[Paper Review] Multi-band D-TRILEX approach to materials with strong electronic correlations

Matteo Vandelli, Josef Kaufmann+4 more|arXiv (Cornell University)|Apr 13, 2022

Physics of Superconductivity and MagnetismPhysics and Astronomy160 references29 citations

TL;DR

This paper introduces the multi-band D-TRILEX approach, a self-consistent diagrammatic method for strongly correlated multi-orbital materials with multiple atoms per unit cell. It extends the D-TRILEX framework to handle frequency- and channel-dependent electronic interactions via a partially bosonized dual fermion-boson action, enabling accurate treatment of both charge and magnetic fluctuations without Fierz ambiguity. The method achieves high accuracy in benchmark systems like the Hubbard-Kanamori dimer and captures key correlated physics such as Mott transitions and pseudogaps.

ABSTRACT

We present the multi-band dual triply irreducible local expansion (D-TRILEX) approach to interacting electronic systems and discuss its numerical implementation. This method is designed for a self-consistent description of multi-orbital systems that can also have several atoms in the unit cell. The current implementation of the D-TRILEX approach is able to account for the frequency- and channel-dependent long-ranged electronic interactions. We show that our method is accurate when applied to small multi-band systems such as the Hubbard-Kanamori dimer. Calculations for the extended Hubbard, the two-orbital Hubbard-Kanamori, and the bilayer Hubbard models are also discussed.

Motivation & Objective

• To develop a self-consistent, non-perturbative framework for describing strong electronic correlations in multi-orbital and multi-atom unit cell materials.
• To overcome the limitations of DMFT and TRILEX, such as the Fierz ambiguity and asymmetric vertex corrections, by extending the D-TRILEX approach to multi-band systems.
• To enable accurate description of both local and non-local correlations, including charge and magnetic fluctuations, with proper symmetry and consistency.
• To provide a numerically feasible alternative to high-complexity methods like dual fermion and dual boson by simplifying the diagrammatic structure while retaining key physics.
• To validate the method on benchmark models such as the Hubbard-Kanamori dimer, extended Hubbard, two-orbital Hubbard-Kanamori, and bilayer Hubbard systems.

Proposed method

• Formulates a multi-band effective fermion-boson action in the dual space, using a partially bosonized representation of the four-point vertex function.
• Derives the D-TRILEX diagrammatic expansion in the dual space, relating physical quantities to dual Green’s functions and vertices via a transformation involving the inverse bare propagator.
• Introduces a channel-decomposed interaction representation using particle-hole and particle-particle channels, with antisymmetrized bare vertex functions to avoid double-counting.
• Employs a partially bosonized approximation for the four-point vertex, where collective fluctuations in charge, magnetic, and pairing channels are described via renormalized interaction kernels.
• Uses the Bethe-Salpeter equation to compute the full vertex function in the dual space, with the bare interaction defined to remove double-counting via a subtraction term (¯ur).
• Implements a self-consistent computational workflow involving iterative solution of dual Dyson equations and vertex equations, with numerical fitting of Green’s function tails for accuracy.

Experimental results

Research questions

• RQ1 Can the D-TRILEX approach be generalized to multi-band, multi-orbital systems with multiple atoms per unit cell?
• RQ2 Does the multi-band D-TRILEX method correctly describe both charge and magnetic fluctuations without the Fierz ambiguity present in TRILEX?
• RQ3 How accurately does the method capture non-local correlation effects such as Mott transitions and pseudogap formation in benchmark models?
• RQ4 What is the numerical feasibility and stability of the D-TRILEX approach in multi-band settings compared to other diagrammatic methods?
• RQ5 Can the method reproduce known physical signatures like the reduction of the Mott critical interaction and pseudogap formation in the single-band limit?

Key findings

• The multi-band D-TRILEX method successfully captures the reduction of the Mott transition critical interaction compared to DMFT predictions, consistent with the single-band D-TRILEX results.
• The method reproduces pseudogap formation in the Slater regime of the single-orbital Hubbard model, confirming its ability to describe competing orders.
• In the Hubbard-Kanamori dimer, the method achieves high accuracy, validating its self-consistent treatment of local and non-local correlations.
• The extended Hubbard model on a square lattice shows that D-TRILEX correctly describes the interplay between charge and spin fluctuations, with results consistent with known trends.
• For the two-orbital Hubbard-Kanamori model, the method captures orbital-selective Mott physics and the correct symmetry of vertex corrections.
• The bilayer Hubbard model results demonstrate the method’s capability to describe interlayer correlations and the emergence of collective modes in layered systems.