Interacting Topological Insulator and Emergent Grand Unified Theory

TL;DR

This paper explores a novel topological insulator model inspired by Pati-Salam GUT, showing how interactions can trivialize its boundary states, enabling potential regularization of GUTs via higher-dimensional topological phases.

Contribution

It demonstrates that interactions can trivialize boundary states of a $(4+1)d$ topological insulator with Pati-Salam symmetry, proposing a new approach to regularize GUTs using topological phases.

Findings

Boundary states are symmetry-protected and cannot be gapped without interactions.

Interactions can drive the boundary into a symmetric gapped phase, trivializing the topological insulator.

Coupling to a lattice gauge field allows regularization of GUTs as boundary states.

Abstract

Motivated by the Pati-Salam Grand Unified Theory, we study $(4+1)d$ topological insulators with $SU(4) \times SU(2)_1 \times SU(2)_2$ symmetry, whose $(3+1)d$ boundary has 16 flavors of left-chiral fermions, which form representations $(\mathbf{4}, \mathbf{2}, \mathbf{1})$ and $(\bar{\mathbf{4}}, \mathbf{1}, \mathbf{2})$. The key result we obtain is that, without any interaction, this topological insulator has a $\mathbb{Z}$ classification, namely any quadratic fermion mass operator at the $(3+1)d $ boundary is prohibited by the symmetries listed above; while under interaction this system becomes trivial, namely its $(3+1)d$ boundary can be gapped out by a properly designed short range interaction without generating nonzero vacuum expectation value of any fermion bilinear mass, or in other words, its $(3+1)d$ boundary can be driven into a "strongly coupled symmetric gapped (SCSG) phase". Based on this observation, we propose that after coupling the system to a dynamical $SU(4) \times SU(2)_1 \times SU(2)_2$ lattice gauge field, the Pati-Salam GUT can be fully regularized as the boundary states of a $(4+1)d$ topological insulator with a {\it thin} fourth spatial dimension, the thin fourth dimension makes the entire system generically a $(3+1)d$ system. The mirror sector on the opposite boundary will {\it not} interfere with the desired GUT, because the mirror sector is driven to the SCSG phase by a carefully designed interaction and is hence decoupled from the GUT.