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

Condensed matter systems in the solid state owe much of their properties to the quantum behavior of their electronic degrees of freedom. Despite these dynamical degrees of freedom being relatively simple, the emergent phenomena which appear from collective behavior of the many constituent particles can lead to highly non-trivial characteristics. Two such non-trivial characteristics are topological phases, and the phenomena which result from many strongly correlated degrees of freedom. 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 Su-Schrieffer-Heeger model with local Coulomb interactions which is treated exactly using the technique of dynamical mean-field theory, with the numerical renormalization group as the impurity solver. Observed in the solution is power-law augmentation of the non-interacting density of states. The topological spectral pole becomes broadened into a power-law diverging spectrum, and the trivial gapped spectrum becomes a power-law vanishing pseudogap. At stronger interaction strengths we have a first-order transition to a fully gapped Mott insulator. 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 the numerical renormalization group, both at zero and finite temperature. We find that the auxiliary systems take the form of generalized Su-Schriefeer-Heeger models, which inherit the topological characteristics of those models. These generalized Su-Schrieffer-Heeger models are also formalized and investigated in their own right as novel systems. Finally, we apply the auxiliary field 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.