Exact Topological Quantum Order in D=3 and Beyond: Branyons and Brane-Net Condensates

TL;DR

This paper introduces an exactly solvable 3D lattice model demonstrating topological quantum order with string-net and membrane-net condensates, extending to higher dimensions and revealing complex topological phases.

Contribution

It constructs a new solvable Hamiltonian in 3D exhibiting topological order and generalizes the framework to higher dimensions using colexes.

Findings

Ground state degeneracy depends on 3-manifold homology

Excitations are quasiparticles and fluxes from string and membrane boundaries

Extension to D≥4 dimensions suggests diverse topological phases

Abstract

We construct an exactly solvable Hamiltonian acting on a 3-dimensional lattice of spin-$\frac 1 2$ systems that exhibits topological quantum order. The ground state is a string-net and a membrane-net condensate. Excitations appear in the form of quasiparticles and fluxes, as the boundaries of strings and membranes, respectively. The degeneracy of the ground state depends upon the homology of the 3-manifold. We generalize the system to $D\geq 4$, were different topological phases may occur. The whole construction is based on certain special complexes that we call colexes.