A concrete strategy is presented for generating strong topological insulators in $d+d^{\prime }$ dimensions which have quantized physics in $d$ dimensions. Here, $d$ counts the physical and $d^{\prime }$ the virtual dimensions. It consists of seeking $d$-dimensional representations of operator algebras which are usually defined in $d+d^{\prime }$ 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 Y. E. Kraus et al., Phys. Rev. Lett. 109, 106402 (2012) and predict quantized forces at the edges. We generate a topological system in $(3+1)$ dimensions, which is predicted to have quantized magnetoelectric response.

• Received 20 March 2015
• Revised 18 May 2015

DOI:https://doi.org/10.1103/PhysRevB.91.245104