*Citation:* Yoshida, Beni. "Information storage capacity of discrete spin systems." Annals of Physics 338 (2013): 134-166.

*Web:* https://arxiv.org/abs/1111.3275

*Tags:* Error-correcting-codes, Higher-dimensional, No-go

In this paper, Beni Yoshida discusses the idea of using a fractal spin system to saturate information-theoretic bounds in error correcting codes.

A big piece of context for this paper is the idea of taking Pascal's triangle modulo primes gives fractals. In particular, taking Pascal's triangle modulo 2 gives you the Sierpinski, and for larger primes one gets a p-adic analogue of the Sierpinski triangle. Pascal's triangle can be described by simple boundary conditions, and a simple local consistency rule: every entry is equal to the sum of the two entries above it. This sort of local check rule is exactly what makes for a good LDPC code.

Explicitly, one starts with an initial codeword and adds redundancy by filling in the rest of the grid with the rule that each entry is the sum of the two above it. The intricate fracal nature of the resulting patters give the resulting error detecting code its unusual properties. The most surprising result is a deep connection between the fractal nature of the code and its information-theoretic properties. The code distance associated to the mod-p Sierpinski triangle fractal code grows like the linear system size raised to the power of the fractal dimension. As p is chosen to be increasingly larger, the fractal dimensional gets increasingly close to 2. Thus, the assymptotic information theoretic properties of the code get icreasingly code as fractal dimension goes up. The result is that, up to arbitrarily small polynomial corrections, the optimal information theoretic bound is satisfied. Finding classical codes that satisfy this upper bound is a famously hard problem.

This spin model is not new. It had been studied once before in

> Newman, M. E. J., and Cristopher Moore. "Glassy dynamics and aging in an exactly solvable spin model." Physical Review E 60.5 (1999): 5068.

However, this new information theoretic perspective brought new life to its study.

After publishing this paper, Beni Yoshida went on to study these fractal codes in much more detail. In particular, he made an analogue of his model based on quantum spins:

> Yoshida, Beni. "Exotic topological order in fractal spin liquids." Physical Review B 88.12 (2013): 125122.

He calls these phases *quantum fractal liquids*. There is now a whole little industry surrounding
these fractal spin liquids. A good review and discussion of the properties of such theoretical condensed
matter systems is found here:

> Devakul, Trithep, et al. "Fractal symmetric phases of matter." SciPost Physics 6.1 (2019): 007.

This paper also advocates more generally for the discussion of spin systems on fractals. A succinct treatement of the Ising and 3-state Potts model on the Sierpinski triangle are found here:

> Yoshida, Beni, and Aleksander Kubica. "Quantum criticality from Ising model on fractal lattices." arXiv preprint arXiv:1404.6311 (2014).