Theory of oblique topological insulators

TL;DR

This paper introduces a new class of fractional topological insulators called oblique TIs, characterized by dyon condensation, topological order, and emergent symmetries, expanding understanding of 3+1D topological phases.

Contribution

It develops a theoretical framework for oblique topological insulators using lattice gauge theories and topological quantum field theories, revealing new boundary states and universal effects.

Findings

Oblique TIs exhibit fractional $ heta$-angles and long-range entanglement.

Theories demonstrate a universal generalized magnetoelectric effect.

New boundary topological orders are characterized for these phases.

Abstract

A long-standing problem in the study of topological phases of matter has been to understand the types of fractional topological insulator (FTI) phases possible in 3+1 dimensions. Unlike ordinary topological insulators of free fermions, FTI phases are characterized by fractional $\Theta$-angles, long-range entanglement, and fractionalization. Starting from a simple family of $\mathbb{Z}_N$ lattice gauge theories due to Cardy and Rabinovici, we develop a class of FTI phases based on the physical mechanism of oblique confinement and the modern language of generalized global symmetries. We dub these phases oblique topological insulators. Oblique TIs arise when dyons -- bound states of electric charges and monopoles -- condense, leading to FTI phases characterized by topological order, emergent one-form symmetries, and gapped boundary states not realizable in 2+1-D alone. Based on the lattice gauge theory, we present continuum topological quantum field theories (TQFTs) for oblique TI phases involving fluctuating one-form and two-form gauge fields. We show explicitly that these TQFTs capture both the generalized global symmetries and topological orders seen in the lattice gauge theory. We also demonstrate that these theories exhibit a universal "generalized magnetoelectric effect" in the presence of two-form background gauge fields. Moreover, we characterize the possible boundary topological orders of oblique TIs, finding a new set of boundary states not studied previously for these kinds of TQFTs.