Topologically Linked Polymers are Anyon Systems

0.2 MB·English

by  Franco Ferrari

Topologi
ally Linked Polymers are Anyon Systems
∗
1,2,
Fran
o Ferrari
1
Institute of Physi
s, University of Sz
ze
in, ul. Wielkopolska 15, 70-451 Sz
ze
in, Poland
2
Max Plan
k Institute for Polymer Resear
h, 10 A
kermannweg, 55128 Mainz, Germany
4
0 We
onsider the statisti
al me
hani
s of a system of topologi
ally linked polymers, su
h as for
0 instan
e a dense solution of polymer rings. If the possible topologi
al states of the system are
2 distinguished usNing the Gauss linking number as a to2pNologi
al invariant, the partition fun
tion of
n anensemNbleof
losedpolymers
oin
ides with the pointfun
tionofa(cid:28)eldtheory
ontaining
a set of
omplex repli
a (cid:28)elds and Abelian Chern-Simons (cid:28)elds. Thanks to this mapping to
a
(cid:28)eld theories, some quantitative predi
tions on the behavior of topologi
ally entangled polymers
J
have been obtained by exploiting perturbative te
hniques. In order to go beyond perturbation
7
theory,a
onne
tion betweenpolymers andanyonsis established here. Itis shown in this waythat

the topologi
al for
es whi
h maintain two polymers in a given topologi
al
on(cid:28)guration have both
]
h attra
tive andrepulsive
omponents. Whentheseopposite
omponentsrea
h a sort ofequilibrium,
c the system (cid:28)nds itself in a self-dual point similar to that whi
h, in the Landau-Ginzburg model
e for super
ondu
tors,
orresponds to the transition from type I to type II super
ondu
tivity. The
m s4ig−npi(cid:28)la
atn
e of self-dualityin polymer physi
sisillustrated
onsidering theexampleoftheso-
alled
-
on(cid:28)gurations, whi
h are ofinterest in thebio
hemistryof DNApro
esses like repli
ation,
t trans
riptionandre
ombination. The
aseofstati
vortexsolutionsoftheEuler-Lagrangeequations
a
is dis
ussed.
t
s
.
t ξµ = ( ,t) µ = 1,2,3 = (x1,x2)
a In this letter we
onsider the statisti
al me
hani
s of tor r , where , r and
m x3 = t
a system of topologi
ally linked polymers[1℄, su
h as for plays the role of time. We need also to intro-
- instan
e a dense solution of polymer rings (
atenanes). du
ethetwo-dimensional
ompletely antisymmetri
ten-
d ǫij i = 1,2
n If the possible topologi
al states of the system are dis- ǫsoµνrρ µ,,ν,ρ = 1,a2n,d3its three dimensional
ounterpart
o tinguished using the Gauss linking number as a topolog- , . These tensoǫ1r2s =ar1e uniqǫu1e2l3y=de1-
c i
al invariant,it hasbeen shownin [2℄ that the partition (cid:28)ned by the following
onventions: and .
[ N
fun
tion o2fNan ensemble of
losed polymers
oin
ides We use middle latin letters as two-dimensional spa
e in-
1 with the point fun
tion of a (cid:28)eld theory
ontaining di
es and middle greek letters as three dimensional in-
N
v a set of
omplex repli
a (cid:28)elds and Abelian Chern- di
es. Sum over repeated spa
e indi
es will be every-
Γ
2 a
Simons (C-S) (cid:28)elds. Thanks to this mapping to (cid:28)eld where understood. The traje
tories of the polymers,
0 a=1,...,N
theories, some quantitative predi
tions on the behavior , will be treated as
ontinuous
urves. Usu-
1
1 of topologi
ally entangled polymers have been obtained ally,thesetσrajeΓ
tor=ies(are(pσar)a,mt e(tσriz))ed0bymσeansLofthLeir
a a a a a a a a a
0 by exploiting perturbative methods, see Ref. [3℄ and ref- ar
-length : { r | ≤ ≤ },
4 eren
es therein. If onewishes to go beyond perturbation being the total length of the traje
tory. To make a
on-
0
theory, however, one is fa
ed with the problem of (cid:28)eld ne
tion with anyons, however,it is
onvenientto use the
/ t t
t theorieswhi
h
ontainmulti-
omponents
alar(cid:28)eldsand time as a parameter. Of
ourse, the
oordinate is
a
m gauge (cid:28)elds intera
ting together. Clearly, it is not easy not a good parameter if weΓdo not introdu
e a suitable
a
to study theories of this kind even in the simplest
ase of se
tioning for the
urves . To this purpose, let us
- N = 2 t
d . The idea whi
h we wish to pursue here in or- noti
e that, with respe
t to the −dire
tion, ea
h
urve
Γ
n der to
ir
umvent su
h di(cid:30)
ulties is to establish a link dtaa(hσaa)s=ma0xima and minima de(cid:28)nedsby the
ondition:
o between polymers and anyons [4℄. Anyon (cid:28)eld theories dσa . The number of minima, a, is equal to the
c
: have been in fa
t intensively investigated. It is known numberoΓfmaxima. Letτuspi
kupagivenτpointofmini-
v for instan
e that their
lassi
al equations of motion ad- mum on a and
all it a,1. Starting from a,1 and going
i
X mitvortexsolutions[5℄. Inaspe
ialregionofthespa
eof alongthe
urveafter
hoosingarbitrarilyadire
tion,one
r parameters,
alledtheBogoml'nyiself-dualpoint[6℄, the willen
ouτntersu

essivelyapointoτfmaximum,thatwill
a attra
tive and repulsive for
es between vorti
es vanish. be
alled a,2, apointofminimum a,3 andsoon. Inthis
2s τ ,...,τ
This point
orresponds in the Landau-Ginzburg model way we obtain aset of a points a,1 a,2sa. Now it
Γ s Γ ,...,Γ
forsuper
ondu
torsto the transitionfromtypeItotype is possible to split a into a open paths a,1 a,sa
II super
ondu
tivity. whi
h
onne
tpairwise
ontiguouspointsofmaximaand
The (cid:28)rst obsta
le in this program is that anyons are minima:
2+1
livingin dimensions,while polymer traje
toriesare
τ t τ
a,I a,I+1
intrinse
ally de(cid:28)ned in three dimensions. For this rea- ≤ ≤
Γ = ( (t),t) (τ )= (τ )
saosnt,imone.e
Ao
o
rodridniaRntge3lym, uastpobientsionfgtlehde othurteea-nddimreengsairodneadl a,Ia  ra,Ia (cid:12)(cid:12)(cid:12) rraa,,IIaa(τaa,,IIaa+)1=ra,rIaa,−Ia1+(τ1a,Iaa,)Ia+1 
(cid:12)
Eu
lidean spa
e will be identi(cid:28)ed by a three ve
-  (cid:12) (1)
2
I I 1,...,2s τ
where a is a
y
li
index su
h that a ∈ { a} at the points of maxima and minima a,Ia in order to
I +2s =I (τ ) Γ
and a a a. Thepointsra,Ia a,IA are
onsidered re
onstru
t the
losed
urve a.
as (cid:28)xed points. This se
tioning pro
edure is expli
itly The polymer a
tion A
an be divided into two
ontri-
s=3
illustrated in the
ase by Fig.; 1. By
onstru
tion, butions:
t = +
free top
A A A (3)
τ
a,6
τa,2 The free part Afree is:
τ Γa,1 Γ Γa,5 Afree =Xa=21IX2as=a1Zτaτ,aI,aIa+1dt(−1)Ia−1gaIa(cid:12)(cid:12)drad,Ita(t)(cid:12)(cid:12)
a,4 Γ a,2 ( 1)Ia−1 (cid:12)(cid:12) (cid:12)(cid:12)(4)
a,3 Γ The fa
tors − appearing in Eq. (4) are neg
essary
ττa,5 a,4 Γa,6 rtoelamtaedketAoftrheee lpeonsgittihvseLdae,(cid:28)Ianitoef.tTheheoppaenram
heatienrssΓaaI,Iaaaares
a,3 follows. Letusdis
retizeΓa,Ia
onsideringitasarandom
n
τ
hainswith segments. IfEq.(4)wouldbethea
tionof
a,1
a set of free two-dimensional
hains, in the limit of large
n n/g
valuesof andkeeping(cid:28)xedtheratio a,Ia,onewould
2s−platΓa s=3 obtain the well known result [7℄: La,Ia ∼ n/ga,Ia. Here
FIG. 1: Se
tioning pro
edure for a with .
things arealittle bit more
ompli
ated, be
auseone has
to take into a

ount the third dimension, whi
h in turn
there is a one-to-one
orresponden
e between the points
t [τa,I,τa,I+1] is also the variable whi
h parametrizes the
urves. As a
of the −axis in the interval and the points
Γa,I
onsequen
e, the above relation governing the length of
of the traje
tories . As a
onsequen
e, it is possi-
ble to parametrizethe traje
toriesΓa,Ia by meansof the the traje
tories is repla
ed by:
t n
variable . L2a,Ia ∼|τa,Ia+1−τa,Ia|2+ g |τa,Ia+1−τa,Ia| (5)
We are now ready to
onstru
t the partition fun
tion a,Ia
of a system of topologi
allyentangled polymers. For the
sakeof
larity,itis
onvenienttousesomeofthenomen- The se
ond term in Eq. (3) is:

leature of knot theory. In parti
ular, from thΓe mathe- 2s1 τ1,I+1 dξµ (t)
a 1,I
mati
al point of view, ea
h
losed traje
tory
an be =ıλ dt B ( (t),t)
top µ 1,I
A dt r
regarded as a knot, while an ensemble of two or more I=1Zτ1,I
X
traje
tories de(cid:28)nes a link. Moreover, with some slight
2s
abuse of language, a knot or a link with points of
maxima and minima will be
alled a 2s−plat. The
on- κ 2s2 dξ2µ,J(t)
+ı dt C ( (t),t)
µ 2,J

ept of plats arises quite naturally in the pro
esses of 2π dt r (6)
J=1
repli
ation,re
ombinationand trans
ription of DNA [8℄. X
To start with, let us writ2eSthe partition Sfun=
tionn fosr a Bµ = ( ,B0) Cµ = ( ,C0)
linkwiththetopologyofa −plat,where a=1 a. In the above equation B and BC B
sa aretwoAbelian C-S ve
tor(cid:28)elds. We haveput 3 ≡ 0
The integers are now kept (cid:28)xed, so that thPe link is C C
Γa 2sa and 3 ≡ 0 in order to stress the role of time of the

omposed by knots whi
h are −plats. In order κ λ
third spatial
oordinate. and are real parameters.
to avoid a proliferation of indi
es, we will restri
t our- κ

oin
ides with the C-S
oupling
onstant (see below).
selves to a system of two linked polymers, thus setting
N = 2
The spurious gauge degrees of freedom have been elimi-
. A
tually, this is not a big limitation, be
ause
nated using the gauge
ondition (Coulomb gauge):
the two-polymer problem
ontains all the ingredients of
N ı =√ 1
the −polymer problem. Putting − , the desired ∇ =∇ =0
·B ·C (7)
polymer partition fun
tion is given by:
S
CS
(λ)= B C e−ıSCS (t)e−A so that the gauge (cid:28)xed C-S a
tion looks as follows:
Z Z D µD ν Zboundary Dra,Ia (2) S = κ d3ξ B ǫij∂ C +C ǫij∂ B

onditions CS 0 i j 0 i j
4π (8)
Γ Z
The boundary
onditions on the traje
tories a,Ia are (cid:2) (cid:3)
thoseofEq.(1). Theseboundary
onditionsarerequired The C-S (cid:28)elds a
t asauxiliary(cid:28)elds whi
h impose topo-
Γ a Γ Γ
by the fa
t that the open paths a,Ia, with (cid:28)xed and logi
al
onstraints on the
losed traje
tories 1 and 2.
I = 1,...,2s
a a
, must join together in a suitable way To show this, one has to integrate out these (cid:28)elds from
3
thepartitionfun
tion(2)andthentoperformaninverse dis
uss the steri
intera
tions, in part be
ause we prefer
(λ)
Fourier transformationof Z : to
on
entrate on the pure topologi
al intera
tions, in
2π dλ part be
ause the in
lusion of these repulsive for
es
om-
(m)= eimλ (λ) pli
ates enormously the task of (cid:28)nding non-perturbative
Z 2π Z (9)
Z0 analyti
solutionsofthetwo-polymerproblemunder
on-
sideration. Thus,wewillassumethatinoursystemsteri
The details of this
al
ulation have been already ex-
intera
tions are negligible. This approximation is valid
plainedin Ref.[3℄andwill notberepeatedhere. There-
(m) for instan
e in polymer solutions whi
h are very dilute
sult is the new partition fun
tion Z , whi
h des
ribes
or in whi
h the temperature is near the so-
alled theta-
the (cid:29)u
tuations of two polymer rings
onstrainedby the
point. The hypothesis of very low monomer densities is

ondition:
naturalin thepresent
ontext,be
ausewearedis
ussing
m=χ(Γ1,Γ2) the statisti
al properties of two isolated polymer rings,
(10)
whi
h are supposed to be very long and thin.
χ(Γ ,Γ )
where 1 2 is the Gauss linking number and the To establish the
onne
tion with anyons, we need to
m
topologi
al number must be an integer. A few
om- passinthepartitionfun
t(ito)n(2)fromthestatisti
alsum
ments are in order at this point. First of all, in [3℄ the overthetraje
toriesra,Ia toapathintegralover(cid:28)elds.
topologi
al
onstraint (10) has been derived from the C- Th(λe)details of the derivation of the partition fun
tion
Z in terms of (cid:28)elds in the general
ase and in
luding
S path integral using the Lorentz gauge, while here the
also the steri
intera
tions will be presented elsewhere.
Coulomb gauge (7) has been
hosen. Formally, things
works exa
tly in the same way in both gauges. How- Here we will4restri
t ourselves to links wshi
h=hsave=th1e
1 2
stru
ture of −plats, putting hen
eforth .
ever, one should keep in mind that the (cid:28)nal expression
χ(Γ ,Γ )
of the Gauss linking number 1 2 obtained starting This
ase is parti
ularly interesting for biologi
al appli-
from the Lorentz gauge di(cid:27)ers from the expression that
ations, sin
e most of the l4inks produ
ed by in vitro en-
oneobtainsstartingfromtheCoulombgauge. Of
ourse, zymologyexperimentsare −plats4[8℄.pMlaotrseover,(cid:16)small(cid:17)
the two expressions are equivalent due to gauge invari- links up to seven
rossinngs are all − .
r
an
e,but thewayinwhi
hthewindingofonetraje
tory Introdu
ing a set of
omplex repli
a (cid:28)elds:
around the other is
ounted, is realized in a very di(cid:27)er- Ψ~ = (ψ1 ,...,ψnr )
ent way. Another important point is that our approa
h Ψ~∗a,Ia = (ψa1∗,Ia,...,ψan,rI∗a) (11)
allowsamorere(cid:28)ned
lassi(cid:28)
ationoflinks thanthebare a,Ia a,Ia a,Ia
Gauss linking number. Indeed, two links whi
h have the
same Gauss linking number
an be further distinguished and a
onvenient notation for
ovariantderivatives:
by their plat-stru
ture, i. e. by the number of maxima κ κ
( λ, )=∇ ıλ ( , )=∇ ı
and minima of the knots whi
h are
omposing the links. D ± B ± B D ±2π C ± 2πC (12)
Finally, it is not di(cid:30)
ult to add also the repulsive steri
intera
tionsbetweenthemonomersinthepartitionfun
- 4
the −platpartitionfun
tion
anbeexpressedasfollows:
tion (2). It is interesting to noti
e that, sin
e we have
x t

hosen the
oordinate 3 ≡ in orderto parametrizethe 2 2

urves, the usual δ−fun
tioVn(p,ott)e=ntivalδ(wh)δi
(ht)dves
ribes Z4−plat(λ)=nli→m0 D(fields) O1,I O2,Je−S4−plat
thesefor
estakestheform: r 0 r , 0being Z IY=1 JY=1
a temperature dependent
onstant. Due to the presen
e (13)
δ 4−plat
ofthe −fun
tionin time, thesteri
intera
tionsbe
ome The a
tion S
ontains the free (cid:28)eld a
tion and the
instantaneous in our
ase. In the following we will not topologi
al terms:
τ1,2
1 2 1 2
S4−plat = dt dx Ψ~∗1,1∂tΨ~1,1+ 4g D(−λ,B)Ψ~1,1 +Ψ~∗1,2∂tΨ~1,2+ 4g D(λ,B)Ψ~1,2 +
τ1Z,1 Z (cid:26) 11 (cid:12)(cid:12) (cid:12)(cid:12) 12 (cid:12)(cid:12) (cid:12)(cid:12) (cid:27)
τ2,2 (cid:12) (cid:12) (cid:12) (cid:12)
1 κ 2 1 κ 2
+ dt d Ψ~∗ ∂ Ψ~ + ( , )Ψ~ +Ψ~∗ ∂ Ψ~ + ( , )Ψ~
x 2,1 t 2,1 4g D −2π C 2,1 2,2 t 2,2 4g D 2π C 2,2 (14)
τ2Z,1 Z (cid:26) 21 (cid:12)(cid:12) (cid:12)(cid:12) 22 (cid:12)(cid:12) (cid:12)(cid:12) (cid:27)
(cid:12) (cid:12) (cid:12) (cid:12)
d dx dx ∂ = ∂
x ≡ 1 2 is the in(cid:28)nitesimal volume element in the two-dimensional spa
e and t ∂t. Let us note that in
4
n
r
thepartitionfun
tion(13)wehavealreadyintegratedout propagate anyoni
parti
les with spin and the statis-
B ,C
0 0
the time
omponents of the Chern-Simons (cid:28)elds , ti
sofboson. Onedi(cid:27)eren
efromanyoni
(cid:28)eldtheoriesis
whi
h play the role of Lagrange multipliers and impose duetothefa
tthat,inthe
aseofrealparti
les,one
on-
the following
onstraints: siders the evolution of the whole system from an initial
T T
1 2
2 2 time anda(cid:28)naltime . Here,instead,timeintervals
=2 Ψ~2,1 Ψ~2,2 θ(τ2,2 t)θ(t τ2,1) measure the extensions of the traje
tories Γa,Ia in the
B − − − (15) x
(cid:18)(cid:12) (cid:12) (cid:12) (cid:12) (cid:19) 3 dire
tion. For this reason, in the partition fun
tion
4π (cid:12) (cid:12)2 (cid:12) (cid:12)2
= (cid:12)Ψ~ (cid:12) (cid:12)Ψ~ (cid:12) θ(τ t)θ(t τ ) (13) part of the system evolves within the time inter-
C κ 1,1 − 1,2 1,2− − 1,1 (16) [τ1,1,τ1,2]
(cid:18)(cid:12) (cid:12) (cid:12) (cid:12) (cid:19) v[τal ,τ ] and another part within the time interval
(cid:12) (cid:12) (cid:12) (cid:12) 2,1 2,2 . This inhomogeneity in the time intervals and
(cid:12) (cid:12) (cid:12) (cid:12)
In the above equation B and C are the magneti
(cid:28)elds the use of the Coulomb gauge, whi
h makes the intera
-
asso
iatedtotheve
torpotentialsBandCrespe
tively: tions instantaneous, is the
ause of the presen
e of the
θ
= ∂ B ∂ B =ǫij∂ B Heaviside −fun
tions in Eqs. (15)(cid:21)(16). In this sense,
1 2 2 1 i j
B − (17)
= ∂ C ∂ C =ǫij∂ C the
omplete analogywith anyons
an be re
overedonly
1 2 2 1 i j
C − (18) in the limit
θ(t) θ(t)=0 t<0
and is the Heaviside theta fun
tion if τ1,1 =τ2,1 T1 τ1,2 =τ2,2 =T2
θ(t)=1 t 0 (fields) ≡ (21)
and if ≥ . The symbol D in Eq. (13)
is a shorthand for the (cid:28)eld integration measure:
Throughout the rest of this letter we will assume that
2sa
(fields)= Ψ~∗ Ψ~ theabove
onditionissatis(cid:28)ed. Thθisistoavoidte
hni
al
D D a,IaD a,Ia (19)
ompli
ations in dealing with the −fun
tions.
Z IYa=1
The
onne
tiontotheanyonproblemsuggesttoapply
Finally, the operatorsOa,Ia are given by: tothea
tionofEq.(14)theBogomol'nyitransformations
[6℄. Using these transformation in a suitable way, it is
Oa,Ia = ψa1,Ia(ra,Ia(τa,Ia+1),τa,Ia+1) possible to rewrite S4−plat in the following form:
ψ1∗ ( (τ ),τ )
a,Ia ra,Ia a,Ia a,Ia (20) S4−plat =F4−plat+I4plat (22)
Eqs. (13)(cid:21)(20) establish the desired
onne
tion between
polymers and anyons. In fa
t, the a
tion (14), together Negle
ting surfa
e terms and introdu
ing the notation
D± = D1 ıD2 Di i = 1,2 i
with the
onstraints (15)(cid:21)(16),
oin
ides formally with ± , where , , denotes the −th
the a
tion of2(asm+ultsi-)lay=er4edanyon system. TΨ~h∗e nu,Ψ~mber spa
e
omponents of the
ovariant derivatives (12), one
of layers is 1 2 , while the (cid:28)elds a,Ia a,Ia has that:
T2 2 2
F4−plat =ZT1 dtZ dx"Xa=1IXa=1Ψ~∗a,Ia∂tΨ~a,Ia+
1 2 1 2 1 κ 2 1 κ 2
+ D+( λ, )Ψ~1,1 + D+(λ, )Ψ~1,2 + D−( , )Ψ~2,1 + D−( , )Ψ~2,2
4g − B 4g B 4g −2π C 4g 2π C (23)
11 (cid:12) (cid:12) 12 (cid:12) (cid:12) 21 (cid:12) (cid:12) 22 (cid:12) (cid:12) (cid:21)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
and
I4−plat = λ2 T2dt dx |Ψ~g11,11|2 − |Ψ~g11,22|2 × Ψ~2,1 2− Ψ~2,2 2 − |Ψ~g21,21|2 − |Ψ~g22,22|2 × Ψ~1,1 2− Ψ~1,2 2
ZT1 Z (cid:20)(cid:18) (cid:19) (cid:18)(cid:12) (cid:12) (cid:12) (cid:12) (cid:19) (cid:18) (cid:19) (cid:18)(cid:12) (cid:12) (cid:12) (cid:12) (cid:19)(cid:21)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)(24)
4−plat
It is easy to re
ognize from Eq. (23) that F
on- instead Coulomb-like potentials, whi
h attra
t or repel
Γ
sists in the sum ofΨ~the self-dual a
tions of four di(cid:27)erent themonomersbelongingtodi(cid:27)erenttraje
tories a,Ia. It
families of anyons a,Ia. These families are
oupled to- isdi(cid:30)
ulttoguesswhatisthein(cid:29)uen
eoftheseCoulomb
gether by the
onstraints (15)(cid:21)(16). The se
ond
ontri- for
es on the statisti
al behavior of the polymers. The
4,plat 4−plat λ
bution to S , the intera
tion term I ,
ontains problem is that their strength is proportional to (see
5
λ
Eq. (24)) and is not a real
oupling
onstant with a the self-dual regime. This does not mean however that
physi
al meaning. At the end, we are interested to elim- atta
tions and repulsions between monomers disappear,
λ
inate and to express the polymer partition fun
tion in be
auseCoulomb-likefor
esstillhideintheself-dualpart
m 4−plat
terms ofthe topologi
alnumber , aswedid in Eq. (9). of the a
tion F .
m
is in fa
t the true physi
al parameter whi
h des
ribes Wewouldlikenowtoexploitfurthertheanalogyofthe
the topologi
al state of the system. What is possible to polymerproblemwithanyons(cid:28)eldtheoriesandsear
hfor

on
lude from Eqs. (22)(cid:21)(24) is that topologi
al for
es possiblenon-trivial
lassi
alsolutionswhi
hminimizethe
4−plat
have attra
tive and repulsive
omponents, whi
h inter- polymer free energy S . Of
ourse, one should keep
fere with the steri
intera
tions. This result
on(cid:28)rms at in mind that, even in the
ase of the simplest model of
anon-perturbativelevelapreviousresult obtained using anyons, vortex solutions outside the self-dual point have
perturbative methods [3℄. Also in Ref. [9℄ it has been been found only by means of numeri
al methods. Also
shown in the limit in whi
h (cid:29)u
tuations in the parti
le at the self-dual point, there is no
lear and detailed un-
densities arerelatively small that the C-S (cid:28)elds generate derstanding of the dynami
s of C-S vorti
es [10℄. Here
e(cid:27)e
tive Coulomb intera
tions in anyon models. the situation is further
ompli
ated by the presen
e of
Remarkably, our approa
h reveals that the two- repli
a (cid:28)elds and multiple families of anyons whi
h are
polymersystemunder
onsiderationhasaself-dualpoint. mixed togetherdue to the
onstraints(15)-(16). In view
g
Thispointisrea
hedwhentheparameters aIa areequal: of these limitations, it seems reasonable to restri
t our-
g11 =g12 =g21 =g22 =g (25) stieolvness, wtohit
hhessaetlifs-fdyuathliety
orengdiimtieon(s25∂)tψaaωn,dIat=o s0tafotir
esvoelruy-
a=1,2 I =1,2 ω =1,...,n
r
value of , and . As a
onse-
If the above relations are satis(cid:28)ed, it is easy to see that
4−plat = 0 quen
e, from now on we will
onsider the pure self-dual
I , so that the only surviving terms in the a
-
4−plat a
tion:
tion S are those
oming from the self-dual
ontri-
butions in F4−plat. We note that the existen
e of the S4−plat =
self-dual point (25) is not submitted to the assumption (T2−T1) d D ( λ, )Ψ~ 2+ D (λ, )Ψ~ 2+
of Eq. (21). Indeed, let us suppose that the points of 4g x + − B 1,1 + B 1,2
maxima and minima τa,Ia of the traje
tories Γa,Ia are Zκ (cid:20)(cid:12)(cid:12) 2 κ(cid:12)(cid:12) (cid:12)(cid:12) 2 (cid:12)(cid:12)
not aligned as spe
i(cid:28)ed by Eq. (21). The only di(cid:27)eren
e D−(−2π,C)Ψ~(cid:12)2,1 + D−(2π(cid:12),C)Ψ~(cid:12)2,2 (cid:12)(27)
in the expression of the intera
tion term I4−plat is that (cid:12) (cid:12) (cid:12) (cid:12) (cid:21)
T1 T2 (cid:12) (cid:12) (cid:12) (cid:12)
now the values of and , whi
h give the boundaries (cid:12) (cid:12) (cid:12) (cid:12)
Of
ourse, stati
solutions do not (cid:28)t very well with the
of the integration over time, should be repla
ed by the
physi
alboundary
onditionswhi
hshouldbeimposedto
following ones: T1
the
lassi
al(cid:28)eldsatthepointsofmaximaandminima
T
T =max[τ ,τ ] T =min[τ ,τ ] 2
1 1,1 2,1 2 1,2 2,2 and . Forthisreason,wearesupposinghereimpli
itly
(26) T T
2 1
thatpolymersareverylongandthattheinterval −
The
hoi
e(26)ofintegrationlimitsinI4−plat isdi
tated islarge. Inthiswaytherelevant
ontributionstothefree
θ 4−plat t
by the presen
e of the −fun
tions of Heaviside in the energyS
omefrominstants whi
harefarenough
t=T t=T
1 2

onstraints (15)(cid:21)(16). Physi
ally, Eq. (26) is motivated from the extrema lo
ated at and . At these
by the fa
t that, in the Coulomb gauge, all intera
tions intermediate points the traje
tories (cid:29)u
tuate in su
h a
t
be
omeinstantaneous. Therefore,ifmaximaandminima way that, at ea
h (cid:28)xed instant , they visit more or less
arenotaligned,itiseasytoseethatthetraje
toriesmay with the same frequen
y the same lo
ations of the two
onlyintera
tinthetimeinterval[T1,T2],whereT1andT2 dimensional spa
e spanned by the ve
tors ra,Ia(t).
are given by Eq. (26). Substituting the new integration From the a
tion (27) one (cid:28)nds the following Euler-
limits (26) in Eq. (24), it is
lear that. if Eq. (25) is Lagrangeequations of motion:
satis(cid:28)ed, the e(cid:27)e
tive Coulomb intera
tions in I4−plat D ( λ, )ψω =D (λ, )ψω =0
4−plats + − B 1,1 + B 1,2 (28)
vanish identi
ally, so that the a
tion S
oin
ides κ κ
on
e again with the pure self-dual part F4−plat. Thus, D− −2π,C ψ2ω,1 =D− 2π,C ψ2ω.2 =0 (29)
to have self-duality we only need that the e(cid:27)e
ts of the (cid:16) (cid:17) (cid:16) (cid:17)
steri
intera
tions are negligible, su
h as for instan
e in
To these equations one should also add the
onstraints
the
ase of very low monomer
on
entration or at the (15)(cid:21)(16) whi
h determine the transverse
omponents of
,
theta-point. the (cid:28)elds B C and the Coulomb gauge (cid:28)xing (7). In
terTphreetesdelfa-dsutahleityreq
ounirdeimtioennt(t2h5a)tmthaeytbreajpe
htyosrii
easllΓyai,nIa- Etoqtsa.l,(2a8f)t(cid:21)er(2s9e)p,awreatoinbgtaitnhe12re×alnranedquiamtiaognins,awryhip
har
tsomin-
mustbehomogeneous,inthesensethattheyshouldhave pletely spe
ify the
lassi
al (cid:28)eld
on(cid:28)gurations. It seems
equalpersisten
elengths. Inthis
ase,theattra
tiveand natural to try repli
a symmetri
solutions of the kind:
4−plat
repulsivefor
esdes
ribedbythetermI
ounterbal- ψω =ψ ψω∗ =ψ∗
an
ethemselvesinasortofequilibriumwhi
hestablishes a,Ia a,Ia a,Ia a,Ia (30)
6
a = 1,2 I = 1,2 ω = 1,...,n t=T
a r 2
where , and . Moreover, arealignedattheheight , self-dualityo

urswhen
Γ
we swit
h to polar
oordinates in order to express the all open
urves a,Ia have the same length. Parti
ular

omplex (cid:28)elds: polymer
on(cid:28)gurations of this kind may exist in nature,
ψa,Ia =eıφa,Iaρ1a/,I2a (31) efovreri,nwsteanh
aeveafsteeerntthheatprtoh
eersesqoufirDemNeAntrsepfolir
tahteiopnr.esHeonw
e-
ofaself-dualpointaremu
hmoregeneral: Weonlyneed
The
onsisten
y of Eqs. (28)(cid:21)(29) requires that:
that the e(cid:27)e
ts of the steri
intera
tions are negligible,
c
φ = φ ρ = a su
h as for instan
e in the
ase of very low monomer
a,1 a,2 a,1
− ρ (32)
a,2
on
entration or at the theta-point.
c c Theexisten
eoftheself-dualpoint(25)showsthatthe
1 2
Here and areρarbitrary
onstantsdi(cid:27)erentfromzero. physi
s of intera
ting polymers is mu
h ri
her than pre-
Alsothedensities a,Ia mustnotvanish. Thereareother viouslyexpe
ted. Forinstan
e, the
lassi
alequationsof
waystomakeEqs.(28)(cid:21)(29)
onsistent,buttheyleadei- motionadmitnon-trivialsolutions,whi
haretheanalogs
ther to unphysi
alorto trivial solutions. After imposing of vortex-like solutions in anyon models. By analyti
al
the
onsisten
y
onditions (32) in Eqs. (28)(cid:21)(29), there methods it is just possible to investigate stati
(cid:28)eld
on-
remainstillfourindependentrelations,whi
h,
anbeused (cid:28)gurations. These su(cid:27)er the limitations mentioned after
to (cid:28)x the
omponents of the ve
tor (cid:28)elds B C: Eq. (27) and
annot be de(cid:28)ned if the requirements of
λB = ∂ φ + 1ǫ ∂jlnρ Eqs. (21) are not ful(cid:28)lled. In fa
t, if maximta and min-
i i 1,1 ij 1,1
2 (33) ima are lo
alized at di(cid:27)erent heights on the −axis, the
κ C = ∂ φ 1ǫ ∂jlnρ a
tion S4−plat be
omes expli
itlyθdependent on time due
i i 2,1 ij 2,1
2π − 2 (34) tothepresen
eoftheHeaviside −fun
tionsinthe(cid:28)elds
B and C. This time dependen
e is very mild. Its only
It is easy to prove that the Coulomb gauge requirement e(cid:27)e
t is that the boundaries of integration over time in
φ1,1 φ2,1 4−plat
(7) is satis(cid:28)ed only if the phases and are
on- the various terms
omposing the a
tion S do not
stant. Finally, insertingEqs.(33)(cid:21)(34)inthe
onstraints
oin
ide. Yet, this is su(cid:30)
ient to spoil the possibility of
(15)(cid:21)(16), it is possible to determine the remaining de- having purely stati
(cid:28)eld
on(cid:28)gurations. For these rea-
ρ ρ
1,1 2,1
grees of freedom and : sons, any
on
lusion on the meaning of the stati
solu-
c tions in the
ase of polymers should be taken with some
2
∆lnρ = 4λn ρ
1,1 r 2,1
are. Having in mind these
aveats, it is interesting to
ρ − (35)
(cid:18) 2,1 (cid:19) note that, in the solutions whi
h we have studied here,
it seems to prevail a repulsive for
e between
ouples of
Γ
∆lnρ2,1 = 4λnr(cid:18)ρ1,1− ρc11,1(cid:19) (36) tisrawjeh
attorEieqs. (3a2,I)asbueglgoensgtsin,gbet
oauthseetshaemdeepnosiltyimeseρr.a,1Tahries

onstrainedtobeinverselyproportionaltotheir
ounter-
ρ
a,2
A further study of the above two equations requires nu- parts at ea
h point x.
meri
al methods. It is worthing to point out that, despite their
lose
Con
luding, we have su

eded in establishing a
on- resemblan
es,the me
hanisms used to establish the self-
ne
tionbetweentopologi
allylinkedpolymersandanyon dual regime in our polymer model and in usual anyon
4
(cid:28)eld theories. We have limited ourselvesto −plats, but (cid:28)eldtheoriesaredi(cid:27)erent. Inanyon(cid:28)eldtheories,infa
t,
it is possible to generalize our results to the
ase of any theexisten
eoftheself-dualpointrequiresthatthee(cid:27)e
-
2s
−plat. Inourapproa
h,linkswhi
hbelongtodi(cid:27)erent tive Coulomb intera
tions, whi
h arise after performing
2s
−plat
on(cid:28)gurations are distinguished. This allows a the Bogomol'nyi transformations, are
an
eled against
better
lassi(cid:28)
ation of the topologi
al states of polymers true Coulomb intera
tions of
harged parti
les [11℄. In
than the bare Gauss linking number. In view of possi- thepolymer model, instead, thee(cid:27)e
tiveCoulombinter-
4−plat
ble appli
ations to knot theory, it would be ni
e to ex- a
tions, represented by the term I ,
ounterbalan
e
tendalsotonon-abelianC-S(cid:28)eldtheoriesourte
hniques themselves and the presen
e of other intera
tions is not
based on the Coulomb gauge and on the se
tioning pro- needed. Thisself-balan
ingtransitiontoself-dualitymay

edure of loops des
ribed above. o

uralsoinotherphysi
alproblemswheremulti-layered
The analogy between polymers and anyons has been systems of non-
harged parti
les are relevant. Another
exploited to investigate the statisti
al me
hani
s of appli
ationofourapproa
harepolymerbrushes. Inthat
4
−plats, a parti
ular
lass of links, whi
h is relevant in
ase,C-S (cid:28)eldtheories in the Coulombgaugeareable to
biologi
al appli
ations. It turns out that, as anyon par- take into a

ount the e(cid:27)e
ts related to the winding of
4
ti
les in multi-layered systems, also −plats have a self- neighboring polymer traje
tories.
dual point. Within the hypothesis of Eq. (21), whi
h Part of this work has been
arried out during a visit
τ
demands that the points of minima a,2Ia+1 are aligned to the Max Plan
k Institute for Polymer Resear
h. This
t = T
1
at the same height , while the points of maxima visit has been funded by a grant from the German A
a-
7
demi
Ex
hangeServi
e(DAAD), whi
hisgratefullya
- [4℄ F. Wil
zek, Phys. Rev. Lett. 49, 957 (1982); F. Wil
zek
knowledged. The author wishes also to thank the Max and A. Zee, ibid.51 (1983), 2250.
[5℄ R. Ja
kiw and S.-Y. Pi, Phys. Rev. D42 (1990), 3500.
Plan
k Institute for Polymer Resear
h for the ni
e hos-
[6℄ E. Bogomol'nyi, Sov. J. Nu
l. Phys. 24 (1976), 449.
pitality. He is also indebted to V. A. Rostiashvili and
[7℄ H. Kleinert, Path Integrals in Quantum Me
hani
s,
T. Vilgis, whi
h parti
ipated in the early stages of this
Statisti
s, Polymer Physi
s, and Finan
ial Markets,
work,fortheirinvaluablehelpandtheir
onstanten
our- (World S
ienti(cid:28)
Publishing, 3nd Ed., Singapore, 2003).
agement. [8℄ S. A. Wasserman and N. R. Cozzarelli, S
ien
e 232
(1986), 952; D. W. Sumners, Noti
es of the Am. Math.
So
. 42 (5) (1995), 528.
[9℄ Y.-H. Chen, F. Wil
zek and E. Witten, Int. Jour. Mod.
∗ Phys. B 3 (7) (1989), 1001.
Ele
troni
address: ferrariuniv.sz
ze
in.pl [10℄ G. Dunne, Aspe
t of Chern-Simons Theories, published
[1℄ For a review on topologi
ally linked polymers see for in- in Topologi
al Aspe
ts of Low Dimensional Systems, A.
stan
e: Comtet et al. (Eds), SpringerVerlag 2000.
A. L. Kholodenko and T. A. Vilgis, Phys. Rep. 298 [11℄ Z. F. Ezawa and A. Iwazaki, Phys. Rev. B47 (1993),
(1998), 251. 7295; S. M. Girvin and A. H. Ma
Donald, Perspe
tives
[2℄ F. Ferrari, H. Kleinert and I. Lazzizzera, Int. J. Mod. in Quantum Hall E(cid:27)e
ts, S. D. Sarma and A. Pin
zuk
Phys.B14 (2000) 3881 [arXiv:
ond-mat/0005300℄. (Eds), Chap. 5, Wiley,New York,1997.
[3℄ F. Ferrari, AnnalenPhys. 11 (2002) 255.