Characterizing topological order by studying the ground states of an infinite cylinder

TL;DR

This paper presents a tensor network method to analyze topologically ordered phases on an infinite cylinder, extracting anyon models, edge theories, and universal properties, demonstrated on the Haldane model.

Contribution

It introduces a tensor network approach to characterize topological order, including anyon statistics and edge theories, from ground states on an infinite cylinder.

Findings

Ground states correspond to different anyonic fluxes.

Entanglement spectra reveal edge sector information.

The Haldane model realizes the =1/2 bosonic Laughlin state.

Abstract

Given a microscopic lattice Hamiltonian for a topologically ordered phase, we describe a tensor network 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. In a chiral phase, the entanglement spectrum of each ground state is seen to reveal a different sector of the corresponding gapless edge theory. A quasi-orthogonal basis on the torus is then produced by chopping off and reconnecting the tensor network representation on the cylinder. Elaborating on the recent proposal of [Y. Zhang et al. Phys. Rev. B 85, 235151 (2012)], a rotation on the torus yields an alternative basis of ground states and, through the computation of overlaps between bases, the modular matrices S and U (containing the mutual and self statistics of the different anyon species) are extracted. As an application, we study the hard-core boson Haldane model by using the two-dimensional density matrix renormalization group. A thorough characterization of the universal properties of this lattice model, both in the bulk and at the edge, unambiguously shows that its ground space realizes the \nu=1/2 bosonic Laughlin state.