Virtual Topological Insulators with Real Quantized Physics

TL;DR

The paper introduces a method to realize strong topological insulators with quantized physical effects in lower dimensions by using virtual dimensions, enabling measurement of topological invariants at fixed virtual coordinates.

Contribution

It develops a strategy to generate topological insulators in mixed physical and virtual dimensions, linking higher-dimensional topological invariants to measurable physical quantities.

Findings

Virtual Chern insulator in 1+1 dimensions with predicted quantized edge forces.

New 3+1 dimensional topological system with quantized magneto-electric response.

Bulk-boundary correspondence fully determined by physical coordinates.

Abstract

A concrete strategy is presented for generating strong topological insulators in $d+d'$ dimensions which have quantized physics in $d$ dimensions. Here, $d$ counts the physical and $d'$ the virtual dimensions. It consists of seeking $d$-dimensional representations of operator algebras which are usually defined in $d+d'$ dimensions where topological elements display strong topological invariants. The invariants are shown, however, to be fully determined by the physical dimensions, in the sense that their measurement can be done at fixed virtual coordinates. We solve the bulk-boundary correspondence and show that the boundary invariants are also fully determined by the physical coordinates. We analyze the virtual Chern insulator in $(1+1)$-dimensions realized in Ref.~\cite{KrausPRL2012hh} and predict quantized forces at the edges. We generate a novel topological system in $(3+1)$-dimensions, which is predicted to have quantized magneto-electric response.