Simulation of Anyons Using Symmetric Tensor Network Algorithms

Abstract

In this thesis I develop a formalism whereby a tensor network may be understood in terms of a unitary braided tensor category, and represented in a particularly efficient manner corresponding to the exploitation of this mathematical structure. This approach permits the exploitation of the global Abelian and non-Abelian internal group symmetries of a lattice model, both to facilitate the study of particular symmetry sectors and to provide a considerable reduction in computational cost. It also enables the study of models with non-trivial exchange statistics (fermions, Abelian and non-Abelian anyons) using tensor network algorithms which scale polynomially in the system size.

PhD thesis, 220 pages, 7 tables, 76 figures, LaTeX2e; total size 2.7Mb as a PDF. Section 4.7 contains practical techniques for the implementation of anyonic tensor networks which have not been published elsewhere. v2: Fixed error in Fig. 2.5 which caused it to display incorrectly in Adobe Reader

References in corpus (29)

• Non-Abelian Anyons and Topological Quantum Computation
• Real time evolution using the density matrix renormalization group
• Computational complexity and fundamental limitations to fermionic quantum Monte Carlo simulations
• A class of quantum many-body states that can be efficiently simulated
• Classical simulation of infinite-size quantum lattice systems in two spatial dimensions
• Accurate determination of tensor network state of quantum lattice models in two dimensions
• From density-matrix renormalization group to matrix product states
• Interacting anyons in topological quantum liquids: The golden chain
• Tensor network states and algorithms in the presence of a global U(1) symmetry
• Tensor-entanglement renormalization group approach to 2D quantum systems
• String order and symmetries in quantum spin lattices
• Simulating Strongly Correlated Quantum Systems with Tree Tensor Networks
• Entanglement renormalization, scale invariance, and quantum criticality
• A short introduction to Fibonacci anyon models
• Infinite size density matrix renormalization group, revisited
• Matrix Product States: Symmetries and Two-Body Hamiltonians
• Quantum MERA Channels
• Exact entanglement renormalization for string-net models
• Collective states of interacting Fibonacci anyons
• Quantum phases of bosons in double-well optical lattices
• Unifying variational methods for simulating quantum many-body systems
• Simulation of time evolution with the MERA
• Simulation of anyons with tensor network algorithms
• Tensor Product Variational Formulation for Quantum Systems
• Anyonic entanglement renormalization
• Grassmann tensor network states and its renormalization for strongly correlated fermionic and bosonic states
• Spinful bosons in an optical lattice
• Graded Projected Entangled-Pair State Representations and An Algorithm for Translationally Invariant Strongly Correlated Electronic Systems on Infinite-Size Lattices in Two Spatial Dimensions
• Topological Properties of Tensor Network States From Their Local Gauge and Local Symmetry Structures