Cluster-diagrammatic D-TRILEX approach to non-local electronic correlations

TL;DR

The paper develops a cluster extension of the D-TRILEX method to treat non-local electronic correlations by combining exact short-range physics within a cluster with a diagrammatic description of long-range charge and spin fluctuations. It introduces a basis-rotation strategy to manage the sign problem and off-diagonal hybridization, and applies the framework to a 1D nano-ring Hubbard model, benchmarking against QMC and parquet D$\Gamma$A. The results show that cluster D-TRILEX yields accurate self-energies at the Fermi surface, reduces periodization ambiguity relative to CDMFT, and better captures momentum-resolved behavior at critical $k$-points, especially near the Fermi energy; non-causality artifacts at small clusters are noted and expected to diminish with larger $N_{c}$ or outer self-consistency. Overall, the approach provides a computationally efficient and scalable path toward including non-local correlations and is well-suited for exploring symmetry-broken states and multi-orbital systems in realistic materials.

Abstract

In this work, we extend the theoretical approach known as "D-TRILEX", developed for solving correlated electronic systems, to a cluster reference system for the diagrammatic expansion. This framework allows us to consistently combine the exact treatment of short-range correlation effects within the cluster, with an efficient diagrammatic description of the long-range charge and spin collective fluctuations beyond the cluster. We demonstrate the effectiveness of our approach by applying it to the one-dimensional nano-ring Hubbard model, where the low dimensionality enhances non-local correlations. Our results show that the cluster extension of D-TRILEX accurately reproduces the electronic self-energy at momenta corresponding to the Fermi energy, in good agreement with the numerically exact quantum Monte Carlo solution of the problem, and outperforms significantly more computationally demanding approach based on the parquet approximation. We show that the D-TRILEX diagrammatic extension drastically reduces the periodization ambiguity of cluster quantities when mapping back to the original lattice, compared to cluster dynamical mean-field theory (CDMFT). Furthermore, we identify the CDMFT impurity problem as the main source of the translational-symmetry breaking and propose the computational scheme for improving the starting point for the cluster-diagrammatic expansion.

Figures (10)

• Figure 1: Schematic representation of the one-dimensional Hubbard model. The lattice is tiled by identical two-site clusters, indicated by dashed boxes, consisting of the "left" (l) and "right" (r) sites. The nearest-neighbor hopping amplitude $t$ is considered the same within and between the clusters. The electrons interact via the on-site Coulomb repulsion $U$. The distance between adjacent clusters is ${A = 2a}$, i.e., twice the lattice constant $a$.
• Figure 2: The left panel shows the discretized dispersion $\varepsilon_k$ along the first Brillouin zone (lattice constant ${a = 1}$). The full line represents the dispersion of an infinite one-dimensional chain, whereas the discrete symbols correspond to the finite number of lattice sites $N_{\rm c}$. The right column depicts the ring geometries: a four-site (blue, top), a six-site (green, middle), and an eight-site (red, bottom) rings. In the dispersion plot their spectra are indicated by blue crosses, green circles, and red stars, respectively.
• Figure 3: The imaginary part of the self-energy calculated as a function of Matsubara frequency $\nu$ at two momenta ${k=0}$ (left) and ${k=\pi/3}$ (right). The results are obtained for the case of ${N_{\text{c}}=6}$ at ${U = 2}$ and ${\beta = 10}$ using different methods indicated in the legend. The ladder D$\Gamma$A, parquet D$\Gamma$A, and QMC results are taken from Ref. PhysRevB.91.115115.
• Figure 4: The real part of the self-energy calculated as a function of Matsubara frequency $\nu$ at two momenta ${k=0}$ (left) and ${k=\pi/3}$ (right). The results are obtained for the case of ${N_{\text{c}}=6}$ at ${U = 2}$ and ${\beta = 10}$ using different methods indicated in the legend. The ladder D$\Gamma$A, parquet D$\Gamma$A, and benchmark QMC results are taken from Ref. PhysRevB.91.115115.
• Figure 5: The imaginary part of the self-energy as a function of Matsubara frequency $\nu$ at momentum $k=\pi/2$ for the systems with $N_{\text{c}}=4$ (left) and $N_{\text{c}}=8$ (right). All results are computed at $U = 2$ and $\beta = 10$ using the methods indicated in the legend. The ladder D$\Gamma$A, parquet D$\Gamma$A, and benchmark QMC results are taken from Ref. PhysRevB.91.115115.