Can we find a physical intuitive language to represent the complex mathematical properties of anyon models? Is there a hidden organization in the space of anyon models? Can we systematically construct complex anyon models from other, more simple, anyon models? This work is an attempt to understand and explore the anatomy of the space of anyon models. I develop a construction to systematically generate and interconnect anyon models. The construction is formulated in a very physical, visual, and intuitive manner. The complex mathematical properties of anyon models, usually expressed in the complex language of braided modular tensor categories, are described here using familiar simple physical concepts like bosons in lattices, Fock states, and tunneling Hamiltonians. The construction formalism is conceptually simple and appealing. As in a Lego construction, a set of building block anyon models are assembled to construct towers of 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 constructed anyon model. By assembling sets of the simple Abelian ${\mathbb{Z}}_{n}$ anyon models, the construction succeeds in generating towers of well known nontrivial anyon models. Moreover, novel anyon models (nontabulated, to my knowledge) arise. Furthermore, by applying the construction principle to other building blocks, such as Ising or Fibonacci, further series of anyon models are generated. In an intriguing connection, the boson-lattice construction unveils an alluring duality between anyon models and curved space geometries. Bosons induce curvature in the lattice geometry. I disclose a geography in the space of anyon models, where one is engendered from another by deforming the geometry of space. Moreover, the potential of the boson-lattice construction can be further explored via the quantum simulation of the corresponding boson-lattice systems in the versatile accessible playground of atomic systems.Can we find a physical intuitive language to represent the complex mathematical properties of anyon models? Is there a hidden organization in the space of anyon models? Can we systematically construct complex anyon models from other, more simple, anyon models? This work is an attempt to understand and explore the anatomy of the space of anyon models. I develop a construction to systematically generate and interconnect anyon models. The construction is formulated in a very physical, visual, and intuitive manner. The complex mathematical properties of anyon models, usually expressed in the complex language of braided modular tensor categories, are described here using familiar simple physical concepts like bosons in lattices, Fock states, and tunneling Hamiltonians. The construction formalism is conceptually simple and appealing. As in a Lego construction, a set of building block anyon models are assembled to construct towers of 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 constructed anyon model. By assembling sets of the simple Abelian ${\mathbb{Z}}_{n}$ anyon models, the construction succeeds in generating towers of well known nontrivial anyon models. Moreover, novel anyon models (nontabulated, to my knowledge) arise. Furthermore, by applying the construction principle to other building blocks, such as Ising or Fibonacci, further series of anyon models are generated. In an intriguing connection, the boson-lattice construction unveils an alluring duality between anyon models and curved space geometries. Bosons induce curvature in the lattice geometry. I disclose a geography in the space of anyon models, where one is engendered from another by deforming the geometry of space. Moreover, the potential of the boson-lattice construction can be further explored via the quantum simulation of the corresponding boson-lattice systems in the versatile accessible playground of atomic systems.

arXiv:1804.01605Physical review. B./Physical review. BMay 29, 20190 citations

Bosonic Continuum Theory of One-Dimensional Lattice Anyons

Martin Bonkhoff, Kevin Jägering, Sebastian Eggert +3 more

Anyons with arbitrary exchange phases exist on 1D lattices in ultracold gases. Yet, known continuum theories in 1D do not match. We derive the continuum limit of 1D lattice anyons via interacting bosons. The theory maintains the exchange phase periodicity fully analogous to 2D anyons. This provides a mapping between experiments, lattice anyons, and continuum theories, including Kundu anyons with a natural regularization as a special case. We numerically estimate the Luttinger parameter as a function of the exchange angle to characterize long-range signatures of the theory and predict different velocities for left- and right-moving collective excitations.Anyons with arbitrary exchange phases exist on 1D lattices in ultracold gases. Yet, known continuum theories in 1D do not match. We derive the continuum limit of 1D lattice anyons via interacting bosons. The theory maintains the exchange phase periodicity fully analogous to 2D anyons. This provides a mapping between experiments, lattice anyons, and continuum theories, including Kundu anyons with a natural regularization as a special case. We numerically estimate the Luttinger parameter as a function of the exchange angle to characterize long-range signatures of the theory and predict different velocities for left- and right-moving collective excitations.

arXiv:2008.00003Physical Review LettersApr 19, 202141 citations

A Calculus of Anyons

Alessandra Di Pierro, Federica Panarotto

Lecture notes in computer scienceJan 1, 20141 citation

{\mathbb{Z}}_{n}

anyon models, the construction succeeds in generating towers of well known nontrivial anyon models. Moreover, novel anyon models (nontabulated, to my knowledge) arise. Furthermore, by applying the construction principle to other building blocks, such as Ising or Fibonacci, further series of anyon models are generated. In an intriguing connection, the boson-lattice construction unveils an alluring duality between anyon models and curved space geometries. Bosons induce curvature in the lattice geometry. I disclose a geography in the space of anyon models, where one is engendered from another by deforming the geometry of space. Moreover, the potential of the boson-lattice construction can be further explored via the quantum simulation of the corresponding boson-lattice systems in the versatile accessible playground of atomic systems.