We present a large class of three-dimensional spin models that possess
topological order with stability against local perturbations, but are beyond
description of topological quantum field theory. Conventional topological spin
liquids, on a formal level, may be viewed as condensation of string-like
extended objects with discrete gauge symmetries, being at fixed points with
continuous scale symmetries. In contrast, ground states of fractal spin liquids
are condensation of highly-fluctuating fractal objects with certain algebraic
symmetries, corresponding to limit cycles under real-space renormalization
group transformations which naturally arise from discrete scale symmetries of
underlying fractal geometries. A particular class of three-dimensional models
proposed in this paper may potentially saturate quantum information storage
capacity for local spin systems.Comment: 18 pages, 10 figure
