Instantons in 2D U(1) Higgs Model and 2D CPn-1 Sigma Models

In this thesis I present the results of a study of the topological structures of 2D U(1) Higgs model and 2D CP N−1 sigma models. Both models have been studied using the overlap Dirac operator construction of topological charge density. The overlap operator provides a more incisive probe into the local topological structure of gauge field configurations than the traditional plaquette-based operator. In the 2D U(1) Higgs model, we show that classical instantons with finite sizes violate the negativity of topological charge correlator by giving a positive contribution to the correlator at non-zero separation. We argue that instantons in 2D U(1) Higgs model must be accompanied by large quantum fluctuations in order to solve this contradiction. In 2D CP N−1 sigma models, we observe the anomalous scaling behavior of the topological susceptibility χ t for N ≤ 3. The divergence of χ t in these models is traced to the presence of small instantons with a radius of order a (= lattice spacing), which are directly observed on the lattice. The observation of these small instantons provides detailed confirmation of Lüscher's argument that such short-distance excitations, with quantized topological charge, should be the dominant topological fluctuations in CP 1 and CP 2 , leading to a divergent topological susceptibility in the continuum limit. For the CP N−1 models with N > 3 the topological susceptibility is observed to scale properly with the mass gap. Another topic presented in this thesis is an implementation of the Zolotarev optimal rational approximation for the overlap Dirac operator. This new implementation has reduced the time complexity of the overlap routine from O(N 3 ) to O(N), where N is the total number of sites on the lattice. This opens up a door to more accurate lattice measurements in the future.

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