Exotic topological order in fractal spin liquids

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 stringlike 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.

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

(a) Continuous deformability of ground states. (b) Continuous deformability of logical operators.

Figure 4

(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

Propagation of quasiparticle excitations.

Figure 6

RG transformation as a linear map $\Gamma$.

Figure 7

Incommensurate RG of the Sierpinski triangle model. Spins at $(i,j)=(0,0)$ (mod 3) are picked up. (a) The original. (b) $\lambda =3$. (c) $\lambda =9$. (d) $\lambda =27$. (e) $\lambda =81$. (f) $\lambda =243$.

Figure 8

Limit cycles in RG transformations for a classical fractal liquid with $f=1+x+x^2$ over ${\mathbb{F}}_5$. (a) ${\text{RG}}_{02}$. (b) ${\text{RG}}_{04}$.

Figure 9

Reduction of ${\mathbb{Z}}_2$ spin liquid. Two qubits are grouped into a composite particle which live inside each square.

Figure 10

(a) Propagation of quasiparticles by $f\left(x\right)$ and $g\left(x\right)$. (b) A pair of localized excitations $e_1$ and $e_2$ with elongated excitations $e_1^{*}$ and $e_2^{*}$.