Non-perturbative landscape of the Mott-Hubbard transition: Multiple divergence lines around the critical endpoint

TL;DR

This paper explores the complex non-perturbative behavior near the Mott-Hubbard metal-insulator transition, revealing multiple divergence lines in the phase diagram that serve as precursors to the transition and impact many-body computational methods.

Contribution

It identifies and classifies multiple divergence lines in the phase diagram of the Hubbard model, linking them to the breakdown of perturbation theory and the non-perturbative physics of the MIT.

Findings

Existence of infinitely many divergence lines near the critical endpoint.

Classification of divergences based on eigenvector frequency structure.

Distinction between different types of divergence related to energy scales.

Abstract

We analyze the highly non-perturbative regime surrounding the Mott-Hubbard metal-to-insulator transition (MIT) by means of dynamical mean field theory calculations at the two-particle level. By extending the results of Sch\"afer, et al. [Phys. Rev. Lett. 110, 246405 (2013)] we show the existence of infinitely many lines in the phase diagram of the Hubbard model where the local Bethe-Salpeter equations, and the related irreducible vertex functions, become singular in the charge as well as the particle-particle channel. These divergence lines accumulate around the critical Mott endpoint in accordance with the interpretation as precursors of the MIT. By comparing our numerical data with analytical calculations of increasing complexity, such as for the disordered Binary Mixture and Falicov-Kimball (FK) models, as well as for the atomic limit (AL) case, (i) we identify two different kinds of divergences lines; (ii) we classify them in terms of the frequency-structure of the associated singular eigenvectors; (iii) we investigate their relation to the multiple branches in the Luttinger-Ward formalism. Moreover, we could distinguish the situations where the multiple divergences simply reflect the emergence of an underlying, unique energy scale $\nu^*$ below which perturbation theory does no longer apply, from those where the breakdown of perturbation theory affects, not trivially, different energy regimes. Finally, we discuss the implications of our results on the theoretical understanding of the non-perturbative physics around the MIT and for future developments of many-body algorithms applicable in this regime.