Pairing susceptibility of the two-dimensional Hubbard model in the thermodynamic limit

TL;DR

This paper calculates the pairing susceptibility in the 2D Hubbard model using diagrammatic methods, revealing key pairing mechanisms and the nature of pairing modes across different parameters.

Contribution

It introduces an algorithmic Matsubara integration approach to compute the pairing susceptibility in the thermodynamic limit, identifying dominant pairing diagrams and modes.

Findings

Identification of $d_{x^2-y^2}$-wave pairing in the weak coupling limit.

Discovery of two main pairing components at $(0,0)$ and $( p, p)$.

Quantitative analysis of local and non-local pairing amplitudes and correlation lengths.

Abstract

We compute the diagrammatic expansion of the particle-particle susceptibility via algorithmic Matsubara integration and compute the correlated pairing susceptibility in the thermodynamic limit of the 2D Hubbard Model. We study the static susceptibility and its dependence on the pair momentum $\mathbf{q}$ for a range of temperature, interaction strength, and chemical potential. We show that $d_{x^2-y^2}$-wave pairing is expected in the model in the $U/t\to 0^+ $ limit from direct perturbation theory. From this, we identify key second and third-order diagrams that support pairing processes and note that the diagrams responsible are not a part of charge or spin susceptibility expansions. We find two key components for pairing at momenta $(0,0)$ and $(\pi,\pi)$ that can be well fit as separate bosonic modes. We extract amplitudes and correlation length scales where we find a predominantly local $(\pi,\pi)$ pairing and non-local $\mathbf{q}=(0,0)$ pairs and present the relative weights of these modes for variation in temperature, doping, and interaction strength.