Liouvillian interpolation of the self-energy of cluster dynamical mean-field theories

Abstract

Two widely-used non-local extensions of dynamical mean field theory (DMFT), cellular DMFT (CDMFT) and the dynamical cluster approximation (DCA), both yield self-energies marred by having some unphysical properties: CDMFT yields real-space self-energies that are not translationally invariant, and DCA yields momentum-space self-energies with discontinuities in their momentum dependence. It is often desirable to remove these flaws by post-processing cluster DMFT results, using strategies called periodization for CDMFT and interpolation for DCA -- for brevity, we refer to both cases as interpolation. However, traditional interpolation approaches struggle to capture intricate structures such as hole pockets in the hole-doped square-lattice Hubbard model, as highlighted in Phys. Rev. B 105, 35117 (2022). Further, these approaches interpolate frequency-dependent functions, which may lead to causality violations. Here, we propose Liouvillian interpolation, a novel, intuitive, and robust scheme for interpolating cluster DMFT results. Our key idea is to interpolate frequency-independent matrix elements of the single-particle irreducible part of the Liouvillian, obtained from a continued-fraction expansion of the cDMFT self-energy. We demonstrate that the ingredients of such an expansion possess a more local Fourier expansion than the functions involved in traditional interpolation schemes, and that Liouvillian interpolation inherently conserves causality. We illustrate our method for the one-dimensional Hubbard model using CDMFT, and for the two-dimensional Hubbard model using four-patch DCA. For the latter, we find that L-interpolation can (depending on doping) yield Fermi and Luttinger arcs which together form a closed surface.