Hierarchy of correlations: Application to Green's functions and interacting topological phases

TL;DR

This paper introduces a hierarchy of correlations framework to analyze strongly correlated quantum systems, enabling non-perturbative insights into quantum phase transitions and topological properties across various models.

Contribution

It develops a generalized correlation hierarchy and decoupling scheme for Green's functions, facilitating systematic analysis of quantum correlations and topological invariants in interacting systems.

Findings

Analytical results valid in any dimension

Effective calculation of topological invariants in interacting phases

Insights into quantum phase transitions and topological features

Abstract

We study the many-body physics of different quantum systems using a hierarchy of correlations, which corresponds to a generalization of the $1/\mathcal{Z}$ hierarchy. The decoupling scheme obtained from this hierarchy is adapted to calculate double-time Green's functions, and due to its non-perturbative nature, we describe quantum phase transition and topological features characteristic of strongly correlated phases. As concrete examples we consider spinless fermions in a dimers chain and in a honeycomb lattice. We present analytical results which are valid for any dimension and can be generalized to different types of interactions (e.g., long range interactions), which allows us to shed light on the effect of quantum correlations in a very systematic way. Furthermore, we show that this approach provides an efficient framework for the calculation of topological invariants in interacting systems.