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Numerical Investigations of the Schwinger Model and Selected Quantum Spin Models

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Numerical Investigations of the Schwinger Model and Selected Quantum Spin Models Adam Mickiewicz University Faculty of Physics Poznań, Poland Master’s Thesis Numerical investigations of the Schwinger model and selected quantum spin models Marcin Szyniszewski Supervisors: prof. UAM dr hab. Piotr Tomczak dr Krzysztof Cichy Quantum Physics Division, Faculty of Physics, UAM Poznań 2012 Abstract Numerical investigations of the XY model, the Heisenberg model and the 퐽 − 퐽′ Heisenberg model are conducted, using the exact diagonalisation, the numerical renormalisation and the density matrix renormalisation group approach. The low-lying energy levels are obtained and finite size scaling is performed to estimate the bulk limit value. The results are found to be consistent with the exact values. The DMRG results are found to be most promising. The Schwinger model is also studied using the exact diagonalisation and the strong coupling expansion. The massless, the massive model and the model with a background electric field are explored. Ground state energy, scalar and vector particle masses and order parameters are examined. The achieved values areobserved to be consistent with previous results and theoretical predictions. Path to the future studies is outlined. 2 Table of contents Introduction ...................................................................................................................5 1 Theoretical background ................................................................................................7 1.1 Basics of statistical physics and quantum mechanics ................................................ 7 1.1.1 Mathematical tools of quantum mechanics ....................................................... 7 1.1.2 Basic concepts in quantum mechanics ............................................................. 12 1.1.3 Quantum mechanics of composite systems .................................................... 17 1.1.4 Basic concepts of statistical physics .................................................................. 19 1.2 Overview of selected quantum spin systems ............................................................. 21 1.2.1 The XY model........................................................................................................ 21 1.2.2 The Heisenberg model ......................................................................................... 23 1.2.3 The 퐽 − 퐽′ Heisenberg model .............................................................................. 24 1.3 Introduction to the Schwinger model ......................................................................... 25 1.3.1 Continuum formulation ...................................................................................... 25 1.3.2 Lattice formulation .............................................................................................. 26 1.3.3 Operators in the lattice formulation ................................................................. 28 1.3.4 The free Schwinger model .................................................................................. 31 1.3.5 The Schwinger model with background electric field ................................... 31 2 The details of used methods ....................................................................................... 34 2.1 Exact diagonalisation ..................................................................................................... 34 2.2 Strong coupling expansion ............................................................................................ 35 2.2.1 “Scalar” states ....................................................................................................... 36 2.2.2 “Vector” states ....................................................................................................... 40 2.2.3 Incorporating the constant background field .................................................. 41 2.3 Density matrix renormalisation group ....................................................................... 42 2.3.1 Basic ideas of the renormalisation group approach ....................................... 42 2.3.2 Numerical renormalisation group on the Heisenberg model ...................... 43 2.3.3 DMRG on the Heisenberg model ...................................................................... 44 3 Simulations of the spin models .................................................................................. 47 3.1 Simulations of the XY model ........................................................................................ 47 3.1.1 Results and the analysis of the exact diagonalisation method .................... 47 3.1.2 Results and the analysis of the NRG method .................................................. 50 3 3.1.3 Results and the analysis of the DMRG method .............................................. 51 3.2 Simulations of the Heisenberg model ......................................................................... 52 3.2.1 Results and the analysis of the exact diagonalisation method .................... 52 3.2.2 Results and the analysis of the NRG method .................................................. 54 3.2.3 Results and the analysis of the DMRG method .............................................. 55 3.3 Simulations of the 퐽 − 퐽′ Heisenberg model .............................................................. 56 3.3.1 Results and the analysis of the exact diagonalisation method .................... 57 3.3.2 Results and the analysis of the NRG method .................................................. 59 3.3.3 Results and the analysis of the DMRG method .............................................. 60 4 Simulations of the Schwinger model ........................................................................ 63 4.1 Exact diagonalisation of the free Schwinger model ................................................. 63 4.2 Using the strong coupling expansion method for the Schwinger model .............. 65 4.2.1 The massless Schwinger model .......................................................................... 66 4.2.2 The massive Schwinger model ........................................................................... 71 4.2.3 The chwingerS model in the background electric field ................................. 74 5 Summary, conclusions and outlook ......................................................................... 80 Acknowledgements .................................................................................................... 83 References.................................................................................................................... 84 4 Introduction The Schwinger model [1] is one-dimensional quantum electrodynamics, which was found to exhibit such interesting phenomena as confinement and chiral symmetry breaking [2]. Therefore, is it often used as a test ground for other, more complicated quantum field theories. This model can be solved exactly in two limits [3]: when the fermion mass is very heavy and very light. The behaviour in between those cases is a subject of intense studies [2,3,4]. A solution to this problem presents itself, especially nowadays, when we are experiencing a technological boom in computer sciences. The computer simulations of quantum models are the essential step in expanding our knowledge, particularly where exact calculations exceed our current mathematical aperture. A useful formalism, that will be used in this work, is the lattice representation of the models [5]. Finite lattice techniques are very successful tools in obtaining the results in the continuum and bulk limits. Thus, our objective emerges – it is to use computer simulations to investigate several quantum spin-½ models and the Schwinger model in the lattice formulation. We will be interested in using many different simulation techniques, to observe which one is the most promising approach. The outline of this work is as follows: Chapter 1 outlines the most important theoretical principles that are essential in understanding this project. Firstly, there is an overview of the basic concepts of quantum mechanics. Then, we summarise selected spin models that are studied in this work, i.e. the XY model, the Heisenberg model and the 퐽 − 퐽′ Heisenberg model. Then follows a review of the Schwinger model, where we present the Hamiltonian for various cases of this model (massless and massive, how to incorporate the background electric field, etc.) in various representations (continuum and lattice cases). In Chapter 2 the reader is introduced to the methods used in the computer simulations of the mentioned models. We present the exact diagonalisation method and the strong coupling expansion of the Schwinger model. The renormalisation group approach is also delivered, with examples of the numerical renormalisation and the density matrix renormalisation group applied to the Heisenberg model. The results of using the ED, NRG and DMRG methods on the spin models are presented in Chapter 3. Estimated values are compared to the theoretical predictions and previous results. 5 In Chapter 4 we use the strong coupling expansion of the Schwinger model to simulate its Hamiltonian. We state our results and the analysis of the results to estimate the bulk and continuum limit values. In order to do this, we are using the finite size scaling methods and the extrapolants approach. In Chapter 5 we discuss the conclusions that can be derived from our results and sketch possible future improvements. 6 Chapter 1 Theoretical background 1.1 Basics of statistical physics
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