Editing Dyadic transformation (section)

==Relation to the Ising model==
The Hamiltonian of the zero-field one-dimensional [[Ising model]] of <math>2N</math> spins with periodic boundary conditions can be written as

:<math>H(\sigma) = g \sum_{i\in \mathbb{Z}_{2N}}\sigma_i\sigma_{i+1}. </math>

Letting <math>C</math> be a suitably chosen normalization constant and <math>\beta</math> be the inverse temperature for the system, the partition function for this model is given by

:<math>Z = \sum_{\{\sigma_i=\pm 1,\, i\in \mathbb{Z}_{2N}\}}\prod_{i\in \mathbb{Z}_{2N}}Ce^{-\beta g \sigma_i\sigma_{i+1}}. </math>

We can implement the [[renormalization group]] by integrating out every other spin. In so doing, one finds that <math>Z</math> can also be equated with the partition function for a smaller system with but <math>N</math> spins,

:<math>Z = \sum_{\{\sigma_i=\pm 1,\, i\in \mathbb{Z}_{N}\}}\prod_{i\in \mathbb{Z}_{N}}\mathcal{R}[C]e^{-\mathcal{R}[\beta g] \sigma_i\sigma_{i+1}}, </math>

provided we replace <math>C</math> and <math>\beta g</math> with renormalized values <math>\mathcal{R}[C]</math> and <math>\mathcal{R}[\beta g]</math> satisfying the equations

:<math>\mathcal{R}[C]^2= 4\cosh(2\beta g)C^4,</math>

:<math>e^{-2\mathcal{R}[\beta g]}= \cosh(2\beta g).</math>

Suppose now that we allow <math>\beta g</math> to be complex and that <math>\operatorname{Im}[2\beta g]=\frac{\pi}{2}+\pi n</math> for some <math>n\in \mathbb{Z}</math>. In that case we can introduce a parameter <math>t\in[0, 1)</math> related to <math>\beta g</math> via the equation

:<math>e^{-2\beta g}= i\tan\big(\pi(t-\frac{1}{2})\big),</math>

and the resulting renormalization group transformation for <math>t</math> will be precisely the dyadic map:<ref>
M. Bosschaert; C. Jepsen; F. Popov, “Chaotic RG flow in tensor models”, Physical Review D, 105, 2022, p. 065021.
</ref>

:<math>\mathcal{R}[t]=2t \bmod 1 .</math>