The topological theory of semidefects in ordered media

HAL Id: jpa-00210965

https://hal.archives-ouvertes.fr/jpa-00210965

Submitted on 1 Jan 1989

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

The topological theory of semidefects in ordered media

R. Kutka, H.-R. Trebin, M. Kiemes

To cite this version:

R. Kutka, H.-R. Trebin, M. Kiemes. The topological theory of semidefects in ordered media. Journal

de Physique, 1989, 50 (8), pp.861-885. �10.1051/jphys:01989005008086100�. �jpa-00210965�

The topological theory ^of semidefects in ordered media (*)

R. Kutka (1), H.-R. Trebin (2) and M. Kiemes (2)

(1) ^Siemens AG, Corporate Research and Technology, Otto-Hahn-Ring ^6, 8000 Munich 83, F.R.G.

(2) Institut für Theoretische und Angewandte Physik ^der Universität, Stuttgart, ^D-7000 Stuttgart, Pfaffenwaldring 57, F.R.G.

(Reçu le 2 novembre 1988, accepté le 23 décembre 1988)

Résumé.

Nous introduisons le concept ^de semi-défauts pour des milieux ordonnés, ^{où le} paramètre ^{d’ordre se divise en deux} composantes couplées ^de rigidités différentes. Une

singularité ^{est un} semi-défaut, si elle est singulière pour la composante ^{flexible et continue} pour l’autre. Les méthodes de topologie algébrique ^sont appliquées afin de classer les semi-défauts et d’étudier leurs correspondances ^{avec des} défauts usuels. Le comportement ^{de ces défauts et des} solitons topologiques linéaires, ainsi que les processus de croisement ^et de transformation mutuelle sont ici analysés. ^Cette approche ^nous permet ^de déterminer algébriquement ^les singularités ^{de c0153ur} nématique ^{dans les cristaux} liquides cholestériques, ^et d’interpréter ^les

défauts d’empilement ^comme des semi-défauts.

Abstract.

The notion of semidefect is introduced for ordered media, where the order parameter divides into a rigid ^{and a soft} component. ^A singularity ^{is a} semidefect, ^{if it is} singular only in the soft component. Methods of algebraic topology ^are applied ^to classify semidefects and to study their interrelations with defects in the full and in the partial order. These defects together

with linear topological ^{solitons are} analysed ^with respect ^to their behaviour during phase transitions, their mutual transformation and their entanglement processes. The scheme allows ^us to determine the coreless singularities in cholesteric liquid crystals algebraically, ^{and to} interpret stacking ^{faults as} semidefect walls.

Classification

Physics ^Abstracts

02.40

61.30

61.70P

1. Introduction.

In most instances defects of ^a condensed matter system grow in ^a phase transition. Rarely

does an anisotropic liquid ^{or a} superfluid display ^{a uniform structure of} global ^broken symmetry Hl G, when it nucleates from an isotropic liquid of unbroken symmetry group G. Unless it is exposed ^{to an} ordering field, ^the system appears nonuniform ^{as a} rule, ^and

instead of being characterized by ^a single ^order parameter, ^{it must} be described by ^{an order} parameter field, valued in the set GIH, of minima of the Landau free energy [1]. According

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01989005008086100

to the Kléman-Toulouse principle [2], ^the d-dimensional singularities of this field in n-

dimensional space ^are classified essentially by the elements of the homotopy group

lrk(GIH,), k = ^{n - d -1} [3].

Many phases, ^{however, are not} accessible directly ^{from the} isotropic liquid, ^but only through intermediary ^{states. If a} phase ^{2, of} symmetry group H2, is ^entered by ^{a transition}

from a uniform ^{state of} already ^broken symmetry Hi > H2, the available order parameter space is the ^set of cosets HlIH2 [4]. Singularities appear only ^{in those} degrees ^{of freedom}

which are added to the order parameter ^{in the} phase transition. Our standard example ^{will be}

the transition from a uniform uniaxial nematic liquid crystal ^of symmetry Hl

Dooh ⁰ (3 ),

to the biaxial phase ^of symmetry H2

D2 h. ^{In the} transition, ^{side axes} grow ^out of the

director, and the order parameter obtains the shape ^{of a cross.} While the main axes stay aligned, ^{the side axes form a} planar ^director field, ^{valued in} HlIH2

S1/Z2 - P 1, ^whose ordering transition is much like that of the plane ^rotator of XY-model. A singularity ^{in the}

side _axes, i. e. one where the order parameter ^{field of} phase ^{1 is} uniform, is denoted semidefect [5] (Fig. la). Semidefects are classified with the help ^of homotopy groups

17 k (Hl /H2). ^{As will be} demonstrated later, the notion of semidefect is also well defined, ^{if the}

order parameter ^{field of} phase ^{1 is} nonuniform, but continuous in a neighborhood ^{of the} singularity. ^Since phase ² is attained in two transitions, also its defects develop ^{in two} steps, and the notion of semidefect will turn out to be crucial in investigating ^this sequence.

Though ^{we have been motivated} by ^the study ^of phase transitions to introduce semidefects

[4], ^{we are} going ^to define them in ^a more general way. Given ^an order parameter, ^{that can be} decomposed ^{into two} coupled components, ^one of which is « rigid » (i.e. ^dislikes forming singularities ^{due to} energy reasons), the other is « soft » : then a semi-defect is the region,

where the rigid component ^{forms a} continuous (or relaxable) ^{field, while the soft component}

forms a singularity (Fig. la). Figure ^lb displays ^{a full} defect, since both orders are

discontinuous. In short, ^a field in the rigid component of the order parameter ^{is called} partial order, ^a field in the complete ^order parameter, consisting ^of coupled rigid ^{and soft} component, is called full order. The general definition neither requires ^a subgroup ^relation

between the symmetry groups of full and partial order, ^{nor even an} identification of the order parameter spaces with ^coset spaces.

Fig. ^1.

Disclinations in a biaxial nematic liquid crystal : a) Semidefect : uniform director field with defect in the side axes. b) Full defect : both directors and side axes are singular.

In section 2 several realizations of semidefects are presented together ^{with a} description ^of

their exceptional properties. Semidefects, ^for example, ^can terminate in the bulk, ^if they ^are

bounded by defects of lower dimension in the partial order. For the special case, where the symmetry of the full order is a subgroup ^{of the} symmetry ^{of the} partial order, ^we apply ^the machinery ^of homotopy theory ^to classify semidefects (Sect. 3) ^and relate them to defects in the full order (denoted ^{full defects,} Sect. 4) by ^{the exact} sequence ^of homotopy groups

(Sect. 5). In section 6 the affinity of semidefects and topological solitons is demonstrated :

when a semidefect-line breaks apart, ^the rupture edges ^{remain connected} by ^a linear soliton in the full order. An action of the fundamental group for linear full defects on the homotopy

groups for semidefects is illuminated in section 3.3. As ^a consequence of this action, topological obstruction exists in crossing processes of full defects and semidefects, ^{similar to}

the well-known disentanglement problem of full line singularities [3, 6], but with the variant,

that in the process point ^defects and linear topological ^{solitons are} produced (Sect. 7.1).

Similarly, ^{when a full line} singularity ^{crosses a linear} topological soliton, ^a semidefect line can

be generated (Sect. 7.2). ^After presentation ^{of some} examples in section 8, ^we analyse ^the

« coreless singularities ^» of cholesteric liquid crystals in section 9.

Generalizing ^the concept of semidefects in section 10, ^we interpret stacking ^faults, slip planes, ^and partial dislocations by semidefects. Criteria for the lock and work hardening ^of slip planes ^are presented.

2. Examples ^of semidefects, properties.

2.1 EXAMPLES. - Figure ^{2a shows a} uniform smectic A liquid crystal consisting ^of elongated molecules, ^{whose centers of mass are} concentrated on parallel layers. ^{The order} parameter

can be decomposed ^{into the mass} density ^{wave of} figure 2b and the director characterizing ^the long range orientational order (Fig. 2c). Both orders are coupled by ^the condition, ^that

director and layer ^{normal are locked} parallel. The field of figure ^{2d is} singular in both orders

Fig. ^2.

a) Layer ^{structure of a} uniform SmA liquid crystal. b) Layer ordering ^{of the mass} density (soft

component of the order parameter). c) Orientational ordering of the directors (rigid component). d)

Full defect : both orders are discontinuous. e) Semidefect : only ^the layer ordering is disturbed.

and hence represents ^a full defect. The field of figure ^{2e is a} semidefect, ^since only ^{the order}

of the layers is disturbed by ^a dislocation while the director field is uniform or can be made

uniform by ^a local fluctuation. In general, the transition to the nematic state occurs at a higher

temperature than the transition to the smectic state. Therefore the director is the rigid

component of the order parameter, and the semidefect carries less energy than the full defect.

The order parameter ^{of the} SmA2 ^double layer system (Fig. 3a) ^can be constructed by endowing ^the simple SmA, ^mass density ^wave (Fig. 3b) ^{with a} superstructure (Fig. 3c). ^In figures 3d and 3e both orders are singular, leading ^{to full} defects, figures ^{3f and} 3g display semidefects, ^{since the} rigid part of the order parameter, ^{the basic} SmAl layer system, ^{has no} singularities.

Fig. ^3.

a) ^Double layer system ^{of an} uniform antiferroelectric SMA2 liquid crystal. b) Layer ^{order of}

the mass density ^wave (rigid component). c) Doubly spaced layering (soft component). d) ⁺ e) ^full

defects : dislocation and disclination in both orders. f) ⁺ g) semidefects : the mass density ^{order is} uniform, ^the superstructure is disturbed.

The two components of the order parameter ^{of a} biaxial nematic liquid crystal ^{are the main}

axis and the side axes coupled by the condition to stand orthogonal. ^The justification ^for calling ^{the main axes the} rigid component and the side axes the soft component of the order parameter ^comes from several theoretical and experimental studies of the uniaxial-biaxial nematic transition : the static critical behaviour is found isomorphic ^to that of the XY-model in three dimensions [7], the critical exponents closely resemble those of the À-transition in

superfluid 4He [8]. ^The semidefect lines of biaxial nematics are classified by half-integer winding numbers, ^since 17" 1 (P 1) = 1 Z. ₂

2.2 PROPERTIES. - For the definition of a semidefect it has been decisive to state, that the field of the rigid part of the order parameter is continuous in a neighborhood ^{of the} singularity. ^Two semidefects therefore are termed equivalent only, ^if during ^{the continuous}

deformation of one into the other the partial order remains nonsingular. ^{If this condition is} violated, semidefects can terminate or change ^their type in the bulk. But then lower- dimensional singularities ^{of the} partial order have to bound the semidefects or must form an

interface between two types of semidefects.

In figure ^{4a a} hedgehog point singularity ^{of a uniaxial nematic} phase is shown. After a

transition to the biaxial phase ^{the side axes form a} tangential ^{director field on each} sphere

around the point, ^{which must} display singularities ^{of total} winding number 2. These can be concentrated on one point ^{of the} sphere, giving ^{rise to an n}

2 semidefect line, ^{which is} bounded by ^the hedgehog (Fig. 4b). ^{If there are two} singularities ^at antipodal points, ^{each of} winding ^{number n}

¹ (with respect ^to the surface normal vector), ^{one obtains a semidefect}

line of winding ^{number n}

1, ^{which turns into one} of index n

1 (with ^{respect to a fixed}

orientation along ^the line) ^{at the} point singularity (Fig. 4c).

Fig. ^4.

a) Director field of a hedgehog point singularity ^{in a} uniaxial nematic. b) Tangential ^{field of}

the side axes in the biaxial phase. ^The connectivity properties ^{of the} two-sphere ^{induce a} winding

number of 477", which is concentrated on one emanating semidefect line. c) ^{Same as} b) with the total

winding ^number split ^{into two} 277-semidefect lines.

For the classification of full defects it is not required, ^{that in a} deformation process ^a component of the order parameter ^must stay continuous along the defect set. Therefore, ^if

viewed as full defects, ^some semidefects can decay (in ^{the biaxial case those of} winding

number 2 and integer multiples thereof). ^{In the course of the} decay, ^{however, a} singularity ⁱⁿ

the rigid part of the order parameter ^{has to} appear intermediarily. ^{It forms an} energy barrier,

over which the system ^{has to} climb, ^and provides ^the stability for the semidefect. Relaxation modes of semidefects are investigated in section 6.2.

Singularities ^{in the} partial ^{order can} force the generation ^{of a} semidefect in a phase

transition leading ^{to the} full order. The semidefect lines of figure 4c grow ^out of the

hedgehog, if the uniaxial nematic phase is cooled into the biaxial phase. The smectic A- smectic C transition can be interpreted ^{as a} transition from a uniaxial to a biaxial phase, ^the

side axes being ^formed by ^the projection of the tilted director to the layers. ^{Perez et al.} [9]

have observed, ^{how a} semidefect line similar to that of figure 4c grows ^out of a smectic A

point ^defect (see also Refs. [15, 3] ^{and Sect.} 8.2).

3. Topological theory of semidefects : spécial ^case.

The variety ^{of new} structures, interrelations and processes in the world of semidefects, ^which

we are going ^to analyse ^{in the next} sections, ^is mainly ^{due to the} coupling ^{of the two}

components of the order parameter. ^Without coupling, ^the complete ^order parameter ^takes

values in a product space V x V composed of the order parameter space V r ^{for the} rigid

component, and the order parameter space Vg ^s for the soft component. ^The classifying homotopy groups also factor into those of the components :

and the singularities ^are simply superpositions ^{of those for the} single components, living ^side-

aside without interactions.

In the following ^we mainly deal with the case, where the coupling ^{can be} expressed by ^a subgroup ^relation H2 Hl between the symmetry group H1 ^{of the} partial order, ^{and the} symmetry group H2 of the full order. After some basic remarks in section 3.1 we demonstrate in section 3.2, that semidefects are well defined, ^{even if the} partial ^{order is not} uniform. At the same time the section provides the tools for understanding the action of the fundamental group 7TI(G/H2) for full line singularities ^{on the based} homotopy groups 7T2(HI/H2) ^for

semidefects. This action leads to the truly classifying objects ^{and is} responsible ^{for the} peculiar appearances of topological obstruction in crossing processes between full defect lines and semidefect lines.

3.1 BASIC REMARKS. - Given a realization of phase ^{2 with uniform constant order} parameter gH2 E G/H2, ^{then in a} smooth transition to phase ^{1 we obtain an order} parameter field of constant value gH, ^E G/Hl. Hence, the transition is described by ^the projection

These considerations also apply ^to transitions of nonuniform phases. ^{Given an order} parameter ^field F2 ^over RBL1, where L1 is a set of singularities, ^the resulting ^{field in} phase ^{1 is}

The space G / H2 ^{can be viewed as a} fiber bundle with base space G/Hl ^and projection

p. To each point of the base space (a director, ^for example), is affixed a copy of the fiber

HI/ H2 = p-’(Hl) (the ^{set of} directions of the side axes) [11]. Semidefects arise, ^{if we reverse}

the process and from ^a fixed order parameter ^field F, ^enter phase ^2. Mathematically, then, ^we

have to « lift » the mapping FI ^into F2 ^{such that} FI

F2 o p. In the ^course of the lift new

singularities may arise ^{or even must} arise

the semidefects. If FI ^{is uniform,} say equal ^to Hl ^E G/Hi, ^{the lift is a field valued} only ⁱⁿ H1/H2, ^and the defects in the lift are classified by

the homotopy groups 7rk(HIH2). ^If Fi ^is nonuniform, ^{it is deformed into a uniform field} according ^{to the} prescription ^{of the next} section, ^{and the} classification of semidefects is lead back to the uniform case.

3.2 BASED HOMOTOPY CLASSES FOR SEMIDEFECTS.

To classify ^a semidefect line in

threedimensional _space, we choose an oriented disk D 2 perpendicular ^{to the line as in}

figure ^{5. Then a closed} loop

can be defined, ^{which runs} along ^the (oriented) boundary oD2, starting ^and terminating ^{at a}

fixed base point p

a (0)

a (1 ). ^As above, ^we denote the field of the full order parameter by F2, that of the partial ^order parameter by F 1. ^{The field} F 1 is well defined and continuous all

over D2, the ^field FZ is defined and continuous on oD2. Now we deform the partial ^order continuously ^{into a} uniform field. In case of the biaxial nematic liquid crystals this process

corresponds ^to aligning ^{the main} axes, ^so that the rotation angles of the side axes can be measured in a common plane.

Fig. ^5.

On the oriented disk D2 perpendicular ^to the semidefect line A the main axes are aligned. ^Its boundary ^{aD 2 is} parametrized by ^the path a starting ^and terminating ^at p.

For the deformation process the field FI ^{and the} loop F2 ^{o a are} expressed ^as

where fi ^{is a lift of} Fi ^{into G, and} f2 ^{is a lift of} F2 ^{o a into G.} Hl ^and H2 ^{are taken as} symmetry groups of F1 (p ), ^and F2 (a (0 ) ), respectively, ^{and we} can assume that f 1 (p )

f Z (0 )

^{e. The}

lifts are defined on the simply connected domains V2 and I, respectively. ^{Due to the} coupling

condition H2 Hl for each t E 1 the subset relation

is valid, and hence also the equality

or

The partial order is made uniform by ^a continuous set of local operators

For the ^« homotopy ^{time » s}

^0, T, ^{is the} identity ^{e. For s}

^{1 one obtains :}

where

Due to equation (8) ^all points ^{of the} path ^{h are contained in} Hl. ^{At s}

1, ^the partial ^{order is a}

constant field of value Hl ^E GIH,. ^{The based} homotopy class of the semidefect is that of the

loop

The classification of linear semidefects in threedimensional space is readily generalized ^{to d-}

dimensional semidefects in n-dimensional space. One has ^to choose an (n - d)-dimensional

disk Dn - d transversal to the singularity ^{and has to} align ^the partial ^{order as described above.}

Simultaneously ^one transforms the field of the complete ^order parameter ^{on the}

r = n - d - 1 dimensional boundary aDn - d into one valued only in the semidefect order parameter space HlIH2. ^{The based} homotopy class for the semidefect is

The classification scheme still depends ^{on the} particular rigid ^order parameter F1 (p ), along

which the fields on Dn - d and aDn - d are aligned. ^{In the next section we} discuss, ^{how this} dependence is eliminated.

3.3 SEMIDEFECT CLASSES AS ORBITS OF BASED HOMOTOPY CLASSES UNDER A GROUP ACTION.

3.3.1 Change of ^base point.

It appears ^{as a} rather special case, if for the classification of ^a semidefect line the full order is aligned along ^a value, ^{which the} partial ^{order assumes at a} particular point p ^{E aD 2.} Instead of p, ^{we can} choose any other point q in the medium as

reference. Let us connect p and q by ^a path ^c.

The order parameters at p and q ^differ by ^a transformation k E G :

This transformation can be expressed by

where c2 denotes ^a lift of F2 ^{into G} along ^the path ^c.

To determine the based homotopy classes with respect ^{to the new base} points Fi (q ), ^{we have to} perform ^two steps : first, ^{we have to describe the order} parameter spaces ^as coset spaces of the symmetry groups Ki

kHi ^{k-1 of} Fi (q ), ^{i = 1, 2 ; then we have to} align

the full order along ^the point F1 (q ) ^{of the} partial ^order.

For the first step ^we note, that in the homeomorphisms

the coset kHi corresponds ^{to the coset} Ki, ^{the coset} Hi corresponds to k-1 1 Ki, ^and, generally,

each coset gHi corresponds ^to gk-1 1 Ki. 4>i expresses ^a translation G -+ G, g N gk-1 ^{of the}

group G relative ^to the order parameter space such that the identy ^{element now} lies above

Fi (q) instead of Fi (p ). Therefore, ^{the field} F, (x) ^{turns into a field}

and the loop F 2 (a (t )) ^{tums into a} loop

The lifts of the order parameter fields into G, ^{which are} needed for the alignment, ^{are now :}

and

Along ^the path ^{c we can take the same lift} C2(X) ^k-1 both for the partial ^and for the full order.

After these detailed preparations, ^the partial ^{order is} aligned ^{on the set D’ - D 2 U c,} consisting ^{of disc} D 2 and the points ^of path ^c. D’ is bounded by ^the points ^{of the} path cac -1. According ^to equation (10), the full order for the points ^of loop ^{a is after} alignment k (t ) K2, ^with

Along ^{c, the} aligned full order is constant with value K2, ^{since we} have chosen the same lift for the partial and the full order. Hence, ^the change of the base point ^is expressed by ^a path isomorphism

3.3.2 Action of ’1Tl(G/H2) ^on ’1Tk(Hl/H2).

^The path isomorphism ^of equation (20) ^{is of} particular importance, ^{if we} guide ^{the base} point p ^{about a} full line singularity ^{back to}

p (or, altematively, keep ^{the base} point ^{fixed and} guide the semidefect line about the full line

singularity). ^{Because of} equations (13) ^and (14), k ^{is an} element of HZ, ^{and the} path isomorphism (20) ^{becomes a} loop automorphism ^of irl(HIH2). ^{Thus an} action exists of

’1Tl (G / H2), characterizing ^{full line} singularities, ^on ’1Tl (Hl/ H2), chararacterizing semidefects.

The action is readily generalized and formulated for d-dimensional semidefects in n-

dimensional space. Without providing ^the straightforward proofs, ^{we state} the group action of

IT i (GIH2) ^on ’1Tr(H/H2)’ r = n - d - 1, ^{for the} case, when G is a simply connected Lie group. In the following ^we always ^assume that G has this property, because it simplifies ^the

calculations and is easy ^to obtain by ^use of universal covering groups [3].

With G simply connected, ’1Tl(G/H2) is isomorphic ^to the factor group HzH°, ^where H20 denotes the component ^of H2 ^{connected to the} identity. Let the element k e ’1Tl(G/H2) correspond ^{to coset} eH° ^e Hz/H2, P E H2. The lift of loop FZ a ^c, which is traversed in

G / H2 ^{as one} encircles the singular line, ^starts at f’ and terminates at e or (by right translation)

starts at e and terminates at e’ - 1, ^{where f’ is an} arbitrary element of ÎH20. Thus the element k which enters the path isomorphism (20) equals e’ - 1. The action of ’1T 1 (G / H2) ^on

’1Tl (Hl/ H2) is expressed by

Generally, ^an element of ir,(H,/H2, H2) is represented by ^{the based} homotopy ^{class of a}

mapping

where

denotes a lift of the mapping ^into Hl. I r is the r-dimensional cube, ^{aI r its} boundary. ^{Under the} automorphism ^induced by ^A this element turns into

The well-known action of the fundamental _group 1T l (G / H) ^{of a} homogenous space

G/H ^on 7T2(G/H) == 7TI(H) is derived from equation (21), ^if Hl ^{is set} equal ^to

H and H2 equal ^to the trivial group {e} .

Singularities ^are classified by ^free homotopy ^classes [3]. ^{These are the orbits of based} homotopy groups 7T,(G/H) ^under the action of 1TI(G/H). ^{In close} analogy, ^{the d-}

dimensional semidefects in n-dimensional space ^are classified by ^{the set} of orbits of

7T,(HI/H2) under the action of 7TI (G/H2), ^denoted symbolically 1T,(HI/H2)/ 7TI (G/H2)’

r=n-d-1.

In the fiber bundle language the action factoring ^{the based} homotopy classes of the fiber

HI/ H2 into orbits is partly ^{due to} connectivity properties of the fiber itself, but also due to

connectivity properties of the base manifold. In terms of fiber bundles Jânich [10] ^has interpreted ^and generalized ^{the notion} of semidefects to defected sections of a bundle and has eliminated the base point dependence by the introduction of ^« kinematic groups ^».

4. Classification of singularities in the full and the partial ^order.

The singularities in the full order are classified by ^{the set of free} homotopy ^classes 7T,( G / H2)/ 7ri (G / H2). ^{If we could observe} only ^the partial order, ^its singularities accordingly

would be labeled by 7T, (G / Hl) / 1T 1 (G / Hl). However, ^{due to} the information available on

the soft component of the order parameter, this classification must be refined. The set of

labeling classes consists of the orbits 7T,( G / HI)/ 1TI (G / H2) instead. Given a based homotopy

class [g (a ) HII a el’] (cf. Eq. (22)), ^{a coset} fH20 cz H21H’2’ 7TI (G / H2) ^{acts on the class} by turning ^{it into}

where P’ is an arbitrary element of ÏH20. The action is constructed in complete equivalence ^to

section 3.3.2. In general ^{the orbits} vr,(G/Hl)/7r1(G/Hz) constitute a finer classification than the orbits 7T,(G/Hl)/7Tl(G/Hl); ^for uniaxial-biaxial nematics, however, ^{there is no}

difference.

The action of an element K E 7T 1 (G / Hz) ^{on an element p E} 7T Z (G / Hl) ^{can be} interpreted

as the change ^{of a based} homotopy ^{class for a} point singularity, ^{when the} point is moved about the line singularity corresponding ^{to K.} However, ^{also linear} topological ^{solitons in the} partial

order are described by elements of 7Tz(G/Hl). Therefore, ^{if such a soliton is} guided ^{about a} singular ^line, its based homotopy class is altered due to the action.

5. Relations between semidefects, defects in the full, ^and defects in the partial ^order.

5.1 EXACT SEQUENCE OF HOMOTOPY GROUPS.

Since GIH2 ^{is a} bundle space with base

space G/Hl ^{and fiber} H1/H2, ^the homotopy groups 7T,(G/H2)’ Tr,(G/Hl), ^and ir,,(HIH2)

are related by ^{an exact} sequence of homomorphisms [4, 11], ^{out of which we} consider the

following ^section for illustration :

In three dimensional space the groups describe, consecutively, point singularities in the full

order, point singularities (or ^linear topological solitons) ^{in the} partial order, ^linear semidefects, linear full defects, ^{and line} singularities ^{in the} partial order. In figure 6, ^each

group is symbolized by ^a disk, ^{whose center is} supposed ^to mark the group identity. ^The points of the inscribed smaller disk are mapped ^{to the} identity and therefore form the kernel of the mapping. The sequence is exact, which means, that the kernel of each homomorphism equals ^the image ^{of the} preceding homomorphism. ^More rigorously, ^{it is not the based} homotopy class, that labels the singularities, ^{but an} orbit under the group action of

7Tl(G/H2). However, ^{we can} factor the sequence ^term by ^term by this action to obtain a

sequence of mappings between the sets of orbits, which still is exact, since kernel and image

retain their meaning and relation. For simplicity, ^{however, we continue to use a} simplified language, ^even identifying ^based homotopy classes with defects, ^without doing ^{harm to the} rigorous ^results.

Fig. ^6.

Section of the Exact Homotopy Sequence involving point defects of full and partial ^{order and}

all three types of line defects. Each disk marks a homotopy group, the ^centre of the disk the unit element, and the inner disk the kernel of the following homomorphism which also is the image ^{of the} preceding homomorphism ^{due to the exactness} property.

The homomorphisms ^{have to be} interpreted ^{in the} following ^{manner :}

i) homomorphism j2 describes the transformation of a point singularity, ^if

maybe ^{in a} phase transition

the full order parameter is reduced to the partial (rigid) ^order parameter.

Defects in ker j2 vanish in the transition, ^{defects outside ker} j2 ^turn into their image ; ii) homomorphism a2 describes the break-up ^of point singularities ^{of the} partial ^{order into}

semidefect lines and establishes the relation between linear semidefects and their possible

boundaries. Singular points ^{in ker} 82 ^{do not form} boundaries. Because the sequence is exact,

they ^{are the} only ^ones having ^{an inverse} image, into which they ^return when the full order arises from the partial order. The singular points outside the kernel form boundaries of stable semidefect lines. Semidefects in im a2 ^can be bounded and thus terminate in the bulk. The allowed boundaries of a E im â2 ^are contained in the set â21(«). ^The interfaces between semidefects a and fi are contained in 021 (a 13 - 1) because, if /3 is folded back, ^{one obtains a}

bounded semidefect line a 13 - 1 ;

iii) homomorphism il ^relates semidefects to full defects. Semidefect lines in ker i 1 ^are

unstable as full singularities. They ^{can relax to} the nondefect if a singularity ^{in the} partial

order is allowed to arise intermediarily. Because im a2

ker 1 1 , the relaxable semidefect lines

are exactly those which can be bounded. Semidefects outside ker i T ^cannot terminate in the

bulk and are stable as full singularities. The full defects in im il B {e} ^can be endowed with cores, that are nonsingular ^{in the} rigid component of the order parameter. These defects therefore are supposed ^to be of lowest energy ;

iv) homomorphism jl describes the transformation of line singularities ^{in a} phase ^transition

from full to partial order. ^{In the reverse} transition, ^{defects of} ker j can be created arbitrarily.

Since ker j 1 = im i ^{i these are the full} singularities ^{which can} be realized as semidefects.

The series of homomorphisms jl, al, io, jo following ^{that of} equation (26) ^is interpreted ⁱⁿ

the same way. Only the dimension of the singularities ^{has to be increased} by ^{1. The} interpretation ^{of the} preceding ^series

is more difficult (what ^{is the} boundary ^{of a} point singularity ?) ^{and first} requires ^several

considerations of semidefect-soliton processes.

6. Semidefects and solitons.

6.1 ORIENTATION CONVENTIONS.

6.1.1 Lift singularities.

^{In a certain sense,} the notion of semidefect is also applicable ^{to full} singularities of fields valued in an arbitrary homogeneous space G/H, where G is a simply

connected Lie group [3]. ^Because 7Tl(G)

^0, -U2(G)

0, ^one obtains the above cited

isomorphisms

from the exact sequence. Equation (28) expresses the fact that ^on walking ^{about a} stable line

singularity, ^each lift of the order parameter ^{field into G does} not return to its original ^{value. A} singular ^{line can} therefore be interpreted ^as boundary ^{of a} semidefect wall in the lift. Similarly

a singular point ^{is the} boundary ^{of a} semidefect line in the lift.

The connection between a point singularity /3 ^{and a} singular line in the lift, ^as expressed by equation (29) ^{has to be} interpreted ^{in the} following way : the singular point is enclosed by ^a pointed sphere (SZ, x), ^{and the} mapping (SZ, X) __> (G/H, H) yields ^{its based} homotopy ^class /3 . Now a hole is cut into the sphere, ^{with x} lying ^{on its} boundary, and the field is continuously

deformed so that it takes the constant value H on the hole. Starting ^{from x, one} encircles the hole such that it always stays ^{to the left} (Fig. 7a). The lift into G along this way forms ^a loop ⁱⁿ H, ^whose homotopy ^{class is the} image ^of /3 ^under isomorphism (29).

On the other hand, the field of a point singularity ^{can be} pressed ^{into a linear} topological

soliton (3 emanating ^{from the} singular point (Fig. 7b). ^{We use the} diagram ⁱⁿ figure ^{7b as}

orientational convention [12] : ^{the lift} singularity 82 (/3 ) and the soliton /3 ^{both flow out of the} point singularity.

6.1.2 Semidefect ^lines.

^{Now let us return to the case of two} orders with symmetry groups

Hl ^and HZ Hl. The lift defect leaving ^a point singularity ^{in the} partial ^{order is} characterized

Fig. ^7.

a) ^{Lift of a uniaxial} hedgehog point defect into G. The lift is performed by drawing tripods (one ^axis suppressed) instead of the directors. The axis parallel ^to the director is aligned ^{on the} hole, while the others form a line defect in the tripod ^field flowing ^out of the hole. b) ^{If all} topological charge

flow is pressed ^{into a} tube which contains the hedgehog point defect, ^a linear soliton arises bearing ^the charge ^{of the main} axis. The « lift defect » transports ^the charge of the side axes.

by ^a loop ⁱⁿ Hl, say h (t ). ^{This same} loop characterizes the semidefect belonging ^{to the} point singularity according ^to homomorphism a2 :

Lift defect and semidefect run parallel. ^{If we reverse} the direction of the semidefect line, ^then

it is labeled by the inverse group element, p -1.

6.2 TRANSFORMATION PROCESSES.

The preceding discussion leads to the following

transformation process : ^a semidefect line p -1 E 7Tl (Hl/H2) ^{which can be bounded (i.e.}

p ^E im a2) flows into a point singularity /3 ^E 7TZ(G/H1)’ ^{with p =} az(f3). ^{A linear} topological

soliton of type f3 ^E 7T Z (G / Hl) ^{flows out of the} point.

Now we can think of the following conversions :

i) ^a semidefect line p ^E im a2 breaks into ^a pair ^of point singularities f3 -1, f3, ^where

f3 ^E a2 1 (p ), ^{as in} figure ^{8a. The two} points ^{are connected} by ^{a linear} topological ^{soliton of} type f3 - 1 ;

ii) ^{a linear} topological soliton breaks up into ^a pair ^of point singularities ^{as in} figure ^{8b. If} f3 lies outside ker a 2, ^a stable semidefect line of type ^{p =} az (f3 - 1) bridges the gap ;

iii) ^a semidefect of type ^{p E} 7T 1 (Hl/Hz) changes into another type ^{cr, where} p ¹

possesses ^a boundary (p (T - 1 E im az, ^{i.e. p and o-- are in the same coset of} im a2 ⁱⁿ

7T1(H1/Hz». ^{The exchange} is mediated by ^a pair ^of point singularities f3-1, f3, ^with

f3 ^e a 2 1 (p o- ^which, if created on the line _p and pulled apart, span line cr ^as in figure ^{8c. If}

the partial ^{order is} kept ^{constant outside a tubular} neighborhood ^{of the} line, ^{cr is} accompanied by ^{a linear} topological soliton of type (3 -1.

The semidefect-soliton transformation processes allow ^to interpret the section of the exact sequence involving semidefect point singularities, ⁱⁿ particular homomorphism a3. ^{We add}

the time dimension ^to the three space dimensions. Then ^an element of 7T2(Hl/H2) ^classifies

the singular world line of a semidefect point singularity. ^{Just as in one dimension} less, ^this

world line can change into the world line of a pointlike topological soliton in the partial order,

classified by ^an element of _7T3 (G/Hl ). ^Thus, given ^uniform boundary conditions of the partial

order in three-space, ^a semidefect point singularity ^can be converted into a point-like ^soliton

without violation of the boundary conditions. In condensed matter physics, ^no system ^is known to us possessing topologically ^stable pointlike semidefects. One could speculate, ^that

there are particle transformations in high energy physics, ^{which can} be described as

semidefect-soliton exchange.

Fig. ^8.

a) ^A semidefect line p, which can be bounded by ^a point ^defect f3 ^{can break} by creating ^a point

defect

antidefect pair. Since this rearrangement is local the two points ^{are connected} by ^{a linear} topological ^soliton. b) ^{A linear} topological ^soliton xi ^can break into a defect-antidefect pair ^{of the} corresponding point ^{defect. The two} points ^{are connected} by ^a semidefect line Õ2(f3 -1), which is stable if f3 ^{E ker} 82. c) Point defects of partial ^{order can act as} interfaces between semidefect lines which differ

only by ^an element of im a2. ^A semidefect line cr changes ^{into p via a} point ^{defect /3 E} Õï" 1 (p u- 1).

7. Topological obstruction in crossing processes of line defects.

The topological theory of defects [3] ^has revealed the possibility ^of topological obstruction for line-defects of nonlinear fields [6, 13]. Certain lines cannot cross by changes of the field limited to a neighborhood ^{of the} meeting point, ^{unless a} third line singularity ^{is created} connecting ^{the first two. Similar} obstructions occur in crossing processes of full line

singularities with either linear semidefects or linear topological ^{solitons. But here} again, ^the variety ^of phenomena ^{is much richer.}

7.1 CROSSING OF A SEMIDEFECT LINE WITH A FULL LINE SINGULARITY. - We test the semidefect line of figure ⁹ by ^a loop, which is tied to a fixed base point, ^{and assume it to be of}

type p ^E ir1 (H1 /H2). Crossing ^{with the full line} of type ^{K e} 7T 1 ( G 1 H2) ^is only possible, ^{if the} bridge ^formed by arriving ^and departing section of the semidefect belongs ^{to the trivial}

semidefect class. The testloop measuring the semidefect on the other side af rc yields ^{a based}

class K (p ), ^{which in the} group action ^of 7Tl(GIH2) ^on 7Tl(H1/H2) is ^the image ^of

Fig. ^9.

^{The left} picture ^{shows two} loops encircling the semidefect _p. If we move the base point ^{of the}

left loop p around the full defect line K, ^we arrive at K (p ). Adding ^the loop p - ^one gets ^a loop ^{of the}

same homotopy ^{class as the one} drawn in the right picture, ^{which tests the} bridge defect line. The

topological stability of this semidefect decides whether this disentanglement is obstructed or not.

p under the homomorphism corresponding ^{to K. The two} parallel semidefect lines of the

bridge ^{combine to one of} type ^K (p ) p -1. ^In contrast to the disentanglement problem ^{of two}

full line singularities, ^{there are now three} possibilities :

i) ^K (p ) P - 1 = e: ^the bridge ^can fade away, crossing ^{is not} obstructed (Fig. 10a) ; ii) K (p ) p -1 E ^im d2 B { e} : ^after crossing, ^the connecting semidefect line is one that can be

bounded ; ^{it can break} by ^{creation of two} point singularities. Crossing is allowed but only by pair production ^of singular points (Fig. lOb). One of them divides the semidefect line into two different types, the other is sitting ^{on a} semidefect ring encircling the full defect. The points

are connected by ^{a linear} topological soliton ;

iii) K(P)P-10 ^im a2 : ^after crossing, ^{the two} singularities ^{remain connected} by ^{a stable}

semidefect line, ^which simultaneously ^{is a} stable full line singularity (of im i 1) ; crossing ^is

obstructed (Fig. 10c).

Fig. ^10.

Disentanglement of semidefect with full defect lines : a) ^No topological obstruction. b) ^The linking semidefect can be bounded, ^if K (p) P -1 E im °2. ^{Then the} breaking process of figure ^{8a is} possible. c) ^{After the} disentanglement ^{the two} line defects are connected by ^a stable semidefect line which cannot be bounded, since K (p ) p - ’ 0 iM a 2 -

7.2 CROSSING OF A LINEAR TOPOLOGICAL SOLITON WITH A FULL LINE SINGULARITY. - Now

we replace ⁱⁿ figure 9 the semidefect line by ^{a linear} topological soliton in the partial ^order

and test it by ^{an area D such that the field on the} boundary ^{aD is constant} (Fig. 11). ^{If it} yields

type ^{y E} lr2(GIH,), ^{then, when the disk} is connected with the base point the other way

around the full defect, ^{it turns into} type K ( y ) ^E 7T2 (G/Hl ). The element K ( y ) ^{is the} image

Fig. 11. Fig. ^12.

Fig. ^11.

The upper ^{test area} yields ^{the class} y ^E 7T’ Z (G / Hl) for the soliton. Guiding ^{the base} point ^of

D along ^a circle which encloses the full defect K e 7T’ 1 (G / Hz), ^one obtains the lower test area after deformation. It provides ^{the class} K ( y ) ^E 7T’ Z ( G / Hl), ^{which is} equivalent ^to the action on the point

defect y by the line defect K.

Fig. ^12.

Disentanglement ^{of linear} topological solitons with full defect lines : a) ^No topological obstruction ; the linking soliton is unstable. If the bridge K ( y ) y-1 ^{is stable it can} break into two point

defects according ^to figure ^{8b. The two} points ^{are connected} by ^a semidefect, ^which b) is unstable and

disappears, ^if K ( y ) y -1 ^{E ker} az B {e} ; c) ^is stable, ^if K ( y ) ’Y-l E -ff2(G/H1)Bker 82. ^The point ^defect /3 ^{acts as} interface between the solitons of type ^{y and} K (y).

of y under the homomorphism corresponding ^{to K} in the group action of -ul (GIH2) ^on lr2(GIH,) (Sect. 4). Again ^{there are three} possibilities :

i) K (y) y - 1 = e ^: crossing ^is possible without obstruction (Fig. 12a) ;

ii) ^K ( y ) y -1 ^{E ker} Oz B {e} : ^the connecting ^{soliton can} break into a pair ^of point ^defects /3, 6 - ^The point 8 = K (y) y ^absorbs topological charge ^{from the} remaining soliton and

changes ^its type (Fig. 12b) ;

iii) K ( y ) y -1 ^ker a 2 : ^after crossing, ^{a soliton is} connecting ^{the two} lines. It can break into two point singularities 8 - 1, f3

^K ( y ) y -1 (Fig.12c) ; these, however, ^{are linked} by ^a

semidefect line of type p

a2 (,0 ) cz ^7T1 (H1/H2). Crossing is obstructed.

Thus we have found an entire chemistry of processes, which ^are expressions ^{of an}

interaction between textures of nonlinear fields. Topology ^{is the} only ^{tool to} presort ^and illuminate these processes and thus plays ^{the same} role for nonlinear fields as for example representation theory ^of groups for linear fields.

8. Examples.

Table 1 summarizes symmetry groups, homotopy groups, and sections of the exact sequence for several ordered media. We are going ^{to discuss a few} examples.

8.1 UNIAXIAL-BIAXIAL NEMATIC LIQUID CRYSTALS.

Semidefect lines in biaxial nematic

Table I .

Symmetry groups for several condensed matter systems ^{and exact} sequences of homotopy groups for ^{the relations} between their semidefects ^and full defects.

liquid crystals ^{are labeled} by half-integer winding ^{numbers u. On a} loop ^{about a semidefect} line, ^{the side axes rotate} by ^an angle ^{2wu. When a} semidefect line is guided ^{about a 180°-}

disclination line in the partial order, ^the winding ^{number is} changed ^{to - u.} Thus semidefects

are labeled by pairs {u, - u} . ^{Full line} singularities correspond ^{to elements of the} quaternion

group Q = {::t 1, ::t i , :t j, ::t k }. Here i denotes a disclination with a rotation by ^{7r about the}

main axis. A semidefect line of label u corresponds ^{to a} full line singularity ^{of label} l 2 u. If a winding ^{number u of a} semidefect line is _even, then the line is unstable as full

singularity ^{and can} be bounded by ^a point singularity ^{in the} partial order of index

z

u /2. ^A point singularity of index 1 can separate semidefect lines of winding ^numbers

+ 1 and - 1 (see Fig. 4c).

The action of -r,(G/H2) ^on irl(H1/H2) ^is given by

Full defects of types -+-- 1, --t i, can cross each semidefect line without generation ^of

additional defects and point singularities. ^{There is} topological obstruction for defects of type

± j, ^± k, ^if they ^cross semidefects with odd half-integer winding ^{number u E} {± ^{1/2, ±} 3/2, ± 5/2, ... ) . ^If they ^cross semidefects with integer winding numbers, ^{then two} point singularities ^{of labels ± u, u E Z are} generated.

Semidefect lines in lyotropic biaxial nematic liquid crystals have been observed by ^Galerne

and Liébert [14]. They ^{call them «} disclinations with uniaxial core ». The lines are broken into

zigzags. ^The peaks ^{of the} zigzags consist of point singularities. ^{It is not} known whether these

are stable, i.e. whether the winding number of the line differs at the zigs and the zags.

8.2 SMECTIC C LIQUID CRYSTALS.

Smectic C liquid crystals ^{combine a} rigid layer structure, ^formed by ^{the centers of mass of the} molecules, ^{with an} orientational order of the molecules’ long ^{axes. In} contrast to smectic A liquid crystals, the directors are tilted with respect ^{to the} layer ^normals by ^{a constant} angle. ^Soft component ^{of the order} parameter ^{is the}

projection of the director onto the layer, ^{which is} frequently ^called planar director. We read from the exact sequence (Tab. I), ^{that each stable} point singularity ^{in the} layer ^{order bounds a}

stable semidefect line, because ker a2

^0. Semidefect lines are disclinations in the planar

director of winding angles 21TW, W E Z. Full line singularities ^{are labeled} by elements of the semidirect product ^Z (dislocations) ^with Z4 (disclinations, ^the winding angle resulting by multiplication ^with 1T). Only ^the even-numbered pure disclinations ^can be realized as

semidefects. How a semidefect line is generated ^{in a} phase transition and how a pair ^of

semidefect lines leaves ^a point singularity, ^can be observed in the polarizing microscope [9, 15].

The action of full defects on semidefects is

Only odd-numbered disclinations change ^the winding number of semidefects. Even numbered disclinations cross semidefects without any topological obstruction, odd numbered only by pair generation ^of points.

8.3 CHIRAL SMECTIC C LIQUID CRYSTALS.

The chiral smectic C phase (SmC *) ^differs

from the usual smectic C structure in that the planar ^director

^the projection of the director to the layer

spirals ^{around a} helix axis parallel ^{to the} layer normal. Its symmetry group is

isomorphic ^{to that of SmC} except ^{that inversion is} missing. Therefore the exact homotopy

sequence ^with SmA as rigid ^{order remains} unchanged (apart ^{from the} possibility ^{of wall}

defects separating regions ^of right ^and left handed helices). Full line defects are again ^labeled by Z n Z4, ^but they ^{have to be} interpreted differently. ^A symmetry ^element (t, r)E

Z n Z4, ^{contains a} translation by t layers ^{and a} rotation r E Z4 == C2 ^c SU(2), plus

^{due to}

the helical ordering ^{of the} planar ^director

^an additional rotation around the pitch ^axis by ^an angle ^of 2 -ffp ^{where P} denotes the pitch ^{and d the} layer distance. If this angle ^{is not an} multiple ^{of ir, then} the defect is denoted dispiration [16], ^{and its} rotational part ^{is called} imperfect disclination. Dispirations ^{bound a wall of mismatch which} is, however, topologically

unstable. The semidefect lines correspond ^{to full defects with} vanishing t (cf. ^Sect. 8.2) ^and

are therefore pure, perfect disclinations.

In a different point ^{of view one can stress the} superstructure [17] of translational periodicity

nd of the SmC* order, where n is the smallest integer such that nd/P ^{is an integer}