Quasiparticle statistics and braiding from ground-state entanglement

Topologically ordered phases are gapped states, defined by the properties of excitations when taken around one another. Here we demonstrate a method to extract the statistics and braiding of excitations, given just the set of ground-state wave functions on a torus. This is achieved by studying the topological entanglement entropy (TEE) upon partitioning the torus into two cylinders. In this setting, general considerations dictate that the TEE generally differs from that in trivial partitions and depends on the chosen ground state. Central to our scheme is the identification of ground states with minimum entanglement entropy, which reflect the quasiparticle excitations of the topological phase. The transformation of these states allows for the determination of the modular $\mathcal{S}$ and $\mathcal{U}$ matrices which encode quasiparticle properties. We demonstrate our method by extracting the modular $\mathcal{S}$ matrix of a chiral spin liquid phase using a Monte Carlo scheme to calculate the TEE and prove that the quasiparticles obey semionic statistics. This method offers a route to nearly complete determination of the topological order in certain cases.

Figure 1

Two types of entanglement bipartitions on the torus: (a) a trivial bipartition with contractible boundaries, for which the TEE $\gamma =\log D$, and (b) a bipartition with noncontractible boundaries, where the TEE depends on the ground state.

Figure 2

A torus, with the top, bottom, left, and right sides identified. Subregions A, B, and C are defined as shown. Regions $A$ and $C$ are assumed to be well separated compared to the correlation length. Regions $AB$ and $BC$ correspond to bipartitions of the torus into cylinders in orthogonal directions.

Figure 3

Vectors ${\vec{w}}_1$ and ${\vec{w}}_2$ define a lattice with periodic boundary conditions. That is, points differing by integer linear combinations of ${\vec{w}}_1$ and ${\vec{w}}_2$ are to be identified as the same point. The area of the lattice is $|{\vec{w}}_1\times {\vec{w}}_2|$, where $\times$ denotes the cross product.

Figure 4

Separation of the system into subsystems $A$, $B$, and $C$ and environment; the periodic or antiperiodic boundary condition is employed in both $\widehat{x}$ and $\widehat{y}$ directions. (a) Subsystem $ABC$ is an isolated square and the measured TEE has no ground-state dependence. (b) Subsystem $ABC$ takes a nontrivial cylindrical geometry and wraps around the $\widehat{y}$ direction, and TEE may possess ground-state dependence.

Figure 5

Filled (black) squares show the numerically measured TEE $2\gamma -{\gamma }^{\prime }$ for a CSL ground state from the linear combination $\left|\Phi \right.〉=\cos \phi \left|0,\pi \right.〉+\sin \phi \left|\pi ,0\right.〉$ as a function of $\phi$ with VMC simulations using the geometry in Fig. 4b. The solid curve is the theoretical value from Eq. (14). The periodicity is $\pi /2$. Open (red) circles show the TEE for the same linear combination for a trivial bipartition. In the latter case, the TEE is essentially independent of $\phi$ and, again, agrees rather well with the theoretical expectation [dashed (red) curve].

Figure 6

Illustration of a lattice of the toric code model; links spanned by star and plaquette are highlighted in red and blue, respectively.

Figure 7

Snapshot of the ground state of the cylinder. Closed-loop strings (“$Z_2$ electric fields”) can wrap around the cylinder. Ground states are doubly degenerate, corresponding to even- and odd-winding-number sectors. The total number of strings crossing the cut $\Delta$ equals the winding number, modulo 2. The number of strings crossing at the boundary $\Gamma$ is even in the degenerate ground states.

Figure 8

The four minimum entropy states of the $Z_2$ topological phase corresponding to the bipartition shown, expressed as linear combinations of the four magnetic flux states. The magnetic $\pi$ flux is represented by the thick (blue and green) lines.

Figure 9

Illustration of a square lattice hopping model connected with a $d+id$ superconductor. While the nearest-neighbor hopping is along the square edges with amplitude $t$ [$(-t)$ for hopping along dashed lines], the second-nearest-neighbor hopping is along the square diagonal (bold arrows), with amplitude $+i\Delta$ ($-i\Delta$) when the hopping direction is along (against) the arrow. The two sublattices in the unit cell are labeled A and B.