COMPETENCIA, FORMACIÓN Y TOMA DE CONCIENCIA

In th is section we sh all o u tlin e th e num erical p ro ced u re we a d o p te d for th e stu d y of th e fee, bcc a n d in te rp o la tin g lattice. T h is p ro ced u re was th e sam e as th e one em ployed by Jackso n e t al [9] a n d has th e a d v an tag e over K le b an o v ’s p ro ced u re of in creasin g th e ra te of convergence by a facto r of two.

N u m erical calcu latio n s of th e m in im u m energy of th e Skyrm e L ag ran g ian , as a fu n ctio n of a, have been carried o u t for th e fee a n d bcc co nfig uratio n s a n d for a ra n g e of c ry sta l array s in te rp o la tin g betw een th e m w ith —0.35 < p < 0.32.

T h e c alc u la tio n p ro ced u re em ployed, involved solving a d iscretised fo rm of H a m ilto n ’s E q u a tio n s. In tro d u cin g th e canonical m o m e n tu m

n ‘ =

6 MS

w here L is given by eq u atio n (2.1.17) an d th e field <j>k satisfies th e c o n stra in t

(f>a<f)a — /* , th e H a m ilto n ia n E q u a tio n s re a d = Arf Un,

IT*: = d t ( B kidi<j>i) + (8.4.2)

w here,

Aki = (1 /4 + {dt<j>m}2)fiki — di<f>kdi<f>i,

Bkl = (1 /4 + — d ^ k d ^ i . (S. 4 .3)

T h e R o m an indices ru n from 1 to 3 (sp a tial co o rd in ates) an d th e G reek indices ru n from 0 to 3 (space-tim e c o o rd in a tes). T h e L agrange m u ltip lier, A, h as been

in tro d u ce d to enforce th e field co n strain t. Here we have in tro d u ce d th e dim en- sionless u n its v ia the tra n sfo rm a tio n xM ■-* x llj 2ef^.

In o rd e r to o b ta in s ta tic , m inim al energy configurations we tr e a t th e tim e v ariab le as a ‘pseudo tim e ’ variable an d use a field re la x atio n tech n iq u e th a t ensures as we advance in ‘pseudo tim e ’ steps A t , we always m ove dow n hill in energy. We perfo rm th e m in im isatio n of th e energy in a single o c ta n t of th e face c en tered cu b e for th e d ilu te p h ase and in an o c ta n t of th e h alf sk y rm io n cube in th e condensed phase. T h a t is, an o c ta n t of th e fee rep resen tativ e cell d ep icted in F ig u re (3.3.1). E xactly th e sam e p ro cedu re was used for th e bcc a n d in te rm e d ia te cases, by triv ially changing th e asp ect ra tio of th e cu b e into a re c ta n g u la r region, w ith lattice displacem ents in th e x and y or z directions becom ing ra or a / r 2 respectively, b u t otherw ise keeping th e sam e b o u n d a ry con d itio n s.

As an in itial field configuration we em ployed hedgehog fields sm o o th ed a t th e b o u n d aries of th e cell. We assum e th a t across th e p lane connecting th e centres of th e sk y rm io n, th e pion fields are reflection sym m etric.

T h e m esh size was typically 183 m esh p oints to th e o c ta n t of th e cube in th e d ilu te p h ase an d the sam e num ber, or fewer p o in ts in th e o c ta n t of th e h alf sk yrm io n cube in th e condensed phase, providing su b sta n tia lly g re a ter accuracy. C onvergence was assum ed, w hen th e e x tra p o la te d energy did n o t vary signif­ icantly over several h u n d red iteratio n s, d u rin g w hich th e b ary o n num ber was sta b le to 0 .01%. C onverged non-condensed phase solu tio n s always h a d a baryon d en sity g re a ter th a n 0.96 an d condensed phase solutions a b ary o n density greater

th a n 0.99,

T h e re su lts so o b ta in e d for th e field energies a n d values of < o > w ere e x tra p ­ o la te d to a n infinite n u m b er of ite ra tio n s, as in [9], w here th is e rro r was show n to be ex p o n en tial. T h en th e energies were e x tra p o la te d to zero m esh size. T h is was easy in th e condensed p h ase, w here th e energy a n d b a ry o n den sity are sp read fairly u n ifo rm ly in space so th a t th e energy varies linearly w ith 1/ n 2, w here n3 is th e n u m b e r of m esh points; b u t in th e d ilu te p h ase th e n u m b er of p o in ts covering each sk y rm io n reduces as th e size of th e cube increases a n d th e e x tra p o la tio n is less clean. T h e bary o n density also reflects th is fe a tu re , so th e d e v iatio n of th e n u m erical calcu latio n of th e b a ry o n n u m b er B from 1, is also a m easu re of erro rs d u e to finite m esh size a n d th is provided an a lte rn a tiv e e x tra p o la tio n p ro ced u re. T h e value of 1 — B is p ro p o rtio n al to 1 / n 2 for large n in th e condensed p h ase, b u t th e slope varies w ith a. T h e e x tra p o la tio n s to zero m esh size w as therefore p e rfo rm ed by lin ear e x tra p o la tio n of th e energy w ith resp ect to 1 — B , th e slope d e te rm in e d by varying n for som e ty p ical cases.

T h is p ro c e d u re is very a cc u ra te for th e condensed p h ase, leading to erro rs of a frac tio n of an M eV, b u t in th e d ilu te ph ase ty p ical e rro rs are tw o M eV. To e n su re co n tin u ity a t th e second o rd er p h ase tra n s itio n p o in ts th e energies in th e d ilu te p h ase w ere a d ju sted (w ith in th e ir errors) to agree w ith th e m ore acc u ra te values d e term in e d w ith th e condensed ph ase b o u n d a ry co n d itio n s.

T h e ra te of convergence of th e energy, as in [9], was observed to be a b o u t tw o tim es as fast as th a t for < a > . In o rd er to d e term in e th e lo catio n of th e

100

! f

p h a se tra n s itio n p o in t accurately, m uch use was m a d e of th e value of th e o rd e r p a ra m e te r < a >. By following th e v aria tio n of < a > as th e la ttic e sp acin g w as re d u c ed , we a ccu rately d eterm in ed la ttic e sp acin g a t w hich < cr > vanishes. H ow ever, in o rd er to provide reliable values of < a > , it was n ecessary to o b ta in well converged solutions in th e condensed p h ase. T h u s , w ith in th e condensed p h a se we devoted a considerable am o u n t of c o m p u te r tim e to o b ta in in g well converged solu tio ns. Indeed, w ith o u t reliable values o f < o > , th e second ord er n a tu re of th e p h ase tra n s itio n w ould n o t have b een e stab lish ed .

T h e n u m erical calcu latio n was perform ed using a C ray c o m p u te r an d involved c o n sid erab ly m ore C .P.U . tim e th a n those w hich h a d prev io u sly b een p erfo rm ed w ith cubic array s of skyrm ions. T his was b ecau se a w hole series of differing la ttic es, g e n erate d by varying th e asp ect ra tio of th e fee cu b e, w ere considered. As a re su lt, th is involved o b tain in g a b o u t sixty converged solutio n s w ith in each p h a se , a b o u t twelve tim es as m uch d a ta as was re q u ire d by Jack so n et al [9] for th e ir calcu latio n w ith a re c tan g u la r skyrm ionic la ttic e .

M oreover, unlike th e previous calcu latio n s, we o b ta in e d reliab le e x tra p o la ­ tio n to zero m esh size for all o u r d a ta a t differing values of a. T his involved esta b lish in g how th e energy varied as th e n u m b e r of la ttic e p o in ts was varied a n d also req u ired well converged d a ta . Jack so n e t al on th e o th e r h a n d , only p erfo rm ed th e e x tra p o la tio n for th e m inim um en ergy condensed co n fig u ratio n a t hig h densities.

of in clud in g a pion m ass te rm , (see (2.1.5)), w hich e x p lic itly break s C h iral Sym ­ m e try a n d for w hich we o b ta in e d a considerable n u m b e r of fee (p = 0 ) converged solu tio ns.