Local Interactions and Non-Abelian Quantum Loop Gases

Two-dimensional quantum loop gases are elementary examples of topological ground states with Abelian or non-Abelian anyonic excitations. While Abelian loop gases appear as ground states of local, gapped Hamiltonians such as the toric code, we show that gapped non-Abelian loop gases require nonlocal interactions (or nontrivial inner products). Perturbing a local, gapless Hamiltonian with an anticipated “non-Abelian” ground-state wave function immediately drives the system into the Abelian phase, as can be seen by measuring the Hausdorff dimension of loops. Local quantum critical behavior is found in a loop gas in which all equal-time correlations of local operators decay exponentially.

Figure 1

Four types of moves enacted by the plaquette flip term $F_p$. Term (a) enforces the loop fugacity $d$, (b) enforces isotopy invariance. Collectively they are known as “$d$-isotopy” moves. Terms (c) and (d) are the surgery terms.

Figure 2

The local imaginary time correlation function $C(\tau )$ for the loop gas with fugacity $d$ and surgery term $\epsilon =0.01$. Data shown are for system size $L=16$ and inverse temperature $\beta =16$.

Figure 3

Length distribution of the longest loop in a quantum loop gas with fugacity $d$; data for various system sizes are rescaled by $L^{d_H}$ with Hausdorff dimension $d_H$. (a) Abelian QLG with $d=1$, $d_H=7/4$. (b) Non-Abelian QLG with $d=\sqrt{2}$, $d_H=3/2$. (c) Data collapse for $d=\sqrt{2}$, $d_H=7/4$ with surgery term $\epsilon =0.05$ at inverse temperature $\beta =16$.

Figure 4

Algebraic decay of the plaquette flip correlations between plaquettes at distance $r$ in the QLG $|{\Psi }_0^{(\sqrt{2})}⟩$.