Topological Electromagnetic Effects and Higher Second Chern Numbers in Four-Dimensional Gapped Phases

TL;DR

This paper introduces a 4D topological insulator model with unique electromagnetic responses, revealing phase transitions and boundary phenomena, and suggests experimental probing in cold atom systems.

Contribution

It presents a novel 4D $ ext{Z}_2$ topological insulator model protected by $ ext{CP}$ symmetry with unique boundary states and topological responses, including phase transitions and experimental proposals.

Findings

Boundary modes include Dirac cones, nodal spheres, and Weyl semimetal phases.

Exotic topological responses described by (4+1)D mixed Chern-Simons theories.

Phase transitions involve gap closing and emergence of gapless phases.

Abstract

Higher-dimensional topological phases play a key role in understanding the lower-dimensional topological phases and the related topological responses through a dimensional reduction procedure. In this work, we present a Dirac-type model of four-dimensional (4D) $\mathbb{Z}_2$ topological insulator (TI) protected by $\mathcal{CP}$-symmetry, whose 3D boundary supports an odd number of Dirac cones. A specific perturbation splits each bulk massive Dirac cone into two valleys separated in energy-momentum space with opposite second Chern numbers, in which the 3D boundary modes become a nodal sphere or a Weyl semimetallic phase. By introducing the electromagnetic (EM) and pseudo-EM fields, exotic topological responses of our 4D system are revealed, which are found to be described by the (4+1)D mixed Chern-Simons theories in the low-energy regime. Notably, several topological phase transitions occur from a $\mathcal{CP}$-broken $\mathbb{Z}_2$ TI to a $\mathbb{Z}$ TI when the bulk gap closes by giving rise to exotic double-nodal-line/nodal-hyper-torus gapless phases. Finally, we propose to probe experimentally these topological effects in cold atoms.