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

TL;DR

This paper introduces a universal bulk-boundary relation in topological matter, linking bulk interactions to surface states, and explores emergent phenomena like Majorana fermions and phonon effects on surfaces.

Contribution

It proposes a general scaling relation between bulk and surface interactions in topological matter and applies it to analyze emergent phenomena and interaction effects.

Findings

Bulk-boundary relation (BBR) links bulk interactions to surface states.

Conditions for emergent Majorana fermions on surfaces.

Optimal phonon-mediated interactions occur when Debye frequency matches the bulk gap.

Abstract

In this article, we propose a simple but general scaling relation between interactions in a gapped bulk topological matter and gapless interacting surface states. We explicitly illustrate such a generic bulk-boundary relation (BBR) for a few specific interactions in a topological quantum matter, where we can perform dimensional reduction of a microscopic bulk theory to project out interacting surfaces. We have examined renormalization effects of the gapped bulk fermions on the interacting topological surface fermions. As simple applications, we utilize effective interacting quantum fields implied by BBR to explore feasibility of routes to various fascinating emergent phenomena on surfaces including emergent Majorana fermions induced by spontaneous symmetry breaking. We obtain sufficient conditions for these interacting surface phenomena to take place. We have also found that for given bulk electron-phonon interactions and when \Omega_{D} \geq m, the phonon-mediated interactions on surface are strongest if the bulk Debye frequency \Omega_{D} matches m, the mass gap of the topological matter