Topological properties of static and dynamic defect configurations in ordered liquids

Physica A 182 (1992) 240-278 North-Holland

~

~

Topological properties of static and dynamic defect configurations in ordered liquids* A. H o l z Fachrichtung Theoretische Physik, Universitiit des Saarlandes, W-6600 Saarbriicken, Germany Received 27 June 1991

The topology of linked disclinations is studied in uniaxial nematic liquids and in anisotropic liquids with an order parameter space SO(3)/Pi(3). In these models {Pi(3)} are the finite point symmetry groups of 3-space with applications to helium 3 (Pi(3) = I), a semi-classical approximation to the spin-½ Heisenberg antiferromagnet (SO(3) spin liquid, P~(3) = I), biaxial nematic liquid (Pi(3) = D2), and anisotropic super cooled liquids (P~(3) = groups of Platonic solids). The topological properties are studied via Hopf's invariant of the 0(3) tr model and its relation to the Wess-Zumino term of the SO(3)/P~(3) o- model in an orthonormal drei-bein representation of SO(3). Dynamic processes are topology changing during intersection of disclinations and are studied via "magnetic" N-pole singularities and the instanton number 77 in an "electromagnetic" formalism. The connection with tunneling amplitudes in a SO(3) Yang-Mills theory is indicated. Applications of the theory to topological fluid dynamics is worked out for the uniaxial nematic liquid and indicated for the SO(3) spin liquid.

1. Introduction Entanglement of line defect structures plays an important role for the static and dynamic properties of liquid crystalline systems [1]. In the presence of a high density of defect lines these systems display properties reminiscent of viscoelastic and viscoplastic behaviour observed in polymeric liquids. As a matter of fact in a continuum approximation to line defects of disclination type, the latter objects may be considered to form a "polymeric melt" of fixed topology. Topology changing processes require mutual intersection or annihilation of disclination lines and are forbidden under strict "excluded volume" constraints. In order to achieve a crossing process the disclination loop may open temporarily forming two "magnetic N-poles" at its ends, and later close up either with itself or with some other open ends. It follows from this that a * Work supported in part by Deutsche Forschungsgemeinschaft through Grant Ho 841/6-1. 0378-4371/92/$05.00 (~ 1992- Elsevier Science Publishers B.V. All rights reserved

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topological study of static and dynamic defect configurations in ordered liquids is of some interest. The topological studies will be based on the 0 ( 3 ) o- model and its Hopf index Q, which is defined by [2]

Q({n}) = -(8"tr) -2 f A. B d3x,

(1)

M

where {n} is a unit vector field and A and B are suitably defined vector potential and magnetic field strength of an Abelian gauge theory and M is a closed and simply connected 3-space. For the SO(3) o- model one uses instead of (1) the Wess-Zumino term [3] Fwz _

1 48,.rr 2

f d3x epqr tr[(R t 3pR)(R t 3qR)(R' cqrR)]

(2)

M

where ~pqr is the totally antisymmetric symbol and R E S O ( 3 ) . Summation convention is implied throughout; furthermore p = 1, 2, 3, Op =-O/Oxp, etc. Using the (3 x 3) matrices R in the form R = (n 1, n 2, n3), w h e r e {na}a=l,2,3 is an orthonormal drei-bein field, one can reduce (2) to (1) in the form 3

Fwz ~ ~

Q({n"})-

(3)

a=l

Topology changing processes require a definition of dQ/dt or dFwz/dt. This cannot be done simply using (1) and (2), because during such processes A is not well defined. For example in the presence of magnetic N-poles {A} is the connection of a non-trivial U(1)-bundle. The time evolution of systems may be studied via the instanton number [3]

q

--

1

fd4" a X 1~,,

128~r2 ~=1

* F at~"

(4)

M4

w h e r e / z = 0, 1, 2, 3 and g is a field strength tensor and *g its dual; in simple cases one may set M a = M x R. Defining

1 ~ f d 3x F~,~~*F a~'~= -dFcs dq _ dt 128~ 2 a=l dt M(t)

where instead of (2) one uses the Chern-Simons term [3]

(5)

242

A. Holz / Topology of defect configurations in ordered liquids

f

1 Fcs - (8~)2

b d3x ~ pqrma(~ - - p x - q - - rA a + ~%bc A q ACt)

(6)

M(/) and M(t) refers to the modified 3-space, the time evolution of the system may be studied topologically. In order to justify the denomination disclination for the line defects in the 0 ( 3 ) (7 model recall the following [4, 5]. In the 0(3) tr model the order parameter O assumes values on the 2-sphere S 2 and defects may be classified by the first and second homotopy groups 'rq(S 2) ~ I,

(7a)

rrz(S 2) ~ 77,

(7b)

whereas

"i1"3(82) ~ 7/7

(7C)

classifies the linking of defects of disclination type measured by (1). 7Z represents the infinite cyclic group. Although disclinations in the 0(3) tr model are topologically unstable due to (7a), they can be stabilized topologically due to (7c) by knottedness and linking. Although such disclinations display no core singularity, their mathematical description requires the auxiliar concept of a cut surface (see Kundu and Rybakov [2]), and the bounding curve of the cut surface may be identified with the disclination loop (see also the appendix). In contrast to the 0(3) o- model, which in solid state physics may apply to magnetic material (ferro- and antiferromagnetics in classical approximation) the order parameter O in a uniaxial nematic liquid assumes values on the projective 2-sphere p2; O is represented by a pair of unit vectors (n, - n ) and p 2 ~ $2/7Z2" Defects of the p2-model are classified by the homotopy groups [4-6] %(p2) ~ 7/2,

(8a)

"rr2(P2) ~ Z ,

(8b)

,rr3(P2) ~ Z .

(8c)

From (8a) follows that a uniaxial nematic liquid features topologically stable disclinations of strength s = 1, whereas from (8c) follows that disclination of any type s E ½77 can be stabilized topologically by means of knottedness and linking.

A. Holz / Topology of defect configurations in ordered liquids

243

An important question is if (1) can be suitably redefined in order to be used as a measure of the classes of (8c). This is a non-trivial problem for disclinations of strength s E Z + 1, because those feature line singularities along their cores, and discontinuities along the cut surfaces 'spanned by their cores. For superfluid helium 3 the orderparameter O may assume values in the SO(3)-group space. Here we have SO(3) ~ p3 (projective 3-sphere) and defects and their configurations are classified by the homotopy groups [4-6] "n"1(50(3)) ~ / 7 2 ,

(9a)

"rr2(SO(3)) ~ I,

(9b)

"rr3(SO(3)) ~ / 7 .

(9c)

The covering space to SO(3) is S U ( 2 ) ~ 83 (3-sphere), and p 3 ~ 83//72" As a measure of the classes of (9c) one may use the Wess-Zumino term (2). For the biaxial nematic liquid and the anisotropic supercooled liquids the order parameter O assumes values in the quotient spaces SO(3)/D 2 and SO(3)/Pi(3 ), respectively, where D 2 and {Pi(3)} represent the dihedral group and the point symmetry groups of Platonic solids, respectively. For the fundamental groups of these spaces one obtains ~r1(SO(3)/~)= ¢*, for ~ = D 2, {Pi(3)} and ~* represents the respective binary group. A study of the linking of defect configurations in these quotient spaces will be done by similar methods as employed for the uniaxial nematic liquid with O E p2 in relation to the 0 ( 3 ) or model with O ~ S 2. The plan of the paper is the following. In section 2 the 0 ( 3 ) tr model is studied in the "electromagnetic" representation. The SO(3) tr model and its relation to the 0 ( 3 ) or model is discussed in section 3. In section 4 the topological properties of liquid crystalline systems are studied for the order parameter spaces SO(3)/~, for ~ = D2and {Pi(3)}. In section 5 the topological fluid dynamics of an uniaxial nematic liquid based on an idea of Moffatt [7] is worked out, and a similar analysis of the SO(3) spin liquid is indicated. The results are discussed in section 6 and brought in connection with topological field theories [8]. Some mathematical details are worked out in the appendix. A preliminary account of this and related work has been published in refs. [9, 10].

2. The 0 ( 3 ) ¢r model and its electromagnetic representation

Closed vortex like solitons in the 0 ( 3 ) o- model in 3-dimensional Euclidean space R 3 have been studied by de Vega [11], and Kundu and Rybakov [2]. In

A. Holz / Topology o f defect configurations in ordered liquids

244

the following some of the formulae of the latter authors are used, and extended to Minkowski space-time in R 4 with Lorentz metric ~ = (1, - 1 , - 1 , - 1 ) ; see e.g. Jackson [12] and Misner et al. [13] for notation. The vector potential A~ and field strength tensor F , , are those of an Abelian gauge theory, F.~ = O ~ A ~ - O ~ A .

(10a) (10b)

= 2~. abc O~n a O~nbnc ,

where n = (n 1, n2, n 3 ) , n 2 = 1; and 0, -- O/Ox ~ and x" = (t, x, y, z) are Cartesian coordinates in N4, and the speed of light c is set to 1. Using the representation 0

- Ex

- Ey

Ex

0

B~

Ey

-B z

0

Ez

By

-

-E~

-BxBY

(11)

B x

where E and B are the "eldctric" field strength and "magnetic" induction field, respectively, one may compute the "electric" 4-current J~ via the inhomogeneous Maxwell equations O,~F'~ = 4"rrJf . Due to O,~t3F'~t3=-0 this current is conserved, je

= .f~ el

(12)

Ot3JOe

= 0,

and given by

abc O"(O.n. O~nonc) .

(13)

Inserting (10a) into (12) yields I--1A" = 4,rrJe~ ,

(14)

where the Lorentz gauge condition O~A ~= 0 has been imposed. Besides J~ one has the topological 4-current [2] Jr" = -(128"trz)-'E""°"F~pA,~, which is also conserved, O~,J~ = 0, and gives rise to the topological charge Q = f d3x jo M

A. Holz I Topology of defect configurations in ordered liquids

245

already introduced by (1). In order to compute Q for some given configurations {n}, the vector potential A has to be determined via a solution of (14), subject to suitable boundary conditions. For the sake of simplicity one may take the 3-space M to be closed and simply connected, i.e., M = R 3 U 0o~ S 3. Besides the currents introduced so far one may also introduce a "magnetic" 4-current J~m = (Pro' Jm) via Maxwell's equations V . B = 4~pm , V x E + OrB

=

(15) (16)

-4~rjm.

Usually the right-hand sides of these equations vanish, due to the absence of free magnetic charge. However, from (15) and (16) one obtains 3

Pm : ~

Jm -

(Oxn X Oyn). Ozn ,

(17a)

[C3yn X O~n\ 3 |OznXO~n | 2"tr Otn "

(17b)

\ Oxn ×

ayn/

One easily verifies that j~ is also conserved, i.e., OPm +V'jm=O Ot

--

Furthermore for a smooth field {n}, (17a) and (17b) vanish identically. In the presence of singularities, which arise e.g. during a topological change of field configurations, this is not the case. The 4-current j~ = (Pc, Jc) can be separated into two parts, j~ = j o ) , + j(2)/3, yielding j(,) =

1 {(O~n x 82~n) • n + [(OxO,n) x O t n ] . n }

21x

j(2) = 1 {(O~n x An)- n + [(0,,0,n) x O,n]. n} 2~ V-j(2) =0,

0 (2) = 0 , 1

P(') = Pc = - ~

{(Otn x An)- n + [(O,O,n) x O,n]. n } ,

0tO (1) + V" j(1) = 0 •

Setting Pc = - V . ~, we obtain for the polarization

(18)

A. Holz / Topology of defect configurations in ordered liquids

246

1 ~k = ~ [(O,n X Okn) • n I .

(19)

From (18) follows j(2) =V x 0¢/, and for the magnetization ~ we obtain 1

d/t, = G

(20)

%~(Ojn x Okn). n .

Introducing the magnetic field strength H and dielectric displacement D in the usual fashion one obtains H = B - 4-rr~ = 0 ,

D = E + 4"rr~ = 0 .

(21)

From (19) follows j (1) = 0,~, and thus V.D=0,

(22)

V×H-OtD=O.

Accordingly for smooth field configurations the "free electric" current, usually defined via (22), J = (p, j ) vanishes identically; from (21) follows that the field energy density e vanishes identically, i.e., 1

e= -~ ( E . D + H.B)=-O.

(23)

Eqs. (21) do not represent the most general solution to the problem but excitations of the H- and D-fields may lead to finite field energy. Setting H = -Vt~-

OtC ,

D = -V

x C ,

which is compatible with the former equations, use of (11) and (22) yields

xf d 3 x ( ~ b p m q - C ' j m ) + - ~ if

E~=~

M

d3x(C'atB-OtC'B)

•

(24a)

M

In this expression (qJ, C) compares to the usual electromagnetic 4-potential (O, A) and the related field energy

o

Ef=~

d3x(Op~+a.j~)+-~ M

d3x(a.O,D-Oe4.B).

(24b)

M

In analogy to (10) and (11) one may introduce another unit vector field {m) and define

A. Holz I Topology of defect configurationsin orderedliquids

247

F'it.l* = O,. C. - O~C~. = 2e ~bc Ot.ma Ol*mbm¢ ! 0 - H x -Hy -H~ H xHz 0 - D z Dr IIlg't*vll =

By

O z

0

-O x

- Dy

D.

o

'

and the respective Hopf invariant Q' =

1 Mfd3x C . D (8 r) 2

(25)

obtained from the topological 4-current 1 J*~°'~F' C 1287r 2 • -l*o ~"

J~'=

In addition one may introduce in the presence of singularities of the m-field a non-vanishing free electric current J ' = (p', j ' ) entering the right-hand side of (22) in the usual fashion. It follows then that for Pm =Jm = 0 or p' = j ' = 0 the n- and m-fields are completely decoupled and represent equivalent descriptions of topological field configurations. Consider next the change of Q in time. Here one starts from the instanton number defined by (4) for a = 1,

q=

1

f d 4x F , ,

1

*F "~ =

32,rr2

M4

f d4xE. B ,

(26)

M4

where *F ~'~ = !2 e ~o~.~ rp. is dual to F.l* and the second representation in (26) follows from

* F/zl*

-B x

0

B0 x

-- B y

E z

0

-Bz-Ey

Bix )

By z

-g

Ey -

"

Ex

For a finite sized space-time

M 4

and

0M 4 =

M ( t 2 ) t O - M ( t l ) one obtains

Aq = Q ( t 2 ) - Q(t,), where M(t2) and - M ( t l ) are bounding 3-manifolds of dq dQ _ 2 f d t - d~ (Slr) z d3xE. B. M(t)

(27) M 4.

From (27) follows (28a)

248

A. Holz / Topology of defect configurations in ordered liquids

It should be emphasized that (27) based on (1) applies to the 0 ( 3 ) o- model only for smooth n-fields at t 2 and tl, because otherwise A is not well defined. In the presence of singularities, A is a connection on a non-trivial U(1)-bundle and differentiation may formally be defined via (28a) in the form - d ( A . B ) / dt = 2 E . B. For the m-field defined above a similar analysis of topology changing processes can be made and one obtains dq' dQ' _ 2 dt - dt (8rr)= J d 3 x H . D .

(28b)

M(t)

It is now easy to show using the representation of E and B following from (10b) and (11) and the formula

(a×b'c)(e×f'g)=det

a.e

a.f

a.g

b.e

b.f

b

c.e

c'f

c

that E . B ~=0 for smooth fields {n}, and similarly that H . D-= 0 for smooth fields {m). In topological fluid dynamics [7], e.g. applied to magnetohydrodynamics d Q / d t # O is usually a consequence of dissipative processes associated with Ohm's law, i.e., F~I -= E + v × B

= "r/elJOh ,

(29)

where ~Te~=-l/o-and o-is the conductivity; rtej~0 violates the frozen field condition (see section 5). Here we have slightly generalized the problem considering the order parameter to be carried along by a fluid of local velocity v, where the Lorentz force Fe~ enters. Use of (29) in (28a) yields dQ_ dt

2r/c1 f (87r)2

d3x

Job" B .

(30a)

A similar discussion can be given for d Q ' / d t using " O h m ' s " law for magnetic charge

Fmag = H + v × D

=

Tlmad~)h

yielding dQ' dt

_

2"r/mag

(8at) 2

j d3xj~)h. D . M

(30b)

A. Holz / Topology of defect configurationsin orderedliquids

249

The quantities defined in (30a) and (30b) are not very useful in the present context, because it has been shown that for the 0 ( 3 ) or model p = j = - 0 and therefore Joh ------0. Furthermore Joh ~ 0 is possible during formation of singularities, but D ~ 0 cannot be related to the n-vector field in a canonical fashion, besides that (30b) gives no information on (28a) which is of interest here. The situation may change if one considers two non-trivially coupled 0 ( 3 ) ~r models using e.g. the constraint n • m --- 0. This case will be discussed further in section 3 in connection with the SO(3) or model. In the following an alternative model for topology changing mechanisms is considered. Because Q and Aq assume integer values after the completion of singular processes and it is well known that in that case Q can be represented in the form of Gauss' linking number [2] we may consider the following mechanism. Suppose that Aq refers to the formation of a H o p f link of two disclinations of strength na, n b E 7/, respectively, indicated in fig. 1. Then its Gauss linking number is ~ ( C a, Cb) and

nanb f f dyieiik(X--Y) k Aq= nanb Cl)(Ca, Cb) = ~ dx' ~_ y~ , Ca

(31)

Cb

where C a and C b a r e oriented loops. Suppose that each disclination is formed from a "magnetic" dipole in the fashion illustrated in fig. 1. Then

if

n, = ~

d3x

pro(x, t),

a = -a,

(32)

refers to the charge of the magnetic N-pole forming the dipole a, and 12_+, are small 3-balls enclosing the respective singularities. Observe that magnetic N-poles are topologically stable defects in the 0 ( 3 ) tr model and (32) is a

7' Fig. 1. Formation of a Hopf link over magnetic dipole formation and annihilation, from left to right; unlinking of the Hopf link proceeds from right to left.

250

A. Holz / Topology of defect configurations in ordered liquids

measure of the classes of (7b). The constituents of a dipole are connected by a tube. In fig. 1 only the " c o r e " line of the tube is drawn. Defining r/+a(b ) > 0 an orientation can be given to the tubes. Setting x+b(0

1 2 E" B(x, t) = ~ 32¢r a=-+a

n"nb via(t) f 4ax

x ,~,)

-- y)~ dy j e~/k(X~ ----~-~

IXa -Yl

~(3)(X

--

x,(t)) (33)

where % =- dx~/dt, and integrating over the space-time volume needed for the completion of the H o p f link, yields aq. At the point of completion the magnetic N-poles annihilate and the singularities described by (33) vanish. Obviously (33) can be extended to the case of an arbitrary number of N-poles involved in the formation of links. Formation and annihilation of H o p f links may, e.g., also proceed according to the mechanism indicated in fig. 2, which involves a fusion process of disclination segments. In case the topology changing process disentangles or forms a knot as indicated in fig. 3, Aq measures the self-linking and this requires that the knot is framed. Framing requires that the core of the disclination is replaced by a small ribbon. Formulae (31) and (33) still apply, except that C a and C b refer now to the two boundaries of the ribbon, and Aq measures also the twists of the ribbon (for further details see Witten [8]). The model (33) used to describe the formation of singularities in the n-field and the change Aq is certainly not the only one (see, e.g., fig. 2). In fact the formalism used allows to replace the n-field by any type of N-vector field ( N E ~ 3) not constraint to S 2. In that case magnetic charge and current (Pro, Jm) do not vanish and the 4-potential A" in (10a) is not well defined, and the same applies to (1). Formation of defects in the n-field may then be

ii¸:, :o °-

Fig. 2. Annihilation of two Hopf links of opposite valued Hopf invariants, over a fusion process from left to right; formation of two Hopf links proceeds from right to left.

A. Holz / Topology of defect configurations in ordered liquids

251

) ~ na Fig. 3. (a) Unknotting of a trefoil knot via formation of a magnetic dipole (n+~, n ~) from left to right; formation of a trefoil knot proceeds from right to left. (b) Annihilation of a linked trefoil knot via a fusion process from left to right; formations of the link proceeds vice versa.

realized via the N-vector fields involving also 2- and 3-dimensional singularities with respect to the n-field and may be studied via dQ/dt given by (26). The physical assumption underlying such an approach is that the order parameter of the system is subject to strong fluctuations during topology changing processes. Eventually it should be pointed out that the Gaussian linking number of the present problem is proportional to [2]

2 f

-- (811.) 3

M

d3x

f d3x'B(x,t).[(x-x')×B(x',t)] i x = x t-~

,

(34)

M

which is obtained from (1) using the Coulomb gauge V.A = 0. On account of the fact that Q is the Hopf invariant one expects that Q = Aq for the singularity free case. This is not quite obvious from (34), because the B-field usually has support in the whole 3-space M. Although the B-field has no singularities the two disclinations forming a Hopf link define cut surfaces in a representation of the n-field by polar angles (O, ~p) on S% and the borders of these cut surfaces define C a and C b in (31). Further details are deferred to the appendix. The present studies may be extended to non-simply connected 3-spaces M. Natural examples are the compact Seifert fibre spaces. These are 3-manifolds, which can be foliated by circles. According to Scott [14] they may be considered as a kind of bundle over a 2-dimensional orbifold, where the fibre is a circle. Due to "rr3(S2) ~ Z there are many representatives of Hopf fibrations which may be used as examples of the 3-space M. Such 3-spaces may also be used for the uniaxial nematic liquid studied in section 4.

A. Holz / Topology of defect configurations in ordered liquids

252

3. The SO(3) ¢r model and its relation to the 0 ( 3 ) cr model

The Chern-Simons action is of the form [3] Fc s

_

1

32,rr2

f d3x Eeq, tr(FpqAr

_

2ApAqAr)

(35)

M

where

Fur = OuA,,- a.Au +

[A u , A,,], (36)

F

= F"~,~T~ ,

[T ~,T b] =

eabcT

c .

Here {Ta}a_I 23 are the generators of the Lie-algebra so(3) in the 3 x 3- , , a2 representation; tr(T ) = - 2 , for a = 1,2, 3; tr(TIT2T 3) = - 1 , and all other traces vanish. Fcs will be studied for the simple "A-connections" A u ( a ) = A(OuR) ~ . R,

(37)

where a is some constant. Similar studies as follows can also be made for connections, where A is replaced by a triple of numbers {Aa}a=l,2, 3 and where Aa multiplies the T°-operator in (37). Using the identities A[0uAa(A) - 0.A,(A)] a

b ) A~(A) 2e ~b,.Au(A

=

(38)

one obtains for (36) F~(A)=(I+A)(0

A -0,,A

Fcs(A)= ~ ( 1 +

2A)

)

(39)

and

f

d3xepqrtr[(OpAq).Ar].

M

Introducing the notation

Apa'~C---=2Ap (+ 1)

a,Y~ O p =

~:pqr C3qA

a~ '

(40)

,

one obtains

3

Fcs = - 2 ( 1 + ~A)A2 -(8w) -1 2

2 2~ d3x Aa, Jc • Ba,~ = 2 ( 1 + gA)A a=l

M

Q~,

a=l

(41)

A . H o l z / T o p o l o g y o f defect configurations in o r d e r e d liquids

253

where Qa is defined by (1). On the other hand A~,(-1) is a pure gauge ( F , ~ ( - 1 ) -= 0) for which we obtain 1 f 48,tr2 ]_ d3x

F c s ( - 1 ) = Fwz -

E pqr

AaAbA c ~abc" ~ p~ "q~ ~r

M

1 (A,A2A 3 1 a~V(SO(3)) = 2( 8,'i"I.2 J 8,'ff 2 "

(42)

Here V(SO(3)) = 8w 2 is the volume of the SO(3)-group space and ~" E Z is the winding number measuring the classes of (9c). In a similar manner one obtains Fcs(A ) = 3 ( 1 + 2A)A2N.

(42')

An alternative form of Fcs(A ) is obtained using the representation R= where

( t i 1 ti2, ti3) , {na)a=l,2,3

(43)

is an orthonormal drei-bein field. In that case we can use

A ( + I ) = - d n 1- n2T 3 - d n 3- n i t 2 -- dn 2. n3T 1 and Ap(+l)=

'E

b

"n c ,

(44a)

a F pq( + l ) = 2E~ocn b ,p • ti c,q •

(44b)

- - 2 abcti ,p

This extends in a trivial manner to p, q---~/x, p. It is now a simple matter to derive the identity F . ~ ( + I ) - 2 (O,,n " XO~n a ) . n

(44b')

inserting tia = tib X ti c, where (a, b, c) is a cyclic arrangement of (1, 2, 3) into (44b), and using the identity below (28b). Comparison of (44b') with (10b) shows that F~,v(+ 1) = F~,~(n. .). = . F~,~e ,

2A~a(+1) = Z ~ ( n a ) =-- A ~a,~'

and therefore is consistent with the notation used in (40). Next we evaluate the individual terms in (41). Use of (43) and the Lagrange identity ((a × b ) . (c x d ) = (a . c ) ( b . d ) - (b . c ) ( a . d ) ) yields (no sum over a)

A . Holz / Topology o f defect configurations in ordered liquids

254

A.,yc

•

Ba.yc

=

..i

Aleab

c

b b c c e piql( n . ,p " n ) ( n ,i • n ) ( n ,q " n a )

.

Suppose that (a, b, c) forms a cyclic arrangement of (1, 2, 3), then A.,~e • B . Je ~--- 4 ~ p i q ( n a , p ° n b ) ( n b ,i ° n c ) ( n c , q * h a ) .

Use of 0 " = - d n b. n c implies A " ' ~ ' B "'~c d3x = - 4 0 " A 0 b A 0 c . From this follows that each component of the drei-bein the same contribution to (41) yielding

field {na}a_l,2,3

gives

Cos(a) = 2(1 + 2/~)/~2 ( ~ ) 2 4 X 3 X WV(SO(3)), which is identical to (42'). From (41) and (42') follows then oa

(8Tr) 2 1 fM d3X A a , ~ , n a , ~ = l dV. "

(45)

This is rather interesting result, because it implies that W must be divisible by 2 in order that the drei-bein field used in the SO(3)-bundle is singularity free. This is related to "r¢1(80(3))~7/2 and therefore there exist disclinations of strength s E 77 + ½, which are non-contractible, i.e., which feature core singularities. The present result is consistent with the fact that a smooth SO(3)bundle may trivially extend to a SU(2)-bundle which is necessarily trivial for "rrl(M ) ~ I. Due to V(SU(2)) = 2V(SO(3)) we obtain in a normalization, where FSU(2)( 1] SO(3) arbitrary integers Y{EZ are allowed for - c s ~ - . j , Fcs ( - 1 ) = lrsv(2)t --cs ~,-- 1) = ½Y{, and from (45) Qas o ( 3 ) = ~Y{. Here the subindex is a reminder that the normalization has been changed with respect to (42). This implies that only those SO(3)-bundles, where Y{ is divisible by 4, extend trivially to SU(2)-bundles, in units where arbitrary integers are allowed for SU(2). The present result is consistent with Dijkgraaf and Witten [15]. The property of the SO(3) o- model to be essentially the sum of three 0 ( 3 ) omodels corresponding to the components of the drei-bein field applies also to the action Sso(3 ) = A'

f4

d x t r ( 0 . R t" 0~*R)

M4 -__ ~1 A , f M4

d4x

(O~,n a • O~'n"),

(46)

A. Holz / Topology of defect configurations in ordered liquids

255

where A' is a coupling constant. Similarly one obtains for the Yang-Mills action SyM =

- ~ ~ f d4x ~.Fal~p

(47)

a=l M4

and for the A-connections defined by (37) S M(A) =

3 l

d4x [(Ovti a° 0/xtia) 2 - ((go ti a • 0~tia)2] . + A) Z a=l " M4

(47')

A further covariant action of the SO(3) o" model is S g = 2 ~ ~ f d4x(cqt~A~) 2 a=l M4

and for the A-connection Sg(A) = ½~A2 ~ f d4X(~abcDtib'tlc). a=l M4

(48)

Observe that under a gauge transformation G E ~, where (g is the set of continuous maps M---~ SO(3) such that locally G: A--~ .~. = G A G t + G dG+,

(49)

we obtain G: Sso(3)({na})---~ S s o ( 3 ) ( ( ~ a ) ) , where (ri l, li 2, ri 3) = R ( { n a } ) . G t, and similarly for Sg, whereas SVM remains invariant, i.e., SvM(A ) = SyM(A , {na}). On the other hand under an active transformation {n a } ~ {ri a } all three actions defined above will change and are therefore useful for a SO(3) ¢r model. For non-trivial SO(3) or bundles Fcs(A ) has to be supplemented by an additional term corresponding to core singularities of disclinations. From a A~,(A) one obtains for the additional field strength tensor ~F~u(A) = 1 obc(ti b

- ti b

tic

and for the corresponding electric and magnetic field strengths

(50)

256

A. Holz / Topology of defect configurations in ordered liquids

8 ET ( , ) = - ½,~.°~c ( li~,o, - li~.o ) " lie, 5Ba(A)

= i~EipqEabc(libpq

_

lib,qp ) • lic

and A~(A). 5B~(A) = ½A2IeabclEi p q ltn b ,i" n c )[(n b ,pq __ n b ,qp)" nC]. Suppose that the ll b- and nO-field are singular, then in a local coordinate system we can use the representation /cos 0 ~ nb=~sino0.),

( - s i n oa~ c°sO"0 /J '

.c=

yielding n

b, p q . n c =

aa(l~)

•

oa

~ma(l~)

;pq ,

n

b

,i

• lic = oa

~2 ipq . . . . ~ Z,A E. I ~ ;i ~

;i ,

;pq .

Due to the use of a local coordinate system a non-linear covariant derivative must be taken, indicated by a semi-colon; for the sake of simplicity it will be replaced in the following by a partial derivative. This yields

~Fcs(A) -

8A2 ~ d P q f d 3 x

(8"if) 2 a = 1

oa,i

~)a,pq

.

(51)

M

Setting A a ,i =- 0 a ,i '

5B~ =

ElPqO)apq

one obtains 5Fcs(A ) _

8A2 ~ f d3x A ~. 5B ~ , (8¢r) ~ ~=1

(51')

M

which is again the sum of three Hopf invariants multiplied by the factor 8A 2. Here, however, the 8Ba-fields are singular and have the properties of fluoxoids confined to the cores of disclinations. From the property that the O"'s are angular variables and Stokes' theorem in the form 2"rrsi =

O ,~, dx ~ = OD i

f2

[d x ] ~ ~F a'~

Di

A. Holz / Topology of defect configurations in ordered liquids

257

we obtain

~Ba(x) = 2zr ~ s7 f dx7 a~3)(x i

- xa),

(52)

cf

where s~ E 77 and C~ represent the strength of the disclination and its oriented loop configuration, respectively. Use of the representation

1 f V x, x ~na(x ') ix _ x, I

A"(x) = ~

d3x '

M

(which satisfies V x Aa(x) = ~Ba(x)) in (51') yields (34). Replacing ~n a by (52) yields 3

8Fcs(A ) = A2 •

~', s~s~dP(C~, C~),

(53)

a=l i<.j

where the Gauss linking number has been defined by (31). For i = j a framing of loops has to be done as indicated earlier. A more precise derivation of (53) requires the use of methods developed e.g. by Misner et al. [13] for electromagnetism in spaces with non-trivial connections. Consider now the change in time of Fcs(h ) and ~Fcs(h ). Use of (5) yields 3

d Fcs(Z) = 2(1+ 2AlZ2 a=l 2 ~ 2 dt

dt

f d 3 x E a "B ~

daX ~Ea" ~Ba"

a=l - ~

(54a)

M

(54b)

M

Eq. (54a) can be studied by means of the theory developed in section 2 extended to three copies of unit vector fields coupled by the orthonormality condition. From this follows that dFcs/dt~O arises only during singular processes involving intersections of disclination loops resulting in links or knots or in the annihilation of the latter. The same applies to (54b), because 8E ~ and 8B ~ have support only along the cores of disclinations. From the property that 8E ~ and 8B ~ satisfy Maxwell's equations it follows that 8E ~. 8B ~ is Lorentz invariant. Accordingly transforming to the rest frame of a moving segment of the loop where 8E ~ = 0, implies that contributions to (54b) can only arise at space-time points, where disclination cores cross each other. At such points, however, one cannot simultaneously transform to the rest frame of both loop segments involved in the crossing

258

A . Holz / Topology o f defect configurations in ordered liquids

process and therefore at such points ~ E ° • ~ B a ~ 0 may arise. Because ~ E a and ~B" involve 6-functions, it seems more reasonable that a crossing process takes place qualitatively as described in section 2, i.e., via the formation of magnetic N-poles. It is obvious that for the ~E "- and ~Ba-fields a similar theory applies as developed in section 2. Eventually it is pointed out that the singular fields ~ E a and ~ B a do not arise in the 0 ( 3 ) or model, as a consequence of the definition (10b). As a matter of fact F , ~ ( n ~) vanishes at such singularities of the n°-field due to its planar character there, as follows from (10b). Finally some remarks with respect to the Yang-Mills theory from the perspective developed in section 2 will be made. The equations in Yang-Mills theory corresponding to the inhomogeneous Maxwell equations are b r c r v = 1or O f F ar~ + 2 e a b c A r r

(55)

or

• . E a + 2Eabc(A b " E c ) = pa , OE a

VxB" _ --

+ 2e, bc( - qgbE c + A b × B ~) = j a ,

at

where A ar = (tih a , A a) and j~r = (p,, j , ) has been used, and for the A-connection given by (37) the 4-current may be computed as a function of the drei-bein field. For the A-connections the integrability conditions b

Or * F °r" + 2eabcAr * F cr~ = 0

(56)

for smooth field configurations are automatically satisfied. Because the second term on the left-hand side of (55) does not vanish the situation apparently is different from the one encountered in section 2. However, this is not necessarily the case, because (55) and (56) may be rewritten in the form (e.g. for A = 1) O,F,,,~rv = jo,~

,

Or *Fa'~tr~= JamXV,

(55') (56')

where the right-hand sides are defined via comparison with (55) and (56). From this follows that for each component of the drei-bein field a similar analysis applies as indicated in section 2 for the 0 ( 3 ) tr model, and that all three 0 ( 3 ) tr models are coupled by the constraint of orthonormality. Because for smooth drei-bein fields the "magnetic" 4-current defined in (56') must vanish according to (17a) and (17b), the second term on the left-hand side of

A. Holz / Topology of defect configurations in ordered liquids

259

(56) must also vanish in that case. In the presence of "magnetic" N-poles, where N E ¼Z according to the discussion below (45), the right-hand side of (56') does not vanish, whereas (56) still applies. In this case the problem has to be analyzed using a non-trivial SO(3)-bundle. On such bundles A-connections of the form defined by (37) can still be used except that now transition functions are non-trivial, i.e., I ~ G E SO(3). This implies that during topology changing processes one always has a non-trivial SO(3)-bundle. In Yang-Mills theory topology changing processes are mediated by instantons [3]. These are usually computed for self-dual fields F = * F from D *F = 0 in Euclidean 4-space and where the instanton number q defined by (4) characterizes the bundle. In (3 + 1)-dimensional space-time, an instanton corresponds to a trajectory connecting two states of the Yang-Mills field with fiat connections, i.e., F(Ai) -= 0, i = 1, 2. Although it is possible that this theory has some significance for the present problem, it is unclear how a curved connection A c should be interpreted within an SO(3) or model. The problem is that the holonomy of an At-field is path dependent and therefore

R(x)=Pexp(- f A)

(57)

o

is not unique. Here P is the path ordering operator and 0 some reference point. Note that for the A-connections defined by (37), (57) is also not unique, but this does not matter because A(A) depends explicitly on R(x) and therefore (57) is not needed by definition. On the other hand, formation of singularities of magnetic N-pole type or in the form of disclinations of strength s ~ ' Z can still be interpreted within an SO(3) tr model, because they imply non-trivial SO(3)-bundles. For instance a singular disclination loop of strength s ~ ' Z implies that the order parameter jumps across the cut surface bounded by the loop according to R . ( x ) = G. R ( x ) , where ! ~ G @ SO(3) characterizes the holonomy (57) of the non-trivial bundle. Further details will be worked out in the following section.

4. Topological properties of liquid crystalline systems Consider first a uniaxial nematic liquid, where the order parameter assumes values in the projective 2-sphere p2, and may be described by the pair (n, - n ) . On a 3-space M (closed and simply connected) the trivial U(1)-bundle consists of smooth n-fields containing disclinations of integer strength s E Z, and is equivalent to that used for the 0 ( 3 ) or model in section 2. Non-trivial U(1)-bundles are necessary to describe the geometrical situation in the

A. Holz / Topology of defect configurations in ordered liquids

260

presence of disclinations of strengths s E 7/+ ½. In that case the n-field is multivalued and changes sign upon crossing the cut surface spanned by a disclination loop. The cut surface itself, however, does not contribute to physical observables due to the doubly valued nature of the director (n, - n ) . Similarly A~, and F , , computed according to section 2 are multivalued and only the pairs ( A , - A ~ , ) and ( F ~ , - F , ~ ) can be used as observables. In the following this point of view will be further developed. In the 0 ( 3 ) or model the H o p f invariant of a two-component link L = C~ U C m, formed from two disclinations of strength n, m E 7/, and oriented loop configurations C , and C m respectively (see, e.g., fig. 1), is

(5s)

Q = nmClg(Cn, C m) •

In the presence of many links of this type this formula extends additively due to -rr3(S2) ~ 7/. A simple guess is that for an uniaxial nematic liquid n and m in (58) have just to be replaced by n, m E ½7/. The modulus of the smallest non-trivial H o p f link is then IQ'I = ¼. For a derivation of this result one may use an extension of (34) taking account of the cut surfaces yielding an additional term

4 ~Q -

I d2x"[B(x)xB(x')]

f

(8~r)3

d3x J

M

Ix - x' I

{Xi}

H e r e the area integral is taken over all oriented cut surfaces {£/}, with B(x) taken at the " o r i e n t e d " side of the cut surface £i for x E £/. In the following a m e t h o d proposed by Dijkgraaf and Witten [15] will be used to evaluate Q'. Consider a H o p f link L = C,, U C m formed by two disclinations of strengths n E 7/ and m E 7/+ 2~, where C n and C m are unknotted, respectively. Upon encircling C m twice A~ and F ~ assume their original values. Take therefore two copies of the 3-space M provided with the cut surface £m (O~m= C m ) and glue them together along the oriented cut surface in such a manner that the A,,- and F~v-fields change smoothly a c r o s s ~m and are single valued in the 3-space M* = l~i= 2 1 M ( i ) U ~m (i) • M * is a 2-fold branched cover of M with C m a s branch set (see, e.g., Rolfsen [16]). In M* the strength of the m-disclination has doubled, i.e., (n, m)--+ (n, 2m), whereas the linking remains unchanged. Accordingly we obtain Q,_

1 f d3x A - B = 2nmqb(C,, C m). (8¢r)2 M*

On the other hand, we have

(59)

A. Holz / Topology of defect configurations in ordered liquids

261

2

o*=E if ,=,- (8~)---5

d3xA-n=ZQ ',

(59')

Mi

because each term in (59') provides the same contribution due to A . B = ( - A ) - ( - B ) . Consequently one obtains

Q'= ½Q*=nm,

nE2e,

mET~+ ½.

A H o p f link L = Cn U C m formed by two disclinations of strengths n, m E 7/+ ½ may be treated in a similar fashion except that now one uses two cut surfaces En and ~m (O~n = Cn, O~JLm= Cm) and four copies of M to construct M * = 4 I M(i) U £(/) U ~m (i) • M* is a 4-fold branched cover of M with C, and C m as Hi= branch set. In M* the strengths of the n- and m-disclinations have doubled, whereas the linking is unchanged implying Q* = 4 Q ' = 4nm@(Cn, C m ) and Q ' = nmq~(Cn, Cm). In case that C n and C m represent non-trivial knots one may use the oriented Seifert surfaces [16] of the knots as cut surfaces En and ~m and proceed as above. This yields

Q, = t Q , = nm@(C,, Gin) ,

(n, m) E ½7/,

(60)

where q~(Cn, Cm) is the Gauss linking number defined by (31), and q~ E Z. This result extends in an additive fashion to all types of links, due to ,rr3(P 2) ~ 7/.

In a similar manner one may study dQ'/dt via the instanton number

q, _

1

32w2

f d4x E . B M~

where M ] is a suitable branched covering space of M 4. For the example of a H o p f link formed by two disclinations, with (n, m) C 7/+ ~ one obtains dQ'_ dt

1 f 2(8~r) 2 d3x E . B .

(61)

M*

As a model one may use crossing processes of disclinations involving the formation of "magnetic" N- and M-poles as indicated in section 2. Consider next the case of a biaxial nematic liquid, whose order parameter space is the quotient S O ( 3 ) / D 2 , where D 2 is the dihedral group of ord(D2) =

A. Holz / Topology of defect configurations in ordered liquids

262

4. Within the representation (43) of SO(3) the elements of SO(3)/D 2 may be represented by Fl(n 1, n 2, n3)-matrices supplemented by the three identifications

R(n l, n 2, n 3 ) -~ R(n 1, - n 2, - n 3 ) ~ R ( - n', - n 2, n 3) ~ R ( - n 1, n 2, - n 3 ) , (62) which are compatible with the orientability of the 3-space SO(3)/D 2. In order to derive the topological invariants for this problem the formalism developed in section 3 and at the beginning of this section will be applied. The result derived earlier that each component of the drei-bein field gives the same contribution to Fcs (A) still applies ignoring the singular part ~Fcs (A) given by (51). Due to V(SO(3)/D2)= ~V(SO(3)) it is more convenient to normalize the Chern-Simons action with respect to the 3-space SO(3)/D z, i.e., we use 02 Fcs(A ) = 3(1 + 2A)A2•D2,

(63a)

where ND2 E Z is the winding number of SO(3)/D 2 and correspondingly 3

F cDs2 ( h ) = 8(1 + 2A)A2 E Q ~

(63b)

a=l

yielding

Q;2-

1 f

(8,rr)2

d3x

aa,~ Ra,x ld~,D 2 " "~D 2 " UD 2 -- 8

(64)

M

This result is compatible with respect to the perspective developed at the beginning of this section for the relation between the Hopf invariants of Szand p2-order parameter spaces due to (64). Suppose, e.g., that we consider a Hopf link formed from two disclinations with strengths (n, m) E 77 + ½in the D2 n'-field. Because each field will give the same contribution to Fcs(A ) it is sufficient to study one component of the drei-bein field. Using the same notation and arguments as for the uniaxial nematic liquid we obtain a

Q;2 =4QD2

~

1

~2¢'D2

(64')

implying (64). Because in the computation of Q* in M* the singular part ~Fcs drops out, (64) applies to the non-singular part of Fcs. In order to derive a formula for Q~2 corresponding to (60), the second identity of (64') should be expressed as a sum over products of disclination

A. Holz / Topology of defect configurations in ordered liquids

263

strengths. For that purpose it is more convenient to start out from a normalization of Fcs with respect to the SU(2)-group. Eq. (45) reads in that case

Q~SO(3) = ~X

(45')

where 9{ E 7/is the winding number with respect to SU(2) ~ S 3. The subindex SO(3) is used to indicate that the normalization has been changed. Eq. (45') may formally be written as follows: (k, h) E ½7/.

Qso(3) = k h ,

(45")

A derivation of this formula can be done according to the method outlined at the beginning of this section. Suppose we have a Hopf link in S 3 formed by two disclination loops C k and C h of strengths (k, h) E 7/+ ½. Each disclination loop bounds a cut surface £, where the order parameter (an SU(2)-matrix) changes sign. Accordingly we have a non-trivial SU(2)-bundle in M U ~ a U ~ b , which can be trivialized in the 3-space M* formed by four copies of M as explained earlier. This yields in the same notation as earlier a

Qso(3) = 4Qso(3)=4khC19(Ck, Ch) ,

(k,h)~

7/ + ½ ,

(45'")

and is compatible with (45"). This derivation also shows why non-trivial SO(3)-bundles displaying diclinations of strength s E 7/+ i cannot be extended to the trivial SU(2)-bundle. Similarly one obtains now in $3/Z2 × Dz-space disclinations of strength p E 17/, and for a Hopf link of strengths p, q = -+ ¼one obtains

1

IQ~o(3~,D21 = ~ . Here one needs in fact 16 copies of M and suitable gluings to construct M*. The general formula for the Hopf invariants of the components of the drei-bein field making up the order parameter of the biaxial nematic liquid with N links is N

Qso(3)/D 2 = E kihidP(Ck,, Chl),

(ki, hi) E ¼Z.

(65)

i=1

In addition one has 3 S O ( 3 ) / D 2 ( ~ ) -- -- 8 X 2 ( 1 + 2 a ) ) ~ 2 ~] 0 ° Vcs SO(3)/D 2 a=l

(66a)

264

A. Holz / Topology of defect configurations in ordered liquids

and

d rs°(3)/D2 ~CS dt

dFcs(X ) - 8 - dt '

(66b)

and for the singular part SO(3)/D 2

rcs

(A) = 8

rcs(;0,

d ~rso(3)/D2tA)=8 d dt v - c s ' ~-~ ~rcs (A).

(66c)

The present result can easily be extended to the finite point symmetry groups SO(3) of o r d ( P i ) > 4. For the octahedral group O of o r d ( O ) = 24 there exist 2-, 3- and 4-fold axes of symmetry. The formulae corresponding to those given above for S O ( 3 ) / D 2 are obtained by observing that ord(Y72 x D 2 ) = 8 and therefore that the factor 8 in (66a) to (66 0 has to be replaced by SO(3)/O ord(Y 2 × O ) = 4 8 . In the normalization Fcs ( - 1 ) = W, X ETI we obtain o as o ( 3 ) / o = ~W. Accordingly only such bundles where W is divisible by 96 can be extended to a trivial SU(2)-bundle. The factor 1 in (65) is obtained from 1/ord(7/: x T2) = ¼, where T 2 = 7/2 is the "torsion" of the elements of D 2. Accordingly in (65) we have to use for O

Pi of

(ki, h i ) E

{1 1 ~22,~Z,~77

1

}

,

(67)

in such a fashion that N E g is satisfied. For the icosahedral group [ we have ord(712 × l) = 120, and the factor 8 in (66a) to (66c) has to be replaced by 120 and Q as, o~ ( 3 ) /__ i - ~ 1W , W E Y ; in addition (67) for the group I has to be supplemented by the set of numbers [1/(2 × 5)]7/ due to the presence of a 5-fold axis of symmetry in I. Topology changing processes in the anisotropic liquids with an order parameter space SO(3)/P, may be studied in a similar fashion as indicated in section 2 for the 0 ( 3 ) o- model over the formation of "magnetic" N-poles, where e.g. for Pg = O, N assumes the set of numbers given by (67). They may also be studied via instanton solutions as indicated at the end of section 3. Furthermore they may be studied over transient non-trivial bundles as will be explained below for the SO(3) or model. Observe that in realistic SO(3) or models there exist also 2-dimensional defects (see, e.g., Holz [17]) along which the order parameter jumps. In the simplest case such a defect may be described by fig. 4b as a disclination loop of strength s E ~1 bounding a cut surface along which the order parameter jumps according to I:l+(x) = GFI (x). Consider now the Hopf link sketched in fig. 4a

A. Holz / Topology of defect configurations in ordered liquids

a)

265

~(t) -~_J..J~.JZf

C(-fi, O(t))

Fig. 4. (a) Formation of a Hopf link via cut surface singularities; cut surfaces are drawn dashed. (b) Formation of an unknotted disclination loop by disconnecting the order parameter along the cut surface drawn dashed and rotating top and bottom part in group space indicated by C(tl, @(t)) and C(-~, O(t)). Cut surfaces in (a) can be interpreted in the same manner providing them with two pairs of "screws" C(-+h,, Oa) and C(-+hb, Oh), respectively.

and measured by (45"). Its process of formation in real time t may be measured by

a t

Qso(3)( ) = k(t) h(t) qb(Ck, Ch)

~ (k(t), h(t)) E 1~1 .

(68)

H e r e we take a non-trivial SO(3)-bundle with two linked disclination loops of strength k(t) and h(t), bounding two cut surfaces Ek and Eh, and being characterized by the holonomy matrices Ck(t ) and Ch(t ). These may be characterized by C~(t)~ (O~(t), e~L=k,h, where O and O are rotation axis and angle, respectively, in SO(3)-space, in the compact sphere representation of radius 6} = -rr (see, e.g., ref. [17]). The process of formation of the H o p f link is completed for 6}k(t) = 4"rrk, 6}h = 4"rrh. At each stage of the process, (68) may be computed over a suitably defined non-simply connected 3-space M* as demonstrated above for the finite subgroups of SO(3). Now, however, the surfaces carry an energy which plays the role of an activation energy for the process. The formation of knotted disclinations may be described by a similar m e t h o d using as cut surface the Seifert surface [16] suspended by the knot. The dimension of instanton space being characterized by the instanton n u m b e r ff~ can be estimated as follows. For the sake of simplicity we consider H o p f links formed from infinitesimal loops in an orthogonal orientation. Then there are 4~'{ translational degrees of freedom, 2 ~ orientational degrees of freedom and 6 ~ - 3 degrees of freedom in SO(3)-space. This yields 12~r - 3 degrees of freedom in comparison to 8 Y ( - 3 degrees of freedom of self-dual instantons in Yang-Mills theory [3]. Eventually it is pointed out that the present analysis can also be extended to closed and non-simply connected 3-spaces M. In that case the set of fiat

266

A. Holz / Topology of defect configurations in ordered liquids

connection over M modulo gauge equivalence classes corresponds in a natural fashion to the quotient space of representations (69)

R(M) = Hom('rrl(M)---~ SO(3))/SO(3)

by the conjugation action of SO(3). For further details see refs. [8, 15]. A case of practical interest is to take for M the 3-torus T 3 with 'n'l(T 3) ~ 7/x 77 x 7/. The trivial bundle in R(T 3) corresponds to a SO(3) o- model contained in a cube and subject to periodic boundary conditions. Imposing boundary conditions on pairwise opposite faces of the cube in the form R ( X a , + ) = G a " [~(Xa _ ) and observing the relations among the three elements G a C SO(3), imposed by the three relations among the generators of "n'l(T 3) modulo gauge equivalence classes, yields R(T3). That R(T 3) contains nontrivial connections follows from the observation that if Ga E C(h, ~9) C SO(3), for a = 1, 2, 3, where C(h, tg) is the Abelian group of rotations around the axis h and 0 ~< O ~< "rr, the three relations are obviously satisfied. Gauge equivalence implies the conjugacy classes O. C(h, ~ga) O~ = C(O. h, ~ga),

a=1,2,3,

O E SO(3).

Accordingly h drops out and the set (~9,, ~92, ~93) parametrizes the non-trivial connections in R(T3). Another example is Poincar6's dodecahedral space S~, which is generated by gluing pairs of opposite faces of a dodecahedron after a clockwise twist of ~ of a full turn [16]. The fundamental group of that space is ~rl(S.) = 7/2 x I the binary icosahedral group of ord(7/2 x I)= 120. The space R(S 0 of flat connections may again be generated via boundary conditions as indicated above. Examples of 3-spaces M which may play a role in (2 + 1)-dimensional physics are M = ~g,N X S l, where ~g,N is a Riemann surface of genus g perforated by N holes.

5. Deformation and topological change of knotted and linked structures

Moffatt [7] has recently developed the concept of the "energy spectrum of knots and links". Based on the frozen-field equation OB =v at

--

x (v x B )

(70)

he shows that the field energy J/(t) = 1 f d3x B2(x) in an incompressible fluid

A. Holz / Topology of defect configurations in ordered liquids

267

(V- v = 0) satisfies under suitable assumptions d--T -

k

dax rE(x, t ) ,

(71)

M

where k is a constant. From this follows that for v ~ 0 at t # 0 , ~t is a monotonically decreasing function. Based on the work of Freedman [18] (implying that for non-trivial topology of ( B ( x , 0)}, ~(t)--> ~ E > 0 for t--->oo), he shows that ,~E~m,

O<~mo<~ml<~m2<~ . . . .

The sequence of numbers mi, Moffatt [7] calls the energy spectrum of the knot, with m 0 its ground state energy. The magnitude of m 0 is a measure of the complexity of the knot or link. In the following Moffatt's idea will be applied to some of the examples studied in the preceding sections. Consider first the 0(3) o- model and convective transport of the n-field by fluid motion. Suppose the fluid motion provides the smooth map (x, t = 0)--> (X(t), t) , then convective transport implies n(x, 0)---> n(X(t), t) = n(x, O)

(72)

and d n ( X ( t ) , t ) / d t = O . Similarly we have F u ~ ( x , O ) - - ) F ~ ( X ( t ) , t), and use of (10b) implies Fpq(X(t) , t ) = 2 e abc OX-----0 ~ na(x,O ) ~ 0

rib(X, O) no(x, 0 ) ,

where we have assumed that at t = 0 the time component of F vanishes. Eq. (72) and the chain rule yield Ox 3x t 3 n~(x,O) 0 Fpq(X(t), t) = 2e ~bc -OX - kp OX q OX k - ~ t rib(X, O) nc(x, O) OX k

Ox t

OX p

OX q

Fkt(x, 0 ) .

(73)

From this follows that under convective transport of the n-field, F changes covariantly with respect to its spatial components. For those we obtain from (73)

268

A. Holz / Topology o f defect configurations in ordered liquids

BP(X(t), t) = ~

Ox

OX p ~ Bq(x, 0),

(73')

and O~X X =1

forV. v=O,

and (73') is the solution to (70) (see ref. [7]). Consequently convective transport of the n-field implies convective transport of the magnetic induction field. It follows from this that in case the 0 ( 3 ) o- model is provided with a field energy ~ , the theory of Moffatt [7] applies. In order to apply the theory to a uniaxial nematic liquid, A/must be replaced by Frank's energy [19] •-,~' = ½ f d3x [ k l ( V " n) 2 + k2(n "V × n) 2 + k3(n × 17 × n)2], M

(74)

where { k i ) i = l , 2 , 3 are Frank's constants. Consider now convective transport of the director field { ( n , - n ) } according to (72). Then

f d3Xf'(x(t))= f @x {k~[Vx.n(X(t),t)] M

2+-..},

M

where due to V. v = 0, d 3 S = d3x, and due to (72), f ' depends only implicitly on t. Accordingly

OW*'(t) f d3X df'dt dt M

and use of the chain rule implies

dt

O(Onk/OX i) 2 OX i 2

OX i

\OXJ/

O(OnJOX i) °'Ji

dt

oxi/3

dt dt o'ji O .

Here the last step follows from dXi/dt = OXi/Ot + (v . V ) X i = 2v ~and ~rd is the Erickson stress [19]. The second step in (75) is only justified for ~ra being

A. Holz / Topology of defect configurations in ordered liquids

269

symmetrical, which is not necessarily the case for general coefficients {ki}, but for the sake of simplicity is used here. The acceleration equation of a uniaxial nematic liquid is of the form dv i P dt

Op ~Or~i OOrji OX,+-~-7+--OX j ,

i=1,2,3

where ~r' is the dissipative stress [19]. Eq. (75) can now be written in the form t

dv2

OOr ]i V i

ddt f ' = - ( v "Vxp ) - ½p - - ~ + - ~ 7

•

(75')

From (75') follows the energy conservation law d(,~' + Ekin) dt

dW dt

(76)

where the right-hand side represents the work done by the dissipative stress. In the derivation of (76) use has been made of p ( X ( t ) , t ) = p ( x , O ) implying v . V x p = -Op/Ot and

f dt f d3X Op= f dt -~t0 f d3Xp(X(t), t ) = 0 Ot M

M

Setting dW = T dS, one obtains

d(~'+ Ekin) dt

--

TS,

where S is the entropy production rate. A rate equation of the type (71) can be derived ignoring dissipation and assuming that in the acceleration equation the velocity field is self-consistently determined by p d v / d t = kv (see ref. [7]) yielding dt

-

k

f d 3X

O 2 •

(77)

M

Accordingly the ideas developed by Moffatt [7] may also be applied to uniaxial nematic liquids. As a matter of fact the equilibrium states of polymeric liquid crystals usually have a high density of entangled disclinations and shear thinning in such systems may be directly related to mechanical disentanglement of linked structures by shear flow (see, e.g., Yamazaki et al. [1]). Another application may be the coarsening dynamics of quenched uniaxial

A. Holz / Topology of defect configurations in ordered liquids

270

nematic liquid crystals as studied recently by Chang et al. [20]. In such experiments a pressure jump initiates an isotropic to nematic phase transition, with the formation of a highly entangled network of disclinations and subsequent coarsening in time. The experimental observations are in good agreement with scaling laws which do not take the topological constraints into account predicted in the present paper. The reason for that may be that the coarsening processes studied lead only to a reduction of disclination line density but to no topological change, i.e., the topological entanglement is conserved, if it exists at all in the isotropic phase from which it is quenched. The other reason is, of course, that two Hopf links of opposite sign in (58) may annihilate leaving one unknotted disclination loop which may collapse, as indicated in fig. 2. Consider now briefly the qualitatively different processes leading in different 0 ( 3 ) o- models to a topological change of structure. Consider a ferro(F)- and antiferro(AF)-magnetic 0 ( 3 ) ~r model, whose action in either case is represented by (46) for a = 1, and in the ferromagnetic case is supplemented by the electromagnetic action. For the F- and AF-model we take respectively B F =

Hv+

4,rrM v ,

B AF = 4 , r r M AF ,

(78)

where we set M v= an and B A F = BAF(n) according to (10b), and a is a constant. Use of B AF = B F - - - > H v -

4 , r r ( M AF -

M F)

(79)

implies for the respective Hopf i n v a r i a n t s Q A F = QF. Due to (79) we may use and without loss of generality s e t ~.F = 1 (dielectric constant). From Amp6re's equation follows then

E F = E AF

jF

= --~7 X M v .

This derivation applies to smooth n-fields, where dQF/dt = dQAF/dt = O, i.e., j v is not a dissipative Ohm current as in (29). Although this derivation applies to each topological sector, i.e., Aq F = Aq Av, it does not apply during topology changing processes, where (79) does not hold due to V. B AF ¢ 0 and V'B F =0. In the latter case a mapping between the two processes may be done by setting B AF=B F-B m, implying

E AF~-E F - E m ,

(80)

A. Holz / Topology of defect configurations in ordered liquids ~r. B m = 4,n.pm

,

~7 × B m +

OrE m =

47rJ

m

271

,

(81) V.E m=0,

//m=

(Pm,Jm) is

where

fields B m

and

E m,

/_/F = BAF

_

0 ,

defined by (17). Solving (81) for the "magnetic" N-pole yields

B m _

4,rrM F

E F = E AF d- E m

(82)

Amp~re's equation, which vanishes identically for the AF-model, yields for the F-model jF

:

-Jm -~7 × M F

-

1

v x

Bm

,

whereas Gauss' equation yields ~7. E F = 4,rrp F = V" E AF = - 4 r r V . ~ I}AF .

This shows that for a given n-field changing from one topological sector to another all relevant quantities in the two models can be computed, and one obtains d Q F __ d Q AF

dt

-

-

dt

+ ~

2

f d3x

(EAF " B m

+

Em

•BA F + E m . B m ) .

(83)

M

Due to Aq F= Aq AF the time integral over the second term above vanishes. Observe that the condition (79) may also be replaced by the condition V x H F = 0, which corresponds to the usual formulation of a magnetic problem [12]. In that case QF expressed in the form (34) can also be expressed through M F simply by the replacement BF--->4"rrMF as can easily be derived, i.e., the magnetic field H F does not enter QF. It is expected that in this case also Q F = QAF arises, although it has not been proven so far. It follows from this discussion that for two models having the same order parameter different topological field theories may be assigned. Qualitative differences essentially arise during topology changing processes. Similar considerations may be applied to the SO(3)/Pi o- models. In Moffatt's [7] approach one may use here the field energy

~/~= l ~a f d3x Ba2 M

obtained from (47) and the hydrodynamic equation

(84)

A. Holz / Topology of defect configurations in ordered liquids

272 du

3

p -~ = -re+

Z ja × B a + ~ A I ) ,

(85)

a=l

where j a is the current defined by (55), and ~7 is the coefficient of viscosity. Within the frozen-field approximation one obtains dB a

dn ~

dt-0,

a=1,2,3

~

dt-0,

a=1,2,3.

(86)

Use of (55), ~/= 0 and p d v / d t = k v implies (71). Due to the representation of the SO(3) or model in terms of the drei-bein field, the ideas developed in the former paragraph for the 0(3) or model also apply here and imply the same consequences. Similar derivations can be made, where (84) is replaced by the field energy obtained from (46), leading to the same result and corresponding to the derivation of (77). The frozen-field condition applies only in the absence of dissipation, e.g. in electromagnetism when the conductivity o- is infinite. In the case or < oo the frozen field condition (70) is replaced by [12] (V. v = 0) dB dt

_ RmlV2B

where R m is the magnetic Reynolds number. In this context the problem of chaotic flow and kinematic magnetic dynamics is of interest and consists according to Finn et al. [21] in the problem: "Given a flow of conducting fluid, will a small seed magnetic field amplify exponentially with time?" It is obvious that similar problems can be posed for the o- models studied in this paper, which are usually dissipative and dynamo effects may be discussed in terms of the fold, stretch and twist deformations of disclination loops. This problem may be of particular importance for turbulent flow in superfluid phases of 4He and 3He, where the order parameter may assume values in the groups G = U(1) and SO(3), respectively. Due to the possibility of "magnetic" N-pole formation in these models, stretching of disclinations may not proceed indefinitely but lead to their fracture. The dynamo problem may therefore also be of interest in Yang-MiUs theory or QCD where the color fields describe gluons and one has the problem of confinement [3]. The SO(3) or model may also play a role as a semi-classical approximation to the spin-½ Heisenberg antiferromagnet (AF) [17]. Due to the itinerant character of the electrons carrying the spins, this o- model may have the properties of a quantum liquid in a metallic system, permitting a coupling of its degrees of freedom to a flow as discussed in this section. This problem may have some significance for high-Tc superconductivity, where short-range magnetic fluctua-

A. Holz / Topology of defect configurations in ordered liquids

273

tions are present. For instance for the doped spin-1 AF the 3-space M due to the presence of holes may not be considered anymore as simply connected or alternatively as dosed. If the holes are considered as the privileged sites of magnetic N-poles the theory developed by Holz and Gong [22] for high-Tc superconductivity may apply. Due to the (2 + 1)-dimensional character of this problem topology changing processes as discussed in this paper play a minor role, except in case that a coupling between neighbouring layers is considered, via hopping processes of holes and spin couplings.

6. Conclusion

Topological properties of defect configurations in anisotropic liquids have been studied via the linking of disclinations. The disclinations considered divide into two classes. In the first class there are the coreless disdinations which due to rrl(S 2) ~ I are topologically unstable if unlinked, but are topologically stable in a linked or knotted configuration due to "tr3(S2) ~ 7/. The second class consists of disclinations with core singularities and are a consequence of non-simply connected order parameter spaces, e.g., "rrl(P2)~2~2 and "rrl(SO(3)) ~ Z 2. They are topologically stable but in fact there exists only one representative in the examples given above, whereas in a linked configuration they are topologically stable in an infinite variety due to -rr3(P2)~Z and "rr3(SO(3)) ~ Z, generating textures of kaleidoscopic nature. In physical applications topology changing processes are most important. For instance if the 0(3) or model is used to describe a classical ferromagnetic material, linking of disclinations determines the ground state properties like remanent magnetization and coercive force, whereas topology changing processes determine the Barkhausen jumps and the structure of hysteresis curves. Computation of such processes requires a certain model. In this paper two models have been suggested. In the first model formation of links or their disentanglement proceeds via the opening of loops and formation of "magnetic" N-poles. Whether such a process realizes depends on its activation energy and therefore on the action one uses for the or model. On the other hand, computation of singular processes usually goes beyond continuum models. In a second model indicated in connection with (68) it is therefore proposed that topology changing processes may also occur via the formation of surface type singularities. In continuum models such defects are forbidden but occur in realistic models, although usually they are not stabilized topologically. The situation in Yang-Mills theory is rather different. Tunneling amplitudes are computed via instantons based on non-flat connections. An

274

A. Holz / Topology of defect configurations in ordered liquids

interpretation of such processes in terms of the variables of a o- model is usually ill defined due to the non-trivial holonomy of the connection but may be possible under the conditions indicated below. The idea of Moffatt [7] on the energy spectrum of knots and links has been applied in section 5 to uniaxial nematic liquids and may be applied to all anisotropic liquids. As this theory makes certain statements about the equilibrium states reached by a defect system it may perhaps also be generalized to systems where topology changing processes are of importance, i.e. driven systems. Such systems are, e.g., polymeric liquid crystals, where the phenomena of shear thinning is a direct consequence of disentanglement of disclinations (see ref. [1]). An important aspect of such systems is that topology changing processes involve a large number of disclinations and therefore collective processes should be treated by topological methods. There exist, of course, qualitative methods to handle such systems, as e.g. the work-hardening model for plastically deformed materials. Solid continua may also be studied by the methods developed in this paper (see ref. [10]). Line defects in these systems are dislocations and disclinations and the relevant symmetry group is the linear group GL(3, E). However, because SO(3) is a maximal compact subgroup of GL(3, E), topological studies are again based on the order parameter spaces SO(3)/P i. Due to the discrete structure of the solids, defects of dislocation type are topologically stable and this requires that in fact the space symmetry group of the solid has to be used in topological studies, increasing the complexity of the problem. The present studies are essentially based on fiat connections, and for simply connected spaces M on the trivial connection. This implies that knot polynomials as computed by the topological field theory of Witten [8] in (2 + 1) dimensions cannot be studied by the methods presented. Non-fiat connections should be interpreted in terms of additional degrees of freedom as applies, e.g., to the ferromagnetic 0(3) o-model introduced in section 5 in contrast to the antiferromagnetic 0(3) o- model. An application to topological field theory based on one of the order parameter spaces studied in this paper, e.g., SO(3)/Pi, requires that a physical interpretation of such a model in terms of a gauge theory exists. Because the singular defects we have studied in this paper have a non-vanishing field strength tensor with support at their cores (see eq. (59) for A -- - 1 ) , they may be used to define non-flat connections. In that case one must introduce continuous distributions of singular disclinations and study the problem on a scale which does not resolve individual disclination cores. A similar method may apply to disclinations without core singularities. In the presence of many of them the flat connection may be a strongly fluctuating quantity, and suitably defined smoothed out connections may be non-fiat. In solid state physics, e.g., one may introduce in this way torsion into an

A. Holz / Topology of defect configurations in ordered liquids

275

originally torsionless connection by representing dipolar pairs of disclinations as dislocations, etc. (see e.g. ref. [23]), or vice versa one may treat torsion for curvature. This approach is also used in solid state physics where continuous distributions of infinitesimal dislocations are used (see e.g. Kr6ner [24]). A gauge theory of such a system may then be based on SvM given by (47) and a topological field theory on (35). For further details we refer to refs. [8, 15]. Non-simply connected 3-spaces M may play a role for doped spin-½ Heisenberg antiferromagnets, which may be described in a semi-classical approximation by a SO(3) o- model [17]. Segregation of holes in bubbles implies that M is not closed, but this case is less likely due to Coulomb repulsion, if it is supposed that the background charge is uniformly distributed. The holes are convenient location points of magnetic N-poles and therefore such a system may contain many open disclination lines. Here again one may distinguish between singular and non-singular disclination cores and conceive a representation of the o- model as a gauge theory based on the Lie algebra SO(3). A (2 + 1)-dimensional version of this theory may be considered as a model for high-T c superconductivity, if Yang-Mills and Chern-Simons action is supplemented by the hole action. The multiconnectivity of the space in the presence of holes implies then that also non-trivial SO(3)-bundles are considered. Due to the weak coupling between layers the problem is in fact (3 + 1)-dimensional and topology changing processes may be described by instantons and the powerful methods developed for them may be used.

Acknowledgements

Useful discussions with H.M. Sauer and S. Kohler are kindly acknowledged.

Appendix

The Wess-Zumino term (2) can be represented in the form Fwz _

1 f tr(R t dR R t dR 48.a.2 Mg

Rt

dR)

(A.1)

where Mg corresponds to M via the map x--+ G = SO(3) and Fwz = X, 2¢"E 77. Setting A = R t dR and B -- d A = d R t d R = - R t d R R t d R

276

A. Holz / Topology of defect configurations in ordered liquids

one has /~WZ ~

1 f tr(AB).

48,rr2

(A.2)

Mg

This shows that Fwz is of a similar type as Hopf's invariant (1), and that its integrand is a closed 3-form; due to d(AB) = BB = 0, however, it is not exact. Fixing some origin in group space we define the "0-form" and "2-form", respectively 5g A and (.fogA) dA, where g E G. Due to g

d[(~A) dA]--AA 0

we can set g

~wz-148,n.2 ftr{d[(~A)dA]} Mg

~n3,

0

Note that the " O - f o r m " .fog A is a multivalued function and therefore d(.[0g A) is not a closed 1-form. In order to apply Stokes' theorem to (A.3) suppose that M = ~3tA 0% and map the point o~E M onto the origin of group space G. Suppose now that the cut surfaces, arising due to the presence of disclinations, are considered as boundaries S in Stokes' theorem, where S is naturally two-sided. This yields g

~wz-148,rr2 ftr[(fdA)dA] S

~a.4,

0

Due to g

g

(j A) dA= ~I(f A) dA]-AA 0

0

and d(R dR t) = AA, and renewed application of Stokes' theorem, we obtain g

~wz-148.rr2 ~tr[(f"td")"td"+"d"'] oS

~ns,

0

H e r e OS represents the boundaries of S observing the two-sidedness of the cut surfaces, where for the Lie algebra SO(3) the last term in (A.5) vanishes.

A. Holz / Topology of defect configurations in ordered liquids

277

The first term in (A.5) vanishes for unlinked disclinations, where the cut surfaces do not intersect mutually. This follows from the fact that for unlinked disclinations 0S is a sum over the boundaries of S, where each boundary is integrated over in two opposite senses, due to the two-sidedness and orientation of each side of the cut surfaces. This applies in fact only to disclination loops which are unknotted. For a knot the cut surface may be identified with the Seifert surface of the knot, which is two-sided but may intersect itself. Accordingly for individual knots Fwz does not have to vanish. For linked disclinations aS has support along the lines, where the cut surfaces intersect (the d a s h - d o t t e d line in fig. 4a). Consider now the case of a H o p f link and set g

g

(A.6) 0

0'

where C is some constant and 0' is located at the intersection line. Independence of (A.5) on the origin 0 of group space implies ~0s R t dR = 0. Setting 3

R t dR

=

-

~

~0 a T a

and use of trace ( T a2) = - 2 , for a = 1, 2, 3, we obtain g FWZ = --~ ~

a~a a=l

OS

f

a ~ a' "

(A.7)

0'

Because each component of the drei-bein field gives the same contribution to Fwz, as has been shown in section 3 in connection with (45), (A.7) can also be written in the form

lff

Fwz = + 8~ 2

g

0q~

oS

~tp'.

(A.7')

0'

Contributions to (A.7') arise only from the two line integrals forming the intersection line of cut surfaces. This agrees with the result of Kundu and Rybakov [2], who also obtain for the H o p f invariant only a contribution along the intersection line of cut surfaces. Explicit evaluation of (A.7') requires a definite model configuration. On the other hand, (A.7') is a bilinear expression in "angular" variables and should therefore be proportional to (2~rn)(2~rm). From Fwz = W and bilinearity

A. Holz / Topology of defect configurations in ordered liquids

278

follows then N = n m . Presumably it can also be shown that (A.7') is proportional to an area integral over a torus of linear dimension 2-rrn and 2~rm, in case that the two disclinations are linked and thus can be expressed explicitly in terms of the Gauss linking number of the boundaries of cut surfaces. In a different approach, where one starts from (45) and uses (34) this is certainly the case using the approach of Kundu and Rybakov [2]. The derivation of (A.7') also applies to knots or say self-linked disclinations. In that case 0S has support along the boundary of the Seifert surface attributed to the knot and its intersection lines. This problem has not been studied carefully so far, but it is expected that the present method corresponds to a framing of the knot (see ref.

[8]). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]

Y. Yamazaki, A. Holz and S.F. Edwards, Phys. Rev. A 43 (1991) 5463. A. Kundu and Yu.P. Rybakov, J. Phys. A 15 (1982) 269. T. Eguchi, P.B. Gilkey and A.J. Hanson, Phys. Rep. 66 (1980) 213. N.D. Mermin, Rev. Mod. Phys. 51 (1979) 592. M. K16man, Points, Lines and Walls (Wiley, New York, 1983). G.W. Whitehead, Elements of Homotopy Theory (Springer, New York, 1978). H.K. Moffatt, Nature 347 (1990) 367. E. Witten, Commun. Math. Phys. 121 (1989) 351. A. Holz, Topological properties of linked disclinations in anisotropic liquids, in: J. Phys. A, to be published. A. Holz, Topological properties of linked disclinations and dislocations in solid continua, in: J. Phys. A, to be published. H.J. de Vega, Phys. Rev. D 18 (1978) 2945. J.D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975). C.W. Misner, K.S. Thorne and J.A. Wheeler, Gravitation (Freeman, San Francisco, 1970). E Scott, Bull. London Math. Soc. 15 (1983) 401. R. Dijkgraaf and E. Witten, Commun. Math. Phys. 129 (1990) 393. D. Rolfsen, Knots and Links (Publish or Perish, Wilmington, DE, 1976). A. Holz, Physiea A 170 (1991) 511. M.H. Freedman, J. Fluid Mech. 194 (1988) 549. P.G. de Gennes, The Physics of Liquid Crystals (Clarendon, Oxford, 1974). I. Chuang, N. Turok and B. Yurke, Phys. Rev. Lett. 66 (1991) 2472. J.M. Finn, J.D. Hanson, I. Kan and E. Ott, Phys. Rev. Lett. 62 (1989) 2965. A. Holz and Ch. de Gong, Phys. Rev. B 37 (1988) 3751. A. Holz, Class. Quantum Grav. 5 (1988) 1259. E. Kr6ner, in: Physique des D6faults (Physics of Defects), Les Houches, Session XXXV, 1980, R. Balian, M. K16man and J.-P. Poirier, eds. (North-Holland, Amsterdam, 1981).