Pascal's Triangle Fractal Symmetries.

Area of Science:

• Quantum mechanics
• Condensed matter physics
• Fractal geometry

Background:

• Fractal symmetries in physical systems, such as Sierpinski triangle models, have been explored.
• Generalizations of these models can reveal new physical phenomena.

Purpose of the Study:

• Introduce a U(1) generalization of fractal symmetries in interacting boson models, termed "Pascal's triangle symmetries."
• Investigate the properties and consequences of these novel fractal symmetries.
• Explore the relationship between symmetry breaking and fractal dimensions in lattice models.

Main Methods:

• Development of a theoretical model for interacting bosons with Pascal's triangle symmetries.
• Analysis of symmetry breaking from U(1) to Z_{p} (where p is a prime integer).
• Calculation of fractal dimensions and exploration of correlation functions at finite temperatures.

Main Results:

• The Pascal's triangle symmetry leads to exact degeneracies and a manifold of low-energy states absent in previous models.
• Symmetry breaking generates lattice models with unique fractal symmetries and a fractal subsystem of dimension d_{H}=ln(p(p+1)/2)/lnp.
• The fractal dimension can be experimentally probed via correlation functions at finite temperatures.

Conclusions:

• The introduced Pascal's triangle symmetries offer a new framework for studying fractal phenomena in quantum systems.
• The study discusses the phase diagram at zero temperature and potential physical realizations of the U(1) model.