Dense skyrmion structures

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DENSE SKYRMION

STRUCTURES

by

PATRICK JONES

A THESIS SUBM ITTED FOR THE D EG REE OF PH.D.

A B S T R A C T

T he Skyrme M odel is used to investigate dense baryonic m atter.

A general review of skyrmionic crystalline arrays thus far investigated, reveals th e existence of a universal delocalising phase tran sitio n as th e density of baryonic m a tte r is increased. At low densities the skyrm ions localise on the lattice points, while a t high densities they become delocalised and an array of half skyrm ions rem ains. This phase tran sitio n is believed to represent th e resto ration of chiral sym m etry of high densities.

N um erical solutions of static arrays w ith fee, bcc and in term ediate sym m etries are considered as a function of density. The fee array is found a t high densities to be th e m ost stable of all the arrays so far considered. As the density is decreased the fee array becomes unstable against deform ations to arrays w ith interm ediary sym m etries. T his instability occurs a t a critical density slightly greater th a n th a t of th e delocalising phase tran sition .

A C K N O W L E D G E M E N T S

I would like to th a n k my supervisor Professor Leonardo C astillejo for his help and active encouragem ent, w ith o ut w hom this work w ould not come to fruition.

I would also like to th an k , for their stim ulating collaboration in th e work contained in C h ap ter 3, Professor Leonardo C astillejo, for his help in b o th setting up and perform ing th e num erical calculations, D r. J. J. M. V erbaarschot, for providing th e co m p u ter program s w ith which these calculations were perform ed, Dr. A. Jackson, for his insights and discovery of the bcc array and Professor A. D. Jackson, for his m any suggestions and num erous discussions concerning bo th the work in C h ap ter 3 and in th e thesis as a whole.

I would also p articu larly like to than k Dr. N. S. M anton who has been a continual source of insp iratio n during the course of this work and w ith w hom I have h ad m any enlightening discussions.

I also wish to th an k th e NORDITA and in p a rtic u la r D r. A. W irzba, for th eir hospitality an d invitation to p articipate in a w orkshop I found stim u latin g , on Solitons in Nuclei.

I should also like to th a n k the SERC and T h e Physics D ep artm en t, University College London, for th e ir financial su p p ort over th e last four years.

C O N T E N T S

C H A P T E R 1

IN T R O D U C T IO N

1.1 G eneral Introd u ction 7

References 18

C H A P T E R 2

IN T R O D U C T IO N TO TH E SK Y R M E M ODEL

2.1 T he Skyrme Model 20

2.2 T he B = 2 Sector Of T he Skyrme M odel And Beyond 32

2.3 A Simple M odel Of Dense Skyrmionic M atter 37

References 46

C H A P T E R 3

D E N S E SK Y R M IO N C M A TTER IN A CRYSTALINE FORM

3.1 M ulti-Skyrm ion C rystals 49

3.2 Skyrm ion A rrays At Low Densities 71

3.3 T he High Density Phase Of Skyrmionic M atter 85

3.4 N um erical C alculations 98

3.5 N um erical R esults 102

3.6 A pproxim ate Param eterised V ariational Forms 113

3.7 Conclusions 116

C H A P T E R 4

S K Y R M I O N I C M A T T E R O N A C L O S E D S U R F A C E O F V A R Y IN G C U R V A T U R E

4.1 A G eom etrical P icture Of Skyrmions 123

4.2 Varying The M etric Of Syhy 133

4.3 A Hedgehog On An E lliptical Three Surface 138

4.4 N um erical Results For A Hedgehog O n An Ellipsoid 147

4.5 A Skyrm ion On A More G eneral Closed Surface 182

References 205

C H A P T E R 5

C H I R A L B A G S W I T H I N D E N S E S K Y R M I O N I C M A T T E R

5.1 In tro d u ctio n 207

5.2 T he C hiral Bag Model 217

5.3 C hiral Bags W ithin A Dense Skyrmionic C rystal 229

References 246

C H A P T E R 6 C O N C L U S IO N S

6.1 G eneral Conclusions 248

C H A P T E R 1

IN T R O D U C T IO N

1.1 General Introduction

In th is thesis, I will present th e results of my investigations involving dense skyrm ionic m a tte r system s, which are believed to p e rta in to dense baryonic m a tte r system s. We shall see th a t th ere exist a num ber of differing forms of skyrm ionic m a tte r and th a t I have employed a variety of differing techniques in order to establish an u n d erstan d in g of each.

T he m ost direct of these approaches involves th e b ru te force technique, of num erically obtaining results for dense skyrm ionic m a tte r in its crystalline form and I shall present num erical results for infinite crystalline arrays of skyrm ionic m a tte r. In p a rticu lar, I consider a fee array, a bcc array and a series of ar­ rays of in term ediate sym m etries. The discovery of th e existence of these arrays of skyrm ionic m a tte r and th e ir respective num erical results, have already been rep o rted in reference [1] .

shaped space as an exam ple, I dem onstrate the significance of th e curvature and th e sym m etry of physical space in a dense skyrm ion environm ent. This model is co n stru cted so th a t its relation to dense crystalline skyrm ionic m a tte r can be revealed.

A final approach to u n d erstan d in g of th e n a tu re of skyrm ionic m a tte r will be outlined, in which th e C hiral Bag M odel is invoked in order to include an explicit quark content w ithin crystalline skyrmionic m a tte r and to establish th e physical relevance of such dense baryonic m atter.

In the rem ainder of this ch ap ter, we shall briefly outline historically, th e developm ent of the Skyrme M odel and finally, th e m an n er in which th e rest of this thesis has been organised.

T he Skyrme model was originally proposed by Skyrme [3] in 1961, as a phe­ nomenological description of th e nucleon and this model has recently a ttra c te d m uch interest. T he m odel describes the nucleon as a topological soliton of an extended version of the Non-Linear Sigma M odel. This revolutionary approach to nuclear physics, a tte m p ts to describe th e fermionic n a tu re of nucleons in term s of bosonic pion fields.

to an energy d istrib u tio n, w ith a r.m.s. of ~ 0 .5 /m , he also deduced th a t the correct asym ptotic form for th e one-pion exchange p o ten tial betw een nucleons em erged. He proposed th a t th e topologically conserved w inding num ber of these hedgehog skyrm ions, be identified w ith th e baryon num ber.

T hough S kyrm e’s work was generally ignored during th e 60’s an d 70’s, it has recently received a revival of interest. Indeed, m uch of th is new interest stem s from th e realisation w ithin theoretical physics, th a t such a topological non­ trivial solution of th e non-linear bosonic field theories, can be used to describe fermionic particles. Indeed, w ithin oth er sim pler m odels [4], th e relationship betw een ferm ion num ber and topological charge, originally predicted by Skyrme in 1961, was dem o n strated .

A ttem p ts to u n d e rstan d hadronic physics from a Q .C .D . stance, had n atu rally led to a series of effective meson models w ith sim ilarities to th e Skyrme model. Indeed, T ’Hooft [5] showed, assum ing confinem ent, th a t th e 1 / N C expansion of Q .C.D. leads to an effective L agrangian in th e low energy sector, containing only mesons. W itten [6] later argued th a t baryons th en emerge as solitons in th e large

te rm in a tru n c a te d infinite series of mesonic term s, which extends th e Non- Linear Sigm a so as to su p p o rt non-trivial solitonic solutions. T h u s, th e Skyrme M odel is now viewed as an effective m odel for low energy Q .C .D . m esonic physics. However, no derivation of this exists.

T he bag approach to nuclear physics presents an instructive altern ativ e to such effective low energy Lagrangians. This model directly a tte m p ts to u n d er­ stan d the relatio n sh ip betw een such effective L agrangians and Q .C .D .. In the m odel qu arks an d gluons are governed by Q .C.D. and placed w ithin a spherical cavity. In th e C hiral Bag M odel [7][8], th e exterior of th e b ag ’s mesonic degrees of freedom are governed by th e Skyrme or sim ilar low energy effective m odels. The quarks are assum ed to be confined to the interior of th e bag an d m esons to th e exterior by th e bou n d ary conditions on the bag surface. Im position of conser­ vation of axial current across the surface of the bag, connects th e two dynam ical regions and leads to th e so called, chiral bo u n d ary conditions. A ssum ing the existence of a spherically sym m etric mesonic field, leads to a hedgehog form w ith a spherical cavity cu t ou t a t its centre, in which spherically sym m etric, massless Dirac ferm ion wave functions, obeying th e D irac E quation, have been obtained [7] [9]. This object is th en viewed as a baryon w ith th e sh o rt distance quark degrees of freedom explicitly present. T hus, this elegant, effective, low energy m odel has developed as a generalisation of th e pure skyrm ionic description of a baryon.

tion of baryon num ber w ith w inding num ber of th e soliton was indeed correct. In order to m ake physical predictions for properties, A d k in ’s e tal[12] developed a projection procedure to obtain physical N and A states from th e classical hedgehog solution. T hey showed th a t such an approach produces a reasonable description of th e NA m ass sp littin g , nucleon m agnetic m om ents, nucleon charge radii and th e coupling constant ratio for i r N N and n NA.

M ore recently, a tte m p ts have been m ade to u n d e rstan d th e in teractio n be­ tween two skyrm ions [13] [14]. By considering th e interactions betw een two hedge­ hogs w ith relative spin-isospin degrees of freedom , they num erically stu d ied the skyrm ion-skyrm ion p o ten tial a t finite separations. Invoking th e quark hedgehog m odel, they d em o n strated th a t mesons which couple to hedgehogs, are precisely those which are known to have couplings to nucleons. This m odel for th e two nucleon sy stem was thus shown to successfully describe th e nucleon-nucleon po­ tential.

In C h ap ter 2 we also briefly introduce M an to n ’s sim ple m odel of baryonic m a tte r on a th ree sphere. T he results of this m odel b ear a striking sim ilarity to those which we discuss in C h ap ter 3 for skyrm ionic m a tte r in a crystalline form . B o th m odels reveal, th a t as th e average baryon density of skyrm ionic m a t­ te r is increased, it generally undergoes a phase tra n sitio n from an uncondensed phase, in w hich skyrm ionic m a tte r’s baryon density is well localised in space, to a condensed phase in which skyrm ionic m a tte r’s baryon density is m ore evenly d istrib u te d th ro u g h o u t space.

For a skyrm ion on a th ree sphere we see th a t this phase tra n sitio n occurs as we reduce th e volum e of physical space an d in the uncondensed phase th e skyrm ion is localised ab o u t a point in space, while in th e condensed phase it becomes com pletely delocalised over the whole of space. For skyrm ionic m a tte r in its crystalline form, as th e average baryon density of th e cry stal is increased, this phases tra n sitio n is shown to correspond to an uncondensed array of spherical skyrm ions centred a t th e points of an infinite lattice, becoming a condensed array of half skyrm ions.

bouring skyrm ions w ithin the lattice, which have d ate gone unnoticed. We show th a t these new sim ple cubic arrays of skyrm ions represent a sim ple generalisation of th e rectan g u lar array which Jackson et al [15] investigated as an altern ativ e to K lebanovs [16] original cubic array of skyrm ions. We see th a t th is general­ isation of th e Jackson et al array, corresponds to a sim ple continuous relative isospin-rotation of the skyrm ions w ithin successive parallel planes. C om paring th e asym ptotic binding energies of this generalised array to th e Jackson et al ar­ ray, by sum m ing th e net asym ptotic p o ten tial of an individual skyrm ion w ithin the array over increasingly d ista n t nearest neighbours, reveals th a t e l s th e relative

isospin-rotation angle of skyrm ions w ithin successive planes is varied, the n et th e asym ptotic a ttra c tio n of progressively increasing num bers of n earest neighbours does n o t change. Hence we ap p ear, a t legist asym ptotically, to have discovered a genuine zero m ode of the Jackson et al array.

We also present in C h ap ter 3 th e arrangem ents of th e fee array, bcc array and an array of in term ediate sym m etry. N um erically we show how th e binding energy of these arrays increases as th eir densities are increased. T h e m an ner in which these calculations were perform ed is first described and th e n we show th a t th eir results reveal th a t a condensed fee array of skyrm ions, corresponding to a sim ple cubic array of half skyrm ions, has an energy which is ju s t 3.8% above the energy lower bound and th u s, th a t it represents th e m o st bou n d crystalline array of skyrm ions so far considered.

skyrm ionic m a tte r by varying th e length of th e fee cube in one direction, while keeping th e o th e r lengths fixed. T h e results reveal th a t th e fee lattice becom es u n stab le w ith respect to these bulk deform ations a t a density close to th a t at which th e p h ase tra n sitio n to a condensed array of h alf skyrm ions occurs and th a t a t high densities th e fee array is stab le, while a t low densities it is u n stab le to bulk deform ation.

in our flat space array calculation.

energy fu nctio n al w ith respect to variations in th e shape of physical space. In C h ap ter 5, we a tte m p t to extend th e crystalline skyrm ionic calculations of C h ap ter 3 by inco rp oratin g explicit quark degrees of freedom w ithin th e m inim al energy condensed fee array. T hus, initially we outline th e fundam entals of th e C hiral Bag M o d el’s description of a baryon. We th en propose the m an n er in which th e C hiral Bag M odel’s two phase description of baryonic m a tte r, w ith its explicit q u ark content, can be incorporated consistently w ithin th e condensed fee h alf skyrm ion array. We d em o n strate th a t th ere exists th e possibility of cubic bag being cut out of th is field in a consistent m an n er, which is in itself, a non -triv ial resu lt due to th e discontinuities a t th e edges of th is cube. Having established th a t cubic bags can exist and having argued th a t these bags are the m ost relevant to th e condensed fee array, we th e outline a calculation which could be perform ed num erically an d which would involve solving th e D irac equation for th e ground s ta te up an d down quark wave function w ithin th e cubic bag. T he results of th is calculation w ould th en reveal th e particle content w ith in this cubic bag and suggest th e physical in te rp re ta tio n th a t should be placed on skyrm ionic crystalline m a tte r.

an integer value, since clearly this is a physical requirem ent of any reasonable m odel of baryonic m a tte r. For th e spherical bag, this n et co n trib u tio n to th e baryon num ber is indeed an integer value [17] and th u s we show th e m an n er in w hich this cancelation occurs in this case. We th en a tte m p t to d em o n strate th a t a sim ilar cancelation should occur for our cubic bag. T hese suggestive argum ents are appealing, though inconclusive.

We also note in this chap ter, th a t since th e condensed fee array is com posed of te tra h e d ra l four skyrm ion u n its, these m ight represent a particle like stru c tu re s w ithin dense skyrm ionic m a tte r. This suggestion is fu rth e r ju stified by com par­ ing th e stru c tu re s of these te tra h e d ra l u n its w ith th e tru e m inim al energy four skyrm ion field configuration [18]. Recently it was proposed, th a t th is m inim al energy four skyrm ion field configuration should represent th e Skyrm e M odel’s description of an alp h a particle.

R e fe r e n c e s

[1] L.C astillejo, P .S .J.Jon es, A .D .Jackson, J.J.M .V erb aarsch o t, A .Jackson Nucl. Phys. A501 (1989) 801

[2] N .S.M anton, Com m un. M ath. Phys. I l l (1987) 469

[3] T .H .R .S kyrm e, Nucl. Phys. 31 (1962) 556

T .H .R .S kyrm e, Proc. Roy. Soc. London 262 (1961) 237 T .H .R .S kyrm e, Proc. Roy. Soc. London 260 (1961) 127 T .H .R .S kyrm e, Nucl. Phys. 31 (1962) 550

[4] A .S.G oldhaber, Phys. Rev. D16 (1977) 1815

[5] G .t’H ooft, Nucl. Phys. B72 (1974) 461 G .t’H ooft, Nucl. Phys. B72 (1974) 461

[6] E .W itte n , Nucl. Phys. B160 (1979) 57

[7] A .C hodos and C .B .T h o rn, Phys. Rev. D12 (1975) 2783

[8] C .G .C allen, R .F .D ash en and D .J.G ross, Phys. L ett. 78B (1978) 307 G .E .B row n an d M .R ho, Phys. L ett. 82B (1979) 177

G .E.B row n, M .R ho and V.Vento, Phys. L ett. 84B (1979) 383

[10] A .P .B alach an d ran , V .P.N air, S.G .R ajeev and A stern , Phys. Rev. L ett. 49 (1982) 1124

[11] E .W itte n , Nucl, Phys. B223 (1983) 422 E .W itte n , Nucl. Phys. B223 (1983) 433

[12] G .S.A dkins, C .R .N appi and E .W itten , Nucl. Phys. B228 (1983) 552

[13] A .D .Jackson and M .Rho, Phys. Rev. L ett. 51 (1983) 751

[14] A .Jackson and A .D .Jackson, Nucl. Phys. A457 (1986) 687

[15] A .D .Jackson and J.J.M .V erbaarschot, Nucl. Phys. A484 (1988) 419

[16] I.K lebanov, Nucl. Phys. B262 (1985) 133

[17] J.G oldstone and R .L.Jaffe Phys. Rev. L ett. 51 (1983) 1518

C H A P T E R 2

IN T R O D U C T IO N TO T H E S K Y R M E M O D E L

2.1 The Skyrm e M odel

It is now believed th a t Q u an tu m C hrom odynam ics, (Q .C .D .), provides th e fun­ d am en tal description of stro n g interactions. T hus, in principle the s tru c tu re of baryons and m esons should be derivable from this S'17(3) colour gauge theory of quarks an d gluons. Due to its non-abelian n a tu re , th e Q.C.D. vacuum is p aram ag netic and gives rise to stro n g infrared forces. At sh o rt distances, colour anti-screening takes place, resu ltin g in an effective coupling c o n stan t ten d in g to zero. In this p e rtu rb a tiv e phase quarks and gluons are asym ptotically free [l][2]. However, a t large distances th e coupling co n stan t grows and only colour singlet states exist asym ptotically. This n o n -p ertu rb ativ e regim e a t low energies, w ith colour confinem ent, is relevant to nuclear physics. Since here p e rtu rb a tiv e m eth ­ ods are in ap p ro p riate, lattice calculations aim ed a t num erically solving Q.C.D.

have been developed in an a tte m p t to uncover th e n a tu re of th is phase. How­ ever, these calculations are far from m aking physical predictions and contain m any difficulties yet to be resolved.

developed. At low energies th e dom inant quark degrees of freedom of Q.C.D.

are th e lightest quark fields. These are the up and down quarks, w ith cu rren t qu ark m asses of ab o u t 10M e V . Neglecting these m asses, m assless, two flavour

Q.C.D. is chiral invariant, w ith the global sym m etry

U(2)l x 17(2)* - S U ( 2)l x S U ( 2)r x U ( 1)v x U( 1)a , (2.1.1)

w here L an d R denote th e left th e righ t sym m etry groups, V refers to th e vector and A to th e axial vector sym m etry groups. At th e q u a n tu m level, it is well known th a t th e U( 1)^ sym m etry group is explicitly broken by th e A -B -J anom aly [3] [4] leaving th e global invariance group

S U ( 2)l x S U ( 2)r x U ( l ) v . (2.1.2)

T he C hiral Sym m etry group S U ( 2) l x S U ( 2) r is th e n spontaneously broken

down to S U ( 2 ) v , via th e N am bu-G oldstone m echanism . T he resultin g three massless pseudo-scalar particles are th e pions. T hus, th e vacuum carries axial charge an d pions are able to decay to th e vacuum . T his leads to th e G oldberg- Teim ann relatio n betw een the axial form factor for th e nucleon ga and th e pion decay c o n sta n t f K. This physically appealing scenario is consistent w ith th e sym m etries of Q.C.D.. However, due to th e n o n -p ertu rb ativ e n a tu re of Q.C.D.

a t low energies, little inform ation can be deduced from Q.C.D. ab o u t ga an d f T.

to hold approxim ately on physical grounds.

T h u s, one is led to believe th a t the N on-Linear Sigm a M odel [5], which in­ co rp o rates all th e essential features of chiral sym m etry breaking, would provide a good m odel of th e low energy mesonic physics as do m in ated by pions. M ore­ over, th e expansion of Q.C.D. in the num ber colours N e, developed by t ’Hooft, suggests th a t a t low energies, assum ing confinem ent, Q.C.D. becomes a theory described by an effective mesonic field [6]. Since a t low energies th e dom inant chiral degrees of freedom are pionic, th e N on-Linear Sigm a M odel should provide a first ap p rox im atio n to this effective low energy theory, even tho u g h N c is not infinite b u t three.

T hus, S k y rm e’s proposal th a t th e N on-Linear Sigm a m odel be used to de­ scribe low energy m esonic interactions [5] has been justified m any tim es over. It provides a m odel w hich describes C hiral Sym m etry breaking an d is also consis­ te n t w ith soft pion th resh o ld theorem s [7]. Its L agrangian reads

L 2 = ^ { d ^ c r d ^ c + d ^ n d ^ n ) , (2.1.3)

w ith th e no n -lin ear co n strain t,

<72 + if2 = f l , [ 2. 1. 4)

w here /*- is th e pion decay co n stan t, 7r is the iso-triplet pion field and a th e scalar sigm a field of th e [1/ 2 , 1/ 2] rep resen tatio n of S U ( 2) l * S U ( 2)r . T hus th e theory is m anifestly chiral invariant.

a tta in s a co n stan t value /*. th ro u g h o u t space. T he in tro d u ctio n of an explicit chiral sy m m etry breaking term , generates m asses for th e pions which are viewed as flu ctu atio n s of th e sigm a field along th e valley of a ‘m exican h a t ’ po ten tial. Such a m ass term is

L m = - m \ f l a, (2.1.5)

w here m T is th e pion m ass and th e coefficient has been chosen so as to generate a pion m ass te rm to leading order in th e weak pion field lim it. T he inclusion of this te rm also leads to th e correct form for th e P.C .A .C . [8] relatio n for the divergence of th e axial cu rren t. Not only does this m odel satisfy th e low energy requirem ents of C hiral Sym m etry, b u t it has also been found to describe well, th e self in teractio n and p ro pag atio n of pions in th e exterior of a nucleus [7].

T h e configuration space of th is m odel possesses a non-trivial topological stru c ­ tu re resu ltin g from its non-linear n a tu re . This observation led Skyrm e to propose th a t i t ’s non-trivial field configuration be identified as classical baryons [5].

It is convenient to introduce th e quaternionic rep resen tatio n of 517(2), in order to u n d e rstan d th e topology of this configuration space. T hus we w rite

U (t ,x ) = -^-(a(t,x) -f i f . n ( t , x ) ) , ( 2.1.6)

J 7T

w ith co n strain t (2.1.4) becom ing

U ( t , x ) U ( t , x ) + = 1, (2.1.7)

rep resen t th e N on-Linear Sigm a M odel Lagrangian as:

L2 = f- f T r ( L , . L % (2.1.8)

4

w here

Lp = U( t , x ) + ( 2. 1. 9)

A t a fixed tim e t0, th e field U(t0,x ) , m aps physical space R3 into S U ( 2), whose group m anifold is isom orphic to a th ree sphere, S 3. For a sta tic field configuration th e energy is given by:

e2 = -

J

O n physical grounds it is n a tu ra l to consider only finite energy fields. T h u s, the derivative of U(x) m ust vanish outside some finite region, in order th a t (2.1.10) be finite. Hence, we are led to consider s tatic configurations satisfying th e n a tu ra l b o u n d ary con d itio n th a t U(x) takes a co n stan t value at spacial infinity, which on physical gro u n d s, we choose to be its triv ial vacuum value. T h a t is, we have th e b o u n d a ry condition

U(| x |—► oo) = 1. (2.1.11)

This b o u n d a ry condition compactifi.es physical space R3, to a th ree sphere, S 3

an d so topologically th e field U (x) is a m ap

It is well know n th a t such m appings are classified topologically into hom otopy classes. T h e set of m appings (2.1.12), is classified by the th ird hom otopy class of a th re e sphere,

n

(S3) =

which is iso-m orphic to th e additive group of integers Z, th e w inding num ber of th e m ap p in g . T h u s, if we consider only continuous fields, th en th e configuration space is disconnected an d th e tim e evolution of a configuration can be view as a hom otopic deform ation. M oreover, th is disconnected topology of configuration space, m eans th e w inding num ber of a configuration is a conserved q u an tity independent of its dynam ics. T he w inding num ber of a static field configuration,

U ( x), is defined to be

B —

j

A t the origin we require th e condition

I7 (| x \ ) = - l (2.1.15)

be satisfied. T his, to g eth er w ith condition (2.1.11), results in a continuous m ap

U(x), m ap p in g th e whole of physical space onto th e whole of th e ta rg e t space

current,[18]

B ° = ~ ^ ~ r T r ( L llL vL f ), (3.1.16)

which is conserved in dependently of th e dynam ics an d th e w inding num ber (2.1.14) and is its corresponding charge.

Skyrm e proposed th a t this cu rren t be identified w ith th e baryon c u rre n t and hence the baryon num b er w ith th e w inding num ber of a configuration [5]. T hus, baryons are described as topological solitons of th e sigm a an d pion fields. Fur­ therm ore, he proposed th a t th e fermionic n a tu re of baryons in this soliton picture, results as a consequence of th e sk y rm io n ’s non -triv ial topological n a tu re . This was shown later to be possible by Finkelstein et al [9]. M oreover, m uch of the resurgence of interest in th e Skyrm e M odel, stem s from W itte n ’s observation of the mesonic fields [10], th a t in an effective mesonic theory of Q.C.D. in th e large

N c lim it, baryon num ber should be identified w ith th e w inding num ber.

F in ite energy static field configurations of th e N on-L inear Sigm a L agrangian (2.1.8), are however, u nstab le. This can be seen if we consider rescaling a static field, U(x). T he energy of this field will be reduced by shrin k in g it to a point, since th e static energy functional e2, scales w ith length. T h u s, finite energy configurations are u n stab le to rescaling to a point.

con-figurations ag ain st rescaling to a point. T hus, Skyrm e considered an ex ten d ed chiral sym m etric m odel, w ith L agrangian given by [11]

L = L 2 + L 4, (2.1.11)

w here th e ad d itio n a l Skyrme Term ,

L * = j ^ T r \L M '

has a dim ensionless p a ra m ete r e, characterising th e size of th e finite energy con­ figurations. T h e energy of a static field configuration is now given by

E = e2 + €4, (2.1.19)

w ith

€4 = — J

T his energy is m inim ised w ith respect to scale tran sfo rm atio n s, w hen b o th th e e2 te rm (2.1.10), a n d th e e4 term s (2.1.20), are equal.

To th is o rd er, th e add itio n al fo u rth order term is th e unique te rm yielding a positive definite H am iltonian, th a t is only second order in tim e derivatives of

U( t , x ) .

n u m ber B, is b o u n d ed from below by

f

E > 6tt2— \ B \ . (2.1 .21) e

T hus, finite energy configurations are locally stabilized as a resu lt of th e u n d er­ lying global topology of configuration space.

T h e d etailed s tru c tu re of these skyrm ionic solutions is d eterm in ed by the highly n o n -lin ear E uler E q u a tio n , which results from functional m in im isatio n of the Skyrm e M o del’s energy function w ith respect to th e field variables. E xcept for fields possessing a high degree of sym m etry, finding analytic solutions to this eq u atio n poses a v irtu ally in trac ta b le problem .

Skyrm e [5] was th u s led to consider the spherically sym m etric hedgehog a n sa tz ,

U( x) = e x p (t-p rr f ( r )) (2.1.22)

I x I

for th e fields. T his field configuration, in term s of th e sigm a an d pionic fields, takes th e form

<r(r) = fn cos / ( r ) ,

n( x ) = 7-^rr s i n / ( r ) . (2.1.23)

I x I

The an satz g reatly simplifies th e resulting equation of m otion, which becomes an o rd in ary no n -linear eq u atio n for th e profile function f ( r ) .

to ta l an g u lar m om entum g en erato r an d I is th e isospin g e n e ra to r), b u t n o t w ith

J or I separately. T his leads one to suspect th a t K will be a conserved q u a n tu m n um b er for th e quantized hedgehog an d th a t it is a scalar in g ran d -sp in space. Since this configuration has positive parity, i t ’s q u a n tu m num bers are 0 + .

S u b stitu tin g th e hedgehog an satz (2.1.22), into th e energy functional (2.1.19), gives th e energy for a hedgehog field

/ 2 • 2 sin2 / , 1 sin 2 / . s i n 2 /

E = 4 * J C°° . ?dr r * { ± ( ? + 2 — ± ) + — — i L ( — J - + 2 F ) } , si n f. 1 si n / .sin / ,(3.1.24)

w here we have perform ed th e triv ial an g ular in teg ratio n an d / denotes th e deriva­ tive of f ( r ) w ith respect to th e rad ial co o rd in ate, r.

U pon functionally m inim izing th e energy for th e hedgehog configuration (2.1.22), w ith respect to its profile function / ( r ) , th e E uler E q u atio n is given by:

, 1~9 . 9 ..x", 1 -v . , 1 . „ sin 2 f sin 2 / ,

( ~ r + 2 sin / ) / + ~ r f + s i n 2 / / 2 - - sin 2 / --- —--- = 0, (2.1.25)

4 2 4 r l

w here we have introduced th e dim ensionless variable r = ef ^r, of reference [12]. T he hedgehog solutions of this equation are also solutions of th e general E u ler E qu atio n , o btain ed from a general field v ariatio n of th e energy functional (2.1.19), subject to th e non-linear co n strain t (2.1.4). T his was shown to be th e case [13] for any field configuration possessing a high degree of sym m etry, such as th e hedgehog ansatz.

Integer w inding num ber configurations can be o b ta in e d by num erically inte­ g ratin g this equation subject to th e b o u n d ary conditions:

f ( oo) = 0, (2.1.26)

to give a baryon num ber B configuration. We n o te th a t for a field tak in g its trivial vacuum value a t sp atial infinity, th e w inding num b er of such fields is dependent only on the b o u n d ary co n d itio n a t th e origin. T h e in terp o latin g function betw een the two points is d eterm in ed by th e E uler E q u atio n (2.1.25).

For hedgehog configurations, th e resulting energy was found to be 864M e V

for B = 1 and 2523M e V for B = 2 [19]. Here we have tak en th e param eters of th e m odel to have values /*- = 64.5M e V a n d e = 5.45, as used by Adkins et al [12]. T hey were d eterm in ed by em ploying a sem i-classical q uan tizatio n technique, which enables physical baryon sta te s to investigated. T h e collective spinning modes of th e Skyrm e L agrangian g en erate classical a n g u lar m om entum . U pon quantizing these collective co o rd in ates, sta te s of definite sp in a n d isospin and hence physical baryon sta te s, can be investigated. T h e p a ra m ete rs were subsequently fixed so as to reproduce, independently, th e nucleon an d delta(1236) masses. This approach fails to rep ro d u ce th e N N tt coupling co n sta n t, giving a value gNMn approxim ately 30% too sm all.

For th e B = 1 hedgehog, from eq uatio n (2.1.16), th e bary o n density is given by:

I*-1 - * 7)

classical b ary o n , thus leads to a reasonable m ass an d a r.m .s rad iu s for th e nucleon.

Since th is is a m odel of a baryon, th e hedgehog field should be quantized as a ferm ion for it to m ake sense. However, th e Skyrm e fields are bosonic. It was proposed by Skyrm e [5] an d later d e m o n strated by Finkelstein et al [9], th a t topological ex citation s of non -triv ial field configurations are capable of carrying half integer a n g u lar m om entum . For such an effective theory of Q.C.D. w ith an o dd nu m b er of colours, W itten [10] fu rth er confirm ed Skyrm e’s proposal, th a t a skyrm ion field has a fermionic n a tu re . T hus Adkins et al procedure involved quantizing th e B = 1 hedgehog as a ferm ion.

as a ferm ion. T his am ounts to quantizing th e hedgehog like a rigid ro to r, w ith th e hedgehog being a sym m etric to p . T h e resu ltin g hedgehog states are p a r­ tially c h aracterized by th e conserved q u a n tu m num ber K , which takes integer values from zero upw ards. T hese sta te s have equal h alf integer values of S and / , w ith th e nucleon sta te s having S = 1 /2 an d I = 1 /2 . T h e tow er of grand- spin sta te s th u s contains th e nucleons, d eltas and higher resonances. T h e masses of the nucleons an d d eltas, as alread y s ta te d , are th e n used to fix th e p a ra m ete r e.

2.2 T he B —2 Sector O f T he Skyrm e M odel and B eyond

T his elegant p ic tu re of a skyrm ion as a generalized baryon was fu rth e r extended to the B — 2 sector. T h e physics of this sector are ch aracterised by th e nucleon- nucleon p o te n tia l an d the existence of a B = 2 b o u n d s ta te , th e d eu tro n . In the initial stu dies a p ro d u c t an satz, of two B = 1 hedgehogs, Ujj(x), was employed (5),

u { z

1,X2) =

UH{xi)UH(xt ).

(2.2.1)

It has been shown [11] th a t for such a p ro d u c t, th e baryon num b er is additive and hence, such a configuration has B = 2.

as th e skyrm ions approach each o th er a t finite sep aratio n s, this overlap becom es significant and results in a large d isto rtio n of th eir fields w hich this a n satz is n o t able to describe. T h u s, as th e hedgehogs approach each o th er, th e p ro d u c t a n sa tz becomes an increasingly less valid app ro xim atio n for th e tru e B — 2

m inim al energy configuration. Hence, th e p ro d u c t an satz as been used to describe asy m p to tic nucleon-nucleon interactions. Indeed, Skyrm e [5] him self originally ad d ressed this problem an d deduced an analytic expression for th e asy m p to tic p o te n tia l betw een two hedgehogs by employing th is an satz.

For two hedgehog skyrm ions centred a t th e points Xi an d £2, which are well se p a ra te d , th e field configuration is described by two u n d isto rte d hedgehogs w ith a relative isospin ro ta tio n of th eir pion fields, which can be represented as

U ( x u x 2) = Uh(xM U h(2 2)A + , (2.2.2)

w here A = ao -f a.r and Uh(x) is a B = 1 hedgehog solution. T h e resu ltin g p o te n tia l, defined as th e difference betw een th e energy of th e B = 2 p ro d u c t configuration and twice th e energy of th e B = 1 skyrm ion,

V = E B = t - 2E B = u (2.2.3)

in th e asym p to tic lim it is given by

v { F l t ) = 2 C { H - - n f . - ^ h { e e 4 )

I r 12 I

zero pion m ass lim it considered here, goes w ith th e inverse sep aratio n distan ce cubed. T his is indeed sim ilar to th e form of th e nucleon-nucleon ten so r p o te n tia l.

E q u atio n (2.2.4) for th e p o te n tia l, shows th a t the te n so r force betw een two well sep a ra te d hedgehogs is op tim ally a ttra c tiv e when th e conditions a 2 = 1 and a .r 12 = 0 are satisfied. To u n d e rs ta n d th e non-isotropic n a tu re of th e p o te n ­ tia l, one should note th a t th e p ro d u c t an satz (2.2.2), rep resen ts two hedgehog skyrm ions relatively ro ta te d a b o u t th e axis a, by an angle 9, w ith | a |= s i n ( 0/2). T h us, th e optim al relative isospin o rien tatio n condition, corresponds to th e two hedgehogs being relatively ro ta te d a b o u t an axis p e rp e n d icu la r to th e ir line of centre, by an angle 7r. T he resu ltin g configuration has th e n fields of th e hedgehog c en tred on pointing radially o u tw ard s, w hile those of th e hedgehog centred on x 2 p o in t rad ially inw ards, having been ro ta te d th ro u g h an angle nr, a b o u t an axis p erp en d icu lar to x\ — x 2. T his configuration of hedgehogs is such th a t th e ir respective fc fields m atch sm o o th ly along th e ir line of cen tre an d th is non­ fru stra te d arran g em en t m inim ises th e 7r field g rad ien ts a n d hence optim ises th e a ttra c tio n .

fields an d is thus strongly anisotropic.

S kyrm e’s asym ptotic result (2.2.4), em ploying th e p ro d u c t a n sa tz (2.2.2), has since been extended using num erical techniques to give th e p o te n tia l (2.2.3) a t all sep aratio n s [15]. These studies revealed th e skyrm ion-skyrm ion p o te n tia l to give a reasonably successful description of th e phenom enological nucleon-nucleon p o te n tia l.

T hese ad d itio n al sym m etries are im p o rta n t in them selves, due to th e ad d i­ tio n al restrictio ns they place u p o n collective co o rd in ate sem i-classical q u an tiza­ tio n of th e zero m odes of skyrm ions an d resu lt in selection rules for th e ir q u a n tu m s ta te s. T h u s, in th e B = 2 case they were shown to resu lt in th e ap p ea ra n ce of a non-degenerate gro u n d s ta te w ith the q u a n tu m num bers of th e d e u tro n [18]. However, th e binding energy of this gro u n d s ta te was fou n d to be u n realistic, as was expected, due to th e naive sem i-classical q u an tiz atio n of only th e zero m odes of th e classical solution, which neglects th e im p o rta n t degrees of freedom describing nuclear break up. In ord er to overcom e th e re stric te d n a tu re of this q u an tizatio n procedure, it is necessary to identify th e degrees of freedom for nuclear sep aratio n . However, due to th e n o n -renorm alisability of th e Skyrm e M odel, a sem i-classical q u an tizatio n procedure is still required.

num ber sectors and th e ir co n stru ctio n was based on th e num erically discovered sym m etries of th e tru e field configurations. B u t it does now seem possible th a t a realistic ev alu atio n of th e binding energy of th e B = 2 ground s ta te will be possible a n d th a t in th is sector th e tru e economy of th e Skyrme M odel descrip tio n will be realised, in co rp o ratin g a description of b o th th e bound s ta te sy stem an d nuclear collisions.

R ecently C arson et al [20] found num erical m inim al energy B = 3 ,4 ,5 and 6 solutions an d found these also to have a significant binding energy relative to isolated skyrm ion configurations of th e sam e baryon num ber. T hese fields also possess a high degree of sym m etry, as revealed by th e baryon density plots given by C arson e t al. O ne thus expects th a t these fields m ay lead to a d escrip tio n of th eir corresp o n d in g bary o n num ber light nuclei.

2.3 A Sim ple M od el O f D ense Skyrm ionic M atter

isospin content, it is in teresting to see w h at the Skyrm e M odel predictions are for dense baryonic m a tte r. D ue to econom ical dual description of b o th baryonic s tru c tu re an d baryonic in teractio n s, th e m odel presents a m eans of describing th e rad ical changes which occur w hen the density of baryonic m a tte r is increased. T h e su bject of C h ap ter 3 will be to review th e resu lts of these calculations an d to present our own resu lt for an fee arrangem ent of skyrm ionic m a tte r.

An im p o rtan t insight into th e n a tu re of dense skyrm ionic m a tte r has been gained w ith th e realisation of a sim ple altern ativ e an d com plem entary m odel of dense baryonic m a tte r to th a t of th e flat space arrays.

T h e flat space static a rra y calculations we shall later review, require one to solve num erically for th e m inim al energy field configuration a t a given average baryon density. For these array s, a set of tw isted periodic b o u n d a ry conditions are im posed and th e resu ltin g m inim al energy fields can be g en erated by la t­ tice tra n slatio n s from th e solution w ith in a rep resen tativ e u n it cell. Even so these calculations are tedious, it being technically difficult to o b ta in converged solutions.

for flat space array s on Sphy( L), is given by B / 2n 2L3 for a s ta tic solution of w inding n u m b er B an d th u s for a given b ary o n n um b er we can find solutions a t all average b ary o n densities by varying th e rad iu s L, of th is sp h ere. A single

B — 1 sky rm io n on S^hy(L) will have a self in te ra ctio n due to th e c u rv a tu re of space a n d th is is believed to give a good m odel of th e in te ra ctio n s of skyrm ions w ith in flat space cry stalline baryonic m a tte r. T his will also be tru e in th e higher b ary o n n u m b er sector on S^hy(L).

T h e resu lts of calculations w ithin b o th th e B = 1 a n d B = 2 sectors have indeed been show n to have rem ark ab le sim ilarities to those o b ta in e d num erically for flat space array s. O f course, even th o u g h a n aly tical resu lts are possible w ith in th e B = 1 secto r, d irect physical resu lts are n o t possible w ith th is sim pler m odel since real physical space is flat n o t spherical.

For th e m om ent we sh all describe th e B — 1 secto r resu lts on S^hy(L) a n d leave th e draw ing of parallels w ith th e physical flat space array s u n til we p resen t re su lts in th e n ex t ch ap ter.

T h e Skyrm e M o d el’s sta tic energy fu n ctio n al on Sj)hy(L) in M a n to n ’s dim en- sionless units[21] read s,

E = f s \ (l) d V ^ i i K i ’ + (*-9.1)

w here gij are th e com ponents of th e m etric on Sphys(L), w ith t , j ru n n in g from 1 to 3 and

w here the dim ensionless fields satisfy th e co n strain t (f)a4>a = 1. In eq u atio n (2.3.1)

d V is th e m easure on S^hy(L). T he dim ensionless u n its of length an d energy can be related to physical u n its by rescaling th e dim ensionless energy u n it by /*■/2e = 5.92M e V and th e dim ensionless length u n it by l/e /V = 0.561 ferm i.

Using th e sta n d a rd extension of p o lar coordinates for physical space,

0 < / u < t t , O < 0 , < £ < 2 t t (2.3.3)

th e m etric com ponents, , are diagonal w ith elem ents,

g = (L 2, L 2 sin 2 /i, L 2 sin 2 / i s m 2 0). (2.3.4)

T h e B = 1 field configuration we shall assum e to be of the hedgehog form . T his gives,

(jP = C O S f(fl)

(j)1 = sin f(fj,) cos 0 <f>2 = sin f ( n) sin 0cos <j)

<f>3 = sin f ( n) sin 0sin <f>. (2.3.5)

In ord er to ensure th e hedgehog has w inding num ber one, we have th e b o u n d a ry conditions for th e profile function

/ ( o ) = 0, / ( 7r) = 7r. (2.3.6)

T h e m a trix Kij is now diagonal w ith elem ents

w here / denotes th e derivative of /(/x ) w ith respect to fj, and hence th e static energy fu n ctio n al takes th e form

9 sin 2 / , sin2 f . • - sin 2 / , , . ,

E ( f ) = 4tt / d/xsin /x { £ (/ + 2 ^ — ) + . -5 (2/ + )}, (2.3.8)

Jo sin /x L s in fi sin /x

w here we have perform ed th e triv ial 9, <f> integrations. T h e E uler E q u a tio n for th e m inim al energy hedgehog field is o b tain ed from a functional v ariatio n of E ( f) w ith respect to / ( a x) an d is

, » sin 2 / , " oSin2u* s in 2 / ■, s i n 2 / . , , sin2 / , , ,

(L +

—r -)f

(Z-M

sin /x sin /x sin fi sin /x sin fj.

E x am in atio n of this E uler E q u atio n , which is an o rd in ary second o rd er differential eq u ation , reveals th a t th a t th ere exists a triv ial solution

/ ( A x ) = M (2.3.10)

a t all values of L. T his solution is th e triv ial m apping of S 3ky(L) onto 5 t? 0(l). T h e energy (2.3.8) of th e trivial m ap is th en given by

E = Qn2(L + y ) . (2.3.11)

L

T his a tta in s a m inim um energy a t L — 1, w hen it is sim ply th e id en tity m apping,

of two th ree spheres of rad iu s one, which s a tu ra te s th e lower b o u n d w ith a value of th e energy 127T2. T his com pletely delocalised solution has th e full 0(4) sym m etry of th e Skyrm e M odel and is an isom etry of S 3.

T he baryon density for a hedgehog skyrm ion is given by

^ n / v 1 sin2 f •

B°(n)

ir-2 TZ - \

{e-s.ie

w hich for th e triv ial m ap (2.3.10), reduces to 1/27r2L 3 and is a c o n stan t over phys­ ical space. T h u s, on a th ree sphere th e triv ial skyrm ion represents a com pletely delocalised baryon over the whole of space.

As th e rad iu s of physical space is increased beyond one, th e energy of the triv ial solution (2.3.11) grows, becom ing linear a t large L. A t infinite volum e th e triv ial solution th u s has infinite energy. At some in term ed iary volum e, th is solution, which is an absolute m inim al energy solution a t L — 1, m ight be ex­ p ected to becom e un stab le w ith respect to an o th er solution which does have a finite energy in th e infinite volum e lim it an d w hich would correspond to th e flat space hedgehog as th e cu rv atu re of physical space becomes zero in th is lim it.

T his is indeed th e case and it as been d e m o n strated th a t th e triv ial m ap becom es u n stab le to a hedgehog solution of eq u atio n (2.3.9), above a volum e corresponding to a critical value of L = L c = \ / 2 . T h e tra n sitio n a t L = L c is of second o rd er an d in stab ility of th e triv ial m ap has been show n to be due to th e existence of infinitesim al conform al m odes which are th e p a th s of steep est descent in energy from th e triv ial m ap a t L = L c. T h u s, up to a volum e of

L — \J2, th e triv ial delocalised solution is a local m inim um of energy an d beyond this critical rad iu s it is a local m axim um . T he local conform al m odes resu lt in th e m inim al energy hedgehog solution beyond L c, th is becom es increasingly localised ab o u t one of the poles of Sphy(L) as L increases. T h u s, th ere are two hedgehog solutions in th is region of L corresponding to localisation cen tred upon eith er pole. These two solutions are degenerate in energy an d tran sfo rm ed into

each o th e r by th e tra n sfo rm a tio n

f {f i ) ■-> 7T - f ( w - ti). (2.3.13)

In th e lim it of infinite volum e th e hedgehog skyrm ion is cen tred on one of th e poles, being well localised a n d having a finite energy co rresp o n d in g to th a t of th e flat space hedgehog [22].

T h e in sta b ility of th e triv ia l m ap w ith resp ect to infinitesim al conform al m odes, for values of L g re a te r th a n \ / 2, in general p ro d u ces a localisation of th e sk y rm io n a b o u t any p o in t on S*h (L). However, th e hedgehog fo rm (2.3.5) is only co n sisten t w ith th e skyrm ion localising a b o u t eith e r pole a n d hence gives a two-fold degeneracy of localised solutions.

M an to n [21] fu rth e r d e m o n stra te d th a t th e triv ial m ap solution on S 3phu(L)

is an ab so lu te m in im u m of energy for volum es of physical sp ace less th a n or equal to th a t of th e ta rg e t th re e sphere. B eyond these volum es, for an a rb itra ry space, th is is no longer th e case, as we shall d e m o n stra te in C h a p te r 4 w hen we investigate th e effect of generalising th e sh ap e of th is sp h ere. T h e global sta b ility of th e triv ial m ap up to th e critical volum e for th e sp h ere is however, still a realistic possibility w hich up un til now has n o t been d e m o n stra te d .

th e p o te n tia l energy surface g en erated by varying th e average b ary o n den sity a n d have n o t included kinetic effects. Since all o u r resu lts are th u s a t zero kinetic energy, th e re is no te m p e ra tu re involved a n d s tric tly speaking this is n o t a p h a se tra n s itio n .

Sim ilar resu lts were also found for th e B = 2 [23] sector of th is m odel, w ith a second o rd er p h ase tra n s itio n occu rrin g a som e critical ra d iu s an d th e low density p h ase having two sp h erical skyrm ions cen tred on opposite poles of th e sp h ere.

T hese p h ase tra n sitio n s have been in te rp re te d as rep resen ting th e re sto ra tio n of C h iral S y m m etry a t high densities [24], in th e sense th a t for b o th th e B — 1 a n d 2 system s th e average values of th e sigm a field < a > an d pion field < n > , are zero in th e high density phase. T his in te rp re ta tio n was fu rth e r ju stifie d by th e o bservation th a t for th e B — 1 solu tio n s a t th e tra n s itio n p o in t, th e fo rm atio n of th e high density p h ase is associated w ith th e d isap p e a ra n ce of th e th re e G oldstone m odes.

T his sim ple m odel th u s provides us w ith an insight in to th e n a tu re of th e different form s of baryonic m a tte r a t high densities w hich th e Skyrm e M odel p red icts. Indeed, for flat space arrays of sk yrm ions, sim ilar delocalising p h ase tra n sitio n s have been discovered as we shall describe in C h a p te r 3.

effective degrees of freedom of dense skyrm ionic m a tte r an d gives th e possibility of perform ing finite te m p e ra tu re calculations for dense skyrm ionic m a tte r.

i Ii t

\

t

t

R e fe r e n c e s

[1] H .D .P o litzer, Phys. Rev. L ett. 30 (1973) 1346

[2] D .J.G ross and F.W ilczec, Phys. Rev L ett. 30 (1973) 1343

[3] S.A lder, Phys. Rev. L ett. 117 (1969) 2426

[4] J.B ell an d R .Jackiw , Nuov. C im ento 60A (1969) 47

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C H A P T E R 3

D E N S E S K Y R M IO N IC M A T T E R

I N A C R Y S T A L L IN E F O R M

3.1 M u lti-S k yrm ion C rystals

In C h a p te r 2 we have seen th a t th e Skyrm e M odel provides a topological de­ scrip tio n of a generalized classical baryo n w hich is b o th elegant a n d physically reaso n ab le, w ith m ax im u m discrepancies of a ro u n d 30%. M oreover, th e large

N c ex pan sio n of Q . C . D., suggests th a t one loop co rrectio n s to th ese classical resu lts will be of o rd er 1/ N C an d w ith th e physical value of N c being 3, these d iscrepancies seem reaso n ab le a t th e classical level.

T h ese successful developm ents in th e a p p lic atio n of th e Skyrm e M odel to light nu clear sy stem s have been m irro re d by progress w ith in th e re a lm of infinite b ary o n ic m a tte r system s. Indeed, it is n a tu ra l to believe, th a t w ith in th is sp h ere th e m odel m ay provide an econom ical d escrip tio n of infinite bary o n ic m a tte r an d th a t it m ay provide a d escrip tio n of heavy nuclei o r n e u tro n s ta r m a tte r.

T h e econom y of th e d escrip tio n offered by th e S kyrm e M odel for such sy stem s, follows from th e m a n n e r in which b o th th e s tru c tu re a n d in te ra ctio n s of b ary o ns are placed on a sim ilar footing from th e o u tse t. For infinite bary o n ic m a tte r, th is provides th e o p p o rtu n ity of d escribing th e rad ically different form s of baryonic m a tte r ex p ected to exist a t high a n d low densities.

In C h a p te r 2 we also discussed M a n to n ’s [5] re su lts for a sk y rm io n on Syhy(L)

a n d th is revealed two d istin c t form s of skyrm ionic m a tte r t h a t ex ist a t hig h an d low average b ary o n density. A t low densities th e sk y rm io n ’s b ary o n d en sity is well localised a b o u t a p o int in space, w hile a t high densities it is com pletely de­ localised over th e w hole of space. T h e econom y p ro v id ed by th e S kyrm e M o d el’s d e sc rip tio n of bary o n ic m a tte r, re su lts here in a single d e sc rip tio n of b o th phases of m a tte r. T h is w ould n o t have been th e case h a d tw o differing m odels been re q u ire d to describe these tw o differing phases.

b o th th e crystalline an d Syhy(L) environm ents.

T h e b e a u ty of th e Syhy(L) m odel of skyrm ionic m a tte r is its sim plicity, en­ abling one to o b tain analytical resu lts which have subsequently revealed th e existence of a second order phase tra n sitio n an d this is due to th e presence of a local conform al instability. An analogous conform al in stab ility in flat space has also been proposed, in order to explain th e existence of an analogous tra n sitio n from a condensed to an uncondensed array of skyrm ions a n d we shall see later th a t th is proposal would seem to have been num erically justified.

K lebanov [6] pioneered th e a tte m p ts to co n stru ct infinite array s of skyrm ionic m a tte r an d recently th is field has a ttra c te d increased a tte n tio n . His m odel was based on th e belief, th a t a classical array of crystalline skyrm ions m ight describe th e dense n e u tro n m a tte r which exists a t th e core of a n e u tro n sta r.

d istin g u ish betw een th e two. However, th e high q u a n tu m zero p o in t energy, associated w ith a crystalline array of localised n eu tro n s, will w ork ag ain st th e fo rm atio n of such crystalline m a tte r.

K lebanov was th u s led to consider an infinite cubic crystalline array of skyrm ions, b ased in p a rt on th e cubic array of n eu tro n s proposed by S m ith e t al. It is n o t however possible w ithin this m odel, to e stim ate th e effect of th e zero point m otions of skyrm ionic m a tte r, due to th e n on-renorm alisability of th e Skyrm e M odel.

T h e asy m p to tic form of th e tensor p o ten tial betw een two skyrm ions is highly an iso tro p ic an d th e ir topological n a tu re enforces stro n g repulsive forces a t high densities. Hence we see, th a t th e two dom inant nucleon-nucleon in teractio n s in n eu tro n m a tte r have analogies w ith in th e Skyrm e M odel a t th e classical level.

T h e arg u m ent th a t led K lebanov to deduce th e s tru c tu ra l form th a t an in­ finite a rray of skyrm ions should take, was based upon th e insights w hich had previously been gained from th e u n d erstan d in g of th e in teractio n betw een two

B = 1 hedgehogs a t large sep aratio ns.

an angle n.

T h u s, in order to p ictu re this arran g em en t, we im agine th a t b o th B = 1 hedgehogs are s itu a te d on th e x axis an d located a t x = ± 00. P resu m in g th a t th e one a t x = —00 is u n ro ta te d and th u s has its pion field pointing radially outw ards from its centre, th e o p tim a l field configuration has th e pion fields of th e hedgehog located a t x = + 00, iso -ro tated a b o u t an axis lying in th e (x, y) p lan e th ro u g h an angle 7r. W ith o u t loss of generality, we can choose this axis to be th e z axis, th en th e hedgehog a t x = + 0 0 has its pion field in th e (x ,y ) plane poin tin g radially inw ards, while along th e line parallel to th e z axis, which passes th ro u g h its centre, its pion field po ints outw ards. T h u s, on tran sv ersin g th e ar-axis from th e centre of th e hedgehog located a t x — —00, to th e centre of th e hedgehog located a t x = + 00, th e pion field on th is axis flows in th e positive x direction. T hus, th is o p tim al arran g em en t has th e pion fields of these two hedgehogs relatively n o n -fru strated . All o th e r n o n -fru strated o ptim al arran g em en ts of tw o hedgehogs are re la ted to th is configuration by a com bination of c o n sta n t global space and iso-space ro ta tio n s. T hese ro ta tio n s leave th e Skyrm e L agrangian invariant and are th u s sy m m etry tra n sfo rm a tio n s of a skyrm ion field.

In o rd er to c o n stru ct o p tim al arrays of skyrm ions, it is useful to im agine placing hedgehogs a t th e points of an infinite lattice. In th is way, K lebanov deduced th a t in order to m axim ise this binding, one should d em an d th a t nearest neighbouring hedgehogs be optim ally o rien tated . T h e pion fields of neighbouring hedgehogs will ru n sm oothly in to each o th er an d neighbouring hedgehogs will have th e ir pion fields relatively n o n -fru strated .

To illu stra te th is, let us first consider a sim ple cubic a rra y of skyrm ions. We choose a p a rtic u la r lattice point to define our origin and th e lattice spacing to be

a. A t each point on th e lattice th e re is situ a te d a hedgehog, w hich a t th e origin we choose to be u n ro ta te d so th a t its pion field points rad ially ou tw ard s. We shall now consider those hedgehog skyrm ions located on p o in ts w ith in th e plane

z = 0 . If we move to th e hedgehog located a t th e point (a, 0 ,0 ), we should require t h a t th e pion field of th is hedgehog be iso -ro tated a b o u t an axis lying in th e (y, z)

p lane by an angle 7r, in order to pro du ce a n o n -fru strated pion field arran g em en t along th e x axis. W ith o u t loss of generality, we choose a t th is p o in t th a t th e hedgehog be ro ta te d ab o u t th e z axis by an angle 7r. O n moving along th e line