Exotic topological order in fractal spin liquids

TL;DR

This work introduces quantum fractal liquids—three-dimensional topological spin liquids whose ground states are condensates of fractal objects rather than conventional string-like or membrane-like excitations described by TQFT. It develops a formalism based on polynomial representations over finite fields to construct both classical and quantum fractal liquids, revealing discrete scale symmetry and limit-cycle renormalization group behavior. A no-string criterion is derived, tying the absence of string-like excitations to algebraic unrelatedness of generating polynomials, with numerous explicit examples including a Cubic-code correspondence. The study expands the taxonomy of gapped, robust quantum phases and points toward potential high information-storage capacity in local spin systems, while leaving open questions about extensions to chiral, non-Abelian, and symmetry-protected cases and experimental realizations.

Abstract

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.

Figures (10)

• Figure 1: $\mathbb{Z}_{2}$ spin liquid (the Toric code). (a) The Hamiltonian. (b) Loop states on a dual lattice. (c) Condensation of loops. (d) Logical operators.
• Figure 2: Discrete scale symmetries and imaginary scaling dimensions in three-point correlation function.
• Figure 3: (Color online) (a) Continuous deformability of ground states. (b) Continuous deformability of logical operators.
• Figure 4: (Color online) (a) The Sierpinski triangle from $f=1+x$ over $\mathbb{F}_{2}$. (b) The Fibonacci model from $f=1+x+x^{2}$ over $\mathbb{F}_{2}$. (c) The generalized Sierpinski triangle from $f=1+x$ over $\mathbb{F}_{3}$.
• Figure 5: (Color online) Propagation of quasi-particle excitations.