The anyonic quantum walk is a dynamical model describing a single anyon propagating along a chain of stationary anyons and interacting via mutual braiding statistics. We review the recent results on the effects of braiding statistics in anyonic quantum walks in quasi-one dimensional ladder geometries. For anyons which correspond to spin-1/2 irreps of the quantum groups $SU(2)_k$, the non-Abelian species $(1
The anyonic quantum walk is a dynamical model describing a single anyon propagating along a chain of stationary anyons and interacting via mutual braiding statistics. We review the recent results on the effects of braiding statistics in anyonic quantum walks in quasi-one dimensional ladder geometries. For anyons which correspond to spin-1/2 irreps of the quantum groups SU(2)k, the non-Abelian species $(1
We study the effects of braiding interactions on single anyon dynamics using a quantum walk model on a quasi-1-dimensional ladder filled with stationary anyons. The model includes loss of information of the coin and nonlocal fusion degrees of freedom on every second time step, such that the entanglement between the position states and the exponentially growing auxiliary degrees of freedom is lost. The computational complexity of numerical calculations reduces drastically from the fully coherent anyonic quantum walk model, allowing for relatively long simulations for anyons which are spin-1/2 irreps of SU(2)_k Chern-Simons theory. We find that for Abelian anyons, the walk retains the ballistic spreading velocity just like particles with trivial braiding statistics. For non-Abelian anyons, the numerical results indicate that the spreading velocity is linearly dependent on the number of time steps. By approximating the Kraus generators of the time evolution map by circulant matrices, it is shown that the spatial probability distribution for the k=2 walk, corresponding to Ising model anyons, is equal to the classical unbiased random walk distribution.
"Anyonic quantum statistics is an exotic phenomenon of identical particles in quantum mechanics. When particles are confined in two spatial dimensions, exchanges of identical particles can induce phase factors in the wave function of Abelian anyons or matrix valued transformations of the wave function of non-Abelian anyons. As a result, systems of anyons may have richer properties than those of bosons and fermions in three spatial dimensions. There is strong theoretical support for the existence of anyons in some engineered two-dimensional systems such as 2D electron gases, strongly correlated spin lattices and as edge modes of nanowires. In the future, anyons could be used in topological quantum computation to perform highly efficient information processing with very small error rates. The phases of matter described by dynamically interacting anyons have recently been studied in chains where anyons interact via Heisenberg-type exchange interactions. In this thesis, new kind of anyonic interactions are studied, induced purely by braiding during free evolution. Such interactions are of topological origin, and the information about the interactions is stored non-locally. A quantum walk model is used to study the effects of these braiding interactions on the dynamical behaviour of anyons. The anyonic quantum walk is a quasi-one-dimensional generalization of the discrete-time quantum walk which allows the simulation of anyonic dynamics analytically. The moving anyon and a chain of stationary anyons interact via braiding statistics, and the behaviour of the anyon is studied in three cases. Striking differences are found between particles with conventional boson or fermion statistics and non-Abelian anyonic statistics. The random walk is a dynamical model that describes the motion of a particle on a lattice. In physics, it is used to describe Brownian motion of fluids and gases. In such systems, energy transport is diffusive, and the order of the system approaches a highly mixed state without any information content. The quantum version of the random walk, the quantum walk, has nonintuitive properties. Generally, the information content in a unitarily evolving system is fixed, which leads to unexpected transport phenomena. An initially localized particle does not propagate diffusively, but escapes the starting point with ballistic speed. Most of our results for non-Abelian anyons use the Ising model anyon which is most likely to be measured in experiments. First, when each anyonic site is occupied by one Ising anyon, the propagation of the anyon becomes diffusive. More precisely, the variance of the spatial probability distribution of the particle depends linearly on the number of time steps. This is in stark contrast to bosons, fermions and Abelian anyons which propagate ballistically, and the variance depends linearly on the square of the number of time steps. The essential reason for this slowdown is that the non-Abelian anyons possess an extra degree of freedom called the fusion Hilbert space. This space can be viewed as an environment for the normal degrees of freedom of the particle, inducing decoherence in the quantum walk. This thesis opens the line for studies of this novel kind of decoherence mechanism in quantum walks. In the second case the study is extended to more general anyon models by losing some of the information about the history of the evolution. In this case the system is subject to decoherence, and the total system evolution is not unitary. The behaviour of the particle in this model is found to be diffusive for all the non-Abelian anyon models studied, while Abelian anyons behave ballistically also in this model. One peculiarity of quantum systems is their behaviour under disorder. Quantum mechanical particles moving in random local potentials are known to freeze and not move at all. Such a phenomenon is known as Anderson localization. Studies have shown that Anderson localization happens also in quantum walks with random spatial fluctuations in the coin parameters. In the third case, the transport properties of anyons are studied under topological randomness, allowing the occupations of anyons change between experiments. Bosons and fermions would propagate ballistically under such randomness, and Abelian anyons are shown to localize. The results show that non-Abelian anyons behave diffusively at short time scales, and it is argued that they do so in the long time limit as well. In all cases, non-Abelian anyons are shown to have very different dynamical properties than bosons, fermions and Abelian anyons. A possible simulation of the anyonic quantum walk in Fractional Quantum Hall systems is also discussed.
The one dimensional quantum walk of anyonic systems is presented. The anyonic walker performs braiding operations with stationary anyons of the same type ordered canonically on the line of the walk. Abelian as well as non-Abelian anyons are studied and it is shown that they have very different properties. Abelian anyonic walks demonstrate the expected quadratic quantum speedup. Non-Abelian anyonic walks are much more subtle. The exponential increase of the system's Hilbert space and the particular statistical evolution of non-Abelian anyons give a variety of new behaviors. The position distribution of the walker is related to Jones polynomials, topological invariants of the links created by the anyonic world-lines during the walk. Several examples such as the SU(2)k and the quantum double models are considered that provide insight to the rich diffusion properties of anyons.
Non-Abelian anyons can exist as pointlike particles in two-dimensional systems and have particle exchange statistics which are neither bosonic nor fermionic. Like in spin systems, the role of fusion (Heisenberg-like) interactions between anyons has been well studied. However, unlike our understanding of the role of bosonic and fermionic statistics in the formation of different quantum phases of matter, little is known concerning the effect of non-Abelian braid statistics. We explore this physics using an anyonic Hubbard model on a two-legged ladder which includes braiding and nearest-neighbor Heisenberg interactions among anyons. We study two of the most prominent non-Abelian anyon models: the Fibonacci and Ising types. We discover rich phase diagrams for both anyon models and show the different roles of their fusion and braid statistics.
Anyons exist as pointlike particles in two dimensions and carry braid statistics, which enable interactions that are independent of the distance between the particles. Except for a relatively few number of models, which are analytically tractable, much of the physics of anyons remains still unexplored. In this paper, we show how U(1) symmetry can be combined with the previously proposed anyonic matrix product states to simulate ground states and dynamics of anyonic systems on a lattice at any rational particle number density. We provide proof of principle by studying itinerant anyons on a one-dimensional chain where no natural notion of braiding arises and also on a two-leg ladder where the anyons hop between sites and possibly braid. We compare the result of the ground-state energies of Fibonacci anyons against hardcore bosons and spinless fermions. In addition, we report the entanglement entropies of the ground states of interacting Fibonacci anyons on a fully filled two-leg ladder at different interaction strength, identifying gapped or gapless points in the parameter space. As an outlook, our approach can also prove useful in studying the time dynamics of a finite number of non-Abelian anyons on a finite two-dimensional lattice.
We discuss how to construct models of interacting anyons by generalizing quantum spin Hamiltonians to anyonic degrees of freedom. The simplest interactions energetically favor pairs of anyons to fuse into the trivial (“identity") channel, similar to the quantum Heisenberg model favoring pairs of spins to form spin singlets. We present an introduction to the theory of anyons and discuss in detail how basis sets and matrix representations of the interaction terms can be obtained, using non-Abelian Fibonacci anyons as example. Besides discussing the “golden chain”, a one-dimensional system of anyons with nearest neighbor interactions, we also present the derivation of more complicated interaction terms, such as three-anyon interactions in the spirit of the Majumdar-Ghosh spin chain, longer range interactions and two-leg ladders. We also discuss generalizations to anyons with general non-Abelian SU(2)k statistics. The k→∞ limit of the latter yields ordinary SU(2) spin chains.