Boson-Lattice Construction for Anyon Models

Abstract

This work is an attempt to unveil the skeleton of anyon models. I present a construction to systematically generate anyon models. The construction uses a set of elementary pieces or fundamental anyon models, which constitute the building blocks to construct other, more complex, anyon models. A principle of assembly is established that dictates how to articulate the building blocks, setting out the global blueprint for the whole structure. Remarkably, the construction generates essentially all tabulated anyon models. Moreover, novel anyon models (non-tabulated, to my knowledge) arise. To embody the construction I develop a very physical, visual and intuitive lexicon. An anyon model corresponds to a system of bosons in a lattice. By varying the number of bosons and the number of lattice sites, towers of more and more complex anyon models are built up. It is a Boson-Lattice construction. A self-similar anatomy is revealed: an anyon model is a graph that is filled with bosons to engender a new graph that is again filled with bosons. And further, bosons curve the graph that the anyon model is: I disclose a geography in the space of anyon models, where one is born from another by deforming the geometry of space. I advance an alluring duality between anyon models and gravity.

arXiv commentThis manuscript is best viewed in two-page setting. Readers who would like a rapid approach to the Boson-Lattice theory, are directed to my short manuscript http://www.theorie.physik.uni-muenchen.de/agparedes/publications/bl-construction-corto.pdf, which condenses the essential ideas and results of the Boson-Lattice formalism, including further generalizations of the theory

References in corpus (32)

• Fault-tolerant quantum computation by anyons
• Non-Abelian Anyons and Topological Quantum Computation
• 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
• New directions in the pursuit of Majorana fermions in solid state systems
• A quantum gas microscope - detecting single atoms in a Hubbard regime optical lattice
• String-net condensation: A physical mechanism for topological phases
• Single-Atom Resolved Fluorescence Imaging of an Atomic Mott Insulator
• Beyond paired quantum Hall states: parafermions and incompressible states in the first excited Landau level
• $2n$ Quasihole States Realize $2^{n-1}$-Dimensional Spinor Braiding Statistics in Paired Quantum Hall States
• Incompressible paired Hall state, stripe order and the composite fermion liquid phase in half-filled Landau levels
• Condensate induced transitions between topologically ordered phases
• The Quantum Double Model with Boundary: Condensations and Symmetries
• A Class of $P,T$-Invariant Topological Phases of Interacting Electrons
• Floquet engineering of correlated tunneling in the Bose-Hubbard model with ultracold atoms
• Anyon condensation and its applications
• Quantum groups and nonabelian braiding in quantum Hall systems
• A theory of 2+1D bosonic topological orders
• A theory of topological edges and domain walls
• Projective Construction of Non-Abelian Quantum Hall Liquids
• Condensation of achiral simple currents in topological lattice models: a Hamiltonian study of topological symmetry breaking
• Hopf symmetry breaking and confinement in (2+1)-dimensional gauge theory
• Realizing non-Abelian statistics
• Anyon Condensation and Continuous Topological Phase Transitions in Non-Abelian Fractional Quantum Hall States
• Broken quantum symmetry and confinement phases in planar physics
• $U(1)\times U(1)⋊Z_2$ Chern-Simons Theory and Z_4 Parafermion Fractional Quantum Hall States
• Topological order from quantum loops and nets
• Effective Field Theory and Projective Construction for the Z_k Parafermion Fractional Quantum Hall States
• On the classification of fusion rings
• The modular S-matrix as order parameter for topological phase transitions
• Local interactions and non-Abelian quantum loop gases
• Non-Abelian Topological Order from Quantum Organization in Indistinguishable Groups