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Submitted on 1 Jan 1977

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Applications of topology to the study of ordered systemsR. Shankar

To cite this version:R. Shankar. Applications of topology to the study of ordered systems. Journal de Physique, 1977, 38(11), pp.1405-1412. �10.1051/jphys:0197700380110140500�. �jpa-00208711�

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APPLICATIONS OF TOPOLOGY TO THE STUDY OF ORDERED SYSTEMS (*)

R. SHANKAR (**)

Lyman Laboratory of Physics, Harvard University, Cambridge, Massachusetts, 02138, U.S.A.

(Reçu le 14 avril 1977, accepté le 22 juillet 1977)

Résumé. 2014 Cet article est une version détaillée d’un preprint récent dans lequel le problème étaitanalysé d’une manière très concise. On donne ici d’abord une introduction pédagogique à la théoriede l’homotopie puis ses applications à l’étude de configurations non singulières dans les systèmesordonnés tels que 3He-A, les cristaux liquides nématiques, etc. On examine aussi brièvement le rôlede l’hamiltonien pour décider de l’observabilité et de la stabilité de ces configurations.

Abstract. - This paper is a detailed version of a recent preprint wherein the above topic wasanalysed rather concisely. A pedagogical introduction to homotopy theory is followed by its applica-tions to the study of non-singular configurations in ordered systems such as 3He-A, nematic liquidcrystals etc. The role of the hamiltonian in deciding the observability and stability of these configura-tions is briefly examined.

LE JOURNAL DE PHYSIQUE TOME 38, NOVEMBRE 1977,

Classification

Physics Abstracts02.40 - 64.60 - 61.30

1. Introduction. - Several years ago Finkelstein [ 1 ]pointed out and illustrated with several examples,how a branch of topology, called homotopy theorycould be profitably used by physicists. Only recently,following the invention of monopoles, instantons etc.have particle theorists come to appreciate this fact.Even more recent is the work of Blaha [2] and Thou-louse and Kléman [3] wherein homotopy theory wasused to classify singularities or defects in orderedsystems such as ferromagnets, nematics etc. In arecent and somewhat concise preprint [4] 1 hadreverted to the approach pioneered by Finkelstein,namely to the study of topology stable but non-

singular configurations in ordered systems and discuss-ed the conditions on the energy functional R fortheir observability. The present work is an expandedversion of reference [4] and contains several detailsomitted therein, in addition to an introduction to

homotopy theory provided for nonexperts.

2. Introduction to homotopy theory. - The orderedsystems in question will be described by a field y(x)defined over the points x of some spatial domain X.The symbol y can stand for a unit spin vector s,if the system is a ferromagnet, the director n, if it isa nematic etc. Let us denote by Y, the manifold of

(*) Research supported in part by the National Science Foun-dation under Grant No. PHY 75-20427.

(**) Junior Fellow, Harvard Society of Fellows.

possible values for y. For example Y = S1, a circle,if y is a two component vector of fixed length ; Y = S2a 2-sphere, if y is a three component vector of fixedlength ; Y = solid sphere of radius 2 7r if y is a 3-vectorof length 5 2 TE and so on.

Let us begin with X, a one dimensional interval,parametrised by 0 x , 2 n. Consider some fielddistribution y(x). So each point xi e X, there is animage point y(xi) in Y. As x varies from 0 to 2 n,the image point traces out a curve in Y starting aty(O) and ending at y(2 n). Let us restrict ourselvesto fields obeying y(O) = y(2 7r) = yo, where yo iscalled the base point. A compact way to write this isy(DX) = yo, where DX denotes the boundary pointsof X. In this case the image point traces out a closedcurve anchored at yo. In every configuration of thesystem there exists such an image loop and vice versa.The study of the system thus reduces to the studyof the corresponding closed loop in Y.

Suspending for a while the question why, letconsider the classification of the configurations ofthe loop into classes such that :

i) any two loops belonging to the same class canbe continuously distorted into each other, and

ii) loops belonging to different classes cannot

be continuously distorted to each other. The classesare called homotopy classes and members of a givenclass are said to be homotopic to each other.

Consider for example the case where Y = E’the two dimensional Euclidean space which we may

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0197700380110140500

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take to be the x y plane. Let yo be any point, say (1,1).It is evident that there is only one homotopy class :any loop can be deformed into any other and inparticular to a point loop at yo. In terms of the fieldy(x), it means that any configuration y(x) can becontinuously deformed to the constant map y(x) = yo.

Consider next Y = E2 - (0, 0). (The exclusion1 of the null vector from the allowed values can occurfor example if y is the magnetic field in a plasma,see Finkelstein and Weil [5].) Let yo be any point in Y.It is clear that while any loop not enclosing the origincan be collapsed to yo, those which do enclose theorigin cannot be. The latter can be further classifiedby means of an integer m, whose magnitude and signspecify the number of times and the sense in whichthe origin is encircled. It is also clear that mapsbelonging to distinct values of m are non-homotopic.Consider the set {yO, y 1 y2, ... 1, where ym standsfor all maps in which the loop encircles (0, 0) m times.It turns out that this set forms a group, that is to say,there exists a law of combination of the élémentswhich obeys the group axioms. The group is calledthe first homotopy group of Y, and denoted byHi(Y). The subscript 1 tells us that X acquires theform of a closed one dimensional surface, i.e., a loop,which topologically is equivalent to a circle of 1-sphere SI. Our interest here will be limited to theset of elements in H,(Y) and not involve their groupstructure. For an introduction to the latter, see forexample reference [6] and additional referencescontained therein.

Now, we have seen that the elements of

H,[E’ - (0, 0)] are in one to one correspondencewith the integers m. A study of the group also revealsthat their law of combination is identical to theaddition of the corresponding integers. One denotesby Zoo the set of all integers forming a group underaddition. These results may be summarized as

Notice that the choice of base point yo plays a veryminor role. It will not be explicitly discussed in thefuture. If our region X is two dimensional and allpoints on the perimeter ôX map onto a single point yo,X will assume the form of a closed two-dimensional

surface, topologically equivalent to a 2-sphere S2.The group in question is il2(Y). Likewise if X werea three dimensional region of space, the group tostudy is I13(y)-A word about the condition y(ôX) = yo. For

infinite systems, the requirement that y - a constantyo at infinity ensures the finiteness of the energy,assuming the existence of gradient terms. In finitesystems this boundary condition may be forced byan external field or the walls of a container. In anyevent it is only with this condition that X becomesa closed surface in Y and the groups become the wellstudied il n(Y)’ We now turn to the question that has no doubt

tested the patience of the reader : why the homotopyclassification ? There are two reasons as far as weare concerned :

i) As long as the time evolution is a smooth andcontinuous change, it follows that the homotopyindex, such as the integer m, is a constant of themotion. We thus have a conservation law without

discussing the hamiltonian in any detail, except toassume it produces smooth changes in the state ofthe system.

ii) We can anticipate a family of metastable configu-rations one from each class. Consider the energyfunctional, which we shall refer to as the hamilto-nian X. In each class, there must be some configu-ration that minimizes it. This is a metastable state :it’s stability within the class is a result of its beinga minimum, while passage to other classes is topolo-gically forbidden. We are reminded of the followingsituation in one dimensional quantum mechanics :while finding the absolute minimum of the energyyields the grounds state, we can get the first excitedstate by minimising in the odd parity sector (assumingJe is parity invariant).With this motivation we proceed to study some

of the groups that will interest us.

3. The groups nm(sn) and nm(pn). - We will beinterested here only in the groups llm(sn) and Ilm(P"),where P" is the projective n-sphere, that is the n-sphereS" with diametrically opposite points identified.Both m and n will be less than or equal to three.Although the elements of these groups are listedin text books, let us consider one concrete examplethat is amenable to intuitive analysis : the groupH,(S,).The aim is to map a closed loop X onto a circle

Y = S’ which is also parametrised by 0 y 2 n.There are first of all maps in which X never completesa tum around Y. An example is given in figure 1.All such loops may be deformed to a point, say aty = 0. We label these maps by an integer m = 0which counts the number of times X covers or wrapsaround Y. There is another equivalent definitionof m. Pick any point y of Y and consider all the

FIG. 1. - a) This map is not a constant, but is homotopic to theconstant map y(x) = yo. It has m = 0 ; b) This map has m = 1.

The identity map y = x is a special case.

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points x e X that map onto it. To each one assigna number ± 1, this sign being that of dyldx at thatpoint. Then m is the algebraic sum of these numbers.Yet another definition is

It is a very practical way of finding m when ourintuition breaks down. (These recipes are easilygeneralised to the case where X and Y are n-spheres,with dy/dx being replaced by the Jacobian J( y/x).If Y is Pn ; the rule can be applied after disregardingthe identification of antipodal points). In figure la,if we choose a point y untouched by the loop X,m is clearly 0, while if we choose a point to whichtwo points of X are mapped, the opposite signs ofdy/dx cancell to give m = 0.

Consider next the case where X goes around Yonce clockwise (Fig. 1 b). A special case called theidentity map, is y(x) = x. It is clear m = 1, followingany of the three definitions. (Note that the identitymap y = x is a nontrivial map and must not beconfused with the identity element of the groupwhich corresponds to trivial m = 0 map.) Likewisem = - 1 if X goes around Y counterclockwise.

Clearly m can be any integer and maps labelled bydifferent m are nonhomotopic. We may summarizeour results as :

Strictly speaking what we have established is notthe equality of the two groups, but only the corres-ponding sets.

It can be similarly shown that ll2(S2) = Zoo’For example in the identity (m = 1) map, X covers Yas the skin of an orange covers it. If m > 0, X wrapsaround Y m times.

Having illustrated the general principles by meansof the preceeding discussion, let us accept withoutproof the following results true for n > 2 :

where 1 is the trivial group with just the identityelement. The exception to the rule, n = 1, will notbe discussed in any detail here.

Among the maps with n > m ; the only nontrivialone with n 3 is il 3(S2). Hopf [7] showed that

and provided representative maps from each classcalled Hopf maps. Since X and Y are spheres ofdifferent dimensionality, the integer m has a some-what complicated interpretation, which we will discussin due course.Our mathematical preparation is complete. We now

tum to the applications.

4. Applications to configuration with singularitiesor defects. - A singular point x E X is characterizedby the fact that as we approach x from many direc-tions we get many limiting values for y. Althoughsingular maps are not our subject, we review brieflythe interesting work of Thoulouse and Kléman [3]because our machinery makes it easy to do so andbecause the relationship between this work and theirsis worth understanding.

Let X be a two dimensional region. Let us sur-round it by a closed loop or circle S 1 and study thevariation of y on it. If the map on SI is a nontrivialelement of lll(Y), (that is, not homotopic to theconstant map) a defect or singular point lies within.To see this, imagine slowly shrinking the loop SIin size. The map on it must also change smoothly,and hence remain in the same class. Ultimately theloop must shrink to an infinitésimal one surroundinga point. The distribution on this infinitesimal loopis non constant by assumption. As a result, yapproaches different values as we approach this

point from different directions. If X is a three dimen-sional region a similar analysis in terms of 112(y)is applicable.

Notice the analogy between the way in which adefect in the interior manifests itself via the mapon the boundary and

i) Gauss’ law, which relates the charge in a volumeto the flux leaving a surrounding surface and

ii) Cauchy’s theorem, relating the integral of ananalytic function on a closed contour to the residuesto the singularities within.

Now, just as opposite charges or poles with oppositeresidues may neutralize each other at the boundary,so may two (or more) defects annul each other toproduce a trivial map on the boundary. To see howthis happens however, we must know how to combinethe maps, that is, the group structure, which is unfortu-nately behond the scope of this paper.Although defect analysis also involves homotopy

theory, there ends the similarity with our presentanalysis. A defect theorist is interested in mapswith nontrivial behaviour at the boundary and adefect inside, while we deal with maps trivial at theboundary y(DX) = yo, with no defect or singularityinside. We now turn to our subject proper.

5. The nematic liquid crystal in two space dimen-sions. - Let us first take as the two dimensionalregion X the entire x-y plane. The order parameteris the director n which we parametrize as

n = (nx, ny, nz) = (sin j8 cos a, sin fl sin a, cos P),where a and p are the azimuthal and polar angleson S2. Since n = - n physically, Y = P2 and notS2. This distinction is however irrelevant for the

topological group we will consider. We impose thecondition that at spatial infinity n = (0, 0, 1), that

is fl = 0. This makes the relevant group H2 (p2) = Zoo’

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Given an energy functional or hamiltonian Je, onemetastable configuration from each class may befound by minimising within that class. This non-trivial task has been done by Belavin and Polya-kov [8] (here-after referred to as BP) for the case

which describes a nematic with all three elastic cons-tants equal. It was shown by BP that :

a) in each class JC >, 8 na 1 m 1b) the minimum is attained by the configurations

where z = x + iy ; and  is an arbitrary scale para-meter. The restriction L mi > Y, ni ensures that as1 z 1 -+ 00 ; P --+ 0. The homotopy index is m = E miand it measures the number of points in the z planethat yields the same w. Woo [9] has shown that theseminima are the only metastable configurations.

Consider next the more realistic case of the nematicin a cylinder of radius R, axis perpendicular to thex-y plane, and walls such that n likes to be parallelto them, that is n = (0, 0, 1) at the walls. The topo-logical classification is then the same as in the BPcase. One hopes that there will exist a one to onecorrespondence between the BP configurations andthe metastable configurations in the finite cylinder.The reasoning is that as one adiabatically shrinksthe infinite cylinder (the x-y plane) the configurationswithin will evolve adiabatically. But upon examiningthe equations 1 find that all configurations are point-like, unlike in the BP case. Such configurations areof no physical interest. We are thus faced with animportant fact : even if the configuration space ofa system admits interesting homotopy classes, thèconfigurations that minimise Je in each class may bephysically unobservable or even non existant. Wewould now like to trace the origin of this disasterin the case of thé je in eq. (5.1).Our analysis follows closely that of Derrick [10]

and is dimensional in nature. We start with the factthat in eq. (5.1) is scale invariant, that is to say,two configurations n(x) and n(x/Â), related by a

scale transformation have the same energy. That is

Consequently, the minimum in each class is attained,not by one configuration, but a family related byscale parameter  (see eq. (5 . 2)). (The family is actuallyeven bigger, but that does not concern us here.)The BP solutions thus represent a system in neutral

equilibrium with respect to scale changes. This is

why the adiabatic arguments are invalid : no matterhow smoothly the walls of the cylinder are broughtin (from infinity to R) the BP configurations collapseunder the push to zero size, with no restrainingforces.

What we need is an Je with terms that grow as the

configuration shrinks. One such example is

The (grad)4 term averts the complete collaps, sinceit grows as 1/d2, d being the size of the configuration,where by size 1 mean the spatial distance over whichthe field differs appreciably from yo. With the wallspushing in and the b term pushing out, there will besome size do which minimises the energy. Severalremarks are in order :

i) Since the system is in reality discrete, there isa fundamental length 10, which is of order moleculardimensions ; the size do of the continuum solutionshould be much greater than 10 for it to have a sensiblecounterpart in the discrete case.

ii) The (grad)4 term is only an example and anyterm of that dimensionality, such as (Ox n)4 is equallygood. These will be referred to as b-terms.

iii) The domain X must be a finite cylinder. Butfor the walls, the 1/d2 nature of the b-term willcause the configuration to expand without limitin the process of minimising its energy.

iv) If X were a 3-dimensional region, the a andb terms, which contribute as d and 1/d respectively,will lead to an equilibrium size do, even if X is allof space.

v) Configurations larger than the equilibrium onemay also be observed under some conditions. Thesewould of course shrink towards the size do. If howeverthe shrinking is slow, they may possibly be observedin the process. This is not a far fetched possibility.Certain closed loop structures in nematics have beenphotographed in the process of collapse, which takesa few seconds [11].

vi) There are no general principles, as in quantumfield theory, that forbid b-terms or even highergradients, though we cannot vouch for their size.That there must be some b-term can be seen as follows.

Although we treat the system as a continuum, weknow there is an underlying length 10. Fields that

vary appreciably over the scale must be excluded,that is, we must impose a cut off. The b-terms, whosecontributions to Je grow rapidly for such configura-tions effect such a cut-off. One cannot say moreabout the size of the b-terms in general.

vii) The above points i)-vi), dimensional in nature,apply to other systems as well and not just nematics.

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There is one BP configuration that can be observedin part in a finite cylinder, quite independent of theb-terms. It is the m = 1 map

p and ç being cylindrical coordinates on the plane.Notice that at p = /), fl = 7C/2 and ç = a, that is,the director lies in the x-y plane and is everywhereradial. It is clear that the introduction of a cylinder,whose axis is along the z-direction, and whose wallslie on p = /L, will not disturb the configuration within,provided n likes to be normal to the walls. Such aconfiguration has in fact been observed and studiedby Cladis and Kléman [12], William, Pieranski andCladis [13] and Meyer [14]. To the best of my know-ledge none of the other BP configurations can beobserved in this manner.

6. The nematic liquid in three space dimensions. -Let X be all of three space, parametrised by thespherical coordinates r, 0 and (p. Given thatn - (0, 0, 1) as r -+ oo, X becomes equivalent to

S3 and the relevant group is 113(P’) = Z,,,,. Let usdiscuss briefly how representative maps from eachclass, called Hopf maps, are obtained and the signifi-cance of the index m.

Let us first trade the coordinates r, 0 and ç forthose describing S3. Rather than use the angularcoordinates that span the surface of S3, we preferfour cartesian coordinates xl, x2, X3 and X4 cons-trained by

We have chosen the radius of the sphere S3 repre-senting X to be unity. The relation between the twosets of coordinates is

Note that the north and south poles of S3, xi = ± 1respectively, go to r = oo and 0 respectively. Ourmap is the three dimensional version of the familiar

stereographic projections of S2 on to a plane. Themaps W(Xl, X2, X3, X4), where w = e"’ cot (flf2) aremore easily described in terms of complex coordi-nates Zl = xi + ix2 and Z2 = X3 + ix4 constrainedby 1 Zl 12 +1 z2 12 = 1. The Hopf maps have thefeature

It can easily be verified that the allowed range forthe variable z is the entire complex plane. Now,we already know that if X is the entire x-y planeand Y = p2 (or S2), there exist maps which fallinto classed labelled by an integer m, the BP configu-rations being representative elements from each class.In fact the Hopf map of degree m,

is just the BP function of index m with no poles andall m zeros at the origin. Retracing our steps backto r, 0, and ç, we get

The broad features of wm are as follows. Centeredat the origin is a circle of unit radius lying in thex-y plane called the core on which fi = n and aroundwhich the field varies most and the energy is mostconcertrated. The core is surrounded by tori of

increasing size and decreasing fi. Their cross sectionsin the half plane (p = ço are circles, whose centersmove away from the core as they grow in size (Fig. 2).The cross sections are like the closed lines of magneticfield that would surroûnd the core if it were a currentcarrying loop. The largest of these tori with p = 0is bounded by the x-axis and the sphere at infinity.If we move once around the circular cross sectionof the torus fl = Po in the ç = 0 plane, am will rangebetween 0 and 2 nm (see Fig. 2). In other words,as one describes this circle in space, the image pointcompletes m turns around the sphere Y = P2 startingat the point a = 0, B = Po and staying on the lati-tude fl = B0. In any other plane a = (po, the only

FIG. 2. - A cross section in the (p = 0 half plane of the Hopfmap. The core (fi = 1t) comes out of the paper as a point. Eachcircle corresponds to a given fi. On any one circle fl = Po; a varies

from 0 to 2 xm.

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difference is that the image point starts out at a = mqJo,fl = Po and completes m terms around the latitudefl = Po. The natural coordinates for these mapsare the toriodal ones, /1, 11 and ç [15], in terms ofwhich

The Hopf maps are merely representatives fromeach class. To find metastable configurations onemust find the minima of R within each class. If aand b terms are present and b is large enough obser-vable metastable configurations can be expected. Thetheory also predicts that Hopf maps larger than thestable one may be observed collapsing if the processis slow enough. Now such configurations with a core,have been seen in nematic liquids [11]. Although theystart out with arbitrary shapes they become circularas they shrink. It is tempting to identify these withHopf maps or more generally with maps homotopicto them. A more detailed experimental study is

needed, especially near the core, to confirm this

hypothesis. On the theoretical side remains the

question of minimising X in the presence of a and bterms. Enz [16] has solved the problem approximatelyand obtains maps homotopic to the Hopf map,following a route very different form ours.

7. 3 He-A in three space dimensions. - Let X beonce again all of 3-space. The order parameter is atriad of orthonormal vectors 1, x, and x2, where 1

is the orbital angular momentum, while x 1 and x2determine the complex phase in the plane perpendi-cular to 1. Our boundary condition is that at spatialinfinity the triad assumes a standard form with xl,x2 and 1 pointing along the x, y and z axes respectively.The triad at any interior point with position vector rmay be obtained by performing an SO(3) rotationon the standard triad at infinity. Let the vector anglen(r) specify by its magnitude and direction, the angleand axis of the rotation. Since n is a vector of arbi-

trary orientation and magnitude 5 2 n, Y = solid

sphere of radius 2 n. Since all points at the boundaryrefer to the same (identity) element, the topologyof Y is S3. The careful reader would have noticedthat our Y actually double counts the elements ofSO(3), in the following sense. Although the SO(3)rotation angle can range from 0-2 n about any axis,a rotation by an amount 0 about any axis is physicallyequivalent to a rotation by 2 n - 0 with the axisreversed. Thus we need only a solid sphere of radius n.Our sphere S3 thus represents SU(2), for which

0 , 1 Q 1 4 n. It can be readily shown that Yfor SO(3) is equivalent to p3 (see appendix). Weshall continue to use Y = S3 since the distinctionis inconsequential : H3 (p3) = il 3(S3).

Notice that the manifolds X and Y are both solid

spheres with points on the surface identified, and

are thus equivalent to S3. At least one nontrivialmap suggests itself readily : we rescale X to a sphereof radius 2 n, and then do the identity mappingon to Y. The general solution to this problem is

One specific example of the smooth function f(r) is :

where  is an arbitrary scale. The homotopy index ism = 1 for this identity map, since to each point ofY ; there corresponds just one point of X. For thesame reason, we can get a map of index m by takinga sphere Y of radius 2 nm (wherein each distinctelement of SU(2) appears m times) and mapping Xon to it :

The configuration may be visualised as follows.At the origin since fi = 0, the triad is in standardform. As we move away in some direction ; the triadrotates about an axis parallel to that direction, andby an angle that goes from 0 to 2 nm as r goes from0 to oo. Figure 3 shows the situation along somespecial directions for the case m = 1.

FIG. 3. - The m = 1 identity map in 3He-A. As we leave theorigin along any ray, the triad twists along that axis and completes

a 2 x revolution as we reach infinity.

As always we have only presented examples fromeach class. The determination of metastable statesinvolves minimising the hamiltonian in each class.

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8. A word of caution. - We conclude our dis-cussion with the following words of caution : theentire preceeding analysis is an approximation. Thisis a reflection of the fact that although we representour media as continuum, they are in reality discrete,characterised by some length 1., which may be alattice spacing, molecular dimension etc. Recallthat the continuum field y(x) is usually defined asthe average of a microscopic variable over a smallbut macroscopic volume surrounding the point x.This volume is much larger than 10. If the field y(x)so defined varies very smoothly, that is, its scalefor variations is much larger than 10, the medium maybe reliably described as a continuum, and our topo-logical analysis applied to it. If however the systemwanders off into configurations where it cannot be

represented meaningfully by a continuum function,we must abandon our analysis, and in particularthe topological conservation laws.

Consider the following example. Imagine a configu-ration that is very large and smooth and is describedby a continuum function y(x) that belongs to a classdistinct from the vacuum. Imagine rescaling this

configuration to smaller and smaller sizes. Thecontinuum approximation will break down at onepoint. If we keep going till only a few lattice sitesor molecules are involved we have essentially reachedthe vacuum. (This would not happen in a real conti-nuum, for a configuration, howevermuch reducedin scale, will contain all the details of the originalmap and hence the homotopy index.) The fact thatthe homotopy index may thus be destroyed is a fatalblow to our analysis in principle, but not necessarilyso in practice if one of two things happens :

i) In the course of performing the above rescaling,we encounter states of very high energy. This iswhat would happen if we had a sufficiently largeb-term : once we scale down to a size do, furtherreduction in size would cost a lot of energy, therebypresenting an effective barrier. It is of course necessarythat do be much larger than 1. for our arguments,based on the continuum estimate of energy, to bevalid.

ii) Even though there is no barrier, the time takenfor the system to go through the above mentionedstages is large. In such a case we have an approximateconservation law and a homotopy classificationvalid over sizeable periods.

The shrinking configuration is a special case ofa general phenomenon of the system going into

configurations that vary so rapidly that the conti-nuum description breaks down. Our topologicalanalysis is useful if such excursions are either slowor suppressed by energy barriers due to the gradientterms in je. Even if one picks on a specific systemwith a well defined hamiltonian, estimating the timescales for these processes is a very difficult task.The purpose of this digression is not to attack this

problem but only to point out its existence and theassumptions implicit in our analysis.

9. Summary and discussions. - We employedhomotopy theory to classify the possible configu-rations of several ordered systems in two and threespace dimensions, since the analysis yields infor-mation on metastable configurations and topologicalconservation laws. But we saw that our analysiswas only as good as the continuum approximationto the system. Thus a static configuration of size doanticipated by the continuum theory had any bearingon the real system only if do > 10, which in turndepended on the b-term in the hamiltonian. Similarlythe topological index was seen to be conserved onlyas long as the system did not wander off into a rapidlyvarying configuration where the continuum des-

cription failed. The general analysis was thus inte-resting only if the excursion into such states was

either inhibited by an energy barrier or for someother reason, slow. Although we studied only nematicsand 3He-A, our topological analysis can be appliedverbatim to any system in two or three space dimen-sions for which Y = P" or S", n = 2, 3. For examplewe can expeçt a Hopf map in a ferromagnet if Xis two dimensional. Even for a different Y, once thegroup nn(Y) is found (say by referring to a book)the rest of the analysis is the same.

Acknowledgments. - 1 acknowledge my indeb-tedness to Professors D. Mumford, B. Halperin,and R. Meyer, and R. Rajaraman for their guidanceduring the course of this work. 1 am particularlygrateful to my colleagues David Nelson and GordonWoo for numerous profitable discussions.

Appendix. - We shall prove here that Y[SO(3)]=p2.Let us begin by considering Y[SU(2)]. If we startwith an arbitrary 2 x 2 complex matrix

and require that it be unitary (UU ’ = I) and hasdet = + 1 we get the general element of

Thus the elements of SU(2) can be visualized as

points on a unit sphere S3. Now, to every two elementsof SU(2) differing by a sign, there corresponds oneelement of SO(3). A simple example is given by theSU(2) matrices representing 2 n and 4 n rotationsabout, say, the z-axis

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both of which correspond to the identity elementof SO(3). Thus antipodal points (Zl, Z2) and

(- zi, - Z2) on S3 correspond to the same elementof SO(3), and Y[SO(3)] = P’. As this identificationof antipodal points has no impact on the topo-logical classification or stability of maps in whichthe sphere X wraps around the sphere Y m times,we set Y = S’ in our analysis.Although in this appendix it was more convenient

to visualize Y[SU(2)] = S3 as points in a four dimen-sional space constrained by

we found it convenient, for obtaining the maps, touse the alternate equivalent description of S 3 : as asolid sphere in 3-space with points on the surfaceidentified, since X itself has that form.

References

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