Interactions and Topology in Quantum Matter: Auxiliary Field Approach & Generalized SSH Models

TL;DR

This thesis explores the interplay of strong correlations and topology in quantum matter, presenting exact solutions for generalized SSH models with Coulomb interactions and developing auxiliary non-interacting models that capture complex dynamics.

Contribution

It introduces a novel auxiliary field approach that maps strongly correlated systems to generalized SSH models, revealing topological aspects of Mott transitions.

Findings

Exact solution of an infinite-dimensional SSH model with Coulomb interactions showing a Mott transition.

Development of auxiliary non-interacting models capturing the full dynamics of correlated systems.

Identification of the Mott transition as a topological phase transition involving domain walls.

Abstract

Presented in this thesis are a set of projects which lie at the intersection between strong correlations and topological phases of matter. The first of these projects is a treatment of an infinite dimensional generalization of the SSH model with local Coulomb interactions which is solved exactly using DMFT-NRG. Observed in the solution is power-law augmentation of the non-interacting density of states, as well as a Mott transition. This calculation represents an exact solution to an interacting topological insulator in the strongly correlated regime at zero temperature. The second set of projects involves the development of methods for formulating non-interacting auxiliary models for strongly correlated systems. These auxiliary models are able to capture the full dynamics of the original strongly correlated model, but with only completely non-interacting degrees of freedom, defined in an enlarged Hilbert space. We motivate the discussion by performing the mapping analytically for simple interacting systems using non-linear canonical transformations via a Majorana decomposition. For the nontrivial class of interacting quantum impurity models, the auxiliary mapping is established numerically exactly for finite-size systems using exact diagonalization, and for impurity models in the thermodynamic limit using NRG, both at zero and finite temperature. We find that the auxiliary systems take the form of generalized SSH models, which inherit the topological characteristics of those models. These generalized SSH models are also formalized and investigated in their own right as novel systems. Finally, we apply this methodology to study the Mott transition in the Hubbard model. In terms of the auxiliary system, we find that the Mott transition can be understood as a topological phase transition, which manifests as the formation and dissociation of topological domain walls.