Monte Carlo Studies of Two Dimensional Field Theories

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

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.

To my parents, my sister ancl my brother

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 obtained 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 introduction 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 statistical 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 semiclassical 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 sufficiently 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.

1.1 The generating functionals

Generating functionals and the correlation functions. Consider a translation invariant, local action, ^9[d], in a d-dimensional Minkowski space, that is a function of a scalar field

3 / and its derivative. This means that the Lagrangian density does not depend on space explicitly but only through the field, /. Using the language of sources due to Schwinger[8] one may define the vacuum-to-vacuum transition amplitude for a free scalar field in the presence of a source J(r) as

dd*ró} zlJl: ltoøl e¿{stót+Ï , (1.1) where S[/] is the action in presence of an external source "I. The functional ZlJl is the generating functional of correlation functions, G(æ1, ...,,ïn), in the sense that ç@)(æt. . . an): (rlþ(rt). . . ó(**)l) : (-i)' z-'@#) #rlrllr=,, (1.2) where Tló@t) . . .ö("")l is the time ordered product of fields /'s. For a free field where sldl : -t, I ddxþ(æ)lz + mzlþ@) (1.3) one can perform the functional integration for ZolJlwith respect to þ(r), obtaining an explicit form of ZolJ) as a functional of J(r):

zo[J] : ï r(æ)a(æ-s)r(s)ddrddsl [ ó(a+*2)ó ad,f (1.4) "lt I VOpl+ , where the propagator is defined by

- (¡ + m2)L,(x,y¡ : td(r - fi. (1.5)

L(*,A) is called the propagator because it represents the propagation of the particle from r to y. The integral in Eq. (1.a), involving /, is just a number and we set that to a constant C. Thus ZolJl: Celi I ddcddv r(ø)a(ø's)r(s)] . (1.6)

Then from Eq (1.5) it is easy to see that A(r, y) has the following Fourier representation

| "-ik,(x-y) L(*,ù:Jadn ffi, (1.7) which is the Feynman propagator for a free field /. Also note that

(-i), 62 zolJl Gr(r,y) : (01"[ ó(*)ó@)]l0l : : -i\(r,a). (1.8) zol0l 6J(æ)J(y) J=O

4 Connected conelation functions. The generating functional for connected Greens functions is defined as WlJl: -iln ZlJl', (1'e) with the connected Greens function being

G!ù qn,. . . *n) : (rrg(rt) . . . : (-.i)* (1.10) ó(*,)n " # h*rrll^o, where Zl0lis set to 1 and c denotes connected. In terms of Feynman diagrams any part of the diagram is joined to the remaining part by at least one propagator. It can be shown that the disjoint pieces of the correlation function G(") do not contribute to the S matrix and hence it is more convenient to work with the connected correlation function. Connected correlation functions have important properties in Euclidian space that will be discussed later.

Generating functional for fermion fi,elds. The generating functional for a Lagrangian density L(rþ,.þ), involving only fermions is defined by

zlrt,,Tl: z(0,0Yt r'.þ¡,'t')+'t'n+iÚ), (1.11) Idbþ,Ú]elilaat where ,þrrþrn and 4 are ??-component anticommuting c-numbers obeying the Grassmann algebra [see Appendix A]. The parameters 4, I arc analogs of J@) and act as fermion sources. For a free fermion field, where the action is quadratic in the field variables, Zlq,îl can be evaluated exactly. In this case the Lagrangian density is

L.(rþrþ) : irþlP 1rtþ - m.þ1þ. (1.12)

Then one can consider Zl|,nl as the continuum limit of

- i D¿,, ø|M iP,þtp+ i D i,.(î.nî +ñ?,þL) zlrt,,,t|: I d,lrþrþl e a,þ ( 1.13) where a and B arc Dirac spinor indices and

Mî;P --D0,)",P16¿,j+, - 6¿,j-r] - m6¿,¡6o,P. (1.14) l.r

Here j * ¡z is j/s neighbor site in the direction of unit vector p. The integral is Gaussian and results in Zolrt,îl: det(iM)"-'Ð',,niu;|ni, (1.15)

b where the spinor component indices are suppressed. In particular Ma|, is the fermion propagator. The free propagator for the fermionic field is defined by

(,þ"(*)úp(ùl : !'1",1!:!1, t-=n=o6o,þ ' - 6q(*)6n@)',

: -;6#l-, I rt@)s(r - y)a@)d*da]l=r=o6o,p - is(r - u)6o,þ, (1.16) where S(, - y) is defined as the continuum limit ol M;,1 and is identical to the free fermion propagator obtained in the canonical formulation of QFT..

Wich rotation and, Euclidean action In Minkowski space-time where the axis are labeled â,s, rs, ï1,¡...,tíxd-r with rs denoting the time, ro:t, one can rotate the time axis by an angle r 12. This is also known as Wick rotation. This space with an imaginary time axis is d-dimensional Euclidean space. Defining

x¿: ixo - 'if,, (1.17) the invariant interval is

d d,s2:(d*o)" -(drt)' (d*o-')': -D( dr þ (1.18) p=l

In addition, in Euclidian space the momentum becomes

lea: -ilco, ( 1.1e)

SlVlng k' : -(k? + kZ + ...+ kÐ : -l*ø, (1.20) and dokt - dd-rka: -iddlc. (r.2r) Using this equation and noting that (ôpl)' : -(ôËó)2, the Euclidean generating functional ZslJl is zoølJl: llaOÞ(-[adcltøló)'+^'ó'-ó4). (r.22) The propagator in Euclidean space is

A(r):år¡r,r,ffi, (1 23)

6 which does not have its poles on the real axis (compare with Eq. (1.7)). Explicitly in d" : 4 dimensions the poles are:

leq: ltri(k2 * m2¡i (Lz4)

Generating functionals and the comelation functions in Euclidel,n spl,ce. trq. (1.22) gives the generating functional for the free scalar field in Euclidean space. In general, for an action, one has ^9[/] ao,t.ö zø[J]: ltaøl e-stôt+[ G.25) which is the generating functional for the correlation functions in Euclidian space, i.e., zg)@,. . . **): (d(",). . . ó(**)) - z-lu:0r åÐ ht"trrL=,. (1.26) In comparison with Eq. (1.2) they differ by a factor of (-i)"-1. Also note that there is no time order product in Euclidean space. From a statistical point of view ZlJl is jæf the partition function for a statistical system with the field / as the order parameter. For example, ZBIJI can be the partition function for an Ising model with an external magnetic field J. Similarly to the the previous section the generating functional for the connected correlation functions can be defined as

Wø[Jl : IuZø[J]. (r.27) which is the analog of the free energy in statistical mechanics. From here on the subscript -E denoting Euclidean will be dropped and quantities will be Euclidian unless otherwise stated. One of the important properties of the connected correlation function is the cluster property. That is the connected correlation functions decrease rapidly when two non- empty sets of space-time points {*r . .. ro} and {Ar . . . yr} arc separated by a large distance [e],

ç(^)(a...tpiUp+t...U,) --+ 0 as min l ro - y¡ l* oo. (1.2S) Furthermore as v¿e will show, in a field theory containing only massive fields, the connected correlation function decreases exponentially when the points are separated, with a minimal rate that is the inverse of the smallest mass in the theory.

(ó(*)ó(y))" ,N e-m(o-s) as l* - yl -) oo. (1.29)

I The relationship between the correlation functions and the connected correlation functions can be easily obtained by taking successive functional derivatives of Eq. (L.27) with respect toJatJ:0: (d(')) : (d(,))" (1.30) (d(')d(')) : (d(,))"(ó(,)).+ \S@)S@))" (1.31)

(ó("r)ó("r)d("r)) : 16çrr)). + (ó(*r)1"+ (d("r))"

+ (ó(r r)) (ó(r r) ó(" + (ó(r r)) (ó(r r) ó(* ) + ( d ('r)l . (ó(" r) ó(, ùl " ")) " " ") " " + (ó(*r)ó(rr;41r'))"

( 1 .32)

Proper uerter functions. From the generating functional of the connected correlation functions WlJl it is possible to derive by a Legendre transformation a new generating functional f [d"], in which S. is a classical field argument of l. This quantity, f , is called the generating functional of proper vertices . The Legendre transformation is defined as : rld"l + wlJl: | ó.{*)t{*)o*,, (1.33) where ó"(*):ffi (1.s4)

Take the functional derivative of both sides of Eq. (1.33) with respect to S.(a) : ffi: -1 ffiYfl ¿a + l 6@- z)r(z)* l o"raffi : r@) (1 Bb) f[/] is called the generating functional for the one-particle irreducible diagrams which can not be made disjoint by cutting one internal propagator. t.2 Scalar Quantum Field Theory on the Lattice

As we saw in the previous section, the n-point correlation function in Euclidean space for a scalar field theory is defined as

G(*r,...,ïn): (ó(*r),...,ó(*òl - I ldölÓ(t')"' Ö(*") e-søtól t ldól e-setÓt

8 To give the above path integral a precise mathematical meaning we introduce a space-time lattice with lattice spacing ø. Every point on the d-dimensional lattice is then specified by d integers which we denote collectively by n - (rr,nr,. . . ,nd). By convention the last component,, nd - ns, denotes the Euclidean time. Define the lattice derivative, by the finite difference operation

ot"ón:'¿tO,*, - ó^), where p is the unit vector in the direction of ¡^1. Then, one has

D o ró"ô ró^ : D ónE' ón, ntþ n where the dimensionless lattice d'Alembert operator E' acts on field / according to d I a'ö("): D (zó"-ó^+î-ó"_¡) ( I .36) p=l a2

Now consider the Euclidean action for a free scalar field sølól= sldl :+ I dduþ(a) l-o * *'ló@). The transition from the continuum to lattice is effected by making the following substitutions I p -+ npa,) I dad - oo+, [dó] - \dþ(na), (1.37) The next step is to rescale the mass puiurn"t", and the field / with their conanical dimension n1, ITÙ: -o, and write the action in terms of the dimensionless variables m" stól: -Ð4"4"-p + tld + la'", ntþ n 2 where ¡r extends only over positive directions and we have imposed a periodic boundary condition. From now on we drop the hat on the d field. We write the discrete action as s[d] : IDr,*-,^ó^, (1.3s) I with Mn,* : - D l6*+t,n * 6^-r,,n - 26n,^f - ñ' 6n,^ (1.3e) l.L We define the asymmetric discrete Fourier transform as õ* : D"'{r'ó,,

\ \- 2rik.n , ó" * Q*', 'lY^tdLe n (1.40) and the discrete ó-functions are defined as

1 -2rih,(n-m) 61,* Nd fe ^¡ k 1 64^ (1.41) k,q Ð"=t4*, ¡vrd n where lú is the number of lattice sites in one direction (we have considered a hypercubed lattice) and I and ,i are dimensionless momenta. We have used the fact that the Fourier transform of a function on the lattice is periodic in momentum space with period of lú and therefore all momenta and coordinates are restricted to the first Brillouin zone (BZ): -¡r 2< î,5 {, ¡/ -¡/2

The latticefreefield propagator G*,n, is defined to be the inverse of Mn,*,i.e.,,

1 6t,^. (1.43) DM,,,Gn,m - n ¡{d,

The solution of this equation is obtained through the Fourier transformation:

1 Gn (r.44) tn'ì' ¡úd I k

Replacing the ó functions in Eq (1.39) by their Fourier representation, Eq. (1.41), one easily arrives at

I z (t - cos îr¡ +ñ' Gî: t It

10 Therefore -1 C,1 ,(1 cos î,) K lf - +a,f lu=t I t d. r-1 lÉ - +ñ'f : - Lp=t"in,i " ) l+î1**'l (1.45) with ñ :2sin*. Since the summation in above equation is restrictedto BZ, the integral wiil be dominated by momenta which are small compare to the inverse lattice spacing; hence \Me can set Î, --+ fr, to get : (1.46) lg3CA (t' + *')-' : ã&). Thus this formulation of the free scalar field on the lattice has the correct continuum limit. The generating functional for the lattice connected correlation functions, WlJ,a]: WlA, and the generating functional for the proper vertex functions, llö,"1: f[-/], can be obtained in a similar fashion to the continuum Euclidean case. That is, for Wlîl we have w[1]: tn zfîl and WlÎl and f [þ] are related through the Legender transform:

6 :awJA.', Yn - ai. G.4T) rtd-l + w[î]:D6*î". (1.4s)

Similarly to the above, the proper vertex frn"tilrm can be defined from the continuum Euclidean case. Take the derivative of Eq. (1.aS) to obtain

ôtl-ól , aÎ^ : \-awÚloî* s--7 : în, (1.4e) @; - Ð uï:n * Ð6^ffi + Ð î*6n,^ By differentiating Eq. (1.a7) with respect to$* one obtains ç _ arwlîl _ sa ozw[a ale 'nt',n: affi;: + aJ"oJrW, ( 1 .50) AJIAó can be obtained by differentiating Eq. (1.a9): a,wÛl aTl6l t :6n,^ p aî,aî, a6ra6^

11 We know that A'WÛ]I amlr=o Gn,r, and defining the two point vertex function as ô,îl-ól : tn(2) aóoaó^ - p,m) =0 we see that l[2à, is the inverse of the connected 2-point function, Gl2,)^ (we have suppressed the index c on G.). The above equation can be generalised by defining aTr6r r*r,...,*,,:mlr_o' _ r

In general one can express the rz-point connected function for Iú ) 2 as

G9Ì,...,^, : - Gß),,r, . . . Gll),,r,1*),...,r. I Q|,f ,...,^,, where the Q(")s are the one-particle reducible diagrams. It is useful to derive the above relation in the momentum space. Note that due to translational invariance

Gf,rî, : h,î,Gf,), f(2)^ : ó..d3). kt,kz K7 'K2 le1 ( 1.51)

Then one arrives at

**,,, * ef,,, î.: -Gfl "å?rf] e î. L.z.L À/4 theory on the lattice.

The À/a is the simplest QFT model on the lattice. It is also an important part of the standard model for weak and the electromagnetic interactions. The theory of electroweak interaction has an SU(2) x U(I) [10] gauge symmetry with corresponding gauge vector bosons and various families of quarks and leptons. The scalar sector is assumed to be in the phase of a broken O(\ theory which leads to a spectrum of one scalar massive boson and three massles Goldstone boson. Taking the gauge symmetry into consideration, one obtains the SU(2) x U(1) Higgs model and three of four massless gauge vector bosons annihilate the three massless states which leads to the well known spectrum of three heavy vector bosons and a massless photon. The only remaining massive degree of freedom is

t2 the Higgs boson. The Higgs mechanism is non-perturbative as it involves infinitely many degrees of freedom in a strong coupling. The results mentioned above are based on the tree level perturbation and some assumptions about the vacuum. One can do a non-perturbative lattice simulation in the bosonic sector of the theory in order to find out whether these results survive quantisation. In the MC calculations one might try to find the phase structure of the theory and find the location of the critical point corresponding to the continuum limit. In the lattice calculations the fermions are not taken into account since the light fermions do not play a crucial role in the symmetry breaking and mass generation [11]. MC studies of the t/(1) x SU(2) Higgs model have indicated that the mass spectrum may be understood from the /a theory in the broken symmetry phase, adding the gauge theory as a perturbation in the normalised gauge coupling [12]. From renormalisation group analysis [13] it can be concluded that in d< 4, the continuum defines an interacting field theory. The renormalised coupling À, is bounded (0 < ), ( constant). In d> 4 the theory is trivial with mean field behavior. The renormalised coupling cosntant vanishes at the continuum limit. There has been a strong indication of triviality of Àþa theory in d :4 (the renormalised coupling vanishes in the continuum limit). The triviality of the lattice þa lheory in four dimensions has not been established rigorously , but the perturbative renormalisation group analysis [14] together with evidence from high temperature series expansions [15] , numerical simulations [16], exact inequalities [17, 18, 19] and block spin renormalisation group studies [20] and combination of high temperature expansion with renormalisation group analysis 12L,22,23] does not leavemuchdoubt that the theory is trivial. Here we are only concerned with the one real component sector in two dimensions. By starting with the Lagrangian of single component /a:

L(t): a,o-'l#r#r + (1 b2) I -'utoor -irffif o' - i,ø^ r(ùóf and by continuation to the imaginary time ú -> -ir f c the action becomes

s : bB) + | r,çqat: -* I o*ollror, +lrfftø, *].ø^ - r(ùó] (1

A lattice version of the action can be written as

st6, î) : r,,, - ó,)' - ö'^+ î^ô-], (1.b4) ln i+ñ' T âdå - + where as before we have defined the dimensionless quantities $ : otalÐ-t ó,, ñ : ma and. j - ¡oa-a and Î - q(d/z)+r¡. We adopt natural units, ie, h, : c: L. The sum over

13 ¡.1 is over the d Euclidean directions. We also impose the appropriate periodic boundary condition on the fields:

Ón¡6u: Ón for all ¡'r , where fr, : (0,... ,Nr,...,0),is a d-dimensional vector with /ü, being the number of lattice sites in the direction ¡^r. The Sa theory in two dimensions is known to exist in two phases, one where the reflection symmetry ó - -ó is spontaneously broken and the other where it is not. Classically speaking, the symmetric phase with ñ ) 0 and (d) : 0 is separated from the broken symmetry phase with ñ < 0 and (ól + 0 by a line of second-order phase transitions where fi and Î urro*" critical values ñ" and Î". At ) : 0 we have a Gaussian fixed point, corresponding to the free field. Any continuum limit in the vicinity of this fixed point is trivial in a sense that it has a vanishing renormalised coupling À,' : 0. In order to find a continuum limit representing an interacting theory one needs to look for a fixed point. The question whether such a fixed point exists and where it is located could be answered once the renormalised. mass, fi", and renormalised coupling, Îr, are known as functions of bare parameters (we will come back to this later).

Lattice perturbation theory

The main purpose of the lattice cutoff is to provide a regularisation which allows application of non-perturbative method. Nevertheless, it is sometimes necessary to perform perturbative calculations with a Ìattice cutoff. In particular it is so, if quantities evaluated by non-perturbative methods are to be related to quantities calculated perturbatively. Furthermore, some quantities, like finite-volume effects, that are of interest in numerical calculations, can be estimated in lattice perturbation theory. In perturbation theory, in order to obtain the expansion of Greens functions in powers of the bare coupling À, the interacting part of the Lagrangian is factored out in the path integral as a differential operator. That is

ztit:""p (l.bb) ' l-itL 41* *1t,ta,6J/"J with zolîl: r î. G,,^î*]. (1.b6) ltaø1""r l; n The operator can then be expanded in the po\¡/ers of the bare coupling. This leads to the well known Feynman rules which in continuum space-time are given in many text

T4 books. On a lattice, the derivation of the Feynman rules proceeds in much the same way. Therefore here we emphasise the differences in the presence of a lattice cutoff. In momentum space, with n external lines, for a lattice with l/ sites in each space-time direction, the Feynman diagram rules for a diagram contributing to a Greens function are as follows: (i) Each line is associated with a free field propagator G@. (ii) trach vertex is associated with bare coupling, Î. (iii) At inner vertices momentum conservation holds modulo 2zr. (iv) The loop momenta has to be summed over the first Brillouin zone and divided by a factor of I/d. (v) Finally there is a symmetry factor as in continuum version of the theory. Note that 1 G(k) : with k,- :', ¿rinfu øh 2 As an example we evaluate the Feynman diagrams contributing to the two-point and four-point vertex functions (the external legs have been taken off) in ìdn rp to two loops.

l.a I.b Ic

I.d I.e I.f I'g Figure 1.1: The one-loop and two-loop diagrams contributing to ¡(z) u1r¿ ¡(+).

These diagrams are shown in the Fig. 1.1, with their associate expression

Io ;¡>crø ^, -ì 1 I6 Dc'(q) 4Nd q ^t I" + #ÐÐ G (q')G (q')G(q' + q')

lll _ ^t tdr, -3Àl--:+Ðc,(ø) 2^ q 3À1=3 ,ef: 4Nd q _ 3À1=3 tÍr D: Gu(qr)G(qr) Zñ q, =3 1 Is : tI G'(q,)c(q,)G(n + 4ù (1.57) Ë 3Ð where À :6). The physical and the renormalised n'¿uss. Consider the Fourier transformation of the correlation function in space coordinates only:

c(nr,Ê) : D (ro, n)). n "-;Î'"ç"1g,

The behavior of the C (n6,k) for large ns is dominated by the complex singularity of the propagator in momentum space. Consider G(îo,k) at constant spatial momentum and assume that its singularities closest to the origin are simple poles at

ko: LiE(k). ( 1.58)

Wedefinethephysicalmassbympny,:E(0). If wedenotetheresidueof thepolelry Z, the propagator near the poles behave as

[õ1î,,0)] :ry+o(îZ+ñ\'). (1.5e)

Application of the Cauchy's theorem gives

C(no,q J-¿-mena"lnol ¡ ... (1.60) = tTt,phgs where the dots indicate terms which fall off faster at large rzs. Thus the exponential decay of C(ns,O) is governed by the physical mass. In practice it is more convenient to consider the wave function renormalisation constant Z, the renormalised mass ræ, and the renormalised coupling Î,. Th" renormalised mass mrris defined by means of the small momentum behavior of the propagator:

qî):+. (r.61) ñ?+k2+o(k4)

16 In the vicinity of the critical point ffipha" and m, turn out to be almost equal [9]. The wave function renormalisation, Z, is defrned through the renormalised field

Órn: Z-t/2Ón, (1.62) whose propagator is normalised to

ã,(î): (1.6s) ==ñ7+k2+o(k4) : :,u=

The bare coupling also has a renormalised counterpart which we denote by )". It is defined in terms of the four-point vertex function aI zero momentum.

¡, : ¡fa)1o, o, o, o).

In the continuum limit we require that the renormalised propagatot, n'1, and À, remain finite. In the perturbation theory one finds that this can be achieved by a suitable tuning of the bare parameters ñ and Î as o, + 0. This is the central statement of renormaiisation theory which we will discuss later in this chapter. The renormalised mass ñ,, can be estimated by

1^ol aG1î¡ -1 -1 (1.64) €t:*;=l ãfî'l ¿îz

On the lattice \rye can write

ñ7 lõtoì;?{all-' (1 6b) = L nzr1fo¡ I ', with îr:(T.0.....0).\¡¿or"r"')"/) (1.66) where we have approximated the derivative with a difference equation. iir th" dimensionless renormalised correlation function. Also notice that we have taken the derivative only in one direction because of the translational invariance. Now take the Fourier transform of ó¡r,...,j, iû the direction of j1

ó(îù : t ój,,,...,jo"i#o, h,...,i¿

: | Ót'""'"1t""(#) +;'i"14;] ' hr...ria (1.67)

77 0.50

0,45

0.40 E

035

030

025 000 0.1 0 0.20 0.J0 0.40 À

Figure 1.2: The renormalised mass, ñ,., calculated by MC methods, (stars), and 2-loop calculations, (solid line), versus the bare Î.

Then one has ãG') \óGìóeî,)) :(o,) +(p,) õ(o) h') (1.68) where d \, óir,...,irrin1ff¡ J7 ,...¡J d, p

'l t ói,,...,io Jl ¡...¡ld ( 1.6e)

Then it can be easily shown that -> z" l(t\ - þ') - (þ\l+ ñ,:W¡-- l ft.zol Eq (1.70) can be used to calculate the renormalised mass using a Monte Carlo algorithm. The renormalised mass calculated using (1.70) is compared with two loop results, which includes the diagrams I.a, I.b and I.c of Fig. 1.1, and the results are shown in Fig. 1.2

18 1-.3 Renormalisation

As mentioned in the previous section, the momentum on the lattice is restrictecl to the first Brillouin zone and therefore the momentum cutoff is r f a. This means there are no divergences in the perturbative loop expansion. In order to transit to the continuum limit the lattice spacing, a, has to be set to zero which means the cutoff is set to infinity. The normalisation is necessary to obtain finite results as the cutoff goes to infinity. Renormalisation introduces renormalised fields, renormalised mass, renormalised coupling etc. The À/a theory is a super-renormalisable theory in two and three dimensions (see Appendix C) and a renormalisable theory in four dimensions. At one loop level and in four dimensions the primitive divergent vertex parts containing /'s are f(2) and l(4), corresponding to diagrams I.a and Ld in Fig. 1.1, respectively. The degree of divergences of these functions as a function of the cutoff is If a2 for l(2) and ln(l/ø) for f(a). Thus they should be renormalised in order to make them finite as ø --+ 0. We take ñ, as the scale and by this \4/e mean that the continuum limit corresponds to ñ,, --+ 0. Here without proof we mention the following theorem :

Theorem L There erist three functions ñ', î and Z which depend on neu parameters ñ}, and, î, such that

rl") çîn;ñ,?,î) : z"/2(i,(ñ,),ñ)l(") (î0,ñ,r,î). (1.21)

The above theorem states that one calculates the bare ¡(n) ¡1 the usual way. Then write ñ',î and, Z. as power series in Î", in which the coeficients are functions o1ñ,1. One then finds that there is a way of choosing these coefficients so as to make the resulting expression on the right hand side of Eq. (1.71), which has become a po\.ver series in Î,, convergent at every order in Îr. This prescription leaves the renormalisation functions undetermined to a large extent. One way to fix these functions is by normalising the finite vertex, i.e,

rf'z)çîc,: o;ñ?,î,) ^tmí, ð ^ r1)rî u;mi,À,)ñ.Ã2 1 6pti''\rc:- tl4)(ît: o;ñ,,,î,) À, (r.72) which is sufficient to fix m2, À and Z. This is very convenient since the external momenta are set to zeto. This renormalisation is fine if there is no massless field. For a

19 massless field such a prescription can not be used. The reason is that the renormalisation constants, such as î, Z, etc., which should be finite in regularised theory, will develop infrared divergences. This will happen even when the Greens functions are calculated with finite external momenta. This problem can be avoided if we impose the following renormalisation conditions :

r[2)10, î,) : o : ,, ftrl"tçî;r",ln=^, lÍ4)(Îc;Î,)ln=,,, : Î,, (1.Tg) where rc is a mass parameter, that gives the scale of the momentum at which the renormalisation is fixed.

L.3.1 Counter terms

The entire renomalisation procedure can be formalised by insertion of counter terms in the originally Lagrangian in such a way that cancels the divergences. This scheme which involves the multiplicative and additive renormalisation, is completely equivalent to the previous renormalisation scheme (the multiplicative scheme). Introduce a Lagrangian density of the form

¿' :f,{ó,+, (1.24) - ó^)2 *T^'o' * ìr^ - J'ö, where ¡t - 7tlz¡. We write the Lagrangian L'in term ol ó, - Z-t/2ö;

: (1.75) ,' {@,nrt, - ó,-)2 +f,2n"6: *Sæ - J',. Define r,,:f,@,n*tt - ó,^)'l!^lr: * Iø, (1.76) and Lct:t rt - t)(ó,^+, - ó?.t +r;zn' -ñ,)ö? +f,tz'î -î)ó1 to put Eq. (1.7a) into the form

L':Lr*L"t-Jó (t.77)

The renormalisation scheme follows as: Given a Lagrangian

(1.78) r - f,{ó,*, - ó¡)' **^' * Iø^

20 which can be replaced by

t r : - ö¿)' * *Io^, (1.7e) f,{öu*, ^: and add a counter term

Lct :T"'@n*, - ön)" *T"ró' * ló^. (1.s0) Then the Greens functions generated by the partition function can be made to have a meaningfut limit as ¿ -+ 0 by choosing the parameters c1(ñ,,î),cr(ñrÎ) and ca(ñ,Î) in an appropriate way to render the theory finite.

I.3.2 The Renormalisation group equations

The effort to investigate the underlying physical theories, invariant under different renormalisation procedures, leads to useful partial differential equations involving the Green's functions. These differential equations, called renormalisøtion group equations, lead to some very useful results which are beyond the reach of perturbation theory. Consider lhe Àþa theory with the bare dimensionless parameters ñ and Î and the renormalised parameters ñ, and Î,. Th" cutoff is characterised by the dimensionless quantity ll(ñ,,), and by a change of cutoff we mean a change inñ,,. Then Eq. (1.71) becomes

g@/z)y(n) rÍ") (Î, î,(ñ.,) ,ñ,) : çî ,î,ñ,¡ ( 1 . s 1 )

We have indicated, at the left hand side, that the renormalised coupling is determined by the scale, ñ,. Now suppose that ñ', is used for the renormalisation condition

lz) t[") 1r, î,(a.',),ñ!,) : 7@ ¡(n) çk,^,ñ!,).

It follows that

z (") ( I rÍ") r, î (Aò, ñ,) : z lñ,, ñ;, î, ît 1" ¡ çk, (ñ',), ñ!,), (1.82) ^ where z(^(ñ, z[ñ,,,ñ!,,î,,Îl] : ,ñ, z(^(ñ,,),ñ',) and Àf In the limit ñ, -+ 0, Z[ñ,,,ñ!,,Î, Î'1 ir finite because it is the ratio - ^,(ñ',). of two renormalised vertex functions. It is evident that a change in scale, at which the

2I theory is normalised, is equivalent to finite rescaling of. Ztlz and a transformed coupling constant. Bare RG equations. Since the bare theory is independent of scaleñ,, then from Eq. (1.81) one can arrive at an identity by differentiating with respe ct to ñ, at Î fixed. a (ñ,#)17-n/zY(n) (Î, Î, ñ,'¡1 : s.

This equation can be written as t^ f,h+ P(î,,^)#,-|n{î,,t"1] tf"' (k,î,,ñ,) = 0, (1's3) where the functions B and n are defined by;

p(î,,ñ,,): (1.84) ^, Hl^, r(î,,ñ,) - ñ, Wlr. (1.8b) According to above equations a change in ñ, can be compensated by changing î or Z at the same time. These changes are globally valid in the sense that B and q are independent of momenta and the index n of the vertex function. It can be shown 19,241that close to the critical region, B and,7 depend only on Î up to scaling violations. This is a very important fact which allows us to solve the renormalisation group equations and to extract information about the behavior of the bare vertex function near the critical region. Integrating Eq. (1.84) and Eq. (1.85) gives the integrated renormalisation group equations: d^, '(ñ") lnñ,,, p(^,) 'ffn{î') h z(ñ,,), (1.86)

From E.q (1.86) one can arrive at number of facts. If B has a zeto at Î;, then Î(ñ,) has Î; as an asymptotic limit. The zeros of the B function play an essential role in the critical behavior. Those which correspond to a negative slope, arc ultraai,olet repulsion (UVR), meaning the renormalised coupling moves away from this zero, except if the original coupling is just sitting on the zero. On the other hand, those which correspond to a positive slope arc infrøred attractiue (IRA). Let suppose that we are at a IRA

p6) _ o, p:p,(î;), > 0 (1.s7)

22 and linearise the first equation of the renormalisation group in integral form, Eq. (1.36), around the IRA fixed point: ¡î,(ã,) d^, ,^ J ed:lt -tnñ'' (1.Sg) Performing the integration in the limit of small we obtain

lÎ,(ñ,) - - ñ4 (1.8e) ^;l Now consider the second equation of Eq. (1.86). The integral on the l.h.s is dominated by small values of. ñ,. Then in the limit where ñ, is small we obtain

InZ(ñ,) - ¡¡Lnñ', (1'90) with 7 : ri(Î.). On the other hand from a simple dimensional analysis (Appendix B) we have

r[")1îu; î,ñ,) - ffid-(n/z¡(d-2) Zø/2)(î,,ñ,¡rf) (î,1ñ,;î., t). (1.91)

Combination of Eq. (1.90), Eq. (1.91) and Eq. (1.71) gives the strong scaling : y(")çñ,îr;î,ñ.,) - ñ,;d+("/z)(d-z+n)ytu)1î¿; î., t¡ as ñ,, --+ 0. (1.92)

Note that in this limit under the renormalisation equations only the scale ñ, changes and the renormalised vertex functions remain unchanged. The scaling laws for other cases such as the Callan-Symanzik equations, where one fixes the bare coupling rather than fixing the renormalised coupling, can be driven in a similar fashion. As we mentioned, as one approaches continuum,ñ, + 0, then Î -r Î.. Thus, In a region close to Î*, scaling region, one expects that the lattice results to be close to the continuum results. Scaling relations, such as Eq. (1.92), also suggests that in the scaling region the dimensionless ratios of vertex functions do not change by a change of scale and become constants. One can also derive similar scaling behavior for models involving fermions. As an example see Appendix D.

L.4 Fermions on the lattice

As we showed previously , the naive formulation of scalar field theory has no problem and the correct continuum limit is straightforward. While defining scalar fields on the

23 lattice is relatively straightforward, the naive procedure for discretising the continuum fermion action results in a lattice model with many more low energy modes than one originally expected. This problem is the notorio:us doubling problem. The most popular scheme for handling this problem is the Wilson projection operator method, where the chiral symmetry is broken with the hope that it will recover at the continuum limit. In this section, \ry'e will discuss Wilson formulation of fermions on the lattice. An alternative scheme for putting fermions on the lattice is the Kogut-Susskind formulation where each lattice site carries only a single component of Dirac spinors, doubling the effective lattice spacing.

t.4.L The Doubling problem

Consider the free fermionic Euclidian partition function in the continuum space:

z : e-st'r"tt, Ild,þlld,g) where s : JI dd xl$(x)1,0,,þ(*)'' + M$(æ)ç(x¡]. The Euclidian 7-matrices 1, = ú0t - 1, . .. ,d), satisfying the algebra

{1r,^1,} = 6r,,'

A transition from continuum to lattice can be done after following substitutions fu:aM, û"Ø) : o#rþ"(r), î" @ _ o#ú"@), Au : oT 0u, (1.e3) where the variables with "caret" are dimensionless variables and the dimensionless derivative is defined as 1 ãuû": i[û"r"+ p)-û"@- ù]. (1.e4) By introducing a space-time lattice the fields r/ and tþ live on the lattice sites nø, and the measure is given by

[dnþlldff] : fI dú" (no,)LdrþP (mø) , d¡f, þ,n

24 where a and B are Dirac component indices. Thus the lattice version of the action becomes

s t î.Ø)R,,8@,m)$B@), nrmrarB where K is the fermion matrix given by

Ko,þ : Ðlr{'rr)-,8 - 6^,n-p] + fu6, ,n6o,0f . t"z f6^,n*, At this stage it has to be mentioned that the form of the lattice action for both fermion and scalar fields is not unique but is chosen to be the simplest one. The correlation functions are obtained from

s {n¡ +î' (n), 2 : e- +D''' î' {ùî' tu)î' lî, ol I toõlto $1 by appropriate differentiations with respect to the dimensionless Grassmann sources, f and fr. The integral is Gaussian and can be calculated exactly. That is

x-' ÇBþn) 2lî,11: det ¡ç "'P(n'm) . "D^,^,',0îþ) Therefore the two point function is given by

(û-,ûpl=- : K!-B')Ø,*),

We proceed in the same way as the scalar case. That is, by finding the propagator, Go,p(n,m): NL-ut)Ø,rn), through a Fourier transformation. After some straightforward algebra one obtain GG) : -itrsin:þ, + !. (l.eb) Drsin2î+fu' Masses of particles can be identified with the poles in G(Î). If we suppose that fu - 0 ( at the continuum limit) and d : 4, then, besides the poles at lc : (0,0,0,0) there are fifteen other poles at lc: (tr,O,0,0),...,lc = (n,r,r,r). That is, a pole at each corner of the Brillouin zone (BZ). This means that our naive model is for 2d fermions, not a single one. It has to be mentioned that the doubling problem is not a consequence of having taken the continuum limit in a naive way, but is connected with the structure of the action. It has been shown by Nielson and Ninomiya [37] that one can not resolve the doubling problem without breaking chiral symmetry. We will discuss the most popular methods of eliminating the doubling problem, namely, the Wilson fermions .

25 L.4.2 The 'Wilson fermion

To overcome the doubling problem one might consider modifying the action such that the poles of the propagator at the edges of the BZ arc lifted by an amount proportional to the inverse lattice spacing. In the construction of the lattice action, one should include the symmetries of the classical theory. However in the construction of a lattice fermion action one is forced to break explicitly the chiral symmetry that the original theory possesses. Looking at the argument of the sine term in the scalar propagatot, ãçî¡,in Eq. (1.a5) and the fermion propagator, Eq. (1.95) they differ by a factor of.2. While for the scalar field the continuum limit \ryas correct and straightforward, for the fermion case this factor of 2 give rises to the doubling problem. In other words, the doubling problem arises because Dirac's equation is a first order equation where as the Klien-Gordon equation is of second

order. This suggests we add a second derivative term to the action [38]

sw) - s -;Dõ- E' ûn,

where r is called the dimensionless Wilson parameter. Considering that û = od-'rþ ur.d. 1' : a2a then it is evident that the Wilson term vanishes linearly in a in the continuum limit. Then the Wilson action can be written as

S(w) : D D,p;qi:f,,,þf,, (1.e6) ntm ørp

where

Qi:h: (M + d,î)6',,^6..,8 -ï+l{t - tr)"'þ6^,n+î,+ (Ê+ ^tr)o'þ6^,n-¡1, (1.97)

and it is called the fermion matrix. We define the fermion propagator as the inverse of the fermion matrix. That is DGn''Q''*: 6n'^

By the momentum representation of the fermion matrix, the momentum space propagator G(Î) becomes

(1'e8) ã''P(î) -ljilY:Yl)"'o ) D,Ê +T[@¡z ' where 1 le ,: =sin(alcr),

26 and k MG): M +2Êf sin2( ) It , Then it can be seen that for any value otî,Mçî¡ approaches M when ¿ --+ 0. However, near the corners of.BZ,MG) diverges as d -) 0. Thus in the continuum limit these states become infinitely massive and decouple, and thus the doubling problem is eliminated. However where the continuum theory has a chiral symmetry, that is the action is invariant under the global transformation (with M :0): -- û "ttrþ ú - ""õ, ,Þ-e¿,"eç ú+e*ier¡r, with 7" : ^1t...1¿,the lattice formulation of the model breaks chiral symmetry. As we mentioned before the Nielson and Ninomiya theorem proves that it is not possible to remove the fermion doubling problem without breaking chiral symmetry. The Hopping parameter. It is more common to write the action in terms of the dimensionless H opping pl,rz,nx eter î,,

ft: aa 2M + 2dî

Then the Wilson action becomes

w rc -,þ" + ^9 Dúiûf - ! **,(î 1 òûß = û,w:,ßû*, f,tQ lr ] ä where Wfi,fl is the fermion matrix and given by

Wi,ß: 6n,^6o,8 - .+6^,n+r(î +.lu)o'?.

It is straightforward to write the propagator, Eq. (1.98), in terms of Ê ã^_ | fu+îlz)sin2î-h.8, Gî:zo ffi¡,ur, (1'99) where we have suppressed the Dirac Components. This concludes our short, introductory discussion of fermions on lattice. A thorough discussion of the topic can become quite technical. However, the materials presented in this section are a sufficient introduction for our later needs.

27 28 Chapter 2

The Lattice Effective Potential

Those who dominated the circle of learning and culture- In the cornpo,ny of the perJect became lamps o,rnong their peers. By daylight they could not escape from the darlcness, So they told a fable, and went to sleep. Omar Khayyam

2.L Introduction

An understanding of the underlying vacuum structure of a quantum field theory is essential for understanding its physical content. This analysis is conveniently carried out by calculating a quantity known as the effective potential Í39,42,43], denotedby U($) and the minimum of which gives information as to the nature of the lowest energy eigen- state of the theory. This makes U($) very useful, particularly in studies of spontaneous symmetry breaking (SSB). The effective potential determines the one particle irreducible (lPI) vertices [39] at zero momenta and reflects any non-trivial dynamics. It is also widely used to study radiative corrections in quantum field theories [43]. Truncating the loop expansion of the effective potential often gives it a complex and nonconvex character, in spite of the fact that on general grounds the effective potential must be real and of convex character [44]. It has been pointed out that the loop expansion for the effective potential fails for the fields in just those regions where the classical potential is nonconvex; the most familiar case corresponds to a double-well potential [45]. Therefore, it is important to carry out nonperturbative studies which can be used even where the loop expansion is not applicable. One convenient nonperturbative approach is to employ a

29 discrete version of the theory, i.e, lattice field theory. Lattice field theories have an ultra- violet (UV) regulator (the lattice spacing) and an infrared (IR) cut-off (the lattice size) and are conveniently studied using Monte Carlo (MC) methods. We investigate calculation of the effective potential for ),þa model using two established methods. The variation of source methods (VSM) 146, 471which introduces an external magnetic field and the effective potential can be calculated from the response of the system to this external field and a version of the constraint effective potential (CEPI) [49] where the effective potential is calculated from the distribution of mean field, /. As we will show, the standard method of calculating the renormalised coupling in À/a theory, by calculating two and four point correlation functions at zero momentum, suffers from large statistical errors, especially where the coupling constant is not very strong. CEPI is also suffers from the same problem. We will show that how the VSM can be used to obtain much more accurate and precise results for the renormalised vertex functions. In addition to above methods we will show how the renormalised quantities can be calculated from the effective potential by calculating appropriate correlation functions in presence of a constraint, d (*" call this method CEPII). Most of materials in this chapter is currently in publication [51]. As we mentioned in previous chapter, both in numerical MC studies and the analytical calculations it is important to find the renormalisation group trajectories (RGT). Along these curves and close to an Infra Red (IR) fixed points (the scaling region) the physics describecl by the lattice regularised quantum field is constant and only the value of the cut-off ( lattice spacing) is changing. It is in the scaling region that the ratio of dimensionless renormalised vertex functions are constants and one expects that the scaling region to be in vicinity of the critical. However we perform our calculations away from the scaling region. The main reason is to examine the accuracy of effective potential methods to the fullest extend. Since the correlation length is large in the scaling region, one expects that the finite size effects can be considerable. By performing the calculations in a region away from the scaling region, where the correlation length is smaller, we tried to reduce the finite size effects on our calculations of renormalised quantities. The other fact is that away from critical point most of Monte Carlo methods perform well. However, near the critical point where the correlation length becomes larger, the autocorrelation length rapidly diverges. This well-known phenomena, to be discussed in details in chapter 4, is called the critical slowing down and causes some well-known complications. Even though, one can overcome the problems arising from the critical slowing down phenomena, for simplicity, we prefer to perform the calculations away from the critical. In the weak cou-

30 pling regime we compare our results with the lattice perturbation results to exclude the finite size effects on our results furthermore. In the strong coupling regime we compare the effective potential results with the results obtained from strong coupling expansion on the lattice, extrapolated to a larger correlation length. However, it is not hard to perform the calculations in the scaling region. As a matter of fact, the effective potential methods, discussed in this chapter, can be accurate tools for finding the scaling region, since they

can provide accurate values for the dimensionless ratios of renormalised parameters, as it would be evident soon. It the case of Àþa the fixed points can be calculated perturbatively [13] in the weak coupling regime. Non-perturbatively, the parameter points on the second order phase transition critical line which separates two phases (ö) :0 and (ól + 0 are good candidates for the IR fixed points with the point ñ,2 : Î : 0 is the trivial fixed point and any scaling region corresponding to this fixed point represents a free field theory.

In Sec. 2.2 we review the lattice effective potential, showing how the all the renormalised vertex functions can be calculated if one knows the full structure of the effective potential. the effective potential. In Sec. 2.3 and Sec.2.4 we perform the calculations for both the symmetric and the spontaneous symmetry breaking cases in the weak coupling regime and we compare our results with those obtained from lattice perturbation theory. We also perform calculations in the strong coupling regime and compare these with the strong coupling expansion.

2.2 The lattice effective potential

Consider a lattice lagrangian density on a d-dimensional cubic lattice with the total number of lattice sites .lúd, 1 L^:Ð (ón,r- ö^)'+v(ó") (2.r) p 2 The classical vacuum ( ground state) is at the minimum of V(þ). The vacuum expectation value (d) of the quantum field is not necessarily identical to the classical vacuum. The vacuum expectation value of the field in the presence of an external source, .f(ø) is given by ó.^[r]=W. e.2) The vacuum expectation value (d) ir the limit of ö,n as ./ + 0. Hence we can ask for what value of "/ can one obtain a given d". One can choose to treat /" as the independent

31 variable instead of J and define the "effective action" f [d"] by a Legendre transformation

f[d"] : Dó"nJ^-WlJl, (2.3) n where /" is defined in Eq. (2.2). It is easy to verify that

r^ló"1=W. Q.4)

In the case J - 0, by translational invariance it follows that /" must become constant (i.e, independent of the label rz). Hence the vacuum expectation value is given by (/) and satisfies 419'11 =Q. (25) 49" lo"--lo,l Similarly for any constant "I we must have ó" : 6 also constant. Define the effective potential, U(ô),bV fldl : Nou(6). (2.6)

The effective action is the generator of proper (i.e, one-particle irreducible) Green's functions and in particular we can Taylor expand the effective action to give

oo1 rI fld"l : t ¡(u)(r21, ...,n*r)ó"nr...ó"n*. (2.7) #-ut fll ¡.,.tflM

(M) Here f ( TtL¡ . . . ,n¡a) is the proper M-point Green's functions in presence of the source Jn a*f.lì ¡(ivr)(n1, ...,nM). (2.8) ðó"n,'''0ó"n* - In terms of its Fourier transforms we have oo 11 r[óJ: i(M)(È,...îòó¡,...õ_î*, (2.e) Ð Mt NMd D M=O ^kt ,'..,kM where here / is the Fourier transform of /" The vacuum proper Green's functions are obtained by setting J :0. If the source is a constant (Jn : J for all n), then the translational invariance is restored and \rye can factor out an overall normalisation and a 6-function to define

f(M)(ir,...,î*)=Ndóo¡*...iÈ*iy)(îr,...,î*) (2.10)

32 Where in the limit / -r 0 we recognise that l!ivr)1îr,. . . ,î¡a) is the dimensionless, lattice equivalent of the proper M-point Green's function in momentum space. For constant J we also have /.,, --+ þ and hence õ: Noh,oþ and Eq. (2.g) gives

r(õ) : No D firf5wço¡{,, (2.11) where ñ!Ml1o¡ :fl(ivr)10,...,0). (2.r2)

Comparing Eq. (2.6) with Eq. (2.9) gives u(6): F="hur)@)6*. (2.13) It immediately follows that

: ñ!iø)10¡ , (2.14)

where here it is understood that we are working in the unbroken symmetry sector, (d) : 0. In the unbroken sector we see from Eq. (2.13) that the dimensionless, proper Greens functions with vanishing momenta can be easily obtained from the effective potential, U(ó), by differentiation. We see that $ minimises U($) and in the limit -/ -+ 0 the minimum Ó - (Ó).Also note that Eq. (2.4) gives an expansion of ,/ in terms of the þ's and l(0)'s r(õ):*oi,è"llÍ'r(o) 6*-' (2.15) In the broken symmetry sector, (ó) + 0, it is more appropriate to use the shifted field

x@)=ó(*)-(ö). (2.16)

The one-particle irreducible (lPI) vertex functions lfff) u." linear combination of the l(M)'s, and can be obtained from the shifted version of Eq. (2.19)

u(6)=uoíÐ:uøt(6- (d)): W#rffltol e.r|) M=Oå As is usually done in lattice field theory studies ï¡e renormalise at the renormalisation point .where all external momenta of the Greens functions vanish. The renormalised quantities can be obtained directly from the effective potential. For example in the À/a theory we have: du(Ðl j:l :_ u^ (2.1S) dQ lo=(o)

33 Z : zÍ|'z)ço¡ f(') :ñ? (2.1e) I - tó=þl

z2 : szff)1s) - fÍ4) - 1, . (2.20) ó=þl where Z is lhe field wavefunction renormalisation constant (Ó, : tßÓ). From the first two conditions above and requiringñ,|> 0 it follows that (/) is at the minimum of U($). Also note that ñ, and Î, defined as above are not the physical mass and coupling, which are defined in the pole of the propagator in the complex energy plane and the on shell four-point function, respectively. However in the scaling region (close to the critical line) these values are a good approximation to the physical mass and coupling [48].

2.3 The MC effective potential.

In this section we will examine three MC methods for calculation of the lattice effective potential. The renomalised coupling constants obtained by these methods are compared with analytical results. From this point on we work exclusively on the À/a model in two dimensions (d:2) where its lattice action is

sl6, il: ö*)', - ó',^ + - r"ô"] (2.2r) [; D(ó,,, - i+ñ', + ió1 + and we have adopted the notations and conventions introduced in the previous chapter.

2.3.L The variation of source method (VSM)

Eq. (2.15) suggests that in the Monte Carlo calculation one can calculate the mean value of the fields, þ, for different values of the source and as a result one obtains / as a function of J. This function can then be inverted to obtain,/ as a function of /, i.e., J(õ).Then using Eq. (2.15) u¡e see that the derivatives of "I with respect to / would give the proper Green's functions at zero momentum. From Eq. (2.2) one also concludes that /.¡ is antisymmetric in J. That is

ó¡ ó-t . (2.22)

Fig. 2.1 shows J(þ) as a function of þ for the symmetric case (Fig. 2.la) and the broken symmetry case (Fig. 2.1b). Note that for the broken symmetry case, /.¡, as a function of J is discontinuous and so the relation in Eq. (2.15) can not be inverted for all õ(J).

34 (b)

0.4 o.4

0.2 0.2

J(0) o J(0) 0

-0.2 -0.2

-0.4 -0.4

-1.0 0 1.0 2.0 -4.0 -2.0 0 2.0 4.0 õ'Þ Figure 2.1: The plot of Î, ,o"rr,r, Î io th" broken symmetry sector Î using LPT (solid line) and VSM (diamonds) with ñ,2 :0.1 and Iú - 20.

Whenever it is possible Eq. (2.15) has to be inverted to obtain the source J as a function of /. Then the derivatives of -/ with respect to / would give the vertex functions at zero momenta and consequently the renormalised masses and couplings can be calculated. The mean value of the field in the presence of a source has a small statistical error. This is expected since it is an analog to the reduction of fluctuations of a spin system in the presence of an external magnetic fietd. As the source becomes smaller the fluctuations become larger. Thus one needs to perform the calculations for large enough sources that the error is small and then extrapolate the results to J : 0. This method will be referred to as the variation of source method (VSM) and has a number of advantages. The vacuum expectation values of the fietd /(J) are the simplest quantities to compute on the lattice and their "I-dependence can be exploited to get the first derivative of the effective potential. Since the source effectively causes the boson field to become more massive, the finite size effects generated by the lattice become exponentially small provided that the lattice is large enough. Since the data become noisy for small values of "I we need to restrict the analysis to a safe region of ,I, which can introduce some errors in the results through uncertainties in the extrapolation.

35 2.3.2 The Constraint Effective Potential (CEP I)

In the last section the effective action and the effective potential U (S) werc defined through introduction of a source "I. There is a different method which does not require such a dynamical symmetry breaking source. The constraint effective potential was first introduced by Fukuda and Kyriakoloulos [a9] as an alternative way of obtaining the explicit expression for the effective potential. It was further analyzed by O'Raifertaigh, Wipf and Yoneyama [50]. In this approach one obtains an explicit expression for the effective potential, without introducing external sources, but instead through the introduction of a ó-function in the functional integral. In the constraint effective potential approach one first definet Ú@) : U(N',õ)

0.4

0.2

gOOOOQg rc0) 0.0 Qooooo@

-0.2

-o.4

-4.0 -2.o 0.0 2.0 4'O

0 Figure 2.22 Anexample of J($) versus / in the broken sector using the constraint effective potential. For these results N :20.

as e-N'ú(6): l[ao]o(#ç ,"--r)e-stot e.zs) and then uses the fact that as.lú2 -) oo we have Ú@) - U(þ) and the effectivepotential is recovered. It is easiest to demonstrate this result in Minkowski space, where Eq. (2.23) becomes

: Q.24) "-;wzt16¡ llaorct*r+ë,-6)""tt. We can replace the ó-function in Eq. (2.2$ by its integral representation to obtain (up to an irrelevant constant) osll+röl-iN26r C al I "-;xzÛ1$¡ = I I VOl",

36 : C' I al e;(wÍ4-N2r6) Q.25) Note that in the integrand of Eq. (2.25) we have / fixed and "I arbitrary. In the limit, N2 - oo the dominate contribution to the integral comes from the stationary point of the integral which is the value of J at which dwlJlldJ :6. Recall that l(/) : (Jõ - W[JDl6=o*14/dr, from which we see that up to an irrelevant overall constant _ e-¿N2Ú(6) + e-ir(6) e-;N2u(6) as.l[2 _¡ oo, e.Z6)

as claimed.

We can also arrive at this result directly in Euclidean space by multiplying both sides of Eq. (2.23) by ew"'tö with ,I arbitrary and then integrating over / to obtain

a6 I "-*'It(Ð-r-ö) = ltaøl e-stöt+rD,ó,. e.zl) As 1/2 -) oo the left hand side of Eq. (2.27) becomes entirely dominated by the stationary point of the one-dimensional / integration given by dlr@)ld6: J, while the right hand side is recognised as ewll for a constant source, ,I. Hence up to an irrelevant overall constant we find e-N"lt(Ð-¡-ø) --+ ewlrl as -fú2 -r oo, (2.2g) and so find that (up to a constant)

e-N'Ú@) --+ ewl4-N2J6 = e-N'u(o) as .fú2 -r oo, (2.2g)

as required. It is important to note that the e-N"Ú(ã) ."1u1", to similar definitions in statistical mechanics and spin systems [52] and that

t-*"0(!),= P(õ)t:¡¡;@ : '. (2'30)

can be interpreted as the probability density for the system to be in a state of "magneti- zation", þ. Then it can be seen that the probability for a state whose average field is not a minimum of tr(õ) then decreases as -fú2 -, oo. This suggests that one needs to study the probability distribution of the order parameter /. Using a Monte Carlo algorithm one generates a Boltzman ensemble of configurations, {/}, weighted by e-s[ø]. Let dN be the number of configurations with average field values in an interval dþ about d-. Then

dN(ø-C"-N"t(Ð¿6, (2.31)

37 with C some constant. Then one can write t@):-fir" W, e.rz) up to an irrelevant additive constant. Eq. (2.30) suggests that one can generate a large number of configurations weighted by e-s[óì, calculate þ for each configuration and construct a normalised histogram. The histogram can be fitted to Eq. (2.32). The most probable average field values are near the minimum of the effective potential. In order to determine Úç$¡ from its minimum, i.e., to sample a range of relatively improbable "*^y values of /, one can introduce a small source. Then a simple generalisation of Eq. (2.30) allows a nonzero external source [54]

1 úro> t6 ln (2.33) - N2

Thus one can check whether such an ansatz gives a good approximation for the effective potential, and so construct the effective potential by performing a simultaneous fit of several histograms corresponding to different values of J. By the expression "simultaneous fit", we mean that the chi-squared values corresponding to each J are summed and this sum is then minimised. This method can be applied easily on the lattice. Note that in Eq. (2.30) we have assumed that for sufficiently large ,fy'2 the finite volume effects "" tr1$¡ can be neglected, i.e.,lhat the lattice volume is sufficiently large. The constraint effective potential method summarised in Eq. (2.33) will be refered to as CEPI.

2.3.3 The One Loop constraint effective potential

Here we will derive an expression for the one loop contribution to the constraint effective potential. In order to calculate one loop correction, we expand around the classical configuration. We will shift the fundamental field /: ó-ó+( (2.34)

Now substitute Eq. (2.21) into Eq. (2.23) to obtain e-N'u(õ): eXp (,t(a;) - 46 c:+ #} {-*,rrrl ltaa"*o -å T "*o {à e fftr} (2.35) If the coupling is small, one can ignore the terms in the last bracket. Then the remaining term is Gaussian and can be integrated exactly. One can diagonalise this terms in the

38 momentum space and using the trace representation for the determinant of an operator and obtain the one loop correction to the effective potential uØ(õ):#Dn[r'* w], (236) where î -2sin(îrlZ) and is constraint to be inBZ. Consider Eq. (2.35). If ô2VlôSis positive matrix the Gaussian term is exponentially convergent and the loop expansion makes sense as an asymptotic series. But if A'zW A6 is negative the Gaussian terms can become complex and the loop expansion does not make sense even term by terms. At the one loop limit, complexity occurs if, and only if, the classical potential is not convex. It does not imply that U(þ) is complex, rather that the conventional loop expansion fails. As a matter of fact, U($) is real and convex for all /.

0 a 0 0 0 0

õ 0 0 o 0 Figure 2.3: An example of diagrams contributing to the one-loop effective potential.

Eq. (2.36) can be easily written as

uo,(6):#rh lî,+a,l+#T;å ç,r,¡ kk lm]" where Vr(õ) :Ìõt lZ. The last term in Eq. (2.37) is the contributions of one loop diagrams including n propagators and n vertices dvr(6)lA6 with all external momentum being set to zero. Some of these diagrams are shown in Fig. 2.3

2.3.4 The Constraint Effective Potential (CEP II)

Now return to Eq. (2.23) and perform a shift of field, ö(") - 6@) +õ. Since the measure is translationally invariant we obtain

e-N"t@) : e-stó+6t (2.ss) ltaoló(uþ +óò

39 Taking the derivative with respect to / we get ry"-.'t(ù - # lvatr#lLffi e-sÍó+øt. (2.3e) Only the potential part of the action is affected by the shift of field since / is constant and so dsld,6: N2dvld,6. Using this fact and shifting the field back to its original form then gives d,t(õ) Ë:\Tlr'ld,v(ó)\ (2'40) where we have introduced the shorthand notation (o(ó))a:7"N'trtø)l I taot 6(# Dó^ -6) o(ó) e-stót. Q.4r)

In the À/a theory being considered here we find

au@) :ñ,2õ+(YT (2.42) -6- - t'L lt¿rl,6\v lÓ'

Expressions for some of the higher derivatives of U(õ) are given in Appendix D. These equatìons are very useful in the Monte Carlo (MC) calculations since they relate the derivatives of the effective potential (and consequently the zeto momentum vertex function) to the averages of quantities that can be calculated directly from the lattice. This method will be referred to as CEPII. There are two ways of calculating the renormalised quantities using CEP II. The first one applies the constraint on the lattice, fixing þ', then calculates (/3), and finally uses Eq. (2.a\ to obtain the first derivative of the effective potential. Higher derivative are evaluated from fitting a curve to the dfl ld6 versus þ results. This has some similarities with the variation of source method, however there is a difference between these two methods. In VSM one sets the source J to constant and (d) : / up to fluctuations due to finite lü2, whereas in CEPII we have (/) - / exactly by construction. In the broken symmetry sector there is another difference between this method and VSM in the broken sector. Since changing the sign of J just tilts the potential towards the other minimum, VSM fails to give any value of (d) between the two minima. When using VSM we are not able to obtain any value of (d) in the region between the two minima, whereas both CEP methods are suitable for probing this region. One can always fix þ to any value including the values between the two minima to get the full shape of J(6) (see Fig. 2.2). However as far as the practical calculation of renormalised

40 quantities is concerned, this method is almost equivalent to VSM and so from here on we disregard this approach. The second approach to CEPII is through the equations shown in Appendix D and is more direct. These equations relate the derivative of the effective potential to the averages of some derivatives of the classical potential. All these averages should be taken in the presence of the constraint which fixes (/) : /. We imposed the constraint using the Hybrid Monte Carlo. The constraint can be taken into account by integrating over one of the site variables, þ¡,. Start with a configuration with the field average being þ. Each time a site is updated by a value, ó, that is

Ó'¿:Ó¿*6,, then the site /¿ must be updated simultaneously by ó'* : ön- ó. This procedure is carried out for all the sites, which completes a s\ryeep; the next sweep then starts again from /1. We will discuss the Monte Carlo methods in chapter 4. The advantage of this method over the VSM is that one does not need to run a Monte Carlo routine several times with different sources, and no curve fitting is required. One disadvantage of this method is that for calculation of the renormalised coupling one needs to add and subtract many average terms as has been shown in Appendix D. Although the statistical errors might be small for each term, the overall errors contributing to the renormalised coupling can be large. However, the renormalised mass in the symmetric phase of the l/a theory obtained using this method is very accurate. We also would like to comment on Fig. 2.2. It has been shown by a very general argument that U"($) > 0 for att$ l++1, (primes denote differentiation with respect t" ó). This general property is known as the " convexity" of the effective potential. Looking at Fig. 2.2 it is clear that this condition is violated tor lrl$). This can be understood by noting that convexity holds only in the thermodynamical limit, i.e., N2 -> oo. To conclude this section it should also be mentioned that the proper vertex functions can be obtained directly using the standard Monte Carlo (MC) method. For example for the )/a four point vertex function one obtains

-1-2^ þ l"-3(ó I; ñ!4)10¡ : - _2 (2.43) (ó l"

Here, for exampl", (õnl"is the connected part of vacuum expectation value of fourth power of the Fourier transform of the field at zero momentum. As we will show, in the weak coupling regime this method suffers from very noisy signals giving rise to large statistical

4T errors. The errors are due to the large fluctuations of correlation functions in this regime as well as the subtraction of the disconnected pieces. However in the strong coupling regime this method gives a relatively good approximation for ñlallO¡ and the statistical errors are reasonably small [59]. However the higher order vertex functions calculated with this approach can be very noisy even in the strong coupling regime, primarily due to subtractions of noisy disconnected pieces.

2.4 The numerical results

In this section \¡/e present our results for the calculation of renormalised coupling, Àr, in two dimensions. It includes the symmetric and broken symmetry sector in the weak coupling regime as well as the strong coupling regime. In the case of the weak coupling regime the results are compared with 2-loop results and the direct calculation of À, using the standard MC method in Eq. (2.43). In the strong coupling regime we also compared the results of each method with the strong coupling expansion results. The details of the numerical simulation are included at the end of this article.

Case 1: The symmetric sector in the weak coupling regime (\MCR)

The aariation of the source method (VSM): Here we study the model in the symmetric sector where (d) : O. As we will see, all methods presented in this paper require the calculation of renormalised mass ñ,, and the wavefunction renormalisation constant, Z. In general, the boson propagator extracted from the lattice has the form Z G1î¡ (2.44) - î, + a,?(îr) ' where ñ, zñ,(ñ,?) is the mass-pole of the scalar particle, i.e., the renomalised mass. In particular at zero momentum z õ^i(k-o):fü, (2.45) where we make the standard approximation that ñ} x ñ,1(0). The renormalised mass, ñ,, is then given as the reciprocal of i, the lattice correlation length,(. In previous chapter we showed how ( can be calculated. There are two different ways of calculating Z. One is to use Eq. (2.45) and the fact that õ10¡ : N'(õ'l to calculate Z. The second way of calculating Z comes from combining

42 0.40

ö 0.30

<<' 0 20

0.1 0

000 000 0.1 0 o20 o30 0.40 0.50 À

Figure 2.4: The plot of Î" .rr".rrr, Î itr th" symmetric sector using LPT (solid line) and Eq. (40) (stars) and the VSM (diamonds) with N :20.

Eqs. (2.a5) and (2.20) which gives d,UJþI :I. Q'46) d6' l¡=(r,, C(0) fnus õ(O) can be directly calculated from the fit and the calculation of Z follows as before. An accurate calculation of ñ, is crucial for both methods. We found that in the weak coupling regime, the second method 1ryas more precise. We compared our results with the 2-loop lattice perturbation theory calculations (LPT) of the renormalised parameters. This means that finite size effects may be present in our comparisons at some level. The comparison is shown in Fig. 2.4. The values for À, seem to be accurate even in the very weak coupling regime. In this regime the effective potential results are in good agreement with the lattice perturbation calculations. The MC results begin to deviate from the perturbative calculations as Î, in"r"ur"s. This is expected since a loop expansion in À/a theory is an expansion in Î, and as this is increased the contribution from higher loops becomes more significant. The VMS can be expensive in CPU time but the cost can be reduced to some extent. For a value of J it is possible to calculate D"(g): d,63ldPJ - N,(õ")" during the calculation of öt, for each value of ,I. From these derivatives one can expand þ.¡ around ./ and then use a curve fitting routine to calculate Àr, as we did before. The statistical errors can become larger for the higher derivatives because of the subtraction of the disconnected

43 pieces of D"(S). In Table 2.1 we have shown a comparison of our previous results for $¡ and results obtained by expansion around three J values, namely, J : 0.075,0.25,0.4 for ñ,2 : 0.1, Î - 0.1. We see that the calculated values of þ are reasonably close to the previous results. However, the price for reduced computational time is a slight increase in uncertainties. 'We have also calculated Î, for Î:0.055, ñ} = 0.1 using Eq. (2.a3) and the result is included in Fig. 2.4. The statistical errors are extremely large and it suggests that the calculation of the 4-point vertex function in this region is impractical with this method. The constraint effectiue potential method I (CEPI): This method is the easiest to implement. We generated the Boltzman ensemble of independent configurations. For every configuration we measured 6: GlN2)f¿/¿ and computed the histograms for the probability density P(õ) for several values of J. We also noticed that the anzatz of Eq. (2.33) only worked well for very small "I in this region. We did a simultaneous fit to Eq. (2.30) of a few histograms corresponding to J :0 and small J's using a three-parameter anzatz tor tr($) of the form

u(õ):o16'*oró-n +osf . Q.47)

Although there was no systematic discrepancy between the data and the fit, the statistical errors were very large. We unsuccessfully tried more histograms and higher powers of þ in the fit. The statistical errors remained large and we concluded that even a reasonable estimate of renormalised parameters in this region was not feasible with this method. The constro,int effectiue potential method II (CEPII): In this method Eqs. (E.a) and (E.8) can be used for calculations of ñ," and Î, respectively. All averages shown in these equations are to be taken with the constraint of ó :0. Although the statistical errors for each term are small, the overall error can be large. However in the symmetric case in in the weak coupling regime most of the terms either vanish atS :0 or are small enough to be neglected. For example for ñ, orily three terms need to be considered. But the computation of Î, suffers from larger cumulative errors. The field wavefunction renormalisation constant can be calculated in two different v/ays. One can use Eq. (2.45) and relatio" õ10¡ : N'(õ'), as in the previous case, or can use Eq. (2.45) and Eq. (2.46) where ryl can be found from Eq. (8.4) dó' lø-=(o) The results are compared with the VSM results and are shown in Table 2.2. We also compared the calculation of renormalised mass using Eq. (1.70) with the CEPII

44 Table 2.1: The comparison of the calculations of. S¡ for different values of ,I's and perturbative calculations of þ.¡ around J : 0.1, J : 0.225 and ,I : 0.425 wilh ñ,2 - 0.1, Î - 0.055 and I{ :20.

J ót error Idr]o* error 0.050 0.4052 0.0032 0.409 0.0057 0.075 0.5841 0.0031 0.5880 0.0052 0.100 0.7648 0.0031 0.7648 0.0031 0.125 0.9253 0.0027 0.9320 0.0042 0.150 1.0840 0.0027 1.095 0.0048 0.r75 I.2LL2 0.0027 1.218 0.0047 0.200 1.3399 0.0027 1.4510 0.0039 0.225 1.4505 0.0026 1.4505 0.0026 0.250 1.5590 0.0026 1.5646 0.0038 0.275 1.6659 0.0026 1.6680 0.0043 0.300 r.7620 0.0026 r.7720 0.0049 0.325 1.8564 0.0026 1.8706 0.0048 0.350 1.9411 0.0026 1.945 0.0038 0.375 2.0166 0.0026 2.0209 0.0033 0.400 2.0963 0.0026 2.0963 0.0026 0.425 2.r704 0.0026 2.r7t4 0.0029 0.450 2.2430 0.0026 2.2470 0.0029

45 Table 2.2: The comparison of the calculations of À, using the VSM and the CEPII in the symmetric sector and weak coupling regime. Î;o"" d"r,otes the renormalised coupling calculated by CEPII. Hercñ'2:0.1 and N :20.

cons À î; error î r error 0.02 0.0191 0.0003 0.018 0.0007 0.04 0.0386 0.0003 0.0363 0.0008 0.055 0.0518 0.0008 0.0510 0.0017 0.07 0.0670 0.0009 0.0657 0.0019 0.1 0.0891 0.0008 0.092 0.0016 0.13 0.112 0.0013 0.121 0.004 0.19 0.165 0.002 0.175 0.0061 0.24 0.216 0.0023 0.22 0.007 0.35 0.313 0.0035 0.321 0.018

calculations in Table 2.3. The comparison indicates that in this sector the CEPII method can provide an accurate calculation of the renormalised vertex functions.

Case 2: The broken symmetry sector

In this section we consider the calculation of the renormalised mass and renormalised coupling in the broken sector, (ó) + 0, in the weak coupling regime. The VSM procedure is exactly the same as for the symmetric sector. For fixed fi2 : -0.1 and 0 < Î < O.tZ we calculated.ñ, and Î, for difierent values of Î. The error on õ10¡ it larger than the symmetric case due to the subtraction of the disconnected pieces. Thus we used Eq. (2.a6) to calculat" õ10¡ and subsequently extracted Z as previously discussed. In order to calculate the renormalised quantities using lattice perturbation theory we followed the standard approach to treating the broken sector. That is, in the bare Lagrangian we shifted the field by the vacuum expectation value of the field z, which can be easily calculated using MC methods, such that x@):ó(r)-r. (2.48)

After this translation the mean value of the field, (¡), vanishes and the perturbative calculation proceeds in the standard manner, keeping in mind that a non-symmetric ¡3 interaction has been generated. In lattice perturbation theory one then needs to also

46 Table 2.3: The comparison of the calculations of ñ}using the VSM and the CEPII in the symmetric sector and weak coupling regime. Î;o"" d"notes the renormalised mass calculated by CEPII. Herc ñ.2 = 0.1 and -ð/ :20.

À n'¿r errol ñ,1"n" error 0.02 0.324 0.001 0.323 0.007 0.04 0.334 0.002 0.330 0.007 0.055 0.340 0.0008 0.332 0.008 0.07 0.343 0.0014 0.339 0.006 0.1 0.345 0.0023 0.347 0.008 0.13 0.350 0.0023 0.357 0.009 0.19 0.375 0.0025 0.372 0.009 0.24 0.398 0.003 0.384 0.010 0.3 0.408 0.003 0.399 0.009 0.35 0.427 0.0035 0.410 0.010 0.40 0.433 0.004 0.428 0.010

consider vertex functions with a three point interaction. Note that y carr be different from the classical value of the vacuum, ,", : 1@fî. As an example for ñ2: -0.1 and Î - 0.1, u :2.187 + 0.002, compare with z"¡ :2.449. The comparison between the two-loop results and the results from the VSM method are shown in Fig. 2.5. In applying the CEPII method to the broken symmetry sector, evaluation of all the terms in Eq. (E.8) is necessary. This renders this method impractical. As one might expect from the symmetric sector results the calculation of the renormalised parameters using CEPI also suffers from large noise difficulties and the signal could not be recovered. It should be mentioned that the spontanously broken symmetry is a discrete one. It is well known that the spontaneous breaking of a continuous symmetry would lead to Goldstone bosons. However, the Mermin and Wagner theorem [40] implies that a continuous symmetry cannot be broken in two dimensions. This has also been extended to the framework of quantum field theory by Coleman [41], showing that in two dimensions it is not possible to construct a massless scalar field. The problem is coming from the fact that, for a massless scalar field in twodimensions, the two-point function suffers from severe infrared divergences.

47 o.20

0.1 5 { I .<. o'lo T

0.05

0.00 0.05 0.'t0 0.1 5 o.20 À

Figure 2.5: The plot of )' versus À in the broken symmetry sector 1 using LPT (solid line) and VSM (diamonds) with ñ'2 : -0.1 and N :20.

Case 3 :Strong coupling regime

The strong coupling expansion. In a weak coupling expansion the interactive term is pulled out of the path integral representation of the partition function as a functional operator. That is

ztî) - ón,,)' *I^'o'-* î^ó^) . Q^s) -*' là +i;] ltaot""o [- ÐLrto^ The remaining functional is Gaussian and can be done exactly. The partition function can then be written in terms of a power series of Î and the standard perturbation theory follows. The strong coupling expansion was introduced by authors of Ref. [55, 56] and later was advanced by authors of Ref. [57, 58]. Later, Authors of Ref. [60] used a lattice as a regulator and introduced a diagrammatic representation. To obtain the true strong coupling regime, they used an extrapolation to the zero lattice spacing. In strong coupling expansion, unlike the weak coupling expansion, the kinetic and the mass terms are pulled out of the path integral as a functional operator. That is

r ztrl-*o (2.50) LÐ #_*^(,,ù#]rUO,

48 where zo[î]: î-ó^f (2.51) ltaøl*r lT ]ø* The remaining functional integral is not Gaussian but can be evaluated as a product of ordinary functions on the lattice.

zs : * r-ó^] I IUoø^*r [Ð ].r^ JnXn : lrII I ay"*pl * - Îi (2.52) Then one can write

zo: N+Fli^î-t/41:,Â/exp r[î^î-vn]], (2.bs) lÐr with F(r): lar"-lîu*"1 . Q.54) and ,Â/ is a constant. The function F(r) is a transcendental function and can be expanded as a power series inø 1' F(x):+i|flf(:.- t ' (2'55) ttþl ?" Qn)t '2 i) Note that f'(J) is an even function. The Eq. (2.50) can be written as

ztît:*'lÐ (256) ftc-'(,,ù#l ".0[Ðr rv*î-r^t]. Then one can expand both exponentials in above equations and multiply the two series together and perform the derivatives with respect to "Î to generate the correlation functions. The generating functional assumes the general form Using this series expansion one can easily expand the both terms in the r.h.s. of Eq. (2.50) to obtain a po\¡/er series expansion for Z[J] which assumes the general form

oo zlÎ1- y, 1+ D î-ht"Atlîl (2.57) k=l where Ao[û are integrals over the source function ,I. Thus the strong coupling expansion is an expansion in powers of î-¡lz. Knowing the above series one can find an expansion for the generating functional for the connected correlation function, WlÎ\,

oo WlJl - tn Z[î] : Bo * Ð^-k/2 BklJl (2.58) ß=1

49 where Bo is a constant. Using Eq. (2.55) and Eq. (2.56) one can find the values for B¡. The derivation of B¡ arc straightforward but they become very long and tedious for ,b > 3. As an example, it is not hard to derive

BrlJl: -A Ð t: + G;:" (2.5e) n with .R : t(3 I 4)lr(1/4).

Graphical etpansion of the Greens function. Using Eq. (2.58) Bender et al. [60, 61] derived a graphical rules for constructing the Ît/2 term in expansion of n-point function which are as follows:

o Draw all connected graphs having a total of n external lines and å - nlz internal lines. Note that we must have k > nlz. Associate with every 2p-Iine vertex the

amplitude Tze. The vertex amplitude has a form

Tzp : î-n/z trr, (2.60)

where L2p are the coefficient in the Taylor expansion of h[.F(ø)/F(O)]

Fl*l *zp Lzp l"I (2.61) Ë (2p)l F[0] P=l

o Compute the symmetry factor. It is done the same as in the weak coupling expan- slon.

o Label the ith vertex by space-time coefficient n¿ and represent the end of every external line by rn¿. Represent every internal line connecting rz¿ to n¡ by 6;,¡. Sum over all vertices n¿.

o Multiply together the symmetry numbeT^the sum over vertices and the vertex amplitude for each graphs to get the (l/y'À)ß term in the expansion of the n-point Greens function.

Using the above rules one can generate the Feynman diagrams for strong coupling expansion. We have shown some of the diagrams of order and î-"1' in Fig 2.6. The terms in expansion corresponding to the figures in Fig ^-r12,Î2.6 arc:

Do : Qz6n,^.

50 (a) (b) o (c) ^\2 rl2 n4 n 2

(d) (e) (Ð (e) n6 "\2 2

Figure 2.6: The strong coupling diagrams for 2-point functions of order diagram (a), order Î-t, diug.ums (b) and (c), and order î4/2, ^-112,diagrams (d), ("), (f) and (g).

Da:

lJcn D¿:

lJen (2.62) D¡: Ds=

(2.63)

Similar to the weak coupling expansion one can arrive at a strong coupling perturbation

expansion. Bender et al. [60, 61] obtainerl aî-*/2. series expansion for g" :î,1ñ1-o of the form _¿t2DL N 9n: A Ð o^t tnyt , (2.64) l=0 ¿=0 where n'¿; 1 ï v-d/2 and î2 (2.65) --), v rnr For fixed ø one has L otÐ (2.66) 9*: A-d/' t @) v' , l=0 where JV oÍ")(")-ta-în. (2.67) r¿=0

This series does not converge for large correlation lengths. Thus the authors of Ref. [61] proposed a scheme to extrapolate the expression for Î" to large y assuming that Î, remains

51 finite in the limit y -+ Q.

15 o 3:, x

10 { 1

5 r

o.oo 0.10 0.20 0¡50 o.4o o.so

Figure 2.7: The plot oI gn - î, ¡A7 versus B : Îi (Î + roo) with strong coupling expansion results (solid line), using Eq. (2.a3) (stars) and the VSM results (diamonds) with ñ? :0.078 *.4V0 and lü : 20.

Raising Eq. (2.6a) to the power or 2Lld and expanding to order -L we find

2L g'i ="" (n Í")(')r')= =r-'åuÍ''(")r'. (2.68)

We then find L (d/2L) -d/2 (2.6e) gR: Y Du,(ùu' l=0 which is equivalent to Eq. (2.64) for small gr and approaches [äfl1r¡1a/2¡in the limit y --+ oo. In this manner the authors of Ref. [62] obtained an analytical series for Eq. (2.69). Since the interesting physics lies in a regime where the correlation length is large, we performed our calculation in this regime. Thus the above extrapolation scheme was necessary. We chose a moderate correlation length € : 3.6 by an appropriate tuning of the bare parameters. This can be done by fixing Î and choosing ñ to be in the symmetric region. As one decreases ñ,, one gets closer to the critical line and the correlation length increases. Using this, one can reach the required correlation length. To apply VSM we followed the same procedure as before. For six different Î's and fixed correlation length { : 3.6 A %,we calculated the values of ó.t for different values of

52 "I. The curve fitting procedure was carried out in the same way as for the previous cases. We noticed that in this regime the inclusion of larger S¡'s can change the behavior of the fit at small /"¡, the region which is of most interest to us. The problem arises due to the curve fitting procedure. In the WCR the data points close to ó : 0 have much larger weighting that the one far away from this point. Thus, calculating the derivatives of t/(þ) at þ :0 seem to be reliable. However, in the strong coupling regime, the data points that are far away from þ' : 0 have much higher weighting and even a small fluctuation might affect the calculated J($) considerably. 'We improved the results by imposing the condition in Eq. (2.46), that is to fixing the coefficient ø1 : IIGQ) where ø1 is defined in Eq. (2.47) ana õ(O): Ir¡,(õ,). thi. improved the results and the inclusion of larger /.¡'s did not affect the results significantly (up to 3%).

15 G" tr G.' x I t 10

o ¿ ti 5

Õ fB

0 0.00 0.1 0 o20 0.J0 040 0.50 0.60 p

Figure 2.8: The plot of sn - 1,lA? versus B : Î71Î + tOO¡ with strong coupling expansion results (solid line), using Eq (2.a3) (stars) and the CPII method results (diamonds) with ñ?:0.078+ 4To and l/:20.

Next we calculated the renormalised parameters using CEPI. Unlike the previous cases the errors in the results were reasonable. For the extraction of renormalised parameters we only used two histograms corresponding to J : 0 and J : 0.005. In the weak coupling regime where the mass term is dominant, one needs to sample the higher values of / in order to improve the calculation of Î,. Thus in the strong coupling regime there might not be a need fo¡ additional histograms. From the VSM results one might expect that

53 sampling very high þ might have a similar problem. This was confirmed from our data for this particular case. We also calculated the renormalised coupling using Eq. (2.a3). Unlike the weak coupling regime, uncertainties in the results in this region were reasonable. All the results in the strong coupling regime are shown in figures 2.7 and 2.8. They also are compared with the strong coupling expansion results. They all seem to be in agreement with each other within errors. This indicates that as the coupling increases the MC results approach the strong coupling expansion results. In the strong coupling expansion, the value of gp with € : 3.6 approaches 14.88 T 0.04 as B --+ 1. This value depends on the correlation length. In order to apply the CEPII method, there are numerous terms in (4.8) which have to be evaluated and consequently the accumulated errors can be very large. However we found that as before the renormalised mass can be calculated accurately.

Details of the simulations

In our MC calculation we chose the Hybrid MC algorithm. In a run we have taken a number of decorrelation MC iterations between two measurements. All the calculations were done on a 202 lattice and the rate of acceptance was kept between 40% and 60%. In all cases (except the broken sector) the calculations of renormalised mass ana õ(O) (where it was needed) and the direct calculation of ñ!a)10¡ were done using 6,800 uncorrelated samples with 50,000 thermalisation configurations. In the broken sector we used 11,000 uncorrelated samples with the same thermalisation configurations. The reason for the increase was to obtain better statistics, since the measured quantities have larger errors due to the non-vanishing disconnected pieces. In applying the VSM to the symmetric case (and in the weak coupling regime), we calculated þ with 0.025 < J < 0.425. We noticed that the necessary number of decorrelation iterations in the presence of nonzero J was smaller than for the J : 0 case. The calculations were carried out using 2,500 decorrelated configurations. We took the number of thermalisation configurations to be 10,000. In the broken symmetry sector we increased the number of uncorrelated configurations to 3,200. In the strong coupling regime only the range of the values for J was different (as mentioned in the previous section). For CEPII we used 5,000 uncorrelated configurations with 50,000 thermalisation iterations. In construction of the probability distribution histograms, trye used 750,000 configurations. The curve fits were done using a standard ¡2 fitting algorithm where the uncertainties on the parameters were obtained from the diagonal of the covariant matrix. For the strong coupling results we also estimated the

54 systematic error due to the fact that ( was fixed to be approximately a% by varying the fixed value within reasonable limits

2.5 Conclusions

In this chapter we studied the calculation of the effective potential for Àó1+r theory using three different methods: the variation of source method (VSM) and two constraint effective potential methods (CEP I) and (CEP II). Using our ne\ry method, referred to as CEPII, we showed how to calculate the vertex function using the correlation functions in the presence of a constraint, This procedure does not require any curve fitting or extrapolation to a zero external field limit, as VSM requires. Also this method does not require a very high statistics that CEPI method needs. The computational time is also dramatically reduced. However, the drawback is that its accuracy in the strong coupling regime is limited. We calculated the effective potential in the symmetric and the broken sector in the weak coupling regime as well as in the symmetric sector in the strong coupling regime. The renormalised quantities, Î, and ñ,,, were then obtained from the effective potential for each case. In the weak coupling regime we compared our results with lattice perturbation theory. We found that in the symmetric case both VSM and CEP II can give accurate results, whereas the CEPI method and the direct Monte Carlo calculation of the (2 and 4 point) vertex functions failed to do so. We also found that in the broken symmetry sector VSM is the most practical and accurate of these methods. We also studied the model in the strong coupling regime and the results were compared with the strong coupling expansion results. In this regime we found that CEPI, VSM, and the results from the direct Monte Carlo calculation of the vertex functions were consistent with each other and with the strong coupling expansion results. In summary then, we have shown that Monte Carlo effective potential methods can be accurate and reliable tools for calculating physical quantities for scalar field theories, but that one should use the method of evaluating the effective potential and its derivatives which is best suited to the regime of interest.

55 56 Chapter 3

Soliton models on Lattice

When úe uere children we uent to the Master for a time, For a time ue uere beguiled wi,th our own mastery; Hear end of the matter, what befell us; I came like Water, and like Wind I go'. Omar Khayyam

Solitons waves were first discovered by J.S Russell [63] in L824 and he referred to them as "The waves of translation". Zabusky and Kruskal [66] introduced the word "soliton wave" to characterise waves that do not dispers and preserve their form during propagation or after a collision. Soliton waves are solution to a certain non-linear differential equations. A system of differential equations can not have soliton solutions since they can be a superposition of plane v¡aves which are dispersive. Soliton ryvaves are regular every where

and have no singularities. Later on the computer experiments [6a] confirmed the existence of wave-like excitations which, rather than disperse their energy, maintain a stable shape

in the course of their propagation and emerged from the collisions unchanged [65]. The other important aspect of solitons is that they are non-dispersive localised packets of energy moving uniformly, and resemble extended particles. The elementary particles in nature are also localised packets of energy, being to be described by some quantum field theory. Because of these features, the soliton might appear as the ideal mathematical structure for the description of a particle. However, there are fundamental differences between the quantum particles and solitons. In the course of evolution of field theory and quantum mechanics the notion of a particle is associated with the elementary excitation of a quantised field. The propagation of a free particle is described by a quantised mode

57 of a linear system, rather than by solution of a nonlinear wave equation. Non linear interactions among the quantised field could be described in perturbation theory. On the other hand soliton emerges as a particle already at classical level. When it was realised that many nonlinear field theories which used to describe elementary particles, also had soliton solutions and these solutions would correspond to particle type excitations, the development of methods for soliton quantisation became important. The quantisation of solitons was done by performing an expansion in powers of ñ, (loop expansion), then the classical soliton solution appears as the term of leading order in the expansion and terms of higher order involve the quantum effects 167, 741. In mid 70's there appeared a number of works [5, 67] which developed the semiclassical expansion in the quantum field theory. In this period, there were schemes being invented to quantise these solitons. The correspondences between classical solitons solutions and extended-particle states of the quantised version was established [5, 6, 68]. Also important problems like renormalisation of quantum fluctuations and the zero mode problems were dealt with [69]. It was shown that one could associate not only a quantum soliton-particle state with a classical solution, but a series of excited states as well, by quantising the fluctuations around the soliton. These solitons were not to be considered as elementary particles of the theory but as additional particle-like excitation that the system possessed due to its nonlinear nature. One of the main questions is whether the solution of the non-linear interacting classical equations of motion of various models, which behave like bound, confined and stable configurations in space-time, survive under quantisation. For weak coupling, a stable time-independent solution plus the small oscillations around it provide a list of classical solutions which are sufficient for the application of the semiclassical quantisation procedure. The fact that a collision might alter the shape of the classical wave is not so relevant to the quantum case because of the quantised set of levels. Thus for the quantised solitons we may asked whether after a collision they maintain their original quantum states or make a transition to an excited state. The other interesting point is that some solitons can be associated with some topological index, from their asymptotic behavior. This index turns out to be a conserved quantity which, in quantised theory, becomes a conserved quantum number characteris- ing the soliton state. The main propose of this chapter is to study quantisation of solitons beyond semiclassical regime using lattice-MC methods. We consider the )/a theory in two dimensions on the lattice which, apart from the trivial solution, has topological solutions which are

58 commonly called the "kink" and "anti-kink". We begin our discussion with semi-classical quantisation of À/a kink solution in continuum, infinite volume. As we mentioned before the semiclassical schemes are only valid when the coupling constants are small. Thus, in order to reach beyond to semiclassical regime one needs to choose a non-perturbative method, such as lattice MC methods. We will discuss the quantisation of the topological soliton on the lattice. We will show that the zero mode problem which exist in the semiclassical regime, can still exist beyond this regime. Then we will calculate the soiiton mass using lattice techniques and compare the results with the classical and semiclassical results. When it comes to MC calculations, one can only perform the calculation on a finite and discrete lattice. In order that a calculated quantity on lattice correspond to its continuum and infinite volume counterpart one needs to consider suitable schemes to approach a well defined continuum limit and the finite size effect must also be taken care of. We address these issues in this chapter.

3.1- Classical solutions

A semiclassical study of solitons requires the classical solutions to the non-linear field equations. These solutions can be used to study the vacuum and other features of the corresponding quantum field theory. In this section we review the derivation of the classical solutions to two 2-dimensional À/a model.

Àó1+, kink

There are many field theories models for which the classical equations of motions have time independent solutions. They also have a finite energy higher than the energy of the vacuum. In the case of the ÀSa in 2-d there are topological excited state which are called the kink and anti-kink. In the weak coupling regime it can be shown [6] that the kink, "baryon", has a mass larger than the mass of normal particle, "meson". In this model the kink (or ant-kink) carries a new quantum number , "topological charge", which ensures its stability against decaying into vacuum. Consider the À/a theory in a finite two-dimensional box with R: L12,,[ being the length of the box. The dynamics of this model is governed by a Euclidean Lagrangian density 1 - L u(ó, (3.1) - ;@,ó)' + R),

59 where u(ó,R):|{0"-r')', (3.2) with /2 : ñ} lî being a dimensionless parameter. The Euler-Lagrange equations gives the equation of motion

oQ: AU aó Let us assume that there exist static solutions, that is 9:N'ttt AU

Multiplying both sides by /' and integrating over ø we obtain 1 : U(ö) (3.3) ,ó'' - "?, where c1 is a constant of integration. This equation conserves the total energy functional Elþl which is given by

Etót : I *lirþ)' + u ç0, n¡]. Let us assume fhat U(þ) has i absolute minima at h¿, which are also its zeroes, and consequently Elfl: 0 if and only ó : h¿. In order to have a local energy density, field must approach one of the minima as Í -) çoo. If t/(/) has a unique minimum at þ - h, as it is shown in Fig. 3.1.a, then the static solution must have Ó(r) - h as x --+ çoo. Let the classical density of field takes more than one minimum, say two minima at h1 and h2 (Fig. 3.1.b) . If there exist solutions satisfying the static field equations, with S@) -+ *,h1 as ¿ -) -oo and ó(") - 1.h2 as r -) oo, to make sure that the energy density being local, then the solutions are topological solutions. The transition between å1 and ä2 is localised in space and the total energy is finite. In the quantum interpretation of the fields å1 and å2 correspond to two vacuum states O1 and O2, in a way that they are the vacuum expectation of the quantum operator O(ø,ú):

(O¿lÕlO¿) :d¿, i:I,2.

If år and h2 are related by a symmetry then O1 and O2 are degenerate even when the fluctuations are included. Going back to Eq. (3.3) and upon integration one arrives at ¡óþ\ dã r-r,O:+ I rö@o) t/zu(iù _ "?

60 u(0) u(o)

0 0 h hr tb

Figure 3.1: The classical potential for Àþa theory in the symmetric phase (a) and broken symmetry phase (b).

In order to solve for soliton solutions we impose the boundary condition þ'p:0, where /¿ is the of field at the edges of the box and from now on \rye set rs : 0. Using Eq (3.a) and the boundary conditions we obtain

À a 2 c?r: Zflç6^¡ : óR) (3.5) 2 U, Then

õ *a dó dó I, (ó) 1", - "? ffr_ór)r_"3 1 dó ,/îr- t: (3.6) (, - f_,) (r, - ó2 + c,¡ (3.7) where : (f' Let Then we have "' - óh), $: õar.

I 1 d, &- P*' (3.8) 12_ lo ú4Ã=æ71 "Zlón '/T+7, where ö'* rc2 f'+"t

61 fix)

Figure 3.2: A PIot of kink solution against the space coordinate.

The solutions to above equation can be expressed in terms of Jacobian elliptic functions

Ó(") : t/¡ Sn(lúø, rc) (3.e) where N - {fm tM The value of the field at the radius R, Ón, can be determined by Í2*cz (3.10) where .Il is the complete Elliptic function of the first kind The minimum value of 1l occurs at ón - 0 which gives 'tt 7t min R^¿n = ft"-ñ,:2*.

Classically speaking for .R ( Ao the kink collapses into a state with Ó(*) : 0 with -R < x I R. For a given radius Æ, Eq. (3.10) can be used to determine the value of /p. Note that as .R -r oo then rc --+ 1 and cz + 0 and the solution in the infinite volume domain can be recovered [82]. That is

ó(*):f/ tanh (rtlþ) . (3.11)

We call the solution with plus sign "kink" and with minus sign "anti-kink". The La- grangian is symmetric, under o <__-l -ø and ó(*) -ö(*) which gives - 4{r)(o) : -6@Ðça¡ : 6@Ð (x)

62 ttlc" where and. "alc" stand for kink and antikink. There are also trivial solutions ó(") : +f which are the vacuum states and the potential U(þ) has two degenerate minima, at ó: If . The energy density D(r) is D(x) : irYr + u(ó) :2U(ó), (3.12) glvlng Ma = ElÓl: I d,nD(x):'++, (3.13) where M.¡ denotes the classical kink mass,

D(X)

Figure 3.3: A plot of energy density versus position.

All the finite-energy solution of this system falls into four topological sectors. We can

label these sectors by pairs of indices, ef , f),U,-f),e1,-f) and (/, /) which represent the values of þ(n) at r : Èoo. Thus kink, anti-kink and the trivial solutions belong to four sectors. Kink provides a description of a particle with an internal structure, distributed over a finite volume rather than concentrated at one point. It possess a conserved quantity which is called the topological charge Q, and is defined as

Q :lóþ)l* - d(')l--l

The just topological charge Q, is the difference between two indices ó(*) I f and þ(-æ) I f , and it is a conserved quantity. Solutions with Q : 0 can exist in theories with a conserved (nontopological) charge and are called non-topologacøl solutions. On the other hand, the kink and anti-kink have non-zero topological indices and therefore they are topological solutions. Formally a classical soliton is a topological stable, finite-energy solution of the Hamil- tonian equations of motion of a classical field theory. In quantum field theory, solitons

63 appear as a sectorsT/,o, in the Hilbert space Tl,of physical states, which are labeled by a topological charge and are orthogonal to the vacuum vector Îlo. Then

7t: ØTts, a where Q is the eigenvalue of the topological charge on Tlq. The kink sector with Q : 1 and the vacuum sector with Q : 0 are not only orthogonal but can not evolve into on another. It takes infinite amount of energy to deform a configuration from one sector to a configuration in a different sector. It has to be mentioned that the topological charge conservation is different from conservation of momentum, energy, etc. The latter come from the symmetries in the Lagrangian. On other hand the topological indices coming from boundary conditions, conserved because the energy is finite.

3.2 Semi-classical Quantisation

In this section we start with a simple application of classical solutions in quantum field theory. That is, quantisation of static soliton solutions to obtain extended quantum particle state. The formalism for the quantisation has been developed through a variety of techniques [5, 70,7L,72) In the semi-classical analysis of our model, one requires that the classical solution ó, to be stable. That is, if in the /a action we set Ón : Ó + ,t" and keeping only the terms in powers of 7, then resulting quadratic Lagrangian should be reducible in a set of i independent harmonic oscillators with frequencies ø¿. The first correction to the classical mass M"¡, corrrês from the zero point energy of oscillations around /. It is important to recall that semiclassical results, which will be discussed next, are only valid if the coupling constants are small. The properties of quantum states such as, their energy, can be expanded in a semiclassical series and the leading order is related to the classical solution. Thus knowing the classical solution of the non-linear system will give informations about the particle states. The important point is that the informations are non-perturbative. The semi-classical regime for ÀSa theory. Consider the full classical action for the l/a model in two dimension. That is

srdl : a*at (3 14) satöt= | | lïr#r + firffY *Ln{ó, - fy]. Note that /2 has dimensions of cã. In the path integral the action appears as e-s.tlt".

64 Thus a change of variable t: æo/c and ( : f þ one obtains o."o.,li,#,.irffr *{re,- -ry: -# I I '),] Thus the semiclassical regime corresponds to the regime where /2 is large. ie, m2 > À. We set h: c: 1 and express the action presented in Eq. (3.14) on the lattice which has a static soliton solution þ, which is an extremum of the action.

1 sldl : t D@" - ó^+r)' -r u(ó"), n 2 p meaning

: îó^lö"^ : o, W I2óL - ö^+r, - ó^+pl + - frl which is the equation of motion on the lattice. We expand around the classical solution / by setting Tn: ón _ ó, (3.15) where s[-d] :0,

and : ^9 ^9. +,9B with

s.[6]:Dõ*ó***tP | ! 4fø' (3.16) n¡lt - 6r'It7' - 4 - Í')'1.

and sB:ÐTn'"rn- 2I' ú- +Dlúfzf - a6\,f6^- nil (r.17)

For small Î orr" might ignore the cubic and quadratic terms in qn:

: 6n,^+î) su l-¡ztn,nL - 6n,^-î - - îr'+ 3Îø] ,f^ + o(q"), (3.18)

where .î is a unit vector in the space direction. Consider the trivial solution (the vacuum) ón : f . Then in the lowest-order quadratic terms we have

SB : qnPn,^T^ (3.19)

with hp : -f26n,* - 6n,^+â - 6n,*-ef + 2îf2 6n,^ The eigenvalues above operator are (,, = VFU'^ with the eigenfunctions eÂ'' where the values of Î,, are înL :2ntr givenby box-normalisation with .[ being the length

65 of the box. Now one can construct a system of harmonic oscillation states around /. Then the lowest energy state has energy

Euo":'r+*:i+lîz +zîrl+ . (8.20)

We call the set of states built around the 6 + f the uacuum sector.

The quantisation of the kink. We can apply the same method to quantised the kink solution. That is one looks at the eigenvalues of the quadratic part of the expansion around the kink solution.

: (3.21) l-Í26^,, - 6*-î- ó-+î] -îl' + lî6n,*of Tn u2nTn,

Of course one still needs to find the kink solution on the lattice. If the (r's are known, similar to quantisation of vacuum, one can construct a system of harmonic oscillators around the kink solution and obtain the energy of the kink, støl E*¿nt ,-r't -| s/-t * o(î), J-n I,'- : Ma *îi,r+ o(î). (3.22)

In order to calculate any physical quantity one needs to study the model in the continuum and thermodynamic limit. Then one might define the soliton mass as

M"- Ek¿nh-Ero". (3.23)

Note that infinite series D,,, ir Eq. (3.22) and Eq. (3.20) in the continuum limit become a quadratic divergent integrals. The energy of vacuum Ero" is also quadratically divergent at continuum and considering that what is important is the difference between the vacuum energy and the kink energy then it is reasonable to subtract Ero"from Ek¿nk, giving

M,ot : E*¿nt - Euo. - Ma ++t '" -ÐIrl+ o(Î). (3.24)

We will come back to this point later. For the case of the continuum kink, The eigenvalue problem, Eq. (3.21) become I ,#* (Btanh2, - r)] n*þ): fin^Q), (s.2b) withz:t6f,ltn.

66 The solutions to Eq. (3.25) are known [73]. The first two eigenvalues are discrete:

c.ro2 : ør2 0 with nLþ): .*#ø and :I^t' wlth r¡t(z): ¿H

The rest of eigenvalues are continuous. For fr ) 2 they are [73] ,î:*'(Iq2i2), with eigen functions:

: x r k2 Siqtanh(u)] . nn(x) "'o' fStanh2 - - -

Note the presence of the isolated eigenvalue c,Jo : 0 corresponds to the "translationalt' mode and has to be present because of translational invariance. For a box of length of tr, with a periodic boundary condition, qn ate obtained from [73]

q-(Ú{2)+ 6(q,,) - 2ntr,, (3.26) where 6(q^):_2tanh-l le] is the phase shift of the scattering states of the Schrodinger equation Eq. (3.25). Then once again one can construct a set of approximate oscillating states around dr in the field space. Then Eq. (3.24) becomes

M"ot_ Er¿nt _ Eoo":r# + ,rl-þ-T>riî\+z¡i I rrî? + zæ¡+ + 116). 2 L\'"t e.2T)

In the continuum, even after subtraction of two divergent integral in Eq. (3.20) and Eq. (3.22), one still ends up with a logarithmic divergent integral and renormalisation is necessary to render the soliton mass finite. The renormalised mass turns out to be (see Appendix F) 2 (2) Mrol = " **È{Ir- (3.28) 3À rtt+o(^). The first term in above expression is the classical value for mass whereas the second term is the leading correction due to quantum fluctuations. Note that the corrections are independent of ). Also note that the leading term is singular for ) : 0 which is a nonperturbative results.

67 3.3 Path integral quantisation

In the previous section we used semiclassical methods to quantise the kink solution without a need to introduce the functional method. This was due to the fact that the kink solution \¡¡as a static solution and the quantisation procedure reduced to just the study of coupled harmonic oscillators. However, when it comes to time-dependent solitons one needs to use the canonical or path integral quantisation. The path integral is the most elegant, convenient and popular approach. In this section, \rye redrive the expression for the kink mass using functional methods. We start with the well known relation between the Euclidean propagation kernel e-Hr and the action S[/],

tte-Hr : løol '-s[ó(x'?)]' where f/ is the Hamiltonian of the system. This trace will be dominated ,when 7 -r oo, by the minimum energy state in that sector. Now consider a in two dimensions with ^Ó4 a classical static soliton solution /, which is an extremum of action. The action can be expanded around / as

s[d(*,¿)] : s[ø] dx dr t[^,t)l#. + o(n"), .t, I #. #lr=rlrtr^,r) and we recall that 7 : ó - þ. Ignoring O(n") and higher terms, the second term in the r.h.s of Eq.(3.29) is a Gaussian integral and can be evaluated exactly

axar tr[e-Hr] N e-s[ó) e-s[-d] I nr'Ía ltaOl = ltorlu : Ce) e-sto)der [M(x, r)]-i (3.30) where azu M(x,T) a2 a2 Aó' AT2 ôx2 ó=ó and C(7) is an Euclidean measure. We can write

trfe-Hrl: C(T"-lstõ1+åt'(u¡41 (3.31)

C g) is yet to be specified and can be adjusted for a correct normalisation of states. For the static case discussed in the previous sector one can obtained the det M through its eigenvalues. Note lhat M(x,T) is a sum of ?-dependent and ø-dependent terms. Thus M(*,?) can be diagonalised separately in both sPace and time. Then l-# . (W),=,.,] €;@) - ,,t¿'

68 Then for the kink solution one has

Hr i *'7 lf trfe- I - C (T) e- ["'a*D, "^tz{' (3.32)

The eigenvalues of -A2lAT2 lu! are easy to be evaluated. Then we obtain ,,hl#*,?]:å ^l#*,?] (33s) For large 7 the sum can be transform into an integral:

ir" d,rtn(r2 +,?):r +#)+ r,'"'] l#*,;] -, I la" [r"rr

Note that the divergent part T/lnr2dr of. the above is separated out. This divergent part can be absorbed in the measure Cg), and we denote the new measure C'(T). The remaining term can be integrated by parts, that is

T a,r" -'#] :r o,&:,nri. I [r I

Thus as T + æ the Eq. (3.32) becomes;

trfe-Hrl + C'(T)e-1s1-6¡1!, ]'¡r¡ (3.34)

By choosing the measure C'(T) to be one, we have

Slóal EN T *DT.o, which is the same result as in the previous section. Renormalisation is required to render the theory finite. The renormalisation procedure is the same procedure shown in Appendix F. That is a mass counter term can be added to the Lagrangian to render the kink mass finite. The zero mode. At this point we would like to discuss the physical consequences of the mode corresponding to Øo : 0, the zero mode. These modes always exist when one quantises a theory with a translational invariant Lagrangian about a solution that is not translational invariant, but having space dependence. Consider the actio" S[d]. The field configuratiott dÍf) and /{f}., where s is a vector in the direction of space, give the same value for action. This means that if the kink solution /(*), is an extremum of action so is all the translated ones. Starting from a kink solution /(t), one can always move along a trajectory such that ^9[/] remains constant. It is along the zero mode where

69 there is no confining force and the wavefunction can be spread throughout this mode. Unlike other modes which are vibrational modes for small fluctuations; the zero mode is not vibrational and means that it is not confined to be near the classical solution. Note that in the calculation of the mass we ignored the zero mode contribution to the kink mass. That means we have ignored the motion of the center of mass contribution to the kink energy which also means we have assumed that the kink energy, in the semiclassical regime, is the same as the mass of quantum kink particle. The semiclassical treatment of these zero modes is done in variety of ways 174,69,75] but we will not discuss these further here.

3.4 Soliton quantisation on lattice

So far, we have discussed analysis of quantum solitons based on semiclassical approximation, where the classical soliton appears as a minimum of the Hamiltonian and is expanded in quantum fluctuations around the classical field configuration. In this section we discuss another formalism using the Kadanoff operators which does not involve any approximation of such kind. The I(adanoff operators. Another approach to the quantisation of the solitons is based on constructing Euclidean Greens functions for soliton fields in terms of functional expec- tations of. order and disorder fields [76]. Kadanoff defines the disorder Ising correlation functions as the correlation functions of spins, in the dual space, with a magnetic dislocation. By magnetic dislocation it is meant that the sign of the couplings between a group of spin variables is reversed while the rest are left intact. It is shown that the disorder correlation functions fall off exponentially as the spin separation is increased. As we will argue, in Àþa theory in two dimensions involving the continuous spin /, a dislocation results in a non-trivial stable topology. We will show how expectation values of disorder correlation functions can be related to the topological soliton mass. We will show how expectation values of disorder correlation functions can be related to the topological soliton mass. Consider the following action in 2-dimensrons.

L,¡p¡zönón+p+ (3.35) ^g - - | f(z + n¡lt fl Çlr'"* þ,

As it is indicated, in addition to lattice sites there are link variables Ln+pl2 which are neighbor sites coupling links. A single cell of the lattice is shown in Fig. 3.4. The unit

70 (ij+l) (i+l j+l)

L(ij+l/2) L(i+l j+ll2)

L(i+tt2j) (ii) (i+l j)

Figure 3.4: A basic cell of the model. The factor 1f 2 arc labels for the dual space lattice sites. vector p,f2 corcesponds to space and time directions in the dual space. In the case where L : I for all L, the action is the usual À/a action. Consider a lattice shown in Fig. 3.5 and take C *,, to be a path in dual space connecting two points rc and z. Let's induce an excitation by letting all links .0, which are crossed by path C^,,, equal to minus one and set the remaining links to one. In Fig. 3.5 we have shown the lattice and two examples of paths, Co,B and C*,,, by light dashed lines, and we have shown the reversed links by heavy-dashed lines. Such an excitation gives rise to a topological configuration. Authors of [80, 76] have defined the bilocal disorder field as

(D*,,) : t+ù, (s.s6) where Z and Z (C are the partition functions of the model with no topological excitation ",*) and with topological excitation along palh C,,*, respectively. z(c*,,): llaallus,*töt1 (3.87) is the partition function for a lattice that has been excited through path C^,, and .9,,, is the action for the dislocated lattice. A very important fact is that the (D,,*) is independent of the path (see Appendix G). Equation (3.36) can be written as

(D*,,):L Ln¡¡,¡zónó*,,**ó*ó^,), IW\J "*pl å where f* takes into account the links with their signs changed and !' refers to the rest

7T a--

I Ò K

x -o-- -r>- q, p

Figure 3.5: The pair of paths C*,, and Co,B, which joins points in the dual space. The dashed dark sign indicates that the sign of that particular link is reversed.

of the links. One can also write this equation as :

(D,,*):L IWrles-2Di''ÓnÓn'p:("-'D)''L''unö'ó^¡ (3.38) In the case of the )/a theory, the introduction of a twist give rise to a topological excitation, with non-zero topological charge. This excited state is the kink state since the kink state is the topological state with the lowest action. The topological excitation appears at some f and is annihilated at f * r and one expects that, at large distances, (D",r) has an asymptotic behavior, governed by

(D,,,1 fo. large rc1, (3.39) = "-M"lu-nl lz - reminding that M" is the kink state mass. As a matter of fact, in a finite set of points,

{rt,...,un} with charge e¿, (Drr,...,r,) turns out to be the Greens function of a soliton fi.eld of charge {q¿} located at {z¿}, i : I,. . . )n [81]. Thus the standard method of calculating the kink mass is to consider a disorder operator with non-vanishing projection on the soliton sector, and then computing the connected 2-point function for large time and extracting the mass from its exponential behavior. However, such a procedure introduces large systematic errors, especially for small masses. The reason is that this procedure is based on non-local quantities, thus the finite size effects can be large. It would be more practical to develop a method of calculating the soliton mass based on a local quantity. Consider the path Co,Bin Fig 3.5. Reversing the sign of links which are cut by this path is equivalent to imposing spatial anti-periodic boundary condition. We imposed

72 periodic boundary condition in a temporal direction. Let S¡ = S(C".B) and z¿ z(c,,u): (3.40) = lÍaól ¿-s,tdl Then, one has: (D,,8) : ?^, (3.41) where to avoid confusion we have defined ,9" and Zu as the action and partition function for this model with periodic boundary conditions in spatial and temporal direction (untwisted). From Eq. (3.a1) and using Eq. (3.39) one arrives at

M"-ry:(E,l-@,|. e.4z) which is the same result as the semiclassical definition of soliton mass Eq. (3.23), with T=la-Bland L (Eù [,r^,c.-_S,(ó)] TZJ LqYJvtv ) (E"l L'L'zuJ' Ir¿¿1 '' s.,"-s(ô). (3.43) A change of variables, ó' - ól\ß,with p: 1/Î, gives zd(r, with 6 : u.¡t (3.44) þ) - A-' lloö'1"-ot[rd.r, where V is the volume and m" ói s,[ó]: -D0;0;,,,+,, * D (2+ )ó""' + (3.45) ntþ n 2 4 Taking the derivative of Z¡ gives 02, V ...'.._:-- (3.46) ap 2p Then one can define the twist parameter [80] O(B) as o(p):_h^Z

Note that O(p) is just the difference between the mean actions. Then the soliton mass can be expressed as

u"a(0): (s.42) -+ I!"tþ'nrp): + Ii"orärs, - s,)

In the À/a theory p" ir th" val e of p af which the parameters are on the transition line. If the parameters are located in the symmetric region where (ö) :0, there is no symmetry breaking and Z¿ - Zu and therefore the soliton mass vanishes.

73 3.5 Lattice Monte Carlo calculation of the kink

mass a

We start this section by a graphical display of kink configurations, in different regimes, on the lattice. Next we will show that even beyond the semiclassical regime the problem of. zero mode exists and the treatment of zero mode on the lattice will be discussed. However, the main purpose of this section is to calculate the soliton kink mass using the

MC methods. Our methodology is motivated from a method suggested by authors of [79]. We will improve the procedure as well as correcting some of their mistakes. The kink on the lattice. The condition for having a topological soliton state is that the

(o)

-¿0 0 to lo t! 20

(b) x x x xx x x

-20 !0 to t5 æ

(") XK x XX x X x x x x x x x r* x x -20 70 t0 ta

Figure 3.6: The plot of time slices of a number of MC generated configurations for three different parameters, .f : 10 (a), "f : 5.5 (b) and / - 1.5 (c). At the right column diagrams the right hand side of each kink solution is plotted with the same scale. The solid line kinks are the classical solutions.

boundary condition at infinity be different from that of the physical vacuum. On a finite lattice, however, one can impose anti-periodic boundary condition in the spatial direction to give rise to a topological soliton. On a large enough lattice and in the semiclassical regime, where ñ2 > Î, on" expects that the classical kink solution, in continuum and infinite volume, to be very close to the MC calculations. Using a MC algorithm we generated a number of configurations corresponding to different values of / and for each value of / we took the average field variable for every site in an arbitrary time slice and

74 the results are shown in Fig. 3.6. As one might expect for large values of /, the classical solution and the MC field configuration are close. As / decreases the fluctuations of fields around the classical solutions increases. As the the figure suggests that the quantum kink does preserve its shape, fluctuating around the classical kink.

1.0

0.5 .. ; , ... t ,

ê 0.0 ,t

-0.5 ii. i t !:.- ;'.... : , :: tj

I - 1.0 -30 -20 -10 0 10 20 30 n

Figure 3.7: The comparative plot of averges of fields on a time slice with : and : "f 0.5 ñ,2 -l for a unconstraint lattice (dashed line) and constraint lattice (dotted line). The constraint fixes the center of the kink to the center of the lattice for all time slices. The solid line is the classical solution.

The other point that we would like to make concerns the zero mode problem. As we discussed earlier in this chapter in the semiclassical quantisation one encounters the zero mode problem with its physical consequences is the motion of the center of the kink. Then, the question is whether this problem persists beyond the semiclassical regime. The main reason for being interested in the zero mode problem is that v/e are interested in the mass of the kink rather than it's energy. Another reason is that, as rrye observed, the movement of kink in the lattice can make the configurations unstable. This instability is more significant if the fluctuations are large. To answer whether the zero mode problem persists beyond the semiclassical regime, one can examine one of the consequences of existence of a zero mode, that is, the kink motion. set : : : We ñ} -1 and î 4 which corresponding to "f 0.5, away from the semiclassical regime. Then for an arbitary time slice we calculated (/") for each site for a number of configurations and an average over these configurations rvas calculated. As

75 Fig. 3.7 suggests movement of kink due to translational mode still persists in the non- perturbative regime. In order to treat the problem, we imposed an additional constraint, that is, that is Ó(M): 0 with M - (rs,Nl2) for all times øs, to fix the center of the kink to the center of the lattice. An average configuration and the classical solution are included in Fig 3.7.

0 3 r

Õ <<2 r -$ --L

Þt-- - t-- \ l..'- 0 t -3 -2 -1 0 f

Figure 3.8: The plot of the transition line between the broken sector and unbroken sector using Monte Carlo methods (diamond) and the light-front perturbative predictions (dashed line). \Me use the symbol î : ñ,2 in above figure.

Phøse spl,ce. As we mentioned in Chapter 1 this model exhibits two phases. CIas- sically, fot ñ,2 ) 0, the minimum configuration energy it (d) : 0 where as ñ'2 ( 0, a spontaneous symmetry breaking occurs and the new minima are (/) : In this +r14.VÀ regime the second order transition line which separates two phases is the line corresponding to the line ñ,2 : 0. By second order phase transition \rye mean that the correlation length becomes infinite, i.", â : Ilñ, -' oo. Beyond the tree level the phase space structure changes. There still are two, previously mentioned, phases and there is a second order phase transition line separating the two phases but the position of the transition line changes. In order to determine the transition line, we choose several values of.ñ,2 < 0 located in the broken symmetry sector where (ól + 0. Keeping ñ,2 frxed, one can increase Î until (/) becomes zero and critical parametrs can be found. Of course there is no phase

76 transition on a finite lattice but by a second order phase transition 'we mean that the correlation length is much larger that the lattice dimensions. Thus, close to this line, the fluctuations become larger and this reduces the resolution of the critical line. However our estimate of the transition line is sufficient for our future calculations. As one approaches the critical line, i b""o-", larger and in order to avoid the finite size effects, once needs to use large lattices. An estimate of the transition line is shown in Fig. 3.8 as it shows, unlike the classical case, the values o1ñ.zcan be negative but still be in the symmetric sector.

The transiton line can also be calculated perturbatively. The one results predict a linear relationship between the critical coupling î",¡ and ñ,2: )., Cñ2

where C - 59.5 for the light-front formulation [78] and C : 54.3 in the equal time formulation 1771. As one approaches the critical line, i b""o*", larger and in order to avoid the finite size effects, once needs to use large lattices. An estimate of the transition line is shown in Fig. 3.8 as it shows, unlike the classical case, the values of. ñ,2can be negative but still be in the symmetric sector.

3.5.1 The scaling region.

Both in numerical MC studies and the analytical calculations it is important to find the renormalisation group trajectories (RGT) (or curves of constant physics [83]). Along these curves and close to an IR fixed point the physics described by the lattice regularised quantum field is constant and only the value of the cut-off ( lattice spacing) is changing. We showed this property for a particular case (bare RG) in Chapter 1 by deriving the strong scaling condition in Eq. (1.92). This equation suggests that in order to find the scaling region one can find a region where the ratio of the dimensionless renormalised vertex functions are constant and one expects the scaling region to be in the vicinity of the critical line. In this model the only non trivial critical region is along the transition line calculated previously. As we mentioned before, À/a theory in two dimension is an interactivetheory. That is, in addition to having a Gaussian fixed point where Î" vanishes, it has another fixed point with Î, being non-zero. Then one might expect that close to the critical line, there must be segments of phase space where the ratios of dimensionless vertex functions remains almost constant. It is very important to find the scaling region corresponding to a non-trivial IR fixed point. That is, one should try to find trajectories away from the Gaussian fixed point.

lt As we showed in Chapter 1 in a scaling region the vertex functions are governed by the critical exponents. The critical exponent corresponding to a trivial fixed point differ from the one corresponding to a non-trivial fixed point. Here, even though in a trivial scaling region the spontaneous symmetry breaking can still occur, the vacuum is governed by a free fieldr. We chose R(ñ,,) : ñ?lî,, to be the dimensional quantity to probe the scaling region. As we showed in chapter 2, this ratio can be calculated accurately using effective potential 'We methods. In our calculations of .R(ñ",Î"¡ *" used the VSM method. found, for a fixed mass ñ2 : -1, a region where values of. R(ñ.,,Îr) *"t" almost constant. Fig. 3.9 shows that this region correspond to a segment where 0.2 < Î < O.g.

0.80

o.70 o { ç- ôt u. I { { r { o.60 { I

0.50 0.0 o.2 0.4 0.6 0.8 1.0 À

Figure 3.9: The plot of R(ñ,,Î,) : ñ?l^, against Î where ñ,2 : -1

From now on we will restrict our calculation to this region of the phase space. Now v/e are in a position to calculate the soliton mass on the lattice. In the previous section we indicated that the mass of soliton can be obtained from the asymptotic behavior of the disorder two point function D,,* where v and, rc being two points in the dual space. We showed that the disorder two-point function can be obtained by calculating (e-2Dn,r"n'"nón+r). For the the path in the dual space we chose the path Co,B shown in Fig. 3.5. which is equivalent to imposing an anti-periodic boundary condi- rThis is one of the mistakes that authors of Ref. [79] have made.

78 0.8 * *

0.6 * oo *,* 0..1 **. * o.2 ***t t 0 5 l0 t5 25

Figure 3.10: The two point functions for Kadanoff operators. Here ñ',î:0.5 and I{:48. tion in the space direction. In such a non-local prescription of the order parameter, the soliton mass has to be extracted from the exponential fall off of the correlation function. To obtain a correlation at a distance r, and to suppress at least the largest finite size effect, the linear size of the system must be at least 2r, which means that, in order to get into the asymptotic regime, the system has to be taken very large. This is the biggest disadvantage in MC simulations. Thus we chose a large lattice, 48 by 48, to minimise the finite size effect. In Fig 3.10 we have shown the disorder two-point functions Ds,^, for Î:0.5 and ñ.2:1. As it is evident the two-point function has an exponential form which should fall off according to Eq. (3.39).

The MC results

In order to calculate the soliton mass, by studying a local quantity, one needs to calculate the twist parameter A(P), defined in Eq. (3.47), which involves calculation of the averages of the actions for twisted and untwisted system. In the symmetric phase where (ól = 0, because the action is symmetric under a/ then changing the sign of .L's does not change the action. Thus the soliton mass is zero as one expects. The procedure for calculating the soliton mass using the disorder operator O(B) is straightforward. For a frxedñ,2: 1, we started from a value of p - ll^:0.2, within the scaling region, and moving towards the transition line we calculated A,SlT, for each value oI B, In general, the statistical errors become larger as one approaches the critical

79 LO o rð ö 0

I 0.6 { 0.4 I o.2

0 2 4 I

Figure 3.11: The plot of LSl?. versus þ : tlî with ñ2 - -1.0 line. However, we noticed that in this region the values of (Sr) suffer from, considerably, larger statitical errors than the values of (S"). . The MC and the classical results are shown in Fig. 3.11. As the figure shows, as one approaches the small B region the quantum contributions become more significant. By integrating the function shown in Fig. 3.11 and using Eq. (3.a7) one can calculate the soliton mass. The MC, the semiclassical and the classical values for kink mass are compared in Fig. 3.12. As the figure suggests, for the case ñ,2 - -1, the MC results for the kink mass lie below the classical values and above the semiclassical values. Also note that as the coupling constant Î b".o-", smaller, the classical, the semiclassical and the MC values for the soliton mass converge. The semiclassical kink mass tends to be negative at some stage which obviously means that the semiclassical analysis is not valid in that regime. We also extracted the soliton mass from the exponential behavior of the disorder two- point function and the results are shown in Fig. 3.13 which are quantitively consistent with our previous results. However, the statistical errors are considerably larger and we expect that the results suffer from finite size effects. In order to see how the soliton mass behaves in different regions of the phase space we set ñ,2 : -2.2, and repeated the same procedure as the previous case and calculated the soliton mass for a set of coupling constants belonging to the scaling region. We have shown the results for soliton mass in Fig. 3.14 where the classical and semiclassical values are also included. As the figure indicates for this case, the values for soliton mass are below both classical and semiclassical values u, Î in.."ur"r. In comparision with the

80 1.0 t r* 0.8

I t_ r* 0.6 rl

I I * 0.4 * *

o,2 * ** 0.0 00 o2 04 06 0.8 't.o À

Figure 3.12: Plot of soliton mass versus B with ñ,2 : -1 using the twist parameter O. The results are compared with the classical (solid line) and semiclassical results (dashed line). results by Authors of [79], our result for ñ,2 : -2.2 is below the semiclassiacal prediction, whereas theirs is above. This is due to number of mistakes, including ignoring the zero mode problem.

3.6 Details of the simulations

All the MC calculations in this chapter have been carried out done using the Hybrid Monte Carlo algorithm with 48 sites in each direction of the lattice. The code has been executed on a vector machine. For the graphical display of the kink solution on the lattice after some thermalisation configurations for each site of an arbitary time slice we calculated the average field over 500 configurations. The same number of configurations were used to produce Fig. 3.7. In order to find the scaling region, we used the variation of source method (VSM) discussed in chapter 2. Calculation of the disorder two-point function (D^,,1 required high statistics. For each point in Fig. 3.10 after 10,000 thermalisation configurations and 25 decorrelation MC iterations rrye sampled 30,000 configurations. For the calculation of the twist parameter A(P), for the parameters relatively away from the critical line, we used 12,000 uncorrelated configurations. We increased this number to 22,000 uncorrelated configurations for the parameters located close to critical

81 1.0

Ë* I 0.8 I I

0.6 t- r{ I l¡ * 04 * * * o2 * 00 00 o2 0¡l 0.8 1.0 À

Figure 3.13: Plot of soliton mass versus B with ñ'2 : -1 extracted from the asymptotic behavior of the Kadanoff operators. line

3.7 Conclusions

In this chapter we discussed the formulation of solitons on the lattice as as applied to the À/a model in two dimensions. We reviewed the derivation of the classical solutions which includes the topological solutions, kink and anti-kink. We also discussed calculation of the contribution of field fluctuations to soliton mass in the semiclassical regime, where the coupling constants are small. The zero mode problem was also briefly discussed. In order to calculate the soliton mass beyond the semi-classical regime and on the lattice, we used the Kadanoff disorder formulation for the topological excitations which gives non-perturbative results. Using this formalism, we calculated the soliton mass using two different methods. One method is to calculate the soliton mass from the exponential behavior of the two-point disorder function at large distances. This method has the disadvantages caused by calculation of a non-local quantity; and thus it is sensitive to finite size effect causing large systematic errors. The second method involves calculating a local quantity, the differences of vacuum energy between the twisted and untwisted systems (twist parameter). By twisted one means that an antiperiodic boundary condition in space direction is imposed. Since the twist parameter is a local parameter it is less susceptible to finite size effects. As we showed, the results from both methods agree with each other within numerical errors.

82 6

4 x

x 2 * * * i * 0 05 t.0 1.5 2.O À

Figure 3.14: Plot of soliton mass versus B with ñ,2 : -2.2 using the twist parameter f,). The MC results are compared with the classical (solid line) and semiclassical results (dashed line).

In order that the calculated soliton mass on the lattice have a well defined continuum limit we computed the soliton mass in the scaling region, In this regime the effect of lattice spacing is small. This region \ryas obtained through the renormalisation group analysis discussed in Chapter. 1. We also showed that the zero mode problem on lattice persists beyond the semiclassical regime and the treatment of zero mode on the lattice was dealt with. This was recognised from the motion of the center of mass. To treat the zero mode in the MC calculations we fixed the center of the kink to the center of the lattice. We calculated the soliton mass using the disorder parameter O(p) for two different fixed bare masses and compared our non-perturbative results for the soliton mass soliton with the continuum semi-classical results up to one loop. We observed different behaviors of the soliton mass for different choices of bare parameters. Fot ñ} : -1.0 and Î in the scaling region the soliton mass apears to be higher than the semiclassical prediction. However, for negatively larger masses such as ñ2 : -2.2 the soliton mass tends to be smaller than the semiclassical results.

83 84 Chapter 4

Monte Carlo Methods

Ah, but my computations, People say, Reduced the year to better reclconi,ng?-Nay, 'Twas only strilci,ng from the Calender unborn to-morrow and d,ead yesterday. Omar Khayyam

4.t Introduction

The main purpose of simulating a field theory on a lattice is to estimate the expectation values of some functions O, of. the field /. That is to compute

(ol:tYffi: z-'1ltaølorøle-stø.. (4.1)

In terms of a lattice field system the number of integration variables is equal to the number of lattice sites

ldó) :40ö", (4.2) and for lattice simulations of physical interest the number of integration variables usually needs to be very large. Performing such high dimensional integrations directly is not practical and one needs to resort to statistical methods. The ensemble average (O) can be well approximated by averaging over a sufficiently large set of suitably weighted, uncorrelated field configurations distributed with a probability distribution determined by the Boltzmann factor exp(-^9[/])l |ldólexp(-S[/]). W" denote this weighted, uncorrelated set of configurations by Cr with I : I,. . . , ¡f with the approximate ensemble average

85 given by N 1 (ol x- lolci (4.3) N I=L with a statistical uncertainty of.Il.rÑ. The approximation becomes an exact relation in the limit.f[ + oo [93, a8]. We use units where h = c: t here for convenience. This set of weighted, uncorrelated field configurations, (C¡ for I : I,"',N), are sampled with a probability distribution determined by the above Boltzmann factor from the set of o// . . possible field configurations (Q for i : L,. , æ). For simplicity, we treat this set of o// possible field configurations as if each member C; can be labelled by the discrete index i. These configurations C¡ can be obtained from sampling the elements of a Markov chain with a sufficiently large number of steps (or links) in the Markov chain (denoted as say M) separating these sampled configurations. The links in the Markov chain are field configurations (denoted C¿ here) generated by a Markov process, with some probability P¿j :- P(C¿ + C¡) fot a field configuration C¿ to be replaced by the configuration C¡ after asingleMarkovstep. If wedenote P!{) ""theprobabilityforanyfieldconfigurationC¿ to be replaced by the field configuration C¡ after M Markov steps, then PIP-ËË Ë P;¿,P¿,¿,...P¿*-,¡. (4.4) it =1 iz=t iU-t=L For a Markov chain with the properties of irreducibility, aperiodicity, and positivity it can be proven [9a] that J'* a;e!{) : n, (4.5) for an arbitrarg initial probability distribution d¡. The probability distribution D¡ is generally referred to as the equilibrium distribution and it should be noted that the existence of the above limit implies that it is unique. It is straightforward to show that D¡ has the properties of a probability distribution and that it is invariant under the Markov process, i.e., it satisfies D¡>0 forall i:I,2,3,"' , ilr:t , Di:ËD;k¡. (4.6) j=l i=1

It is simple to demonstrate that any probability distribution which is invariant under the Markov process must correspond to the limit in Eq. (a.5) and is therefore unique and given by the equilibrium distribution D¡. It can be proven that the deviation from the equilibrium distribution decreases with each Markov step [92]. The property of irreducibility is that a finite number of Markov steps can lead from any configuration C¿ to any other configuration C¡ with finite probability, i.e., there is a finite M such that P!{) l0 for

86 an! C¿ and, C¡. The property of aperiodicity is that p!!') + 0 for any M, i.e., there is a nonzero probability of a configuration returning to itself after any arbitrary number of Markov steps M. Denoting plY) the probability of going from configuration C¿ to itself "" af|er M steps withozt reaching the configuration C; at any intermediate step þ.f., n{)), then the mean recurrence "time" r¿ of the configuration C¿ is given by

oo oo ¡¿:Dr,(M)plP twplY), (4.7) M=l - M=tt where we define here r¿(M) - lltf. Positivity of the Markov chain corresponds to f¿ being finite for all configurations in the space, i.e., for all C¿. If, in addition, we have finite variance of the recurrence "time", / æ \r/z A4=(D¡'(u)-îi'pÍy,.*) (4.s) \'t,rã / , then, as a result of the central limit theorem, Eq. (4.3) follows as a property of the particular Markov process provided we have chosen 4¡ such that it results in the Boltzmann form for the equilibrium distribution

D¡ : exp( -SÍCl)li"*p(-^slcn)) . (4.e) Ë=1

Note that finite variance of the recurrence time for a positive Markov chain requires only that ,!') îV,f*¡l"plY) *"oly)

¿-slcilprr- "-s[cilprr, (4.11) then the resulting D, will correspond to the correct Boltzmann form of the equilibrium distribution. To see this one simply sums both sides of Eq. (4.11) over j, makes use of D¡ P,¡ : 1, and then divides each side by f¿ exp(-,91c*)). This leads to Eq. (a.6) with D¡ being given by Eq. (4.9).

87 In summary, we see that any Markov process which has the properties of (i) irreducibility, (ii) aperiodicity, (iii) positivity, (iv) finite variance of the recurrence time, and (v) satisfies detailed balance, will lead to an appropriate set of weighted configurations C¡ provided the number of Markov steps M taken between the sampled configurations is sufficiently large. The configurations can then be used in the calculation of quantities of the form of Eq. (a.3). Finding a P¿¡ which satisfies the detailed balance condition is a sufficient but not necessary condition for generating appropriate configurations. In addition, the requirement of detailed balance does not determine the transition probability P¿¡ uniquely, and one can use this freedom to invent algorithms optimally adapted to the problem one is studying. There are number of standard algorithms for generating Markov chains which satisfy properties (i) to (v) with different P¿¡, e.g, the Metropolis, Langevin, Molecular Dynamics (in the thermodynamic limit), Hybrid and Hybrid Monte Carlo algorithms [a8]. For a more detailed discussion of these and related issues see Refs. [93, 48] and references therein.

4.2 The Langevin Algorithm

The Langevin Algorithm was originally proposed by Parisi and Wu [85]. The standard Langevin algorith begins by introducing a Langevin lime r¡a - Met, where M is the Markov step number and e¡ is the Langevin time step. In the limit er -r 0 we see that the Langevin time becomes a continuous variable. However, in this section we discuss a generalised version of this algorithm. Introduce an updating rule as follows:

ó;(r¡a*r¡: ó¿(ru) - ,,W# + t/-ktn¿?u),, (4.r2) where ó(*) refers to the value of the field / at lattice site i in the Mth Markov (or Langevin) step and n{r) are random variables with probability distribution P(nîD: ({I #)*o{-\i,:r"t\ (4 1s) and variance htç*¡,1¡ (r¡v)) = 6;,i 6 v ¡'t' Note that we have labeled the random noise with the same label as the label of the configuration being updated. From Eq. 4.L2 we see that the probability of obtaining the field configuration ó(r¡',¡+t) from þ,*r, is equal to the probability that , \ lÓ7u+) :!þu)l (4'14) T;þ*¡ = rrr, * tVras[Ó(ru)],ffi, - 88 for all lattice sites, i. Hence we conclude that

P(nuD: (ç . (4 15) #)",.n { -'r\|"ryÐ +.ffil'} In the standard Langevin algorithm v¡e see that we must have lóþu+r) - ó¿(rw)l: 0(eù as é¿ -) 0 and hence one can show that [93]

-'îo exp ¡u+t) 'ffi {- :tø,r - óJ,òTW} cl +O ') exp-ISlóþ*)-Sló?¡ø+r)lj. (4.16)

In the standard Langevin algorithm detailed balance is satisfied only if 2e¡ --> 0. In this case the finite step size e¡, introduces an error. As an example consider a free scalar field with action

s[d] : ñ,r)+ (4.17) T2r^rr^,,nó^ : -'r^ónön+t" + ]fz + ór^, - ntÌn fl¡lr where ñ is the dimensionless bare mass and p is a vector pointing along the p, direction.

After reaching the equilibrium distribution, the values for two-point functions become [86]

(Ó",Ó^l: I{;,L (4.18) rather than (/,,, Ó^¡ : K;,k. Earlier works using the Langevin algorithm took the approach of running the simulation with several step sizes and then attempting an extrapolation to e¡ : 0. Doing this reduces the systematic errors but such an extrapolation can be very time consuming. Thus, the major disadvantage of this algorithm is that making the step size e¡ too big induces a systematic error, and making it too small, for a fixed computational effort, means that the sum will be done for too small a time, introducing statistical errors. In MC calculations one creates a large number of configurations and a estimate of an expectatio" (X) :7 of. a physical quantity Xlþ) can be obtained by taking an average over all configurations. In the limit where the number of configurations are infinite, the expectation value becomes equal to true expectation value X. lt all the configurations contained in a sample are statistically independent, according to central limit theorem, then the sample average F is normally distributed around mean value X with the standard deviation

2 D^(X^ x2 X" o (x) _ -x)' _ (4.1e) M _L M -r

89 $/ith M being the number of samples. However, a sequence of configurations can be statistically dependent. This correlation in a sequence of generated configurations is called autocorrelation. For a quantity X the autocorrelation is defined as

c(r) : (x*x*+,| - và(x"+*l (4.20) : (X*X^+,| -N' (4.2r)

: . . The autocorrelation (X^X*+,) is only r dependent with X- = Xlö^l (* I,2,. , M). if the sample is very large. Then it is easy to write the variance in Eq. (4.19) in term of autocorrelation can be written as o'(x-):([# þø^-",]') : #:Ð-W'n, (4.22) u*æ,* ]cço¡ lll note that C(0) - f -f - (M - l) o2(X). The autocorrelation time -r, is defined as "Ë g!2 v,'- :L 2,?*c(0) e's)

Comparing Eq. (4.19) and Eq. (4.22), one can see that due to the autocorrelation the effective number of independent measurements is reduced to Ml(2f,). On the other word, in order to sample the uncorrelated configurations one need to have at Ieast 27, configurations between measurements.

Critical Slowing Down

Away from critical point most MC methods perform well. However, near the critical point where the correlation length becomes larger, the autocorrelation time rapidly diverges as

vrx€"', (4.24) where r is the quantity which is being measured and z, is called the dynamic critical exponent and depends on the observed quantity. This phenomenon is called critical slowing d,own. The reason for such a behaviour of the autocorrelation is that, for local updating, the short-wavelength structure tends to evolve far more quickly than the structure of long- wavelength,i.e., typically faster by a factor of (2 (measured in units of lattice spacing).

Since one must limit the size of. a single update to maintain stability at short distance [91], the evolution of large-scale features is greatly slowed. Indeed the number of sweeps of

90 the lattice required for appreciable change at these wavelengths gro$¡s as (2 (2, : 2), which tends to infinity in the continuum limit. Equivalently, one can say that the critical slowing down arises from the fact that the updating procedure is local. Any information about a change in one site is only transmitted to its nearest neighbors. In order to update and generate uncorrelated configuration such information should travel a distance longer than the correlation length. Thus 7, represents the typical amount of simulation time it takes for a local change in ø to propagate a distance longer than the correlation length. For a local updating procedure such as the Langevin algorithm, z - 2. Good updating algorithms avoid critical slowing down as much as possible. For some special cases there are algorithms, such as the cluster algorithm whose updating rule is global updating, and it is possible to achieve z N 0. Unfortunately, these algorithms are not easily generalised from one model to another and have not been implemented for most lattice gauge theory models. Also it has to be mentioned that for a lattice in d dimension with its volume being fixed, the computational time is at least {d. Thus, close to the continuum limit one expects that the computational time to be, at least, ('+d. In order to calculate 7, one can use a truncated version of Eq. (4.23). However in practice it is easier to calculate the autocorrelation using the binning technique. That is to build blocks of subsequent configurations, called bins, and average the quantities in the bins. The average in each bin can be considered as a single measurement and its variance can be calculated using Eq. (a.19). If the bins are large enough then estimate of variance in Eq. (4.19) is correct. Thus if one, starting with a small bin size, increases the bin size until the variance is ¡emains constant, then the autocorrelation length can estimated. An equivalent procedure is to keep the size of the blocks fixed and find the minimum number of configurations to be skipped until the error estimate remains constant. For the case of Langevin algorithm one can overcome the critical slowing down, to some extent, by updating in momentum space [90]. However, the systematic errors due to finite step size still persist. In addition, a momentum updating of gauge fields is only useful if calculations are done in a smooth gauge such as the A0 : 0 gauge [90].

4.3 The Molecular dynamics method

The idea behind 1 the molecular dynamics is that the Euclidian path integral in a quantum field theory in d dimension can be considered as a partition function for a classical lThe foilowing argument is based on the Ref, [93]

91 statistical mechanical system in d spatial dimensions with a canonical Hamiltonian that governs the dynamics in term of the simulation time. In the other words the quantum theory in d-dimensional Euclidean space-time resembles a classical canonical ensemble in d spatial dimensions in a contact with a heat reservoir. Consider a scalar field theory with an action S[{]. As was shown before, the expectation value of an observable O, is

(o) : Llvdol/]e-stó,rr G.25) with z : e-sró'r¡ IldÓl where À is some parameter of the system (e.g. coupling). The important point to note is that Z is not the partition function of a classical Hamiltonian system. However, one can write this in such a form by introducing a set of canonical conjugate momenta zr, and writing Eq. (a.25) in the form

(o) : * I4dtonlolöl "-Hró'r'\t, @.26) where

HlS,n,Àl _ D++ sld,.\1,

z' : i'^ld'trf e-Hló'n'\l' e:7) Notice that Eq.(4.26) should not be confused with the phase space representation of the path integral Eq. (a.25), whose structure is completely different, and involves the Hamiltonian of the system in (d-1) spatial dimensions. Now we want to calculate the expectation value shown in Eq. (4.25). It is well known that in the thermodynamic limit the canonical ensemble average can be replaced by ttenergy" microcanonical ensemble average evaluated on an surface, which is determined from the parameters of the system. By "energy" here \¡¡e mean the energy of statistical system in d spatial dimensions. Note that the action of the quantum mechanical system in Eq. (4.26) is allowed to fluctuate and this is how the quantum fluctuations are built into this method. The expectation values in Eq. (4.26) can be written as

lldó)ldrlolól t dE6(H[ó,"; À] E)e-ø (O)"on = - (4.28) lldó)ldti I dE6(Hló,,r; )l - E)e-ø

92 On the other hand, the microconanical ensemble average (Ol^;", on the energy surface H : E, is given by

(Ol : Z^¿"(8, À)-L ldþl ldrl olól 6(Hló,r; - E), (4.2e) ^¿"(E) I ^l where z^¿"(8,¡): (4.30) It¿/llldrl6(Htó,r;^)- E), is also called the density of state e(S), at energy E. Let us define the "entropy" of the system by o(8,À):InZ^¡(8,À). (4.31)

Inserting Eq. (a.30) and Eq. (4.31) into Eq. (4.28), one can write

dE (o) *;"(E' À)e-@-"(ø)) (o)",,(À) t (4.32) - I dEe-tø-o(E'\)) What comes next is a standard statistical mechanics argument. That is, as the degrees of freedom become very large, the exponentials appearing in Eq. (4.32) are strongly peaked about an energy .E', given by

lôo(E)l _ l (4.33) I an J"-",- " which shows how .E' can be calculated for a particular À and implies that, in the thermo-

dynamic limit [93] (o)"""()) : l(Ol^;"1(E'). (4.34) This equation establishes a connection between the canonical and the microcanonical averages. Note we can write the Hamiltonian equation of motion as:

0H[þ,r, À] ó¿ on¿ ) 0H[þ,r, À) ttz (4.35) ôót

This can be used to generate configurations with constant energy. Since the observable only depend on coordinates, the microcanonical ensemble average can be replaced by a time average over /(r). Then in an ergodic system the following holds;

(o)*'(À) (o)*;"1ø=ø,1À¡ d'ro({s¿þ)}). (4.36) ,n#.,,^,, "**--r" + Ir' The trajectory of the field configuration is then given by the equation of motion. That is drö, asþl Et: -nó, (4.37)

93 Then on a discrete lattice one can use the lattice representation of the second derivative in Eq. (1.41) and write the equation of motion as: ek ASWI Ó;(r¡o*'¡ - ö¿(-'*) ' - îffiò* e^r¡(r¡a)' (4'38) where e- is the time step. The conjugate momentum on the lattice is: n¿(ru)- I (4.39) ern þ¿?nr)-ó;(rna-r¡1. There are two important issues to be addressed. First note that if the momentum {z'¡} is replaced by a Gaussian noise then the molecular dynamics method becomes the standard Langevin algorithm with et : e?^12 where e¡ is step size in the standard Langevin algorithm. Then the distance travelled by the molecular dynamics after n steps is O(ne) while in the standard Langevin algorithm with the same number of steps, the travelled space is O(1fie). Thus the molecular dynamics algorithm moves faster through phase space. The other important point is that in the molecular dynamic algorithm, unlike the Langevin algorithm, ergodicity is not built in exactly. This is because Eq. (a.36) is only valid in the thermodynamical limit and does not hold for a finite lattice. The other major difference between these methods is that the molecular dynamics process is deterministic whereas the Langevin algorithm is a stochastic method.

4.4 The Hybrid algorithm

The slow convergence of the Metropolis and Langevin algorithms come from the fact that they perform a random walk in the field configuration space. The walk through space configurations would be faster if we chose a deterministic algorithm such the molecular dynamics approach. It is possible to combine these two methods to benefit from the good features of both. The Langevin approach builds ergodicity into the microcanonical simulation. The ¡esult still contains systematic errors due to finite step size but a Metropolis test can be included to eliminate these. This is the idea behind the Hybrid Monte Carlo (HMC) algorithm. This was originally proposed by Scalapino et al. [95] and it was later modified by Duane et al. [96]. The first step is to discretise the equations of motions in their Hamiltonian form, Eq. (a.37). This is done using a Runge-I(uúúø integrating scheme. Introducing a step size €¿ and assuming that / and zr are functions of a ficticious time ¡ we expand ó(" + e¿) and r(r * e¿) around r. That is ó(, + eh) : ó(,) + ,nff * +# t oþsn) (4.40)

94 0r n(r+r^):r(r)*en (4.41) a, -+# *oþso) On the other hand we know that

(4.42)

and ôr¿(r) ASWI (4.43) 0r oó¿(r)' Then one can write Eq. (a.aO) as

ót(r * e¡) : ó¿(r) +,nlo,þ) - +rf,h] * r,.t, : e¡r¿(r ö¿(r) + * i) * o(esn). @.44) Taking the derivative of Eq. (a.a3) and using Eq. @.aÐ we obtain

: * oþ¡) (4 45) +P: - ? #ho'(o + lú#õ - #run] Insertion of Eq. (4.45) into Eq. (4.4I) and using Eq. (4.43) gives

+'ù - : - * o(eso) [",(' +ffi] þ,,", +rîi"] -,^ffi - n,(, *Y¡: ¡r¿(r *i) -,^ *ffi+ oþl), (4.46) since n¿(r + +), up to O(e2) is

en ôSlól r¿(rt!)-"n(r) (4.47) 2 0þ¿(r)

We rewrite Eq. (a.aa) and Eq. (4.46) up to O(r") in a form more convenient for numerical calculations.

(4.48)

(4.4e) with n¿(rPr): ¡r¡(ru) -+W (4.50) The Runge-Kutta integration is reversible. Consider a trajectory in the phase space governed by Eq. (4.48), Eq. (a.a9) and Eq. (4.50), with the starting point at r¿ with the

95 initial configuration {/(r¡),"(t¿)} and ending at r¡ and the corresponding configuration {ó(r¡),"(r¡)}. If we start from r¡ and {/(ry),-r(rò} with step -e¡ instead of e¡, one arrives at r¿ with {ó(rr),-o(r¿)}, meaning that

P n (r r (r : P r ¡ø ¡r (r u n (r [{ö(r*), u)} - {ó(ru +t), u +t)} ] [ { d( +t), - +ù} - {ó(r¡,r), - v)}] (4.51) Now, the rules for the hybrid algorithm. Given an initial configuration [/¿(r¡a)]:

1. Generate random momenta [zr1, according to Eq. (a.13)

2. Find [r¡a] from Eq. (a.50)

3. Iterate Eq. (a.a8) and Eq. (4.48) a number, say N, times and let {ó(r¡ø+r),rc(r¡ø+t)} be the last configuration generated by the molecular dynamics method where î¿(r¡rt+t) is obtained from Eq. (4.50).

4. Accept lóîu+r),r(ru+)l as the new configuration with probability

Pl{ó("¡rt),rr(r¡rr))-ló(r¡ø+r),o(r*+r)}:min {r,"-'101"*+r)''("v+r)l+HlÓ(ru)'r(r¡a)l) (4.52)

5. If the configuration {þ(r¡a),"(r*,)} is not accepted keep the old configuration as an initial configuration and start from step 1. Otherwise use [/(r¡a)] to generate a new configuration starting from step 1.

Note is just the Metropolis test. The only difference is that here the action that step 4 ^9 is replaced by the Hamiltonian H. Configurations generated by this algorithm satisfy detailed balance. The proof is rather simple. Let r j rM and r' :- rM+r. Note that there are three probability densities. The Gaussian density Pc, pc - Ce-iDi"?, (4.59) the probability density corresponding to the molecular dynamics PM,,,,,, and the probability density corresponding to the Metropolis acceptance test P¡,,,,, ,

PA,,,,, = min {lr"-'lø1"')'"(')l+rr[d(r)'r(r)]] (4.54) which is equivalent to

e-Hlóþ'),n(r')lp¡,r,,, (4.55) "-HlS(r),r(r)lp',r,r, -

96 Note that using the Hamiltonian H[ó,n|, the configuration [/,n] is changed along the discretised tranjectory in phase space up to the end point lók'),"("')l=T¡alþ(r),"(r)] where T¡a we mean the Molecular Dynamics transformation. Since the dynamics equation is deterministic, this can be express as

PM,,,,, : 6([n(r'), ó(r')] - T¡a(þr(r), d(r)]). (4.56)

Then the total transition probability density Pt,r1r is: p,,,, : lla"t )lÍar(r')lp¡a,,,,,pA,,,,,p¡("(")) (4.57) Multiply both sides of the above relation by e-sldl to get

: pA,,r,r, (4.5s) "-slólpr,", l[a"{r)]ld,r(r')lp¡a,r,r, "-Hls(r)¡r(r)1. Since Pu is reversible (Eq. a.51) , Hló,rl : H[ó, -zr] and the invariance of measure ldtr (r)lldr("')l : [d(-,r(z) )] ld(- r (r'))l

e-sÍó(")1P,,,,: l[anlr)lfd,tr(r')]p¡a,,,,,pA,",,,ê-ulöþ)J. (4.59)

Using Eq. (a.55) and Eq. (4.27), the above equation can be written as

"-slöG)lp,,,, : "-slök')lPr,,r, (4.60) which we recognise as the detailed balance condition. Note that the leapfrog integration is "area preserving". That is

ldr(r)l[dó(r)] : ldr(r')lldó(r')l Vr,r', (4.61)

In the hybrid algorithm there is no systematic error corresponding to finite step size

e¿, instead one is required to deal with an acceptance rate which is a function of e¿. For e¡ being small the acceptance rate is usually high. Thus if the acceptance rate is very high (- 1) then there is a possibility that the trajectory does not travel through a sufficiently large portion of phase space. On the other hand, if e¿ is large one expects a lower acceptance rate. Too low an acceptance rate (- 0) it means that a very large number of configurations has to be generated. For most of our calculations we kept the acceptance rate roughly between 40% and 60%. One of the advantages of this algorithm is that the field configuration is updated in a parallel way over the whole lattice and later the new configuration is either accepted or rejected globally.

97 The HMC is "exact", in the sense that it is a Markov process which converges to the desired distribution with no systematic errors provided that the fictitious time Molecular Dynamic (MD) trajectories are reversible and area preserving, that the computation of the Hamiltonians for the Metropolis accept/reject step is exact, and that we have a sup- ply of perfectly random numbers. All numerical computations carried out using floating point arithmetic are subject to rounding errors, but unless these errors are amplified exponentially we do not normally consider them to be a serious problem. These amplified errors can cause irrevesibility. The amplifications can be caused by either a small error changing the Metropolis accept/reject choice or the MD trajectories exhibiting an exponential sensetivity to initial conditions (or other words they are unstable). One might ask to what extend irreversibility introduces systematic errors into the results of an HMC trajectory. As a matter of fact, it is hard to see significant effects on physical observables unless a large violations of irreversibility are inadvertently induced[89].

Hybrid Monte Carlo and the imposition of constraints

There are numerous occasions where one would like to impose a certain constraint on field configurations. In chapter 2 we encountered such situation, concerning the constraint effective potential, where each configuration /, was required to satisfy llNo D¿ þ¿ -$ with .lüd being the number of lattice sites. It was shown how this can be achieved by introducing a "demon like" site. Here we propose another method of imposing a constraint within the Hybrid Monte Carlo algorithm. One can eliminate r(r¡v) in Eq. (4.48) to arrive at

SÍÓ1, : ehÍi(rM) ú . ö;(r¡ø+r) ó;(qo¡ + - 2 ôþ¡(r¡a)'=A ôsI r¿(ru+t) n;(r¡r,t) (4.62) - +l ó;(ru)

This is a special case of the leapfrog integration which is also reversible and area preserving. Therefore, one can also iterate these equations and use it to calculate the MD trajectories. Now, Introduce the following updating rule

ó;(r¡n*r¡ ó;(r¡r¡ * enege('¡'t)n¿(ru) - L'"'utç*) m, r;(r¡t¡t) n¿(ru)-!"att'*'lm + (4.63)

98 Table 4.1: Comparison of CM and IM results in the symmetric phase with

ñ,2 :0.25 and Î:0.1 .

CM error IM error (ó) 0.995 0.008 1.00002 0.000006 (ó') 1.28 0.02 1.306 0.008 (d') 1.85 0.03 1.90 0.03 þ^l 3.r2 0.08 3.24 0.05

where 0(ru): IIIND;ó;(r¡a) -þl and þ is a parameter to be tuned for the best performance of the routine. Note that if. 0 : 0 the above updating rules are the same as Eq. (a.62). Here, a nerry step size, e!¡ - e90e¡, is introduced. After a sweep over all sites one recalculates 0 and proceeds as before. This is repeated until convergence occus, i.e. 0 -+ 0, which also means that the constraint is imposed. There is no priori reason that the iterations should converge, however, for this simple case B can be easily tuned to achieve convergence. The rest of updating rules goes the same as the hybrid MC, i.e. refreshment of the momenta, the Metropolis test for Hamiltonians, etc. All is changed is that the step size changes from one MD iteration to the next. In this way one moves along the MD trajectories towards configurations with (ól :7. fne integration is still area preserving, however, reversibility is not exact. In general, 0(ru) I 0(r**r) which makes this routine irreversible. However, by choosing e' to be small, one can achieve an approximate reversibility, i.e. 0(r¡rr) x 0(r¡aa¡). Here we impose a constraint, If NdD¿ö;: þ, in 2-dimensional þa theory using the method introduced in Chapter 2, which we call CM for the Constrained Method, and IM, for the new Iterative Method. We compare results obtained from both method for lþ2), (/3), and (/a). Starting from the symmetric phase with ñ2 : 0.25 and Î : 0.1 with ñ and Î being the dimensionless bare mass and bare coupling parameters, respectively. Vy'e calculated the correlation functions, mentioned above, using both CM and IM methods and the results are shown in Table. 4.1 As this table shows the results from both methods are consistent with each other. Note that, in the case of CM, (d) : 0.995 rather than 1, which is caused by the accumulated round off error. The same calculations can be done in the broken symmetry phase. Setting ñ,2 : -I

99 Table 4.2: Comparison of CM and IM results in the broken symmetric phase withñ,2: -1.0 and Î :2.0.

CM error IM error

( ó) 1.510 0.004 1.515 0.003

( ó' ) 2.50 0.01 2.518 0.008 (d') 4.31 0.025 4.29t 0.018 (ón\ 7.96 0.08 8.05 0.06

and Î : 2, without the presence of any constraint one has (ó) : 1.515 + 0.003. Then the field configuratiosn were constrained in such a way that (/) : 1.515. As it was shown in Chapter 2, in absence of an external source, one should have: (d') ñ'(ól 6 -_ 2 -"'_n

The results are shown in Table . 4.2 and they are seem to be consistent. In both cases one has (/3)/6 -ñ'þl12 - 0, within the statistical uncertainties. The calculations are done on a 20 x 20 lattice. For both cases, 2000 uncorrelated configurations were generated. However, it was noticed that for the case IM, the number of thermalisation configurations was less than for the CM case. In the calculations discussed here, almost any value of B such that -0.25 < P < -0.5 was suitable to achieve convergence within 6 to 10 sweeps.

4.4.L Hybrid Monte Carlo and Simulation of fermions

A very useful feature of the hybrid MC is that it is suitable for simulating dynamical fermions, which will be discussed next. Apart from the notorious fermion doubling problem on the lattice at the level of formulation of the fermion action, the numerical simulations are much more difficult than bosons. Unlike the bosons Boltzman factor, the fermion Boltzman factor e-sl'þ"þ\, which is a operator, can not be directly interpreted as a probability so that standard MC methods can not be used. This makes the task of numerical very difficult, however, if the action is quadratic in fermionic fields one can integrate them out. All the algorithms that simulate fermions depend on this property. Consider a theory involving boson and fermion field. The path integral has the fol-

100 lowing form s¡tö"þ'ït, z : [l¿Ó]ld,rþlld,ÚlIr e-std- J r @.64) where r¡re assume the field variables are dimensionless, and .9[/] is the pure boson action and s ¡lö, rþ,úl : DE'"f"¡w!:!í @,m)gri (m) (4.65) where n,rn ate the coordinates, a,B are the Dirac indices and /, f' are theflavorindices. We take W to be the Wilson matrix, with an addition of a fermion to boson coupling term, and it is given by

Wi,ß: 6n,^6..,86¡,¡t - rc6¡,¡,D6^,n*r(î + 1r)"'þ I 6n,^îón (4.66) p where f is the dimensionless Yukawa-type coupling. Assume lhat W is not flavour dependent, that is sló,rþ,Vl : Dú',(")w,,u@,*)rþrB'(m)6¡,¡, (4.67) ntrn Then since ,S¡ is quadratic in fermion fields this term can be integrated out giving

Z_ e- s' t ! tÓl (4.68) I ld óltd+ll¿tti and s"¡¡löl: sldl - N¡ ln tt[w] (4.69) where Iü¡ is the number of flavours. The important fact is that matrix W has a real determinant. Thus one can define a ne\4/ matrix .t as

Llól: wtlólwló1, (4.70) with detrlþl:+[d"tlld]lå, (4.7t) noting that Eq. (4.70) ensures that L is positive definite. Whether detwlþl is positive or negative depends on the hopping parameter. For example, in d, : a detWlfl ) 0 for n < Il8 [100] . Taking detW[þ) ) 0, then s"¡ ¡lól- s[d] - +tn derr[/] (4.72) Note that det[/] is the relevant determinant for two fermions. Fucito et al. [97] and Petcher et. al. cite petcher suggested using so-called pseudofermionic fields y and yt, which are not Grassman but bosonic fields, to calculate detLfþ] detL[g]: l@*l@*\e-D,,ixt't'7]Íøtxi, (4.Tg)

101 Ti :r

Thus, one can rewrite the partition function as

z : vln¡,]totxt . (4.74) J'[l¿ó]ld,ylld.ytle-stól-D¡,¡ 'Jr Then the desired probability distribution for / and ¡ is

p(ó,x) x e-slól-xt¡'-11ólx. (4.75)

This can be done as follows: First generate a random vector ( with a Gaussian weight

P(O o "-e'c. @.76) Then construct x:We, ØJ7) such that X is distributed with the probability

P(x) o (4.78) "-xtL-tx for a fixed /. From a computational point of view updating the use of pseudofermion variables is fast since it only involves a matrix multiplication. If one wants to evolve the system using the Molecular Dynamics method, then the Hamiltonian and equations of motion are given by:

H(ó,tþ,'þ) L*' *stdl + lxlL,,ilól-'x¡, 0ö¿ a, Ti 0o¿ ASþ] (4.7e) d, 0ó, -D,tff,, where the auxiliary field 7 is given by

rt : L-rx. (4.80)

Thus in the molecular dynamics evolution, each time the scalar field are updated we need to modify matrix -L and consequently evaluate L-r which appears in Eq. (4.79). Updating .L is very easy and fast since -t is sparse. In this case, in two dimensions, only 5 of 13 elements of each row of -L depends on the scalar field. On the other hand calculation of .L-1 is time consuming since it involves matrix inversion and unlike L, L-t is far from being sparse. Now we are in a position to discuss the application of (HMC) to simulation of dynamical fermions on the lattice. The implementation of HMC algorithm is as follows:

r02 1. Choose a random scalar field configuration.

2. Generate z'from a Gaussian distribution.

3. Choose ( to be a field of Gaussian noise, Eq. (a.13).

4. Generate the pseudofermion fields ¡ according to Eq. (4.77)

5. Use Eq. (a.79) to evolve the configuration through the molecular dynamic chain.

6. Accept the new configuration with a probability

Pl{ó?),tr(r)y(r)l ló?'),n(r'),x?)}:/!\ ,, min {t,"-ffi#,")ffi\, (4.81) - t ) where Hamiltonian is given by Eq. (4.79).

7. If the configuration {þ(r'),n(r'),X0')} is not accepted keep the old configuration as an initial configuration and start from step 1. Otherwise use [/(r')] to generate a nev/ configuration starting from step 1.

This algorithm generates field configuration which eventually are weighted by e-s¡lö,'þ,'þ)-sþl and the proof is similar to the scalar case. This algorithm like the scalar Hybrid MC algorithm is exact. On the other hand, the fermion matrix has to be squared, which implies that there are two flavors of Wil- son fermions. In order to reduce the number of flavors to one, a factor proportional to number of flavors can be put in front of pseudofermionic term when other fields are being

updated [91].

4.4.2 An efficient method for simulation of Wilson dynamical fermions

Most of Monte Carlo lattice calculations are so large that it is impractical to use ordinary computers. However, the gigantic capacities of present supercomputers, both with respect to computing power and to data storage, offer a possibility to tackle lattice Monte Carlo problems. The efficient use of these machines often requires the design of new numerical algorithms or the redesign of existing algorithms and data organisation in order to meet the possibilities of the new architectures. Suppercomputing is fast becoming the most frequently used technique to explore new questions in quantum field theory. In just the last few years, it has produced results that were inconceivable a decade ago.

103 In general there are two type of supercomputers, namely, vector and parallel machines. 'We Here, we are only concerned about writing efficient codes for vector machine. are not going to discuss the hardware or software design of vector machines since they are not essential for our purposes. Instead, we discuss the rules concerning writing codes which minimise the computational time.

Vectorisation of a Fortrtan Code

A vector machine has two distinct processors: a conventional scalar processoÌ and a vector processor. The vector processor is for fast numerical calculations that can be uectorised. A uector operation acts on a set of elements of an array, the result of which is independent of the ordering of the element operations. There are number of points to make:

¡ A vector operation implies performing the same operation on all the array elements involved.

o The number of constituent operations in a vector operation (the uector length) is determined before vector operation begins.

o Every vector operation has an associated integer index set to label the constituent operations.

On a vector machine the whole vector operations correspond very closely to machine uector instructions that can be executed efficiently. A vectorising compiler can be relied on to generate efficient vector instructions for most vectorisable code segments. However, in many real application it is not always clear from the code which loops are vectorisable and which are not. In theses case programmer can use appropriate vectorising commands to clear any ambiguity, as we will show. The scalar processor performs instruction fetching and decoding and calculations that can not be segmented. The speed of the scalar processor is slower than that of the vector processor. If a computer has two CPU's with different speeds, the overall performance is dominated by the speed of the slower CPU. Thus, for a fast performance one needs to make use of the vector processor as much as possible. Here, we described the basic rules that govern whether or not the vector machine is able to produce instructions that utilises the vector unit of machine. Some of these rules are machine dependent and here, we mention rule and vectorisation commands compatible with the Fujitsu VPP300. What can be uectorised? The basic unit that can be vectorised is the do loop. Only

104 innermost loops can be vectorised, and there must not be any passing of control outside the loop. In general it can not vectorise loops containing:

o Computed or assigned go to statement.

. pause, return and stop.

o Input and output statements.

¡ Call.

o Loops with backward branches.

In addition to the above, if any statement branches into the loop outside it, the loop can not be vectorised. Even though branching backward inhibits vectorisation, the forward branches do not prevent vectorisation. The compiler can not vectorise the following data types:

o two-byte and eight-bite integer.

o quadruple precision, real or complex.

r one-byte logical.

¡ character.

Vectorisation will be performed on the following intrinsic and operators:

\/, log, €xP, sin, cosr î*'Fy.

Recurrences. A recurrence (vector dependence) is an expression (within a loop) that requires a value calculated in a previous iteration of the loop in order to be evaluated. Most recurrences can not be vectorised because vectorisation causes the calculations for successive iterations of a loop to overlap; values from previous iterations are unavailable. For example:

b(1) : a(1) do i=1, N b(i) =b(i-1) +a(i) enddo (4.82)

105 A loop is not vectorised if the compiler detects either a recurrence or even the possibility of a recurrence. If there is no recurrence in a loop one there are commands, which are machine dependent, to force the compiler to vectorise the loop. For VPP300 this can be done as IOCLVECTOR doi:1,N b(i)-a(i) +3 enddo (4.83)

It may seem surprising, but there are some situations in which a loop will execute faster in scalar mode than in vector mode. The reason for this is that there are certain time delays in preparing a loop for vector execution. Normally the time delay is small when compared to the total time of execution of the loop. However, when only a small amount of work is actually performed in the loop, then such delays are not insignificant, The most common example arises when the iteration count of a loop is low (less than about 8 for the VPP300). Other examples include some partially vectorised loops that generate extra memory references, if tests that are always false, and loops that generate certain memory access overheads. Reducing load and store trffic. Consider the two code segments:

do i:1, N c(i):a(i) *d(i) +b(i) do i:1, N enddo and c(i) : a(i)* d(i) +b(i) do i:1, N e(i) - a(i) + c(i) + d,(i) e(i):a(i)*c(i)+d,(i) enddo enddo (4.84)

Although they are equivalent but the loop on the right can be considerably faster. In both loops there are 4 arithmetic operations, two addition and two multiplication. However, in l.h.s loops there are six loads and 2 stores while in the r.h.s loop the are three loads (a,b and d) and two stores. Then for the loop in the l.h.s can be up to 33% reduction in execution time. The are also for short vectors there can be an extra time saving in not paying the loop startup cost for the second loop.

106 Unrolling loops. One of the standard methods of optimising nested loops is loop unrolling. The method involves reducing, by some factor, the number of iterations of one of the outer loop by explicitly including repeated code for that loop index. Loop unrolling also reduces the percentage of time spent in the loop overhead. As an example consider the following:

.do j:1, 100 do j:1, 100,2 do i:1, 10 do i: 1, 100 "(i,i): a(i,i) + Uçt,¡¡ "(i,i): a(i,i) + t(¿,i) enddo c(i,i + r) : o(i,i + t¡ + b(i,, + 1) enddo enddo enddo

(4.85)

Here, the outer loop has been unrolled twice (an unrolling depth of 2 which is indicated by stride two for the j loop. The unrolling can be done either by hand (like above example) or using vectorisation directive unroll (n) which can be applied to a nested loop. If appearing outside the outer loop, n dominates the depth of unrolling of that loop. If appearing immediately before an inner loop, rz should betFU LL' indicating that the loop should be completely unrolled. The unroll directive can be applied to any loop of a loop nest, not just the innermost loop. However, the loop must have no data dependencies across its iterations. General tuning princi.ples. The principle aim in tuning a program for execution is to minimise the total execution time for that programme. This usually amounts to ensuring that a significant portion of the time-consuming code executes in the vector processor. A few basic rules for writing a code are:

o Use do loops with as large as iteration count as possible.

¡ Avoid nested loops as much as possible.

¡ Avoid complicated if and go to logic inside loops

¡ Avoid lots of short and indeterminate length loops within a body.

107 The Model

We consider the linear ø model on a in 2-dimensions lattice, whose dynamics is governed by Eq. (4.64). Lattice labeiling. The first step is to choose a proper lattice labeling scheme. There are number of occasions in Hybrid MC routine that one needs to take some average over the entire lattice or update the entire lattice, i.e calculating the action or evolving the field according to Molecular dynamics. If one chooses a 2 dimensional array' labeling the time and space, then in order to perform an operation over all lattice sites, one is forced to use large multiple nested loops, containing the same number of iterations. Since only the inner most loop is vectorised the computational time is increased dramatically. One can use a helical labeling scheme in which the position of lattice sites are stored in a vector rather than a matrix, to make use of full vector processor units. The labeling scheme is shown in Fig. 4.1 for a lattice with 4 sites in each directions. In addition to that one needs to construct one dimensional arrays which, for each site, indicate the position of the neighbor sites. The details will be presented later. o o o o l3 l6 l5 t4

o o o o 9 l0 ll t2

o o o o 5 8 7 6

o o o o 4 I 2 J

Figure 4.1: An example of helical labeling of a 4 x 4 lattice.

The Wilson fermion matria. In our program we subdivide the fermion matrix W by their Dirac indices a, B as it is shown in Fig. 4.2. Note that there are three distinctive blocks. From Eq. (4.66) one can see that there are only 5 non-zero elements in each row of these matrices. Also note that, in each cycle of the Hybrid MC only the diagonal, scalar field dependent, elements of these matrices change. Thus, we divided each matrix into two part. One deals with the constant elements, being executed only once, and the other one contains the diagonal elements that depend on the scalar field, The non-zero

108 elements of matrices are stored in one dimensional arrays, e.g. ml1and their positions on the lattice are stored in another one dimensional arrays, e.g. iml1. Such an arrangement improves the vectorisation process. We also divide vectors ¡ and r¡ in Eq. (4.S0) into two block-vectors for practical reasons. In addition to above one need to construct matrice

V/I v/"

w" W

Figure 4.2: The block division of Wilson fermionic matrice I,7.

L : WrW in a similar fashion. As before we divide the matrix .[ into four block of matrices. Each of these of matrices has 13 non-zero elements in each row which does not depend on the size of the matrix. We have called these matrices, mtmLl,mtmI2,mtm2l and mtm22. These matrices, also, can be separated into two parts, one part is not being updated and the other depends on the the scalar field. For these matrices apart from the diagonal elements there are 4 elements in each row that depends on the scalar field. In the hybrid MC one needs to find the inverse of tr and in our program we have used the Preconditioned Conjugate Gradient Method (PCGM). The (PCGM) in the code is adopted to our representation of matrices [g9].

Some comments on the program subroutines.

Here we discuss the subroutines in our code. We address the purpose of each routine and and the optimisation considerations and the statistics concerning the routine performance. The input parameters. nsteps: Total number of iterations. itherm: The thermalisation iterations. nblock: The size of the blocks (used to find the autocorrelation length). icode: This parameter determines the type of the initial configuration. For code:l, the initial configuration is set to be random (hot start). If icode:2, then the initial configuration is read from file phijnp. Finally if icode=3 then the value of field on each site is

109 set to a constant, xinitial (cold start). idec: Number of decorrelation configurations to be skipped between two sampled configurations. xr: ar:ñ2, with ñ is the dimensionless boson bare mass. eps: The step size e¿. xl: The dimensionless scalar bare coupling Î. xj: The scalar source J. xka: The Hopping parameter rc. g: The scalar to fermion coupling g. pconv: The convergence parameter for the Conjugate Gradient routine. ns: The number of sites in spatial direction. nt: The number of sites in temporal direction. nn:nsxnt. nsamp: The maximum number of blocks.

o Set-parameter. In this routine the parameters and constants of the programme are defined and initialised.

o Label-sites. Here we label the lattice sites according to the helical scheme that is shown in Fig. 4.1.

o Initialise. Here we initialise the scalar fields phi, psedofermions (chilr chi2), auxiliary fields eta1, eta2 and the conjugate momentum xpi. Note that both auxiliary field and pseudofermions vectors with the length 2 x nn are divided into two nn dimensional vectors.

o Label-site. Here we construct vectors ntI(nn),nt\(nn),nsl(nn),ns}(nn). With the help of. ntl,nt},nsl and nsO we can identify the neighbors of a specific lattice site which are located one step forward in time, one step backward in time, one step forward in space and one step backward in space, respectively. Now, instead of working with a matrix to label the sites we work with vectors. We need to impose antiperiodic boundary condition over the fermion fields located on the border of the lattice in the temporal direction. This is done by creating vectors ntlap, ntOap(nn) which characterise the neighbors of a site in this direction for a fermion field

o Hamilton. In this subroutine the Hamiltonian is evaluated. In the Program we evaluate the action before and after a configuration is updated to be used in the

110 Metropolis test. Note that due to our labeling this routine contains only one, fully vectorised loops.

o MD. Here we perform a stochastic evolution step modifying the boson fields phi, the bosonic conjugate momenta xpi and as a consequence the auxiliary fields etal,eta2.

o Metropolis. Here we accept or reject the configuration generated in MD, using the usual Metropolis prescriptions. The only diference being that instead of a local Monte Carlo updating u/e use a global one in a sense that the whole configuration is accepted or not. In ihis subroutine we also refresh the pseudofermionic fields 'We chil,chi2. also generate the new auxiliary fields etal,eta2 and refresh the momenta xpi. r Defmll, Defm12, Defrn22. He we construct the Wilson fermionic matrix by subdividing it to 4 block matrices in terms of their Dirac's indices. For the Wilson fermion matrix the off-diagonal matrices are identical (see Fig. 4.2). Consider the block matrix W{y We construct two vectors all(5 xnn), iam (5 xnn). The vector amll contains the values of non-zero elements of very sparse matfix W{r. The elements of Wn depending on scalar field / is set to zero. In the vector iam we store the column of the matrix W¡ to which each element of aml1 belong. In the

other words, we have separated the static part of the I,1211 and the part which is updated and consequently have saved some CPU time.

. xm' Here we modify the elements of amll, am12, arn22 that depend on scalar field. This routine, unlinke the previous routine, which is called only once, must be called whenever we generate a ne\ry configuration of the pseudofermionic field.

¡ Defmtm. Here we construct the block matrices associated with the matrix MT M (see Fig. 4.3. As an example, mtml1 (13 x nn) contains the non-zero elements of the sparse matrix (Mr M)r¡. The vector mtm11(13 x nn) contains the column of matrix (Mr M)tt. The scalar dependent is set to zero. This routine is only called once. o xmtml, xmtm2, xmtm3. In this routine we include the scalar field dependent elements of Wilson fermion block matrices. o Sample. Here we start sampling after a number of thermalisation iterations itherm and a number of decorrelation iterations idrop. lVe calculate (/") for n : Lr2,3,4

111 using routine meanfis, (/") for n : I,2,3,4 using routine moment and the correlation functions using correlations. We also divide the measurements into blocks which can be used as tool to find the autocorrelation length for number of quantities. The correlation length is also calculated.

ffi¡wþ,, 'nfq¡uþ9,

v4w¡wþ,, ufJv¿4w,,

nnrnn nn {. nn

Figure 4.3: The block devision of.WrW.

o Stat. Here we process the measurements and find errors in calculated averages.

. cg. In this routine we find the solution of the linear system AX : B (where for our problem we have A: W¿,jWj,ì,X : ry and B : X). A version of the Preconditioned Conjugate Gradient is implemented. The implementation details are given as comments in the code.

multia. Here we calculate the product of a block matrix by the proper block vector of auxiliary field q.

4.5 Conclusions

In this chapter we started with a basic introduction to stochastic processors and Monte Carlo methods. We reviewed early Monte Carlo routines such as the Metropolis and Langevin algorithms. Since most of our calculations are done using the Hybrid Monte Carlo (HMC) we discussed the Molecular dynamics method and showed how this method can be combined with the Langevin algorithm and a Metropolis testing to arrive at an

LLz exact algorithm, Hybrid MC (for a scalar field). In this method there is no systematic error due to the finite step size and a trajectory generated by HMC travels faster through the phase space. We showed that constraints can be imposed on field configurations within the hybrid Monte Carlo updating rules. In this case the reversibility condition is not exact. However, we showed that for a simple the results for some observables did not appear to be affected. We showed that HMC method can be extended to fermion simulations. We chose the model to be a Yakuwa type model and for such a model we discussed the HMC for fermions in details. The method is rather slow since in each sweep there is need for number of matrix inversions. However the flexibility and reliability of this algorithm have made it to be the most popular method for the simulation of dynamical fermions. Finally, we presented a very efficient code (in Fortran) written for vector machines. In addition to make use of the directives provided by the machine to optimise the code we constructed the program in a such a way that we obtained a very high vectorisation rate.

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r20 Appendix A

Grassmann algebra

The elements [rr...r?N are the generators of a Grassmann algebra, if they totally anticommute among each other

{rt.,rt¡} : qiqi } njn¿:0, i,i :1,... ,N. (4.1)

It follows T? :0. A general element of a Grassmann algebra is defined as a power series in 4's but since q? :0, this power series has only a finite series in 7's:

Íþt): /. + Ð f;n;+Dl,¡rt,rÌ¡ +... + h2-.¡vTtTz...T¡,t (4.2) i ;+i

Now we state the Grassmann rules for calculating the integrals of the form

I TIdn'rø), " i=L Since a given Grassmann variable can appear at most to the first power in l, the following rules are sufficient to calculate an arbitrary integral :

I 0; 0 I dqrnr:7. In the calculation of multiple integrals one must further take into account that the integration measures {d4¿} also anticommute among themselves, as well as, all of the 4¿'s

{dqr,dn¡} : {dn¡,dn;} :0, for all i, j

Using these integration rules, let us calculate a Gaussian integral:

ñM:itti IlMl : Dl,=,, .. (A.3) I fir,*,"- r2t We write the integrand in the form

Dlr=, iM;ini M:,ini : fi ,-oD ¡=r . "- i=L Since ,t? : 0, only the first two terms in expansion of the exponential will contribute. Hence ¿-D;,¡î;M;,irj :(1-TrMr,iJlit)...(1-TnMn,inTin) (4.4) where a summation over repeated indices i,¡(I : 1,...,12) is understood. Because of the Grassmann integration rules, the integrand of Eq. (4.3) must involve the product of all Grassmann variables. Therefore we need to consider I{(T,7) - t T¿,T1T¿,î2...T¿.TnMth...Mnín (4.5) i1 r...rit The summation clearly includes only those terms for which all the indices ù,,...,inate different. Now the product of Grassmann variables in Eq. (4.5) is antisymmetric under exchange of any indicies i¡ and i¡,. Thus we can write Eq. (4.5) as:

I{(q,T) : TtTflzT-z...TnT-n. f. e;,...;" Mtü... Mnin N\ "..t'î : [detM] Tûfl2T2...\nTn, (4.6) where eit...in is the e tensor in n-dimension. Then replacing the exponential in Eq. (4.3) by the above expression we obtain ln r l IlMl lf[ / d4¿dn¿n¿r1¿l detM : d,etM. (4.7) - LIJJ I Thus we have proved that i;M:'ini' lW)la'1"-Dl'=' (A'8) Consider the following integral

t ¡ M ; n tD € ¡ *ã'r ¡ z : e- D''t ¡ ¡ ¡(l¡ )' (A.e) l€,8) | Ín'rlloql The sources, { and f u,re also anticommuting Grassmann elements. To evaluate the integral we write the integral as

-n" r't : ni] i u Ê z : e- D''i t :' ¡E | | i, (A.10) le,E) ll Wtd'ql "D where

Tí rt¿ -DMlo'€n, k -tT; T, -DE*M*'' (4.11) h

L22 Using the fact that the integration measure is invariant under above transformation, and Eq. (4.10), one concludes that

zrc,¡jl: detM ¿D,,¡Ewr¡'e. (A.12)

Comparing with the boson case lrye can see that the generating functionals for fermion is proportional to detM where for the boson generating fuctional it is proportional to detM-t/2. Differentiation of Grassmann uariables. Consider a Grassmann function, f þt). Then we define the left derivative, ôf þt)ldrlu as follows : Bring the variable 7; all the way to the left, using the anticommutation relations in Eq. (4.1), and apply the rule: Lqn: t' T¿ We can also bring the variable 7; all the way to the right and hence define the right derivative as ,,L: t. ' 0q¿ Note that such a definition for the differentiation of Grassmann variable is is actually equivalent to the Grassmann integration rules. That is Iorn¡(r):firrrl.

This is a rather remarkable peculiarity of Grassmann algebra.

723 t24 Appendix B

Dimensional engineering

In a quantum field theory after settin1h: c: I, the only quantity remaining with a dimension is the length (or equivalently inverse mass). In addition mass and momentum are no longer distinguished. Thus it is possible to assign a length -L or momenta ¿\ dimension to each variable. Here by [A] we mean the dimension of A. We know that the action is dimensionless or equivalently [4] : Äd. The dimension of / can be obtained from the kinetic part of Lagrangian, i.e.,

lö): ¡Yid-''

We also see that ----- l*'ó'): ^A,d l*'l: L2(r-rd)+d: /\2. (8.1) lÀö'l: [1¡åa-t : Ad [] : ¡a-åa+'. (8.2) -, The dimensions for correlation functions are:

[g(")(xr,. ..,h)]: [ö]" - ¡n(d/z-t). (8.3)

Then the Fourier transform for the correlation functions has a dimension of

[G"(k)] - ¡-na¡çn(r)l : ¡-n(d'lz-r). (8.4)

Integrating over the overall momentum conserving ó functions, we have

[G"(k)] - ¡d-n(d/z-t). (8.5)

The dimensions for vertex functions are

[f"(r)] - [G"(r)][ft-"][C{z)1 r)]-" - ¡n(d'/2tt), (8.6) r25 where O is the volume. Then the dimension for the Fourier transform of vertex functions, after the momentum conserving ó is removed, is

lt(")(k)l - ¡n+d-t'. (B.T)

For the fermion case one can find that:

¡,il: Wl- ¡1G*. (B.B)

Then for the correlation functions one can derive

lGlì(þ'V',. . .,rþ^Ú*)l : ¡n(d-l), : (B.e) lãf,lWr,úkr,...,rþ0,Úu)l ^". For theories with fermion fields coupling to the fermi fields with a coupling constant g one has

lg6'þÓl: [g] : ¡-r/2dt2, ^d -' : (B.10) fc@,^)(rr,. . .¡tn¡uLt...,aàl l\*+^a-^-", where n, rn denote the scalar and fermion fields respectively. Then for the composite vertex function one has

¡(r,,m) @,y) : lG@)@¿)l[o]-"¡ç{z)(*¡,*r)l-"pf}@jlq-^pf)il(*¡,*n)l--, (8.11)

- ¡t"(f+t¡{m(3d-1) : Â1. (8.12) After taking the Fourier transform and removing the overall momentum á-function one obtains ¡(',,"')(k, p) _ ¡t+za-d@!n) (8.19)

126 Appendix C

The strong scaling in the linear o rnodel

Lets consider the linear ø model in two dimensions:

d*r$*f,{}q$ zlJ,Q,n) : e-slÓ"þ"þl+I Ilaølld,rþltd,øl , where the Euclidean action

sló, rþ,61 : sulóJ + s ¡lrþ,Úl

is given by

s¡[d] : *llta,Ð' , | +T*'ó' * Ir^f s¡î,þ,úl: I dxg1.o * M¡ + gúö,þ,þ, Snlórþ,úl: g I or -f rf . The variables 7 and r¡ arc Grassmann sources and the Dirac matrices are defined as

[trt,] - 26p,.

The composite correlation functions can be defined as

J=O,4=j=Q (c.1) Note that in the above definition we have only allowed the even fermion vertex functions generated by Z, Odd functions always vanish when the fermion sources are set to zero.

L27 The Generating functional for connected correlation functi onW[J,r¡l] car. be obtained

AS WlJ,T,7l - ln Z[J,q,T]. (C.2) Then the generating functional for the vertex function l[o, v, rr-] can be obtained through a Legendre transform rlo, v-iúl - wlJ,n1l: I tç*¡oç*)dx * | ar¡,¡1*¡V(") + ?'V(')1, (c.3) with o(') 6WlJ,T,fil 6J v(') 6WlJ,q,\l 6q 6WlJ,q,fll v(r) (c.4) 6rì

The proper vertex functions are defined as 6*'n o, vvl ¡('n'n)["r, . . . ¡ïrniut¡., . runl : 6Õ(11) . . . óiú(y') (y')...óv(y")6V(y") óo(ø-) =Ort1=fi=O (c.5) The Fourier representation of the composite vertex function is

f(^'n)|t r,...,k*iQt,...,gnl= De-;t*'tt+v'oJ¡("'-)¡sl¡...: r,niut¡...,un!. (c.6) ¡fr;¡ ' k,q

The field renormalisation factor for fermions, Z,¡,, can be defined together with renormalised fermion mass, M, by the low-momentum behavior of the inverse fermion propagator G+(k) : zl' (M, + ik'1 * o(kt))' RG and the scaling behaviour Consider the renormalisation condition for the vertex in the linear ø model,

r[-'")1k,; e¿iÀ,m2,,9,M,) : zi(*',À,M,)zi(M,,9)f(-'")(/ú¿iQ¿i ¡ffi2,À,,9,M,), (c.7) where we have chosen the renormalised fermion mass M, as the scale, setting rn2 and À to be constant. Since the bare theory is independent of the scale one has

128 l+ t-'r*'t zr" ffi - nz=* zløo (L . m) ,ù + z=# zt" - À,m2 , M,) 0. (c.8) (#!* h)]tf çkn;v¿i e, = Multiplying the above by Z^ Z$M, and after some straightforward algebra we find

npçu,¡q1u,,g) : 0. l#rr*,) -,T(M,,s) - - #rl fÍ-'') (c.e) Here we have defined a þ =F@,) M, 9, AM, l^, ,l ô n q(M,) M, lnZ(M,), = ÔM, ^2r\ n = rt(M,) lnZ,¡,(M,,,g) (c.10) ,À As in the scalar field case, one can integrate the obove equations to obtain ds' = lnM, I" þ(g') ôln Z : lnM, ôl\z*n lr¡ : rnM,. (c.11) Similar to the scalar field case which was discussed in Chapter 1, if there exist a IR fixed point, namelyg* then at the vicinityof this fixed point, as M, + 0, one has lg-g.l-Ml o=ffr0, lnZ-qlnM, T=ry(g.), LnZ,¡, -\lnM, î:Tþ.). (c.12)

On the other hand, with a dimensional argument (Appendix B) one can see that y(*,")Qt¿;e;jÀ,m2,9,M,) - M1+2d-d(^+ùZ+Zõ"lf,")çkr¡M,lQ;lM,i,m2,À,g,I), (c.13) with 7 - n(d/2 + 1) + m(Sd - 1). Then close to the IR fixed point one has ¡(^'")(lc;;e¿iÀ,m2,g,M,) - 1¡41*2d-d(^*")-\n-mî¡@'n)(lrrltW,;lrf M,;rffi2,À,g*rl), (c.14) giving the strong scaling.

r29 130 Appendix D

Super renomalisable models

Consider lhe Sa theory. By an easy dimensional analysis one can show that a graph of nth order with E external legs and with a cutoff, Á., behaves asymptotically as Â6 with 6:n(d,-4)+(d+E-luo, and d being the dimension. If the primitive divergence of a vertex function increases with the order of the perturbation theory, there is no way the in which the strong r\. dependence can be absorbed in the a finite number of constants. Thus )/a theory is non-renorrnalisableif d > 4 and for cl:4is renormalisable. Finallyfor d ( 4 the degree of primitive divergences decreases as we go to higher orders and the theory is called s up er- ren o rm alis abl e.

Figure D.l

For example, consider þa in 2 dimensions. The only superficially divergent is shown in Fig D.l . One can add a counter term of form ø(Â)l/2 to render the diagram. Actually divergence in the diagram, which corresponds to a self-contraction of the vertex Sa, can be eliminated by replacing the vertex ön by the a normal ordered vertex :/a:

z Sa z (a) : ó4 - 6ó'(ö,(")) + 3((d,)),

131 (ó(r)') is calculated in a free field theory with mass ¡.r:

(ö(")'l

The quantity : þa:1, is such that

1: ó4 :lrl 0

(,ón ,1,ö{rr) ó@r)) 0 (D.1) in which the averages are calculated with a free action with mass of ¡r

s,(ó):'u Id0 [(^(d)), + t,ó,f Then the averages using another mass, say þ', will be different from the averages with mass p but the differences are finite.

t32 Appendix E

CEPII equations

The differential equations relating the constraint effective potential and the classical po-

tential are :

For the first derivative we have

(E.1)

which lor Àþa theory becomes

ry-*'6+f<ø'1. (E.2)

The second derivative is given by

: - *', (E.3) w (w), Ë (ww)u (w)J', which for ),þa theory becomes W : *,+)wl+¡{dl#,r',, *{øt\+*^6,] -*'þ^wø *(tó"û + {to"æt.(øæl (E.4) The third derivative gives æU(Ð : n,¿û[) @ *-gry + zNd (W) - N2¿ W)" * (ffi): (E.5)

133 which lot ÀSa theory becomes

: nlto"l- r,,,|tø, ó,) - wdîþ,õl + ryd,, lOf¡') terms] + lo@t) terms] + [t"r*' that vanish ar$: o] + ... (8.6)

Finally for the fourth-derivative we obtain w: Nd (w)' - +¡v,,w w)' * (d^vÁ!¡),*, (8 7) which for À/a theory gives

ry : î- ¡n'- Nd^ñ2(ó'l - Nd^ñ2þ21 dó-" terms] that vanish at 0] . (8.8) + [o1.lz¡ terms] + lo@^) + [t".*t þ:

t34 Appendix F

Renormalisation of the kink rnass

Starting from Eq. (3.26) we find

M"ot:'#*, irfo * m * i+lrirz + z¡* - &'. + z*"¡i] + o(^) (F.1)

In the continuum limit both two terms in the bracket, become quadratically divergent integrals. In a finite box.L, qn and lcn are related by Eq. (3.26) and the fact that înL: 2nzr. That is Qnrn[' * 6(q") : lenl :2ntr. (F.2) T Then the term in the brackets in Eq. (F.1) becomes

: (F.s) [,t - lr'*r*']' - &'^*2m2)i -rYl&',+2m2¡-ï + o(h)

In the thermodynamic limit, where Dr * Il r I dle , we have: M,., ='+P.irl-fu- * l:*ffi+o(^) (F4) Then the phase shift ó(fr) can be written as:

ó(È): -rlml *r1^¡ (F.b)

Ignoring the O(IlL) we can evaluate the integral in Eq. (F.4) integrating by parts. That ls:

oo * 2m2 l2m2 + o(^) Idkffi:fi¡,r*r -oo *urr,

135 Insertion the expression for phase shift in Eq. (F.5) in above equation gives

6m r l^trqp-@+GøJdk[a*&:3m

The integral term in Eq. (F.6) is logarithmically divergent and renormalisation is needed to render the mass finite. À$a in two dimension is a super renormalisable theory. It has only one superficially divergent diagram which is shown in Fig. C.l Adding a coumter term of the form:

C(m/t') 2 ^, to the Lagrangian density renders the theory finite. Parameter rnÄ is the cutoff. Adding this counter term alters both vacuum and kink spectrum. The parameter C(mÌt) can be calculated from Feynman diagram shown in Fig C.1 which gives

3À ¡Ä c(mn):# (F.7) l^^^anrfu -t4r J-¡^ Then the leading contribution of counter terms to the kink mass is

LEÍ",!)o - LE[::) : c(mtr) l***6-anh'zff) = úcç*¡¡. (F.8)

The insertion of Eq. (F.8) into Eq. (F.7) gives the renormalised soliton mass

M"ot : E*;nt l L4n;nx - (Euo" I LEoo") : 2rt*". l'ß s l BÀ ** lñ,-;æl 3'f2rn k2 +2 1 4r J-¡[n Arl + o(À), (F.e) L ,Æ(kn+4) 'ßæ where Euo" and Er¿n* are defined in Eq. (3.20) and Eq (3.22), respectively. Notethat at large momenta both two terms in the last bracket behave like 1/,b and thus one can cancel the logarithmic divergences. Take the first integral in the above bracket. By a change of variable k : sinhu one can write

: : out (F'10) ¡ orffi, I au.o,nu;ffi¡ I - çfu;'

136 Note that the divergent part of the integral ([ d") is separated out. It is easy to show that the second integral term in Eq. (F.9) is ([ d,u). Thus the divergent integrals cancel out. Then the remaining integral, with the infinite cutoff is convergent and can be evaluated. Then one obtains: g M.^,:24_.",,-' , ,L [-g -.-¿u. 3r + -(ãV i - æ) + o(^). (F.11)

137 138 Appendix G

Path independence of dislocated partition function

Consider the following action in 2-dimensions,

s*,,: -DL^*,¡zónón+u+ *ÇloZ* ]lr^. (c.1) f,¡þ T(zn corresponding to dislocating path, C^,, shown in Fig. G.l.a . Then for the partition function one has: z(c*,,) : llaó1"-t-'"t'l (G.2)

K (a) a (b) K

I I vt

Figure G.1:

139 To see see the path dependence, consider changing the patition function by inverting the the signs of the field located at the positions indicated by circles in Fig. G.1.b. Note that the links appear in the form of Ln¡¡,¡zónón+u. The effect of changing the sign of these summation variables is to invert sign of all links connecting the region outlined with the remainder of the lattice. Under such transformation

Z(C,,,) -+ Z(C'^,,) where Z(C',,,) does not have minus signs on the path Ca,s but now has them on the path Cl,, shown in Fig. G.1.c. Since the change of a summation variable does not change the result of the sum we have Z(C*,,): Z(C'*,,) (G.3) and then the correlation functions has equall values for all possible paths connecting the point rc and u. The generalisation to even correlation functions is straightforward.

140 Appendix H

An efficient vectorised Fortran code for simulation of dynamical \Milson fermions

Here we present a list of our vector Fortran code which simulates Wilson dynamical fermions in two dimensions. The list is generated by a listing software on Fujitsu VP 3000 which in addition to the code and line numbers it provides some information about the vectorisation status of the loops. The statements within a loop which are labeled by v are vectorised whereas the ones labeled with s are scalar. There are also loop statements labeled by m meaning that the loop is a mixed scalar and vectorised. Most of the effort is put into vectorising the loops which are executed numerous time during a run,

t4l 00001 00002 c------00003 C PROGRAM TO STUDY, BY MEÀNS OF THE ¡ÍYBRID MONTE CARLO METHOD, 00004 C THE LINEÀR SIGMÀ MODEL IN 2 EUCLIDEAN DIMENSIONS ON A LATTTCE. 0000 5 c------00006 00007 implicit, real (a-h,o-z) 00008 paramet,er (ls=6, lt=6, 11=ls*lE, Iex=11+4*Is,nsamp=200) 00 009 000L0 conìmon /]-ab!/ eps,xI ,pi 0 00Ll_ common / ]-ab? /ns,nt, nn,nsteps, nevo1, npasso, iEerm, 00012 * nblock, ioption, idiv, idrop, insamp 000L3 common /Iab3/ pconv 00014 cornmon /Iabâ/ ct, c2, c3, c4, c5, c6 00015 cornmon /]-ab5/ iseedl, iseed2, iseed3 0001_6 common/ 1ab 6 / fí, fi2 , fi4 , fí2|', fit, fi4t, fi3 , fí6 , fi8 00017 , fi3t, fi6t, fi8ts 00018 common/labi /rns, rnt, rns2, rnb 00019 common/ 1ab8/henrgt, herrE, henrg, herr, henrg2 00020 common/ 1ab9 / zmoml-, zmott2 , zmom3 , zmom4 , zmom6, zmom8, zmomlE, zmom2t, 00021 zmom3t, zmom t, zmom6È, zmom8t 00022 common/ Iab10 /xj , xj e 0002 3 common/1ab11/f, xka 00024 common/ lab12 /nsum 0002 5 common/ Iabl3 /iceror 00026 00027 dimension xpi (1I), averaqet (11), average (11) 00028 dimension cor(0 :1t,-1),cort(0:1t-1),rowav(0 :1ts-1), 00029 * con(0:1t-1),xcon(0:It-1),rowavs(0:1t-L) 0003 0 *, )cnean (0 : 1E-1-),xerror (0 : tt-1) 0003r. dimension phi (Iex, 0 : 3 ), phisave (11) 00032 dimension smpl (nsamp, 0 : 1t,-1), smp2 (nsamp, 0 : lts-L) 00033 integer nt1(1I),ntO (11),ns1'(11),ns0 (11) 0003 4 integer nts1ap(11),nE0ap(11),ns1ap(11),ns0ap(11) 00035 inEeger iro(11), ico(11),ntdl (11),ntdO (11),nsdl (I1),nsd0 (11) 00036 dimension fisamp(nsamp,4) , fiersamp(nsamp, 4l ,zmer (nsamp,4) 00037 dimens ion coupl ing (nsamp ), hsamp (nsamp ), hersamp (nsamp ), pcor (nsamp ) 0003 I integer iicount 0003 9 00040 real ul (11) ,u2 (11) ,u3 (11) ,u4 (11) ,u5 (1I) 0004 1 rea]- zL(11),roL (11),dsl (11), dvl (11) 00042 rea! z2(11),ro2 (1I),ds2 (11), dv2 (11) 00043 real m1L (1I*5) ,m12 (I1*5) ,¡n22 (11*5) (5*11) (5*11) (5*1I) 00044 inEeger im11 , imL2 'ím22 00045 real mtrmll- (13i11) ,mt¡nl2 (13*11) ,mÈm2L (13*11) 00046 * ,mtm22 (13*1I) 00047 inEeger imEm11 (13*11) , imtmL2 (13*11) , imtm21 (13*11) 00048 * , imtm22 (13*I1) 00049 real eEal ( ll+1t+1s, -1:3 ) 00050 real eta2 ( I1+1t+1s, -L:3 ) 00051 real chiL (I1) , chi2 (11) 00052 integer mid(ls) 00053 00054 00055 open (unit,=15, file='Ekink. inp', sEaÈus=' old' ) 00056 read(15,*) 00 057 read(15,*) nsteps, itherm, nblock, icode, idec 00058 read(15,*) 00059 read(15, *) )snass, eps, xI, xiniEial, xj 00060 read(15,*) 0006L read(15,*) xka , g, pconv 00062 00063 close (unit=L5 ) 00064 00065 c 00066 OPEN (UNIT=I, FILE='phi-out' , ACCESS='SEQUENTIÀ!' , 00067 * BLANK='NULL') 00068 open (uniÈ=6, file='cor.out',access='SEQUENTIÀL', 00069 * blank='NULL') 00070 open (unit=3, file='phisave.out', access='SEQUENIfAL', 0007L * blank='NULL') 00072 open (unit=S , file='sample. out' , access=' SEQUEIirrIÀL' , 0007 3 * blank='NULL') 0 0074 0 0075 cal seElrarameler (míd, cor, ro$tav, rowavs, xconn' conn, corÈ 00076 * ,average,aweraget) 00077 00078 ns=1s 00079 nt=1E 00080 nn= lI 00081 nsum=1ex 00082 insamp=¡gamp 00083 00084 call label_sites (nt1,ntO,ns1,nsO,nÈ1ap,nÈOap, iro, ico 00085 *,ntdl-,ntdO,nsdl,nsd0,nsLap,nsOap) 00086 00087 c- We initialise the scalar fietd and the pseudofermion field 00088 00089 call initialíse (zL, 22, dsL, ds2,ro!,ro2, eEal, eta2,phi, chi1, chi2 00090 *,xpi,nt1,ntO,nsL,ns0,nt1ap,nt0ap, 0009L * ml1,m12,m22,mtm11,mtm12,mtm21,mtm22,n 00092 *, imll, ímL2,im22, imtmL1, imtml2,imtm2L, ímtm22, 000 93 * uL,u2,u3,u4,u5,icode,nsLaprns0ap) 0009 4 00095 00096 00097 call mulE (m11 , im11 ,mL2 , ímL2 ,m22 , ím22 , eEaL 00098 * ,eLa2,dv1 ,dv2,n) 00099 00100 C************************************************************ 0010L c HERE I¡¡E CAI,CULATE THE HÀMILTONIÀN. 0 0102 C************************************************************ 00103 00104 call hamilEon (chi1, chi2,phi,xpi,ntJ.,nEO,ns1,nsO,nt.1ap 00105 , nt0ap, hL ,h2, 0, etal , et.a2, nsLap, nsOap) 00106 hh=hl 0 0r.07 00108 c 00109 C********************************************************** 001 10 C HERE V',E BEGIN THE MOLECULAR DYNAMICS UPDATING. 0011L C********************************************************** 00L]-2 do npasso = l-,nsteps 00113 00114 call MD (zL, 22, dsl-, ds2, roL, ro2 0011 5 * , etal, eta2 , chiL, chi2, phi 001L 6 *,xpi,nt.1,ntO,ns1-,nsO,nt1ap,ntOap,ns1ap,nsOap,n, 00117 * mEmLL,mtm12,mtm21,mtm22, 0 0118 * imtmLL, imtm12, imtm2L, imtm22,mid) 001 19 C************************************************************** 00120 c AFTER THE MD EVOLUT]ON WE CATCULATE THE HÀMILTONIAN. OOL2L C************************************************************** 00]-22 call hamÍlton (chi1, chi2,phi,xpi,ntL,nbO,nsl,nsO,nt.Lap 00123 * ,nt.Oap,}rL,h2,nevo1,eta1 ,et,a2,ns1ap,ns0ap) 00L24 C************************************************************** 00L25 C WE APPLY THE METROPOIJIS TEST. 00126 c* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 00]-27 call Metropolis (phi, etaL,eta2, chi1, c}:.í2,h1,hh 00L28 * , ntL, ntO, ns1, ns0, mL1 ,m72 ,m22, imLL, J-m1-2 , ím22 , 00129 * * nt1-ap,nt0ap, xpí, zt, z2,rot, ro2, dsL,ds2,n, 0013 0 mtm11 , mtmL2 , mtm2 1 ,mLm22, imtm11 , imtml2 , imtm2 L 0013L *, imtm22, iacc, ns1ap, nsOap) 00132 c 00133 C***************************************************************** 00L34 C ÀFTER REFERESHING THE NOISE WE MUST CAJ,CUI,ÀTE 00135 C THE }IAMILTONTAN ÀGÀIN. 00 r-3 6 C****************************************************************** 0013 7 call hamilton (chi1, chÍ2,phi,xpi,nEl,ntO,ns1,ns0,nt,Lap 00138 * ,ntOap,hh,hveI2,0,etaL,e!a2,ns1ap,nsOap) 00139 if ( (npasso .ge itherm) .and.. (mod(npasso, idec) .eq.0) ) then 00140 iicount=iicount+1 00141 C*********************************************************************** 00L42 C HERE AT'TER THE THERMALISATION AND SKIPPING ENOUGH CONFIGURATIONS, 00L43 C TO HA\/E UNCORRELATED CONFIGURÀTIONS, WE CAI,CULATE PHYSICAÍ, 00144 c EXPECTÀTION VÀÍJUES. 00145 C*********************************************************************** 00146 calI sample (phi, cor, corE, average, averaget., xcon, 00147 * rowavs , iicounE, iscounE, fisamp, fiersamp, smpl , smpz , coupl ing 00148 *,hsamp,hersamp,h1,h2, zmer,pcor,ssx, ssy,ssxy,xs,ys, xys 00149 * , ccx, ccy, ccxy) 00150 endif 0015L 00152 enddo 00 153 00 154 C * * * * * * * * * ** * * * * * * * * * ** * * * * * * * * * * * * * * * * * * * * * * t * * * * * * * * 0015 5 c HERE WE SAVE THE LAST CONFIGURÀTTON 00156 C****************************************************** 00157 do 666, i=1, nn 00158 666 phisaveli¡ = phi(i,0) 00159 C* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 00L60 xx=floa! (iicount) 0 0161 xxx=xxl floats (nblock) 00 162 xs=xs/xx 0 01.63 ys=yslxx 00L64 xys=xys/xx 00165 ccx=ccxlxx 001_66 ccy=ccy/xx 00 167 ccxy-ccxy/xx 00L68 xys=xys- (ccxy**2 ) 00169 xs=xs- (ccx**2 ) 00170 ys=ys- (ccy**2) 00171 00L72 corlen= (ntl (2*pi) ) *sqrt ( (xys-xs-ys) / (xs+ys) ) 00173 00t7 4 xnac=f1oat (iacc) /float (nsteps) 0017 5 c*********************************************************** 00176 c HERE THE STATISTICAT À}TAI.YSIS IS DONE. 00I'17 c*********************************************************** 00L?8 call stat (smp1,smp2, iscount,)snean,xerror 00179 *,xx, fisamp, fiersamp,coupling,hsamp,hersamp,zmer,pcor) 00180 00L81 write ( 5 ,*)'DELTA = ',eps,'ACR = ',:(llac 00182 write (5 ,*) 'Lattíce Size : ',ns 0 0183 t¡trite ( 5 , *) 'No of iEeraÈion',nsEeps 001_84 write ( 5 , *) 'St.arE sampling from : ', itherm 00L85 write ( 5 ,*)'Block size:',nblock 00186 wriEe ( 5 ,*)'No of Configuration droped for sampling 1 dcro 001 87 wriEe ( 5, * 00188 * 00189 r¡¡rite (5, ) 00190 xx=fIoat ( iicounts) 0 019 r. do 23, j-=0, nE-1 00192 cort(i) = (cort(i))/(xx)/rns 0 0193 enddo 00194 00195 do j =1, ¡¡r 00196 averaget ( j ) = (averagec(j ) ) / (xx) 00197 enddo 00L98 00199 00200 do j-0, idiv 00201 do í=1,n8 *averagets *nt.+i) 00202 rowav(2*j) = averaget (i) (2*j +rowav (2*j ) 00203 enddo 00204 enddo 00205 00206 do j=9, idiv 00207 do i=1,nE * *averageL((2*j+2) *nÈ-i+l) 00208 rowav ( 2 l +1) = averageÈ (i) 00209 + rowav( 2 *i+1) 00210 enddo 0 021_L enddo 002L2 00213 do i=0, nE-1. 002L4 rowav ( i ) =rowav (i) / f loaE (nt) 0 0215 enddo 002L6 002L7 do i=0,nt-1 00218 con(i) = cor¡ (í) -rowav(i) 002L9 enddo 00220 write (4, *) averaget 0022L 00222 do 534, i=1,ns 00223 v/riÈe (5, *) averagets(i) 00224 enddo 00225 00226 00227 do k=l,nn 00228 write (3, * ) phisave (k) 00229 enddo 00230 0 02 31_ end

00232 00233 00234 *********************************************t************************ 00235 c ******************************t*****************************i********* 00236 c 00237 subroutine label-site (nt1,nEO,nsL,nsO,nÈ1ap, 00238 * nE0ap, iro, ico, ntdL, nEd0, nsdl, nsd0, nsLap, nsOap) 00239 00240 c IN THIS SUBROUTTNE WE DEFINE lHE NEIGHBORS OF A CERTATN LATTTCE 0024L c SITE. WE LABEL THE LATTICE ELEMENTS IN À VECTOR IN THE FOLLOWTNG 00242 c VüAY: *------<------l 00243 c I ----- >______* 00244 c * 00245 c ] ------>______* 00246 c WHERE ] ----->-----* MEANS TIIAT THE COI'NTING GOES FROM T¡IE LEFT TO 00247 c RIGHT. 002 48 c THE NOTATION USED FOR THE NETGHBORS TS: 00249 c E].(T) IDENTTFIES THE NEIGHBOR ON THE RTGHT SIDE OF A SITE T 00250 c EX].(I) IDENTIFIES THE NEIGHBOR ON THE LEFT SIDE OF A STTE I 00251 c EO(I) IDENTIFIES THE NEIGHBOR ABO\/E THE SITE I 00252 c EXO(I) IDENTTFIES THE NETGHBOR BEI,OW THE SITE I 00253 c 00254 0 0255 implicit real (a-h,o-z) 00256 00257 cornmon / Lab2 / n1, n0, nn,nst,eps, nevol, npasso, iEherm 00258 , nblock, ioption, idív, idrop, insamp 00259 integer nt1 (nn),ntO (nn),nsl (nn),nsO (nn),ns1ap(nn),ns0ap(nn) 002 60 integer isgn (nn) , ntLap (nn) , nL.Oap (nn) 002 6r. 00262 integer iro (nn), ico (nn),ntdL (nn),nÈdO (nn), 00263 nsdl (nn) , nsdO (nn) 00264 00265 00266 do i=1,nn 00267 iro(i) = (i-1) /nL+L 00268 if (mod((i-1-) /n\ ,2).eq.0) Ehen 00269 ico (i) = mod( (i-1) ,nl) +1 0027 0 else 0027I ico (i) = ( (i-1) /nL+Ll *nL-i+l- 00272 end if 00273 enddo 0027 4 00275 do i=L, n0-1 0027 6 ntdl(i) = i+l 00277 ntdo(i+1) = i 0027 I enddo 00279 00280 ntdl(n0) = 1 002 81- ntd0(1) = n0 00282 do i=1,n1-1 002 83 nsdl(i) = í+1 00284 nsd0(i+L) = i 00285 enddo 00286 nsdl (n1) = 1 00287 nsdO(L) = nL 00288 00289 do i=l-, nn 00290 if (mod(iro(i) ,21 .eq.l) then *nL+nsd1 0029L nsL (i) - (iro (i) -L) (ico (i) ) 00292 ns0(i) = (iro(i)-1)*nl+nsdO(ico(i) ) 00293 else 00294 ns1 (i) iro (i) *n1-nsdl (ico (í) ) +1 = *nj_-nsdO 00295 ns0 (i) = iro (i) (ico (i) ) + j- 00296 end if 00297 if (mod (ntd1 (iro (íl L2 ) . eq.1) then 00298 nc1 (i) = (ntdL (iro (i) ) -1) *nL+ico (i) 00299 else 00300 ntl (i) = nEdl (íro (i) ) *n1-ico (i) +L 0030L end if 00302 if (mod(ntdO (iro (íll ,2).e9.1) then 003 03 nEO(i) = (ntdO(iro(i) )-L)*n1+ico(i) 00304 else 0030s ntO (i) = nEdO (iro (i) ) *nl-ico(i) +1 00306 end if 003 07 enddo 003 08 003 09 00310 C * * * * * * * * * * ** * * * * * * * * * * * * * * * * ** * * * * *** * * * * * * * * * * * * * * * * * * * * * * * 0 031L C ANTI-PERIODIC BOI'NDARY CONDTTTONS IN THE TEMPORAL DIRECTION 003 12 C * * * * * * * * * ** * ** * * * * t* * * * * * * * * * * * * * * * * * * * ** * * * * * ** * * * * * * * * * * * * * 00313 do i.=n1+1, nn-n1 003 L4 ntLap(i) = nÈ,1 (i) 00315 nts0ap(i) = ntO(i) 003 16 enddo 003 17 do 65,k=L,nn 003L8 nslaP (k) =ns1 (k) 003 19 ns0ap 1¡ç)=ns0 (k) 00320 enddo 00321 00322 do i=l,nl 00323 ntLap(nn-nl+i) = nn+nl+i 00324 ntlap(i) = ntl(i) 00325 nt,Oap(nn-nl+i) = ntlap(nn-n1+i) 00326 nt'0ap(í) = nn+i 00327 enddo 00328 do !=0,n1-2,2 *nl *n1 ( 00329 ns-1a ( ( i+L ) ) = (nn+ ( 2 ) ) + i+1 ) 00330 nsOap ( (i+2) *n1) = (nn+ (4*nl) ) - (i+1) 00331 enddo 00332 00333 do i=1,nn 00334 ns-1a ( (i+l) *n1+L) =ie1 (i) 0033s nsOap ( (i*n1)+1) = ns0 (i) 00336 enddo 00337 reÈurn 00338 end 00339 00340 subrouEine setJarameter (mid' cor, rowav, rohlavs, xconn, conn, corE 00341 * , average, averageL ) 00342 00343 common / Lab2 /ns, nt, nn, nsÈeps, nevol, npasso, itherm, 00344 * nblock, ioption, idiv, idrop, insamP 00345 common /Lab|/ ct,c2,c3, c4,c5, c6 00346 conìmon /l-ab' / i.seed1, iseed2, iseed3 00347 common/lab6 / fí, fL2, fi|, fi2E, fit, fi4t' fi3, fi6, fi8 00348 * , fi3E, fi6t, fiSE 00349 common/ labS /henrgÈ, herrt, henrg, herr, hentz2 00350 co¡nmon/ 1ab9 / zmoml , zmom2, zmom3 , zmom4 , zmom6 , zmomg , zmomlg, zmom2 t, 0035r. * zmom3E, zrnom4E, zmom6t, zmomSE 00352 common/ labtt/g,xka 00353 common/ 1ab12 /nsum 00354 common/ 1ab13 / icerror 00355 dimension averageE (nn), average (nn), cor (0 :nt-1), corE (0 :nt-l) 00356 dimension cort (0 :nt-l), rowav(0 :nÈ-l), con (0 :nÈ-l) 00357 dimension,xcon(0 :nts-1), rovtavs (0 :nt-1),mid(nE) 00358 dimension )snean (0 :nt-l), xerror (0 :nts-1) 00359 00360 00361 00362 g=g*xka* 2 003 63 iacc=0 00364 00365 iseedL =8731- 00366 íseed2 =8731 00367 iseed3 =873L 00368 nevol= 3 00369 pi=3 . t4L5926 00370 icerror=0 0037L 00372 do i=L,ns,2 00373 mid(i) = (i-l) *ns+ns/2 00374 enddo 00375 00376 do i=2,ns,2 00377 mid(i) =L+ i*ns-ns/2 00378 enddo 00379 do i=1,ns 003 80 Phi(mid(i),0)=0.0 0038L enddo 00382 00383 cl- = eps 00384 c3 = 2.00*alinha*eps 00385 c2 = 2.00+0. 5O*:r¡nss 00386 c4 = 2.00*c2 003 87 c5 = xL/24.0 00388 c6 = 4*c5 00389 idiv = (nE-21 /2 00390 rns2=floaE (nn) 0039r. ¡¡s=ff6¿t. (ns ) o0392 rnt,=f loat (nE) 00393 rnb=float (nblock) 00394 003 95 fi=0 .0 00396 fí2= 0.0 00397 fir =0.0 00398 f.í2 t =0.0 00399 fi4 = 0.0 00400 fi4È = 0 00 0040L fí3=0 0 00402 fi3t = 0.0 00403 fi6 = 0.0 00404 fi6r = 0.0 00405 fi8 = 0,0 00406 fi8 E = 0.0 004 07 0040 I henrgt=0.0 0040 9 herrt=0.0 0041 0 henrg=6.9 00411 henrz2=0.0 004L2 herr=0.0 004 13 0 0414 0 041_5 zmom2=0.0 0 0416 zmom3=0.0 004L7 zmom4=0. 0 00418 zmom6=0.0 00419 zmomS=0.0 00420 0042L zmomLt=0 .0 00422 zmom2t=0.0 00423 zmom3t=o.0 00424 zmom4t=0.0 00425 zmom6t,=0.0 00426 zmom8t=0. 0 00427 00428 iicount.=0 00429 ssx=0.0 00430 ssy=o .0 0043 L sxy=o.0 00432 xs=O.0 00433 ys=0.0 00434 xys=O.0 0043 5 pi=3.14L592654 0043 6 00437 do i=0, nt-l- 00438 cor (i) = 0.0 0043 9 rovtav(i)=0.0 0044 0 roh¡avs(i)=0.0 00441, xcon(i)-0.0 00442 con(i)=0.0 00443 cort (i) = 0.0 00444 enddo 00445 00446 do i=l,nn 00447 averaqe(i)=0.0 0044 8 averaqet(i)=0.0 00449 enddo 00450 0045 1 hsum =0.0 00452 iscount = 0 0 0453 00454 return 00455 end 0045 5 00457 subroutine ínit,ialise (zL,22, ds1, ds2 et,aL, et,a2 * ,ro1-,to2, 00458 , phi, chi1, chi2 * 0045 9 , xpi, nt1, ntso, nsl-, ns0, nt1ap, nt,oap, 0 0460 * ml1, m12, m22, mlmLL, mÈm12, mtm21, mtm22, n 0 0461 , im1L, ímL2, im22, imEmLl, intml2, imtm2 L, imtm2 2, * 00462 uL, u2 , u3 , u4, u5, icode, nsLap, nsOap) 00463 00464 ímplicit real (a-h,o-z) 004 65 00466 common /labl/ alinha, eps, grc,pi 00467 cornmon /Iab2/ns,nt,nn, nsteps,nevol,npasso, itherm, 0046 I nblock, iopÈion, idiv, idrop, insamp 00469 conìmon /Lab3/ pconv 0047 0 conunon /Iab/l/ cL, c2, c3, c4,c5, c6 00471 common /lab5/ iseedl, Íseed2, iseed3 004?2 common/ lab13 /xinÍtiaI 00473 co¡unon/ Iabl L / g, xka, >cnu 0047 4 common/ 1abl2 /nsum 00475 0047 6 real xpi (nn) chil (nn) chi2 (nn) , *ns , 00477 real etal (nn+2 , -1 : nevol ) , eEa2 (nn+2*ns , -1 : nevol ) 00478 real phí(nsum,0:nevol) 00479 real m1l (5*nn) ,m12 (5*nn) ,m22 (5*nn) 0048 0 integer im11 (5*nn) , im12 (5*nnl , ím22 (5*nn) 00481 dimension mtmLl (13*nn),mtmL2 (13*nn), mtm21 (13*nn) 00482 * , mÈm22 (13*nn) 00483 inLeger imtm1L (L3*nn) , ímEml-2 (L3*nn) , ímtm2l- (L3*nn) 00484 * , imtm22 (13*nn) 00485 real z1 (nn) , ro1 (nn) , ds1(nn) 0048 6 real z2 (nnl , ro2 (nn) , ds2 (nn) 0 0487 double precísion u1 (nn) , u2 (nn) , u3 (nn) , u4 (nn) , u5 (nn) 00488 inÈeger ntL (nn) , nt0 (nn) , ns1 (nn) , ns0 (nn) 00489 integer ntl-ap (nn) , nt0ap (nn) 00490 integer nslap (nn), nsOap (nn) 0049L 00492 n=0 00493 call rnseE (iseedl-) 00494 call drnun(nn,u1) 00495 k=L,nn,2 00496 v do *dcos 00497 v u3 (k) =dsqrt (-2.d0*dlos(1.d0-u1 (k) ) ) 00498 * (2.d0*3 .L4159265359d0*u1 (k+L) ) 00499 v u3 (k+1) =dsqrE(-2.d0*dIoS(1.d0-ul- (k) ) ) *dsin 00500 * (2.d0*3.14t59265359d0* u1 (k+1) ) 0050 Lv enddo 0050 2v do ix = l,nn 0050 3v >çi ( ix) = sngl (u3 (ix) ) 0050 4v enddo 0050 5 0050 6 if (icode.eq.1) Ehen 0050 7v do i=L,nn 0050 8v phi (i, 0) = (sngl (uL (i) ) ) -' 50+1. 0000 0050 9v enddo 005 L 0 else 0 051 1 if (icode.eq.2) then 0051 2 open(unit=20, file='phi-inp', sEatus=' old' ) 0 0513s do i=l,nn 0 0514s read(20,*) phi(i,0) 0051 5s enddo 005L6 else 005L7 v do i=1,nn 00518 v phi (i,0) =xiniEial 00519 v enddo 00520 end if 00521 end if 00522 rewind ( 2 ) 00523 c 00524 c*****NON DYNA.È4ICAL EÍJEMENTS OF MTM ARE INITIÀLIZED******************** 00525 c 0052 6 call defmlm(mtm11, imEml-l,nE1,ntO,ns1,nsO, 1. 0,nn) 0052't call defmtm(mÈmL2, imtml-2,nEl,nEO,nsL,nsO, 0. 0,nn) 00528 call defmtm(mtm21, imtm21,nEl-,nEO,nsL,nsO, 0. 0,nn) 00529 call defmtm (mEm22 , imtm22 , ntL, ntO, nsL, nsO, 1 . 0, nn) * * * * * * * * * * ** * * * * * * * * * * * 00530 c* * * * *DYNÀMICAI ELEMENTS OF MTM ARE INITIÀIIZED* 00531 00532 call :<¡n21= 1¡**2*phi (i,0)-f-xka*f ) (i,0) 007L0 -xka*f*eEa1 (ns0 (i) , 0) -2*xka*f*et,al (ntOap (i) , 0) 007L1 OO'|L2 v ;g¡¡.22= (f**2*phi (í, 0)+f-2*xka*f) *et.a2 (i, 0) - 00713 xka*f*eEa2 (nt0ap(i), 0) - xka*f*eta2 (ns1 (i), 0) 007L4 v s 2 =:<¡n1 1 +:<¡n 72 + :p'¡2 t + ;g'¡2 2 00715 007L6 v s1= (phi (i,0) *c4 007L7 - (phi (nslap (i) ,0) +phi (ntL (í) ,0) +phi (ntO (i) ,0) + 0 0718 * phi(nsOap(i),0) ) + c6* (phi(í,0)*phi(i,0)*phi(i,0) )+xj) 007 19 00720 w xpi (i) = >rpi (i) -c1* (s1+s2+xje*s1) OQ72L v phÍ ( i,1) = phi (i,0) +xpi (i) *c3 00722 v enddo 00723 00724 00725 ca1 1 rcntml (phi, mtmLL, nt1, nÈ0, ns1, nsO, 11, nn, nevol, ns Lap, ns Oap ) 00726 call :antm2 (phi, mtml-2, ntL, ntO, ns 1, nsO, n, nn, nevol, ns1ap, ns Oap ) o0727 call :sntm3 (phi, mtm2 L, ntL, ntO, ns1, nsO, n, nn, nevo1, ns1ap, nsOap) 00728 call :sntm4 (phi, mtm2 2, nt1, ntO, ns 1, ns 0, n, nn, nevo1, ns 1-ap, ns Oap ) 00729 calI cg( mEml1, imtmL1,mtmL2, imtml2,mtm2L, imtm2l 00730 * ,mtm22, imtm22 , chi1,chi2 00731 * , zL,z2, roL,ro2,dsL,ds2, etal,eta2,0,n) 00732 IOCL VECTOR 00733 do i=1,ns 00734 etal (nt0ap (i), 1) = -eta1 (nn-i+L, 1) 00735 etal (nt1ap (nn-í+L) ,1) = -etal- (i, L) 00736 enddo o0737 IOCL VECTOR 00738 do i=1,ns 00739 eta2 (ntOap (i) ,1) = -eta2 (nn-i+1,1) 0 0740 et,a2 (ntLap(nn-i+1-), 1) = -eta2 (i,Ll 007 41 enddo 007 42 007 43 s 1=0 .0 007 44 s2 =0 .0 007 45 ll= n+1 O0746 v do i=1, nn 00747 v rsnLl-= 1¡**2*phi (í,Ll +f-2*xka*f) *etaL (i, 1) *eEaL - 00748 v xka*f (nt1-ap (i) ,1) - xka*f *etal- (nsL (i) , 1) 00749 v 00750 v rsnl-2= 1¡**2*phi (i,1) -f-xka*f) *eta2 (i, L) 00751 v * -xka*f*eta2(ns1(i),1)-2*xka*f*eta2(ntLap(i),1) 00752 v :an2l-= 1f **2*phí (i,1)-f-xka*f )*etal (i,1) 00753 v * -xka*f*etal(ns0(i),1)-2*xka*f*etal(ntOap(i),1) 00754 v 00755 v :gr.22= (i,1-)+f-2*xka*f) *eta2 (i,1) * 1¡**2*phi - 00?56 v xka*f*eta2 (ntOap (i) , 1) - xka*f*eta2 (nsl- (i) , 1) 00757 v s 2 =:sn1 1 +:snI2 +:r¡¡2I+rer.22 00758 v 00759 v s1= phi (i,1) *c4 00760 v * -(phi(nt]-(í),1)+phi(nslap(i),1)+phi(ns_Oa(i),1)+phi( 0076L v * nt,O (i) ,1) ) +c6* (phi (i,1) *phi (i,1) *phi (i,1) ) +xj Q0762 w 00763 v xpi (i) = xpi (i) -c3* (sl+s2+xje*s1) 00764 v phi(i,2) = phi(i, 1)+xpi(i) *c3 00765 v enddo 007 66 00't 67 00768 call :sntmL (phi, mtml L, ntL, ntO, ns 1, ns 0, n, r1n, nevo1, ns 1ap, ns Oap ) 00769 call rsnLL= 1¡**!*phi (í,2l +f-2*xka*f ) *eta1 (í,2) * - 00789 xka*f*et'al (ntlap (i) ,21 - xka*f*etaL (ns1 (i) ,2't 00?90 00791- v 00792 * -xka*f*e!a2 (ns1 (i),21 -2*xka* f* eLa2 (nE1ap (í'), 2l 00793 v rsn21= 1¡**l*phi (Í,2) -f-xka*f) *eta1 (i,2) 007 94 * -xka*f *et,a1 (nsO (i) ,2) -2*xka* f *eta1 (nt0ap (íl ,2) 00795 00796 v >s¡22= 1¡**l'tphi (í,21 +f-2*xka*f) *eta2 ( í,21 - 007 97 * xka*f*eEa2 (nt0ap (i) ,2) - xka*f*eta2 (ns1 (íl ,2) 00798 v s 2 =rcn1 1+:snL2+:g¡2]-+nt22 00799 v s1= phi lí,2)*c4 * (ír,2)+phi(ns0ap(i),2)+phi(nsLap(i),2)+phi( 00800 -(phi(nE1 *phi 00801 * ntO (i) ,2) ) +c6* (phi(i,2) *phi(í,2) (i,2) )+xj 00802 v xpi (i) = xpi (i) -c3* (s1+s2+xje*s1) 00803 v phi(i,3) = phi(í,21 + xpi(í)*c3 00804 v enddo 00805 00806 00807 call :sntml- ( phi, mtmll-, nt1, nÈ0, ns 1, ns0, rt, nn, nevol, ns Lap, ns 0ap ) 00808 call :<¡nEm2 (phi, mtmL2, ntl-, ntO, ns1, nsO, n, nn, nevol, nslap, nsOap) 00809 call >anean, fixerror, fi2:snean,fí2xetxot 0t290 , f i3:snean, fi3xerror, f i4>sntm (phi, amatd , nt1, ntO, nsl, nsO, n, nn, nevol 0L634 * , nslap, nsOap) 01 635 0163 6 c******************************************************* 01637 C HERE WE MODIFY THE PHI DEPENDENT TERMS IN ÀIVÍAT 0163I C******************************************************* 0163 9 0 1640 impliciÈ real (a-h,o-z) 0 1641 07642 common /J-abL/ alinha, ePs, gc,Pi 016 43 common/ Iabl2 /nsum 0t644 common/ 1ab1 1 / g, xka, rcnu 0L680 0L681 real am1(5*nn) 0L682 ínt,eger nt1 (nn) , nt0 (nn) , nsL (nn) , ns0 (nn) 01683 integer iamjl- (5*nn) 016 84 01685 C**THE DIAGONAI EIJEMENTS (DYNAMICÀI ONES) ARE IMPOSED TO BE = Q.***** 016 I 6 016 8 7v do i=1,nn 0168 8v ind.ex = 5* (i_1) 0 168 9 0L69 0v am1(index+l) = 0.00 0 L69 1w iamjl(index+L) = i 016 9 2 0169 3v amL(index+2| =-2.*xka 0 159 4v iamjl (index+2) = ntL (i) 01_69 5 01 69 6v am1(index+3) = -xka 01697 v iamjl- (index+3) = ns1 (i) 01698 01699 v amL ( index+4 ) = 0.0 01700 v iamjl(índex+4) = ntO (i) 0170 L 0L702 v am1 ( index+S ) = -xka 0L703 v íamj1(index+S) = nsO (i) 01-704 v enddo 01705 01.7 0 6 reÈurn 0L707 end 0L?08 077 09 0L?10 OLTLT subroutine defm2 (am2, iamj2,ntL,ntO,nsl,nsO,nn) 07712 0L7L3 implicit real (a-h,o-z) OL7L4 0171s colnmon /lab1-/ alinha, eps, gc, pi 0L7!6 common/ 1ab1l- / g, xka, :snu 01777 017L8 real am2 (5*nn) 01719 inEeger nt1 (nn) , ntO (nn) , ns1 (nn) , ns0 (nn) 0t720 integer iamj2 (5*nn) OT72L 07722 c**THE DIÀGONAI ELEMENTS (DYNAIÍICAL ONES) ÀRE IMPOSED TO BE = 0 01723 0L724 v do i=1,nn 0t725 v index = 5*(i-1) 01726 0L'127 v am2(index+1) = 0 00 0L728 v iamj2 (index+l) = i 01,'t29 01730 v am2(index+2) = 0.0 0L731- v íamj2 (index+2) = nE1 (i) 07732 01733 v am2(index+3) = -xka 0L734 v iamj2 (index+3) = nsl (i) 0L735 01736 v am2 (index+4) = 0.0 0]-737 v iamj2 (index+4) = ntO (i) 0173 I 0L739 v am2(index+S) = -xka 01740 v iamj2 (index+S) = ns0 (i) 0L747 v enddo 017 42 0t7 43 return 0L'144 end 0I7 45 0L7 46 subroutine defm3 (am3, iamj3,nC1,ntO,nsl,nsO,nn) 0L7 47 0 1748 0t7 49 implicít, real (a-h,o-z) 01750 OL75L co¡nmon / IabL/ alinha, eps, gtc,pi 01752 co¡nrßon/lab11/g, xka 017 53 0L754 real am3 (5*nn) 0 1755 inÈeger nt,1 (nn) , nt0 (nn) , ns1 (nn) , ns0 (nn) 0L756 integer iamj3 (5*nn) 0L757 0175 I C**THE DTÀGONAL ELEMENTS (DYNAI4ICAÍJ ONES) ÀRE IMPOSED TO BE 0. ***** 01759 0L760 v do i=1,nn 01?6L v index = 5* (Í-1) 01162 01763 v am3 (index+l) = 0.00 0t764 v iamj3 (index+l) = i 01765 0L?66 v am3(index+2) = 0.0 0t767 v iamj3 (index+2) = nÈ1 (i) 0L768 01769 v am3(index+3) = -xka 0L770 v iamj3 (index+3) = ns1(i) 0L7'tt Ot772 v am3 (index+41 = -2.O*xka 0L773 v iamj3 (index+4) = nt,O (i) oL17 4 0L775 v am3 (ind,ex+S) = -xka Ot'776 v iamj3 (index+S) = nsO (i) 0t777 v enddo 0r778 ot'|?9 return 0l_780 end 017 81 ll't82 subrouÈine :cn (phi, a¡l, n, al , a2, nn, nevol ) 0r.783 c ot'?84 c************************************************** 01785 c HERE WE MODIFY THE PHI DEPENDENT TERMS IN AM 01786 c aL is 1 for Ml1 and Ì422 and. zero for NIL2 0t7 87 c************************************************** 01788 implicic real (a-h,o-z) 01789 01790 common /Labt/ alinha, ePs, gtc,Pi 0L791 common/ labl1 / g, xka, :snu 0L792 common/ lab12 /nsum 01793 0L794 real phi (nsum,0:nevo1) , am(5*nn) 01?95 v do i=1, nn 01?96 v am(5* (i-1)+1) = aL+f*.5*phi(i,n) +a2*>snu 0t797 v enddo 0L?98 return oL'?99 end 0L800 0180L subrouEine cg ( a1l- , ia j LL , aL2 , iaiL2 , a2L , íaj2L , a22 , íaj22 ,bL,b2 0L802 , zI ,22 , t¡L,h2,dI ,d2, vetl ,veE2 , nindex,n) 018 03 018 04 c**************************************************************** 0L805 c CON'JUGATE GRÀDÌENT ROUTINE 01806 c****************************************************************** 0L807 0L808 impliciE real (a-h,o-z) 0L809 0L810 cornmon /Iab3 / pconv 01811 common/ 1ab1 1/g,xka 0181.2 common /Lab2/ns, nt, nn, nsteps,nevol, npasso, itherm, 01 813 nblock, iopti.on, idiv, idrop, insamp 01814 common/ Iabl3 / icerror 01815 parameter (1 1=36) 01816 real x1 (11) , x2 (11),vet1 (nn+2*ns, -L:3),vet2 (nn+2*ns, -1:3) 018L7 real d1 (nn) , d2 (nn) 01818 real z1 (nn) , bL (nn) , h1 (nn) , 22 (nnl , b2 (nn) , h2 (nn) 01819 real all (l-3*nn),a12 (13*nn),a21 (13*nn),a22 (13*nn) 0t820 inEeger iajLL (13*nn), iajL2(13*nn)' iaj2L(L3*nn)' iaj22(13*nn) 0182L 0L822 01823 ninE=75 0L824 v do i = 1,nn 0L825 v x1(í)= vetl(i,nindex) 01826 v x2(i) = vet2(i,nindex) 0l-827 v enddo 01828 0l-829 v do i=L, nn 0L830 v zI(íl = b1 1 0L831 v z2(íl = b2 ]' 0L832 v enddo 01833 01 834 call mult,ia (xL, x2, aLL, aL2, a2I, a22, iaj L L, i-ajL2, íaj2L' iaj22, lrl, 0183 5 * tJ'zt 01836 v do i = 1, nn 01837 v z2 (íl = h2 I z2 (il 0L838 v z1 (i) = hL I z1(i) 0L839 v enddo 01840 call multinv ( all , iaj 1L, a22 , íaj22 , zL, z2 ,t.L,h2't 0L84L v do i=L,nn 0L842 v d1(i) = -hl(i) 01843 v d2(i) = -h2(i) 01844 v enddo 0l_8 45 0L846 conv = provet ( zL,z2,hl,h2) 0L84't k=1 01848 L0 if( (conv.1t,.prec) .or. (k.eq.nint) )goto11 01849 01850 018 51 call multia (d1, d2, aL]-, a!2, a2]-, a22, iajLL, íaj]-2, íaj2t, íaj22,t,L, 0185 2 * tj-2') 0L853 alfa = conv / provet (dl ,d2,}¡L,}r2) 0L854 v do i=L ,nn 0L855 v x2 (í ) = x2(i) + alfa* d2 1 0L856 v x1 (i ) = x1(i) + alfa* dL 1 0L857 v enddo 0t_85 I 018 5 9v doi L, nn 01.8 6 0v )t (i = zL(í) + alfa*hl(i) 0L86 1v z2 (i = z2(í) + alfa*h2(i) 018 6 2v enddo 018 6 3 01_8 64 call mu1Èinv( a11,iaj11,a22,íaj22, zL,z2,L,L,t^2l 018 65 convd = provet lzl ,z2,hl ,}r2l 0r.866 0L867 beta=convd/conv 0L86I conv = convd 01869 01-870 v do i = l-,nn 01871 v d1(i) = -h1(i) + beta*dl (i) 0L872 v d2(i) = -h2(i) + beta*d2 (i) 0L873 v enddo 0t87 4 01875 k=k+L 0187 6 go to 10 oI877 11 continue 0L87I if (k.ge.nint) then *) , , , 01879 wriTe (4, K, ITERATIONS IN GRDPCD, STEP ' ,n, , 01880 npasso,' Conv= 'rconv, pconv 0L881 icerror= icerror+1 0L882 endif 0L883 v do í=1,nn 01884 v vetl(i,n) = xl(i) 0L885 v vet2 (i,n) = x2 (Í) 01886 v enddo 018 87 return 01_888 end 01889 subroutine mulEinv ( a11,iaj11-, a22,iaj22,y7,y2,wL,w2 01890 01891 C************************************************************ 0L892 C ******* PRODUCT OF THE II{IVERSE OF THE DTÀGONAT, OF A BY Y 01893 C GIVING W 01894 C* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 01895 01896 implicit real (a-h,o-z) 0L897 01898 common/1abLL/g,xka 018 99 co¡ûnon / Iab2 / ns, nÈ, nn, nsÈeps, nevol, npasso, itherm, 0L900 nblock, ioption, idiv, idrop, insamp 019 01 *nn) 01,902 real all ( L3 , w1 (nn) , w2 (nn) , yl (nn) ,y2 (ntr't * 01 903 , a22 (13*nnl 01904 integer iajl-1 (13*nn), íaj22 (13*nn) 0L905 0L906 v do i = l-,nn 01907 v index = L3* (i-1) 0L908 v wL (i) =y1 (i) /a1L (index+1) 01909 v 12 1i¡ = y2(i) / a22 (index+1) 01910 v enddo 0191L OL9L2 return 019L3 end OL9L4 0l.e 1s 0191-6 subroutine mulEia (x1-,x2, alL, aL2, a2L, a22, iaj11, íajL2, íaj2L, OL9L7 * íaj22,dvL, 0 L918 * dv2) 01919 0L920 parameter (11=36) oL92t 0L922 impliciE real (a-h,o-z) 01923 0]-924 conmon /Labt/ alinha, eps, gc,pi 0].925 common /Lab2 /ns, nt, nn, nsteps, nevol, npasso, itherm, 0L926 nblock, ioption, idiv, idrop, insamp 01927 common/labLL/g,xka 0192I 0L929 0193 0 0193 1 0]-932 c*********************************************** 01933 c HERE WE CATCULATE THE PRODUCT OF A AMAT BY A ETA 01934 c* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 0193 5 0193 6 real dvl- ( 1I ) , dv2 (11) 0L937 real aLL (13*nn) , a12 (13*nn ), a2L (1-3*nn) ,x1 (11) ,x2 (11) *nn) *nn) 0193 I inEeger iajLL (13*nn) , iajl 2( 13 , iaj2L ( 13 , íaj22 (nn* 13 ) 01939 rea]- a22(L3*nn) 01940 v do í=1, nn 01941 v auxLlr = 0. 0L942 v auxL2r = 0. 01943 v auxLlc=0.0 01944 v aux2Lc=O.0 0L945 v index = 13*(i-l) 019 46 *x1 0t947 aux11r aux11r+a1l (index+1) (iajL1 (index+1) ) = *x2 0194 8 v aux12r = aux12r+a12 (index+l) (iajL2 (index+l) ) 0]-952 aux21c = aux21c+ a21 (index+1) *x1 (j-aj2l(index+L) 01956 v auxLlc = auxLlc+a22 (index+l1*x2(íaj22 (índex+1) ) 0 r-9 57 *x1 01958 auxLlr auxLlr+a11 (index+2) (iaj11 (index+2) ) = *x2 01_95 9 v auxl-2r = auxl-2r+a12 (index+2) (iajL2 (index+2) ) aux2l_c aux21c+ a21 (index+2) (index+2) 01960 v = *x2 "xL(iaj21 0196L V auxllc = aux11c+a22 (!ndex+2) (iaj22 (index+2) ) 0t962 *x1 0L963 v auxLLr auxll-r+a11 (index+3) (iaj1L (index+3) ) = *x2 0L964 v auxL2r = aux12r+a12 ( index+3 ) ( iaj 1-2 ( index+3 ) ) 0r.965 v aux2Lc = aux2Lc+ a21 (index+3) *xl. (j-aj2]-(index+3) 01966 v aux11c = auxLLc+a22 (index+3)*x2(iaj22 (index+3) ) 0L967 01968 auxLlr auxl-1r+a1L (index+4) *xl (iaj11 (index+4) ) = *x2 01969 auxl-2r auxL2r+a12 (index+4) (iajL2 (index+ ) ) = *x1 0 r_97 0 aux21c aux2Lc+ a2l- (index+4) (íaj2L(index+4) = *x2 OL97L auxLl-c = auxLlc+a22 (índex+A) (iaj22(index+ ) ) 0rg't2 0197 3 auxllr auxl1r+a11 (index+S) *xl (iajL1 (index+S) ) = *x2 oL97 4 v auxl2r aux12r+a12 (index+S1 (iajt2 (index+S) ) = *xL 0197 5 aux2Lc = aux21c+ a2l- (index+5) (ía)2L(index+S) oL97 6 v auxLlc = auxl-1c+a22 (index+S],*x2(iaj22 (index+S) ) 0L977 *x1 0L978 auxLlr aux1lr+al1 (index+6) (iaj11 (index+6) ) v = *x2 0L979 v auxL2r = aux12r+a12 (index+6) (íaj12 (index+6) ) 01980 aux2Lc aux2lc+ a2L (index+6) *x1 (iaj21 (index+6) = *x2 019 81 v auxLlc = auxLlc+a22 (index+61 (íaj22(index+6) ) 01982 v *x1 01983 auxl-lr auxLLr+all (index+7) (iajLL (index+7) ) v = *x2 01_984 auxL2r = aux12r+a12 (index+7) (iajL2 (index+7) ) 0L9I5 v aux2Lc = aux2lc+ a21 (index+7) *x1 (iaj2L (index+7) 01986 v auxlLc = auxLLc+a22 (index+71*x2(íaj22 (index+7) ) 01987 *x1 0r.988 v auxlLr = auxLLr+all (index+8) (iajL1 (index+8) ) 01 989 v auxl-2r = aux12r+a12 (index+81*x2(íajt2 (index+8) ) 01990 V aux2lc aux2Lc+ a2l- (index+8) *x1 (iaj2L(index+8 = *x2 019 91 v auxLlc = auxlLc+a22 (index+81 (iaj22(index+8) ) 01992 v 0L993 auxLlr auxlLr+a11 (index+9) *xl (iajl-1 (index+9) ) = *x2 01994 v aux12r = auxL2r+al-2 (index+9) (iajL2 (index+9) ) 0L995 aux2lc aux2l-c+ a2l- (index+9) *x1 (iaj2L (index+9) = *x2 01996 v aux11c = auxLlc+a22 (index+91 (íaj22(índex+9) ) 0L997 01998 v auxllr auxl-Lr+al1 ( index+10 ) *xL ( iaj LL ( index+l0 ) ) = *x2 01999 auxL2r = aux12r+a12 (index+10) (íaj12 (index+L0) ) 02000 aux2lc aux2Lc+ a21 (index+10) *x1 (iaj2L(index+lO) = *x2 02001 v aux11c = auxLlc+a22 (index+L0) ( iaj22(index+10) ) 02002 *x1 02 003 aux11r = auxl1r+a1l (index+ll) (iaj11 (index+L1) ) 02004 aux12r auxL2r+aL2 (index+11) *x2 (iaj12 (index+L1) ) v = *x1 02 005 aux2l-c = aux21c+ a21 (index+11) (iaj2L (index+11) 02006 v aux1lc = auxl_1c+a22 (index+l_Ll*x2lfaj22 (index+1j_) ) 02007 *xl 02008 v aux1lr = aux11r+a1l (index+121 (iaj1L (index+12 ) ) 02009 v auxL2r = auxL2r+a12 (index+121*x2 (iajl_2 (index+L2) ) 02010 v aux21c = aux2lc+ a21 (index+t2l *x]-(iaj2L (index+12) 020L1 v auxl1c = aux11c+a22 (index+121*x2(iaj22 (index+12) ) 020L2 *x1 020L3 v aux1lr = aux11r+all (index+13) (iajl-1 (index+L3) ) 020L4 v aux12r = auxL2r+al2 (index+l3)*x2(íaj1,2 (index+L3) ) 02015 v aux21c = aux2Lc+ a21 (index+13) *xl (iaj2L (index+13) 02016 v auxlLc = auxlLc+a22 (index+131*x2(iaj22 (index+13) ) 020L7 02018 v dv1(i) = auxl1r+aux¡2r 020L9 v dv2 (i) = aux11c+aux21c 02020 v enddo 0202t 02022 return 02023 end 02024 02025 C******************************************************************** 02025 C******************************************************************** 02026 02027 q2028 function provet ( dlvl-, d2vt,dlv2,d2v2) 02029 02030 C************************************************* 0203L c HERE WE CALCULATE THE PRODUCT OF TWO VEcToRs 02032 C************************************************* 02033 02034 implicit real (a:h,o-z) 02035 02036 conìmon / Lab2 /ns, nt, nn, nsteps, nevol,npasso, itherm, 02037 nblock, ioption, ídiv, idrop, insamp 02038 02039 02 040 real dLvl (nn) , d2v1 (nn) , d1v2 (nn) , d2v2 (nn) 0204I totl =0.0 02042 tot2 =0.0 02043 v do ind=1,nn 02044 v tot.L=tot1+dlv1 ( ind) *d1v2 ( ind) 02045 v EoE2 = tot2 +d2v1(ind)*d2v2(ind) 02046 v enddo 02047 provet = È,otl-+tot.2 02048 return 02049 end 02050