Berry Phases of Vison Transport in ${\mathbb{Z}}_2$ Topologically Ordered States from Exact Fermion-Flux Lattice Dualities

Abstract

We develop an exact map of all states and operators from 2D lattices of spins-$1/2$ into lattices of fermions and bosons with mutual semionic statistical interaction that goes beyond previous dualities of ${\mathbb{Z}}_2$ lattice gauge theories because it does not rely on imposing local conservation laws and captures the motion of ``charges'' and ``fluxes'' on equal footing. This map allows to explicitly compute the Berry phases for the transport of fluxes in a large class of symmetry enriched topologically ordered states with emergent ${\mathbb{Z}}_2$ gauge fields that includes chiral, non-chiral, abelian or non-abelian, that can be perturbatively connected to models where the visons are static and the emergent fermionic spinons have a non-interacting dispersion. The numerical complexity of computing such vison phases reduces therefore to computing overlaps of ground states of free-fermion Hamiltonians. Among other results, we establish numerically the conditions under which the Majorana-carrying flux excitation in Ising-Topologically-Ordered states enriched by translations acquires $0$ or $\pi$ phase when moving around a single plaquette.

arXiv comment9+16 pages, 3+8 figures

References in corpus (30)

• Fault-tolerant quantum computation by anyons
• Anyons in an exactly solved model and beyond
• Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries, and the fractional quantum Hall effect
• Quantum Orders and Symmetric Spin Liquids
• Z_2 Gauge Theory of Electron Fractionalization in Strongly Correlated Systems
• Symmetry Fractionalization, Defects, and Gauging of Topological Phases
• Quantum orders in an exact soluble model
• A classification of symmetry enriched topological phases with exactly solvable models
• Topological odd-parity superconductors
• Classifying fractionalization: symmetry classification of gapped Z2 spin liquids in two dimensions
• Exact results on the Kitaev model on a hexagonal lattice: spin states, string and brane correlators, and anyonic excitations
• Fermions, strings, and gauge fields in lattice spin models
• The sign of the overlap of HFB wave functions
• Majorana Fermions in Nodal Superconductors: Gapless Topological Phase
• Majorana Fermions, Exact Mappings between Classical and Topological Orders
• The Bond-Algebraic Approach to Dualities
• Classification and Properties of Symmetry Enriched Topological Phases: A Chern-Simons approach with applications to Z2 spin liquids
• Tunneling of anyonic Majorana excitations in topological superconductors
• Refined symmetry indicators for topological superconductors in all space groups
• Mutual Chern-Simons theory for Z_2 topological order
• Exact bosonization in two spatial dimensions and a new class of lattice gauge theories
• Unified approach to Quantum and Classical Dualities
• Symmetry-based indicators for topological Bogoliubov-de Gennes Hamiltonians
• Microscopic definitions of anyon data
• Translation-symmetry protected topological orders on lattice
• Arbitrary Dimensional Majorana Dualities and Network Architectures for Topological Matter
• Local Commuting Projector Hamiltonians and the Quantum Hall Effect
• Theory of weak symmetry breaking of translations in ${\mathbb{Z}}_2$ topologically ordered states and its relation to topological superconductivity from an exact lattice ${\mathbb{Z}}_2$ charge-flux attachment
• Translation invariant topological superconductors on lattice
• Anatomy of Z2 fluxes in anyon Fermi liquids and Bose condensates

Cited by in corpus (9)

• Kitaev Quantum Spin Liquids
• Nature of visons in the perturbed ferromagnetic and antiferromagnetic Kitaev honeycomb models
• Emergent glassiness in disorder-free Kitaev model: Density matrix renormalization group study on a one-dimensional ladder setting
• Error-correcting codes for fermionic quantum simulation
• Classification and Quantification of Entanglement Through Wedge Product and Geometry
• Non-Abelian Anyons in Periodically Driven Abelian Spin Liquids
• Anyon polarons as a window into competing phases of the Kitaev honeycomb model under a Zeeman field
• Wegner's Ising gauge spins versus Kitaev's Majorana partons: Mapping and application to anisotropic confinement in spin-orbital liquids
• Jordan-Wigner composite-fermion liquids in 2D quantum spin-ice