Topological Susceptibility in the Schwinger Model

在這篇論文中，我們應用最佳手則對稱之新費米場作用量(optimal domain-wall fermion)，研究Schwinger模型上的拓樸率。藉由拓樸率的定義，我們可以利用整體拓樸荷(global topological charge)來決定拓樸率。同時，透過鞍點近似法(saddle point expansion)，我們也可以在一個固定的拓樸區(fixed topological sector)內，藉由拓樸密度(topological charge density)的兩點相關係數以得到拓樸率。最後我們將證明在晶格上，這兩種方法會得出相同的拓樸率。In this thesis, we study the topological susceptibility in the Schwinger model with optimal domain-wall fermions. We obtain the topological susceptibility from counting the global topological charge. Also, using the saddle pointxpansion, topological susceptibility can be extracted from the 2-point correlation functions of topological charge density in a fixed topological sector.We can demonstrate that the values obtained from this two methods are inood agreement.1 Introduction 3.1 Schwinger Model . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Lattice Formulation of U(1) Gauge Field . . . . . . . . . . . . 6.3 Naive Lattice Formulation of Fermion Field . . . . . . . . . . 7 Dirac Operator Schemes 10.1 Wilson Dirac Operator . . . . . . . . . . . . . . . . . . . . . . 10.2 Ginsparg-Wilson relation . . . . . . . . . . . . . . . . . . . . . 11.3 Overlap Dirac Operator . . . . . . . . . . . . . . . . . . . . . 13.4 Conventional Domain Wall Fermion . . . . . . . . . . . . . . . 16.5 Optimal Domain Wall Fermion . . . . . . . . . . . . . . . . . 18 Monte Carlo Method 22.1 Fermion Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.2 Pseudo-Fermion Field . . . . . . . . . . . . . . . . . . . . . . . 24.3 Hybrid Monte Carlo Method . . . . . . . . . . . . . . . . . . . 25 Topological Susceptibility 29.1 Topological Charge On the Lattice . . . . . . . . . . . . . . . 29.2 Topological Susceptibility On the Lattice . . . . . . . . . . . . 31.3 Fixed Topological Sector . . . . . . . . . . . . . . . . . . . . 31 Data Analysis 35.1 Autocorrelations in Monte Carlo simulation time . . . . . . . . 35.2 Data Blocking . . . . . . . . . . . . . . . . . . . . . . . . . . . 36.3 Jackknife Method . . . . . . . . . . . . . . . . . . . . . . . . . 37 Results 38.1 Topological Susceptibility from Global Topological Charge . . 38.2 Topological Susceptibility from Fixed Topological Sector . . . 42 Conclusion 46ibliography 48 Force in Hybrid Monte Carlo 51.1 Gauge Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Fermion Force . . . . . . . . . . . . . . . . . . . . . . . . . . . 5