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.

Figures (25)

• Figure 1: (a) Spectral function, (b) self-energy and (c) cumulant computed with the TaSK method for a $L=50$ Hubbard model.
• Figure 2: The first six Fourier coefficients of the imaginary part of (a) the self-energy of Fig. \ref{['fig:Hubbard1d_MPS_phsym']}(b) and (b) the cumulant of Fig. \ref{['fig:Hubbard1d_MPS_phsym']}(c).
• Figure 3: (a,b) Matrix elements $\epsilon_{kn}$ and $t_{kn}$ of $\mathcal{L}$, extracted via CFE from the spectral data in Fig. \ref{['fig:Hubbard1d_MPS_phsym']}. (c,d) The corresponding Fourier coefficients $\widetilde{\epsilon}_{rn}$ and $\widetilde{t}_{rn}$ up to order $r=10$. The color scheme of panel (d) applies to all panels.
• Figure 4: Self-energy $\Sigma^{[R]}_k(\omega)$ obtained from truncating the Fourier series of $\Sigma$-, $M$- or $\mathcal{L}$-expansions at $r \le R$ as described in the text. Regions where $-\mathrm{Im} \, \Sigma^{[R]}_{k}(\omega) < 0$ (non-causal behavior) are depicted gray.
• Figure 5: Interpolated CDMFT self-energies for different cluster sizes $N_\mathrm{c}$, obtained using four different interpolation schemes.