# Fractional $S$-duality, Classification of Fractional Topological Insulators and Surface Topological Order

## TL;DR

This paper generalizes $S$-duality to fractional charges in 4D QED, classifies fractional topological insulators into two classes, and explores their surface topological order, providing a topological quantum field theory framework.

## Contribution

It introduces fractional $S$-duality, classifies FTIs into two distinct types, and analyzes their surface topological order with a new topological quantum field theory approach.

## Key findings

- Two classes of FTIs: type-I (odd $t$) and type-II (even $t$).
- Type-I FTIs can be constructed via stacking; type-II cannot.
- Surface topological order characterized for fractional topological insulators.

## Abstract

In this paper, we propose a generalization of the $S$-duality of four-dimensional quantum electrodynamics ($\text{QED}_4$) to $\text{QED}_4$ with fractionally charged excitations, the fractional $S$-duality. Such $\text{QED}_4$ can be obtained by gauging the $\text{U(1)}$ symmetry of a topologically ordered state with fractional charges. When time-reversal symmetry is imposed, the axion angle ($\theta$) can take a nontrivial but still time-reversal invariant value $\pi/t^2$ ($t\in\mathbb{Z}$). Here, $1/t$ specifies the minimal electric charge carried by bulk excitations. Such states with time-reversal and $\text{U(1)}$ global symmetry (fermion number conservation) are fractional topological insulators (FTI). We propose a topological quantum field theory description, which microscopically justifies the fractional $S$-duality. Then, we consider stacking operations (i.e., a direct sum of Hamiltonians) among FTIs. We find that there are two topologically distinct classes of FTIs: type-I and type-II. Type-I ($t\in\mathbb{Z}_{\rm odd}$) can be obtained by directly stacking a non-interacting topological insulator and a fractionalized gapped fermionic state with minimal charge $1/t$ and vanishing $\theta$. But type-II ($t\in\mathbb{Z}_{\rm even}$) cannot be realized through any stacking. Finally, we study the Surface Topological Order of fractional topological insulators.