ParquetIR.jl : efficiently solving the parquet equations using a sparse representation

Abstract

In this thesis, we present and discuss the first solver for the parquet equations (which we call ParquetIR.jl) based on the recently developed intermediate representation (IR) basis and written in the Julia programming language. Two-particle quantities are stored in a sparse representation enabling a significant reduction in memory and computation time requirements. The solver takes as input (an approximation of) the irreducible vertex and the non-interacting one-particle Green’s function. It then employs a state-of-the-art fixed-point solver to find a numerical solution to the parquet equations. This yields the full vertex, the one-particle Green’s function, and the self-energy. We discuss the theoretical background of the parquet equations and the implementation details of the solver and apply it to the Hubbard atom and the 4 × 4 Hubbard model on a square lattice; and show comparisons with a traditional approach. The results exhibit good agreement with the benchmark data and demonstrate the potential of ParquetIR.jl to tackle larger systems.