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