accuracy, opens door computations hitherto inaccessible regimes. We benchmark method Hubbard atom multi-orbital weak-coupling limit, observe expected exponential convergence analytical results. then showcase realistic impurity problem.