t-) A 13.(o.q" b Monte Carlo Studies Of Two Dimensional Field Theories Anutx AnopKANI This thesis forms part of a Doctor of Philosophy done in the Department of Physics ancl Mathematical Physics, Ijniversity of Adelaicle Adelaide, March, 1998 Contents 1 Introduction 1 1.1 The generating functionals 3 1.2 Scalar Quantum Field Theory on the Lattice 8 I.2.1 l/a theory on the iattice. 12 1.3 Renormalisation 19 1.3.1 Counter terms 20 I.3.2 The Renormalisation group equations 2r oa 7.4 Fermions on the lattice L') 1.4.1 The Doubling problem 24 1.4.2 The Wilson fermion 26 2 The Lattice Effective Potential 29 2.L Introduction '2t) 2.2 The lattice effective potential 31 2.3 The MC effective potential. 34 2.3.I The variation of source methocl (VSNI) 34 2.3.2 The Constraint Effective Potential (CtrP I) 36 2.3.3 The One Loop constraint effective potential 38 2.3.4 The Constraint Effective Potential (CtrP II) 39 2.+ The numerical results . 42 2.5 Conclusions ltl't 3 Soliton models on Lattice Ðt 3.1 Classicalsolutions 5t) 3.2 Semi-classiåal Quantisation 64 3.3 Path integral quantisation 68 3.4 Soliton quantisation on lattice 70 3.5 Lattice Monte Carlo calculation of the kink mass t+ 3.5.1 The scaling region, 77 3.6 Details of the simulations . E1 ó.t Conclusions E2 4 Monte Carlo Methods 85 4.L Introduction 85 4.2 The Langevin Algorithm 88 4.3 The Moleculat dynamics method 91 4.4 The Hybricl algorithm 9)¿I 4.4.7 Hybrid Monte Carlo and Simulation of fermions 100 4.4.2 An efficient method for simulation of Wilson clynamical fèlnions r03 4.5 Conclusions .... II2 A Grassmann algebra L2T B Dimensional engineering L25 C The strong scaling in the linear o model L27 D Super renomalisable models 131 1qÐ E CEPII equations I d.) F Renormalisation of the kink rnass 135 G Path independence of dislocated partition function 139 H An efficient vectorised Fortran code for simulation of dynamical \Milson fermions L4L ll Acknowledgements I would like to thank my supervisors, Tony Williams and Linclsay Doc1cl, for theil guidance and attention to my work. I also extend thanks to Jirn Nl'Calthy ancl Tien Kieu who have helped me during my PhD candidature. In aclclition, m1, r,volk has benefrtted from discussions with Don Koks' Andv Rawlinson' I would like to thank my family and all my friends and lelatives in Aclelaicle for their support. Thanks also to University of Adelaide for their- financial suppolt. 1V To my parents, my sister ancl my brother V Chapter 1 fntroduction Neitlter you tTor I lcr¿out th,r: ntrlstries of eternitrl, Neitl'¿er yoLl tlor I reacl tltis enigma, You, ar¿d I on,ly talk tltis side o.f ueil; When the ueil falls, neith,er yol¿ rTor I uill be here. On¿u,r I{hayyctnt Since their introduction by Feynman (19a8) II,2], path integral methocls have become an important tool for particle physicists. Many of the mocleln clevelopments i¡ clua,ntlrl field theories are based on path integral ideas. A iarge nunbel of p¡ysicist have examinecl various methods to evaluate the sums over the paths. HoweveL, llost of these a,r-e 'rethocls based on the existence weak coupling parameters in which the theory can Jre expa¡clecl perturbatively. Thus they are not suited for analysis of phenonena, gover.necl þy large couplings. They also are not suitable for cases whele the clel;enclence on the coupling constant is not analytic. On the other' hand, the lattice formulation of cßrantum fielcl theories leads to a non-perturbative study of such fielcl theolies. The lattice method involves replacing the space-time, infinite volune, Eucliclian yer.- sion of a theory, by a discrete and finite lattice, The lattice acts as a legulator ancl removes the divergences of Greens functions. The regularisation must errentually be 1e- moved by letting the lattice spacing gradually cleclease to zeto ancl in this process the couþling parameters.must be renormalisecl. A well clefinecl continunrl linit can be ob- tained parameteïs if approach a scaling fixecl point, where tire fixecl points can be fincl using renormalisation group analysis, (this will be cliscnssecl in cletails lattel on in this chapter)' The important remaining question is to establish r,vhether.such a fixeci poi¡t exists' If so, then the results which have been establishecl in tire strong coupling regi¡re 1 can be extrapolated to this fixed point. Having done so, establishes that the lattice model corresponds to the continuum theory. The path integral in the continuum involves an infinite number of integrals. By in- troduction of the lattice, the number of integrals becomes finite, but for most cases of interest it is still large. In order to evaluate these integrals one needs to use some sta- tistical methods, principally Monte Carlo (MC) methods. Monte Carlo simulations have been extensively applied to the study of lattice gauge theories for a number of years and several important results have been achieved. Until 1974 all the predictions of QCD were restricted to the perturbative regime. It was the lattice formulation of QCD by K.Wilson [3] (1974) that opened the way to the study of non-perturbative QCD using MC methods. The attraction of the lattice formulation of field theory is the correspondence between the Euclidean field theory, introduced by Schwinger [4], and statistical mechanics. It gives an equivalence of quantum of field theory and statistical mechanics. An elegant way of highlighting such an equivalence is through the renormalisation group. One can consider the quantum field theory being mapped into classical equilibrium statistical mechanics, and transform the problem of the continuum limit into the problem of critical behavior. In this thesis we are at times concerned with the non-perturbative quantisation of some soliton models on the lattice. Solitons have been known since the early 19th century. Soli- ton waves are the solutions to non-linear differential equations which are characterised as localised waves which do not disperse and which preserve their shape after a collision. In the early to mid 70's it was realised that some of the non-linear field theories which were used to describe elementary particle also had soliton solutions and these could correspond to particle type excitations. These quantisation of these solutions then clearly became important. The quantisation of solitons was mainly done using the developed semiclassi- cal methods where one quantises the soliton around the classical solution [5, 6, 7]. The semiclassical methods can only be applied when the non-linear coupling constant is suf- ficiently small. In order to go beyond the semiclassical approximation one may attempt lattice stochastic quantisation methods. In these methods one is limited by the finite size and discreteness of the lattice and care must be taken that the continuum limit is well defined. Also one requires that the symmetries of the continuum model (or the discrete version of the symmetries) hold in the lattice version of the theory hoping that, as one approaches the continuum limit, the discrete symmetries should approach the continuum symmetries. In this work we address all the above aspects of quantisation of the solitons on the lattice. The purpose of this chapter is to discuss some concepts in quantum field theories and 2 their lattice formulations as well as introducing important notations and conventions. 'We start with a brief summary of elementary concepts in the path integral formulation of quantum field theories. The generating functionals and Greens functions (correlation functions) for both bosonic and fermionic fields, in Minkowski space are introduced. The corresponding representation of the generating functionals, Greens functions, etc. contin- ued to imaginary time (i.e., Euclidean) are discussed. The rest of the chapter includes the lattice formulation of the scalar and fermion field theories. There is deliberate em- phasis on the renormalisation group since it plays a major role in the understanding of continuum limit. In chapter 2 we discuss the calculation of renormalised quantities for a scalar field, by calculating the lattice effective potential. As we will show the effective potential can provide an accurate tool to calculate the renormalised quantities. We propose a new Monte Carlo effective potential method of calculating the renormalised parameters and we examine the applications of this method and two other, already established, methods to different regimes. We also discuss the computational details concerning these methods. Our study of the effective potential will be useful in chapter 3. In chapter 3 we study the simplest quantum field theory admitting topological solitons, Àþa in two dimensions. As an introduction we discuss the classical solutions and the semiclassical quantisation of the topological solution. A non-perturbative quantisation scheme using the Kadanoff operators will be discussed. We show that the zero mode problem which is already known to exist in the semiclassical regime, can persist beyond this regime. We treat the zero mode problem on the lattice and we calculate the soliton mass, using Monte Carlo methods and compare them with the semiclassical results. We show that the soliton mass can be higher or lower than the semiclassical predictions, depending on the bare parameters. In chapter 4, Monte Carlo methods for the simulation of scalar and fermi fields are briefly discussed. We propose an algorithm, a modified version of the hybrid algorithm, to generate a series of configurations in Markov a chain with an imposed constraint. In this chapter, we also present an efficient code for simulation of the linear ø model, with dynamical fermions, on vector machines.