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

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 ν=1/2 bosonic Laughlin state.

arXiv comment10 pages, 11 figures

References in corpus (18)

• Fault-tolerant quantum computation by anyons
• Non-Abelian Anyons and Topological Quantum Computation
• Topological entanglement entropy
• Detecting topological order in a ground state wave function
• Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States
• Spin Liquid Ground State of the $S=1/2$ Kagome Heisenberg Model
• Entanglement renormalization
• Classical simulation of infinite-size quantum lattice systems in two spatial dimensions
• Identifying Topological Order by Entanglement Entropy
• General Relationship Between the Entanglement Spectrum and the Edge State Spectrum of Topological Quantum States
• Quasi-adiabatic Continuation of Quantum States: The Stability of Topological Ground State Degeneracy and Emergent Gauge Invariance
• Spin Liquid Ground State of the Spin-1/2 Square $J_1$-$J_2$ Heisenberg Model
• Quasi-particle Statistics and Braiding from Ground State Entanglement
• Fractional Quantum Hall Effect of Hard-Core Bosons in Topological Flat Bands
• Topological Entanglement Entropy in Chern-Simons Theories and Quantum Hall Fluids
• Applying matrix product operators to model systems with long-range interactions
• Entanglement renormalization and topological order
• Variational quantum Monte Carlo simulations with tensor-network states

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