Collective states of interacting non-Abelian anyons have recently been studied mostly in the context of certain fractional quantum Hall states, such as the Moore-Read state proposed to describe the physics of the quantum Hall plateau at filling fract
ion v = 5/2. In this manuscript, we further expand this line of research and present non-unitary generalizations of interacting anyon models. In particular, we introduce the notion of Yang-Lee anyons, discuss their relation to the so-called `Gaffnian quantum Hall wave function, and describe an elementary model for their interactions. A one-dimensional version of this model -- a non-unitary generalization of the original golden chain model -- can be fully understood in terms of an exact algebraic solution and numerical diagonalization. We discuss the gapless theories of these chain models for general su(2)_k anyonic theories and their Galois conjugates. We further introduce and solve a one-dimensional version of the Levin-Wen model for non-unitary Yang-Lee anyons.
Quantum mechanical systems, whose degrees of freedom are so-called su(2)_k anyons, form a bridge between ordinary SU(2) spin systems and systems of interacting non-Abelian anyons. Such a connection can be made for arbitrary spin-S systems, and we exp
licitly discuss spin-1/2 and spin-1 systems. Anyonic spin-1/2 chains exhibit a topological protection mechanism that stabilizes their gapless ground states and which vanishes only in the limit (k to infinity) of the ordinary spin-1/2 Heisenberg chain. For anyonic spin-1 chains we find their phase diagrams to closely mirror the one of the biquadratic SU(2) spin-1 chain. Our results describe at the same time nucleation of different 2D topological quantum fluids within a `parent non-Abelian quantum Hall state, arising from a macroscopic occupation of localized, interacting anyons. The edge states between the `nucleated and the `parent liquids are neutral, and correspond precisely to the gapless modes of the anyonic chains.
