Topological characterization of the non-Abelian Moore-Read state using density-matrix renormalization group

The non-Abelian topological order has attracted a lot of attention for its fundamental importance and exciting prospect of topological quantum computation. However, explicit demonstration or identification of the non-Abelian states and the associated statistics in a microscopic model is very challenging. Here, based on a density-matrix renormalization-group calculation, we provide a complete characterization of the universal properties of the bosonic Moore-Read state on a Haldane honeycomb lattice model at filling number $\nu =1$ for larger systems, including both the edge spectrum and the bulk anyonic quasiparticle (QP) statistics. We first demonstrate that there are three degenerating ground states for each of which there is a definite anyonic flux threading through the cylinder. We identify the nontrivial countings for the entanglement spectrum in accordance with the corresponding conformal field theory. Through simulating a flux-inserting experiment, it is found that two of the Abelian ground states can be adiabatically connected, whereas the ground state in the Ising anyon sector evolves back to itself, which reveals the fusion rules between different QPs in real space. Furthermore, we calculate the modular matrices $\mathcal{S}$ and $\mathcal{U}$, which contain all the information for the anyonic QPs, such as quantum dimensions, fusion rule, and topological spins.

Figure 1

The ES for three topological GSs: (a) Identity GS $|{\mathrm{\Psi }}_{\mathbb{1}}\rangle$, (b) fermion GS $|{\mathrm{\Psi }}_f\rangle$, and (c) Ising anyon GS $|{\mathrm{\Psi }}_{\sigma }\rangle$, where ${\lambda }_i$ is the eigenvalue of reduced density matrix ${\widehat{\rho }}_L$ of the left half of an infinite cylinder. The ES are labeled by the relative boson number $\mathrm{\Delta }{N}_L=N_L-N_L^0$ of the left half cylinder in each tower ($N_L^0$ is the boson number of the state of ${\widehat{\rho }}_L$ with the largest eigenvalue). In each tower, the horizontal axis shows the relative momentum $\mathrm{\Delta }{K}_y=K_y-K_y^0$ in the transverse direction of the corresponding eigenvectors of ${\widehat{\rho }}_L$ ($K_y^0$ is the momentum of the state with the largest eigenvalue for ${\widehat{\rho }}_L$ in each tower). The numbers below the red dots label the nearly degenerating pattern for the low-lying ES with different $\mathrm{\Delta }{K}_y$'s. The black dashed line shows the entanglement gap in each momentum sector. Here the calculation is performed on the $L_y=6$ cylinder using infinite DMRG with keeping up to 3200 states.

Figure 2

The ES flow with inserting flux $\theta$ in the hole of the cylinder: (a) Starting from the GS $|{\mathrm{\Psi }}_{\mathbb{1}}\rangle$ at $\theta =0$ and adiabatically threading a $\theta =4\pi$ flux. (b) Starting from the GS $|{\mathrm{\Psi }}_{\sigma }\rangle$ and threading a $2\pi$ flux. Here the calculation is performed on the $L_y=6$ cylinder using infinite DMRG with keeping up to 1200 states.

Figure 3

Real-space configuration of the accumulated charge $\langle \mathrm{\Delta }{Q}_x\rangle ={\sum }_y\langle \mathrm{\Delta }{Q}_{x,y}\rangle$ (the summation is over all the $2L_y$ sites in each column $x$) with increasing flux $\theta .\mathrm{\Delta }{Q}_{L\left(R\right)}$ is defined by the total charge localized on the left (right) end of the cylinder. Here the calculation is performed on the $L_y=4$ cylinder with length $L_x=24$.