The Hierarchical Parity Model

TL;DR

This paper introduces the Hierarchical Parity Model, a new recursive spin system with exact RG equations and efficient algorithms, revealing a thermal phase transition in the uniform coupling case.

Contribution

It presents a novel hierarchical spin model based on parity coupling, with exact RG equations and efficient algorithms for key computations.

Findings

Exact RG equations derived for the model

O(N) algorithms for partition function and ground state

Thermal phase transition observed in uniform couplings

Abstract

Hierarchical spin-glasses are Ising spin models defined by recursively coupling together two equally-sized sub-systems. In this work a new hierarchical spin system is introduced wherein the sub-systems are recursively coupled together through the parity of their spins. Exact Renormalization Group recursion equations for many correlators may be derived for this Hierarchical Parity Model, even for completely general couplings. Moreover, the model is computationally tractable in that $O(N)$ algorithms exist for the computation of the partition function and ground state energy. In the special case where the couplings are all equal, the model is shown to exhibit a thermal phase transition.