Time Reversal, SU(N) Yang-Mills and Cobordisms: Interacting Topological Superconductors/Insulators and Quantum Spin Liquids in 3+1D

TL;DR

This paper explores the classification of 3+1D topological superconductors and insulators with various symmetries, their field theories, and how gauging symmetries leads to new phases of SU(N) Yang-Mills theories with potential applications to quantum spin liquids.

Contribution

It provides a comprehensive classification of topological phases with specific symmetries using cobordism and constructs explicit manifolds to detect these phases, also connecting to Yang-Mills theories.

Findings

Complete classification of SPT invariants via cobordism.

Explicit 4-manifolds for detecting SPTs.

Identification of distinct conformal field theories at $ heta=\pi$.

Abstract

We introduce a web of strongly correlated interacting 3+1D topological superconductors/insulators of 10 particular global symmetry groups of Cartan classes, realizable in electronic condensed matter systems, and their new SU(N) generalizations. The symmetries include SU(N), SU(2), U(1), fermion parity, time reversal and relate to each other through symmetry embeddings. We overview the lattice Hamiltonian formalism. We complete the list of field theories of bulk symmetry-protected topological invariants (SPT invariants/partition functions that exhibit boundary 't Hooft anomalies) via cobordism calculations, matching their full classification. We also present explicit 4-manifolds that detect these SPTs. On the other hand, once we dynamically gauge part of their global symmetries, we arrive in various new phases of SU(N) Yang-Mills (YM) gauge theories, analogous to quantum spin liquids with emergent gauge fields. We discuss how coupling YM theories to time reversal-SPTs affects the strongly coupled theories at low energy. For example, we point out a possibility of having two deconfined gapless time-reversal symmetric SU(2) YM theories at $\theta=\pi$ as two distinct conformal field theories, which although are secretly indistinguishable by correlators of local operators on orientable spacetimes nor by gapped SPT states, can be distinguished on non-orientable spacetimes or potentially by correlators of extended operators.