Quasi-particle Statistics and Braiding from Ground State Entanglement

TL;DR

The paper develops a framework to extract quasi-particle statistics from ground-state entanglement on a torus by exploiting the nontrivial topology of entanglement cuts. It introduces Minimum Entropy States (MESs) as the ground-state basis that maximizes the topological entanglement entropy for a given cut, enabling the construction of modular matrices $\mathcal{S}$ and $\mathcal{U}$ that encode self- and mutual-statistics and quantum dimensions. Through analytical and variational Monte Carlo analyses of a SU(2) spin-symmetric Chiral Spin Liquid and analytic treatment of a $Z_2$ toric code spin liquid, the authors show that MESs correspond to definite flux sectors and that the $\mathcal{S}$-matrix can be extracted from rotations acting on the MES basis, up to a global phase fixed by the identity particle. They also provide explicit algorithms to obtain $\mathcal{S}$ and related data from a degenerate ground-state manifold without requiring edge states or finite-size extrapolations. The approach offers a practical, ground-state–based route to diagnosing topological order and quasi-particle statistics in lattice models and beyond, with potential generalizations to higher dimensions.

Abstract

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) on 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 quasi-particle excitations of the topological phase. The transformation of these states allows for a determination of the modular S and U matrices which encode quasi-particle properties. We demonstrate our method by extracting the modular S matrix of an SU(2) spin symmetric chiral spin liquid phase using a Monte Carlo scheme to calculate TEE, and prove that the quasi-particles obey semionic statistics. This method offers a route to a nearly complete determination of the topological order in certain cases.

Figures (8)

• Figure 1: A torus (the top and bottom sides and left and right sides are identified). Subregions $A,\,B,\,C$ are defined as shown. Regions $A$ and $C$ are assumed to be well separated as compared to the correlation length. The regions $AB$ and $BC$ correspond to bipartitions of the torus into cylinders in orthogonal directions.
• Figure 2: The separation of the system into subsystem $A$, $B$, $C$ and environment, periodic or antiperiodic boundary condition is employed in both $\hat{x}$ and $\hat{y}$ directions. a: The subsystem $ABC$ is an isolated square and the measured TEE has no ground state dependence. b: The subsystem $ABC$ takes a non-trivial cylindrical geometry and wraps around the $\hat{y}$ direction, and TEE may possess ground stated dependence.
• Figure 3: Numerically measured TEE $2\gamma - \gamma'$ for a CSL ground state from linear combination $\left|\Phi\right\rangle =\cos\phi\left|0,\pi\right\rangle +\sin\phi\left|\pi,0\right\rangle$ as a function of $\phi$ with VMC simulations using geometry in Fig. \ref{['fig3']}b. The solid curve is the theoretical value from Eqn. \ref{['eqntee7']}. The periodicity is $\pi/2$.
• Figure 4: The presence of two fluxes $\theta_{1}$ and $\theta_{2}$ on a torus. In cylindrical cut $A$ that wraps around $\hat{y}$ direction, only $\theta_{2}$ is a measurable.
• Figure 5: Illustration of a lattice of the toric code model, the links spanned by star and plaquette are highlighted in red and blue, respectively.