Field Theory Simulations on a Fuzzy Sphere - an Alternative to the Lattice

TL;DR

This paper introduces a novel simulation method for quantum field theories using a fuzzy sphere instead of a traditional lattice, applied to the 3d b4 b4 model, revealing phase structures and limits to flat space models.

Contribution

It presents a new lattice-free approach to simulating quantum fields using fuzzy spheres, bridging non-commutative geometry and field theory.

Findings

Identified disordered and ordered phases, including uniform and non-uniform order.

Analyzed the model's behavior in large N and R limits, connecting to flat space theories.

Demonstrated the feasibility of fuzzy sphere regularization for quantum field simulations.

Abstract

We explore a new way to simulate quantum field theory, without introducing a spatial lattice. As a pilot study we apply this method to the 3d \lambda \phi^4 model. The regularisation consists of a fuzzy sphere with radius R for the two spatial directions, plus a discrete Euclidean time. The fuzzy sphere approximates the algebra of functions of the sphere with a matrix algebra, and the scalar field is represented by a Hermitian N x N matrix at each time site. We evaluate the phase diagram, where we find a disordered phase and an ordered regime, which splits into phases of uniform and non-uniform order. We discuss the behaviour of the model in different limits of large N and R, which lead to a commutative or to a non-commutative \lambda \phi^4 model in flat space.