Multiplicative topological phases

TL;DR

This paper introduces a new class of multiplicative topological phases of matter, constructed through symmetry enforcement on Hamiltonian components, revealing novel properties and potential applications in topological quantum systems.

Contribution

It develops a framework for multiplicative topological phases based on tensor product Hilbert spaces, extending known phases like Hopf and Chern insulators to new multiplicative variants.

Findings

Introduction of multiplicative Hopf and Chern insulators (MHI and MCI)

MHI exhibits properties of parent phases and non-trivial child phase topology

MCI hosts topologically protected gapless states with unique boundary properties

Abstract

Symmetry-protected topological phases of matter have challenged our understanding of condensed matter systems and harbour exotic phenomena promising to address major technological challenges. Considerable understanding of these phases of matter has been gained recently by considering additional protecting symmetries, different types of quasiparticles, and systems out of equilibrium. Here, we show that symmetries could be enforced not just on full Hamiltonians, but also on their components. We construct a large class of previously unidentified multiplicative topological phases of matter characterized by tensor product Hilbert spaces similar to the Fock space of multiple particles. To demonstrate our methods, we introduce multiplicative topological phases of matter based on the foundational Hopf and Chern insulator phases, the multiplicative Hopf and Chern insulators (MHI and MCI), respectively. The MHI shows the distinctive properties of the parent phases as well as non-trivial topology of a child phase. We also comment on a similar structure in topological superconductors as these multiplicative phases are protected in part by particle-hole symmetry. The MCI phase realizes topologically-protected gapless states that do not extend from the valence bands to the conduction bands for open boundary conditions, which respect the symmetries protecting topological phase. The band connectivity discovered in the MCI could serve as a blueprint for potential multiplicative topology with exotic properties.