We consider a chain of SU(2) 4 anyons with transitions to a topologically ordered phase state. For half-integer and integer indices of the type of strongly correlated excitations, we find an effective low-energy Hamiltonian that is an analogue of the standard Heisenberg Hamiltonian for quantum magnets. We describe the properties of the Hilbert spaces of the system eigenstates. For the Drinfeld quantum SU(2)k×SU(2)k doubles, we use numerical computations to show that the largest eigenvalues of the adjacency matrix for graphs that are extended Dynkin diagrams coincide with the total quantum dimensions for the levels k = 2, 3, 4, 5. We also formulate a hypothesis about the reason for the universal behavior of the system in the long-wave limit.
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.
Quantum spin models have been studied extensively in one and higher dimensions. Furthermore, these systems have been doped with holes to study $t\text{\ensuremath{-}}J$ models of
SU(2) spin-1/2. Their anyonic counterparts can be built from non-Abelian anyons, such as Fibonacci anyons described by
SU(2)3 theories, which are quantum deformations of the
SU(2) algebra. Inspired by the physics of
SU(2) spins, several works have explored ladders of Fibonacci anyons and also one-dimensional (1D) $t\text{\ensuremath{-}}J$ models. Here, we aim to explore the combined effects of extended dimensionality and doping by studying ladders composed of coupled chains of interacting itinerant Fibonacci anyons. We show analytically that in the limit of strong rung couplings these models can be mapped onto effective 1D models. These effective models can either be gapped models of hole pairs, or gapless models described by $t\text{\ensuremath{-}}J$ (or modified $t\text{\ensuremath{-}}J\text{\ensuremath{-}}V$) chains of Fibonacci anyons, whose spectrum exhibits a fractionalization into charge and anyon degrees of freedom. The charge degrees of freedom are described by the hardcore boson spectra while the anyon sector is given by a chain of localized interacting anyons. By using exact diagonalizations for two-leg and three-leg ladders, we show that indeed the doped ladders show exactly the same behavior as that of $t\text{\ensuremath{-}}J$ chains. In the strong ferromagnetic rung limit, we can obtain a new model that hosts two different kinds of Fibonacci particles, which we denote as the heavy $\ensuremath{\tau}$'s and light $\ensuremath{\tau}$'s. These two particle types carry the same (non-Abelian) topological charge but different (Abelian) electric charges. Once again, we map the two-dimensional ladder onto an effective chain carrying these heavy and light $\ensuremath{\tau}$'s. We perform a finite size scaling analysis to show the appearance of gapless modes for certain anyon densities, whereas a topological gapped phase is suggested for another density regime.