Abstract
Given a microscopic lattice Hamiltonian for a topologically ordered phase, we propose a numerical approach to characterize its emergent anyon model and, in a chiral phase, also its gapless edge theory. First, a tensor network representation of a complete, orthonormal set of ground states on a cylinder of infinite length and finite width is obtained through numerical optimization. Each of these ground states is argued to have a different anyonic flux threading through the cylinder. Then a quasiorthogonal basis on the torus is produced by chopping off and reconnecting the tensor network representation on the cylinder. From these two bases, and by using a number of previous results, most notably the recent proposal of Y. Zhang et al. [Phys. Rev. B 85, 235151 (2012)] to extract the modular and matrices, we obtain (i) a complete list of anyon types , together with (ii) their quantum dimensions and total quantum dimension , (iii) their fusion rules , (iv) their mutual statistics, as encoded in the off-diagonal entries of , (v) their self-statistics or topological spins , (vi) the topological central charge of the anyon model, and, in a chiral phase (vii) the low energy spectrum of each sector of the boundary conformal field theory. As a concrete application, we study the hard-core boson Haldane model by using the two-dimensional density matrix renormalization group. A thorough characterization of its universal bulk and edge properties unambiguously shows that it realizes a bosonic fractional quantum Hall state.
- Received 13 August 2012
DOI:https://doi.org/10.1103/PhysRevLett.110.067208
© 2013 American Physical Society