Pascal's Triangle Fractal Symmetries

TL;DR

This paper introduces a novel bosonic model with fractal symmetries based on Pascal's triangle, revealing unique degeneracies and low-energy states, and explores symmetry breaking, fractal dimensions, and phase diagrams.

Contribution

It presents a new $U(1)$ generalization of fractal symmetric models, expanding understanding of fractal symmetries and their physical implications.

Findings

Exact degeneracies and low-energy manifold in the $U(1)$ Pascal's triangle model

Fractal symmetry breaking to $Z_p$ with fractal operators of specific Hausdorff dimension

Finite temperature correlation functions can probe the fractal Hausdorff dimension

Abstract

We introduce a model of interacting bosons exhibiting an infinite collection of fractal symmetries -- termed "Pascal's triangle symmetries" -- which provides a natural $U(1)$ generalization of a spin-(1/2) system with Sierpinski triangle fractal symmetries. The Pascal's triangle symmetry gives rise to exact degeneracies, as well as a manifold of low-energy states which are absent in the Sierpinski triangle model. Breaking the $U(1)$ symmetry of this model to $Z_p$, with prime integer $p$, yields a lattice model with a unique fractal symmetry which is generated by an operator supported on a fractal subsystem with Hausdorff dimension $d_H = \ln (p(p+1)/2)/\ln p$. The Hausdorff dimension of the fractal can be probed through correlation functions at finite temperature. The phase diagram of these models at zero temperature in the presence of quantum fluctuations, as well as the potential physical construction of the $U(1)$ model are discussed.