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Trivializing Maps for the O(2) Model in Two Dimensions
Armstrong, Balin
Armstrong, Balin
Abstract
Particles like protons and neutrons, are constructed from smaller particles called gluons and quarks. Quantum Chromododymaics or QCD is the theory that describes the interactions between these particles. Quantum Chromododymaics can be defined as the continuum limit of a discretized theory on a lattice called Lattice QCD. In Lattice QCD, physical quantities can be computed via the integration of a high-dimension integral using Monte Carlo Integration. However, when using Monte Carlo integration, a slowing down in the computation of the integral occurs as the continuum limit is approached. This work aims to alleviate this slowing down by performing a change of variables (like a u substitution if the reader is familiar with that) which is called a trivializing map. We aimed to construct an approximation of the trivializing map for the O(2) Model in two dimensions. This model can be thought of as a 2D grid of interacting unit vectors that can change the direction they are pointing. Our trivializing map is defined as the solution of a first order differential equation describing the evolution of the spins (the unit vectors) in a new time direction (called the flow-time). We aimed to approximate the trivializing map as a power series in the flow time, restricting ourselves to low-order polynomials. We found that the approximating worked better as a quadratic polynomial than as a linear approximation. We also observed that the effectiveness of our map was inversely proportional to the square root of the number of unit vectors on the lattice, indicating that the effectiveness of our approach will deteriorate with the square root of the system’s volume.
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2023-05-01
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Orginos, Konstantinos
Stathopoulos, Andreas
Griffioen, Keith
Stathopoulos, Andreas
Griffioen, Keith
Department
Physics