The victory project v1.0: An efficient parquet equations solver

Abstract

Victory, i.e.vienna computational tool depository, is a collection of numerical tools for solving the parquet equations for the Hubbard model and similar many body problems. In the current release, we focus on the single-band 2D Hubbard model, based on which generalizations to non-local interactions, multi orbitals and other lattices are straightforward. The parquet formalism is a self-consistent theory at both the single- and two-particle levels, and can thus describe individual fermions as well as their collective behavior on equal footing. This is essential for the understanding of various emergent phases and their transitions in many-body systems, in particular for cases in which a single-particle description fails. Our implementation of victory is in modern Fortran and it fully respects the structure of various vertex functions in both momentum and Matsubara frequency space. We found the latter to be crucial for the convergence of the parquet equations, as well as for the correct determination of various physical observables. In this release, we thoroughly explain the program structure and the controlled approximations to efficiently solve the parquet equations, i.e. the two-level kernel approximation and the high-frequency regulation. Program summaryProgram Title: VictoryProgram Files doi:http://dx.doi.org/10.17632/ym5kscj9sz.1Licensing provisions: GPLv3Programming language: Fortran 90Nature of problem: The parquet equations require the knowledge of the fully irreducible vertex from which all one- and two-particle vertex and Green’s functions are calculated. The underlying two-particle vertex functions are large memory objects that depend on three momenta with periodic boundary conditions and three frequencies with open ones. The coupled diagrams of the parquet equations extend the frequency dependence of the reducible vertex functions to a larger frequency space where, a priori, no information is available.Solution method: The reducible vertex functions are found to possess the simplest structure among all two-particle vertex functions and can be approximated by functions with a reduced number of arguments, i.e. the kernel functions. The open boundary issue of the vertex functions in Matsubara-frequency space is then solved as follows: we solve the parquet equations with the reducible vertex functions whenever it is possible, otherwise we supplement these by the kernel functions.