An Interaction Bulk-Boundary Relation and its Applications Towards Symmetry Breaking and Beyond

Topological insulators are materials known to possess robust boundary states whose low energy excitations resemble half of the massless Dirac fermions, with the other half being localized at the opposite boundary. Hence, an attractively interacting topological surface naturally results in a topological superconducting phase (TSC) that hosts emergent Majorana fermions. In addition, phase transitions between the semi-metallic phase and the superconducting phase of the surface states have an emergent (2+1) D Supersymmetry. In this respect, we studied the interactions between topological surface matter. We put forward a simple scaling relation that connects interactions between gapped bulk topological matter and gapless surface fermions which we have named 'the interacting bulk-boundary (BBR) relation'. Using the BBR phenomenology we developed, we propose that the attractive phonon-mediated interaction between electrons is greatly enhanced if one chooses a topological material such that its bulk energy gap matches its Debye frequency. As an application, we propose a semi-realistic approach to realize attractively interacting electronic ground states on topological surfaces.