Protected edge modes based on the bulk and boundary renormalization group: A relationship between duality and generalized symmetry

Abstract

We propose a theoretical formulation of protected edge modes in the language of quantum field theories based on the contemporary understanding of the renormalization group. We use bulk and boundary renormalization arguments which have never captured enough attention in condensed matter physics and related fields. We revisit various exotic bulk and boundary phenomena in contemporary physics, and one can see the conciseness of our formulations. Moreover, in the systems with open boundaries in general space-time dimensions, we also analyze their implications under general duality implemented by the shift of defects corresponding to generalized symmetries, including higher-form, non-invertible symmetries, in principle. Our formulation opens up a new paradigm to explore the systems with protected edge modes in the established language of the renormalization group.

Figures (2)

• Figure 1: Diagramatic picture of emergent edge degree of freedom. $|\alpha\rangle$, $|\beta\rangle$ corresponds to boundary states of the UV CFT and $(|\alpha\rangle)'$ corresponds to that of IR theory, and each arrow corresponds to the RG flows. For the RG flows represented by the yellow and blue arrows, the monotonic decreasing of the degrees of freedom occurs ($c$-theorem and $g$-theorem). However, for the red arrow, the boundary $g$-value can show nondecreasing, because the relevant boundary operators that trigger the boundary flow $|\alpha\rangle\rightarrow |\beta\rangle$ can become irrelevant.
• Figure 2: An implementation of duality for general $D$ dimensional quantum field theory and its bulk and boundary RG flow. The boundary states live in $\mathcal{M}_{D-2}\times \{ 0\}$ and $\mathcal{M}_{D-2}\times \{ 1\}$ respectively and the $D-2$ dimensional defect which can be thought as a generator of duality moves from the left boundary to the right boundary.