Constructing Non-Abelian Quantum Spin Liquids Using Combinatorial Gauge Symmetry

TL;DR

This paper introduces a spin Hamiltonian with only 1- and 2-body interactions that exactly exhibits non-Abelian gauge symmetry, realizing a non-Abelian topological phase suitable for quantum simulation.

Contribution

It constructs the first explicit spin Hamiltonian with non-Abelian gauge symmetry using combinatorial gauge symmetry, applicable to superconducting circuits.

Findings

Realizes non-Abelian gauge symmetry with simple interactions

Demonstrates a sign-problem-free Hamiltonian for non-Abelian topological phases

Proposes a superconducting circuit implementation

Abstract

We construct Hamiltonians with only 1- and 2-body interactions that exhibit an exact non-Abelian gauge symmetry (specifically, combinatiorial gauge symmetry). Our spin Hamiltonian realizes the quantum double associated to the group of quaternions. It contains only ferromagnetic and anti-ferromagnetic $ZZ$ interactions, plus longitudinal and transverse fields, and therefore is an explicit example of a spin Hamiltonian with no sign problem that realizes a non-Abelian topological phase. In addition to the spin model, we propose a superconducting quantum circuit version with the same symmetry.