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Exchange of Identical Particles in Two Dimensions P Ramadevi
Ramadevi received her MSc (Physics) from IIT, Chennai and her PhD in theoretical physics from the Institute of Mathematical Sciences, Chennai. She then worked as a postdoctoral fellow at TIFR, Mumbai and MRI, Allahabad. Her research interests are in the areas of string theory and topological field theory. She is presently CSIR Senior Research Associate at the Department of Physics, IIT, Mumbai.
E x c h a n g e o f id e n tic a l p a rtic le s is o n e o f th e th e o r e tic a l id e a s in q u a n tu m p h y sic s w h ic h w a s c o n siste n t w ith sc a tte r in g e x p e rim e n ts. T h e re a re su b tle d i® e re n c e s in th e e x ch a n g e p h e n o m e n o n in a tw o d im e n sio n a l p la n e a s c o m p a r e d w ith o u r th re e d im e n s io n a l w o rld . T h is le a d s to th e p r e se n c e o f a n e w c la ss o f p a r tic le s c a lle d anyon s b e sid e s th e tw o w e ll-k n o w n u n iv e rsa l c a te g o r ie s (b o s o n s a n d fe rm io n s) in th e tw o d im e n s io n a l p la n e . 1 . In tr o d u c tio n E x ch a n ge o f id en tica l p a rticles is a sim p le m a th em a tica l tech n iq u e w h ich a cco u n ts fo r th e p resen ce o f tw o classes o f p a rticles in o u r th ree-d im en sion a l w o rld { n a m ely, boson s an d ferm ions. In fa ct, th e a b o ve th eory co n ¯ rm s th e ex p erim en ta l o b serva tio n th a t th e sy stem of id en tica l h eliu m a to m s H e3 b eh a ves d i® eren tly fro m th e sy stem of id en tical H e 4 ato m s (iso to p e) in a sca tterin g p ro cess. S u p p o se w e restrict th e ex ch an g e to b e p erfo rm ed o n ly in a tw o d im en sio n al p la n e, w e w ill see th a t on e m ore class of particles is a llow ed to ex ist. W ilczek co in ed th e n a m e anyon s fo r su ch ex o tic p a rticles (n ot seen in ou r rea l w o rld ). T h e rea d er m a y w o n d er a s to w h y it is in terestin g to lo o k a t tw o dim en sion s w h ich is fa r rem o ved fro m o u r ex istin g w orld . W h a t a ctu a lly h a p p en s is th a t so m e p h y sica l sy stem s, th ou g h th ree-d im en sio n a l, b eh av e a s tw o -d im en sio n a l sy stem s w h en su b jected to ex tern a l fo rces. F o r in sta n ce, electro n s in a cu b ic sem ico n d u ctor g et co n ¯ n ed to a th in tw o-d im en sio n a l la y er b y
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With so many interesting applications of anyon physics, it is very essential to learn how these anyons emerge as a logical possibility only in a two dimensional space.
th e a p p lica tio n of stron g electric ¯ e ld s. In itia lly , an yons w ere co n sid ered to b e o n ly m a th em a tica l fan tasies. B u t th is v iew d rastica lly ch a n ged w h en th ese th eo retica lly p red icted a n y o n s w ere ex p erim en ta lly id en ti¯ ed a s co llective ex c itatio n s in certa in co n d en sed m a tter sy stem s. T h is ob serv a tio n certain ly h a d a g rea t im p a ct resu ltin g in w id esp rea d in terest in a n y on p h y sics. F o r ex a m p le, th e co llectiv e ex cita tio n s ab ov e g ro u n d sta te in sy stem s ex h ib itin g fra ctio n al q u a n tu m H all effect (for d eta ils on H all e® ect, see [3 ]) w ere id en ti¯ ed w ith a n y on s. In fact, th e th eo retica l w o rk o f L au g h lin co n ¯ rm in g th e ex p erim en ta l p red ic tio n s o f a n ew form o f q u a n tu m ° u id w ith co llectiv e ex cita tion s b y S t¶o rm er a n d T su i b a g g ed th em th e N o b el P rize in P h y sics in 1 9 9 8 . A n oth er a rea in w h ich a n y on s a re co n jectu red to p la y a ro le is in th e th eo ry o f h ig h tem p era tu re su p erco n d u ctiv ity (fo r d eta ils see [4 ]) b u t n o co n crete ex p erim en ta l ev id en ce h as b een ob serv ed till d a te. W ith so m a n y in terestin g ap p lica tio n s of a n y o n p h y sics, it is v ery essen tia l to learn h ow th ese a n y on s em erg e a s a log ica l p o ssib ility o n ly in a tw o d im en sion a l sp ac e. T h is is w h at w e elab ora te in th is article u sin g ex ch a n ge p h en om e n o n . W e w ill n o t d escrib e a p p lication s of a n y o n s a s it is b eyo n d th e scop e of th is a rticle. T h e p la n o f th is a rticle is a s fo llow s: In sectio n 2 , w e w ill recap itu la te th e tex tb o o k m a teria l on ex ch a n g e o f id en tica l p articles in th ree d im en sio n s. In sectio n 3, w e sh a ll p resen t ex ch an g e p h en o m en o n in th e tw o d im en sion a l p la n e a n d sh ow h o w a n ew cla ss of p a rticles called a n y o n s em erg es as a lo g ica l p o ssib ility. W e su m m a rise th e resu lts in th e con clu d in g sectio n . 2 . E x c h a n g e o f Id e n tic a l P a rtic le s in T h r e e D im e n sio n s It is w ell k n ow n th at th ere a re o n ly tw o cla sses o f p a rticles in o u r th ree-d im en sio n a l w o rld , n a m ely, boson s a n d
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ferm ions. W e sh a ll u se th e sim p le tech n iq u e o f ex ch a n g e p h en o m en o n to a cco u n t fo r th e tw o cla sses.
A wave function is
L et u s co n sid er a sy stem o f n id en tic al p a rticles. In q u a n tu m p h y sics, th e sta te o f n id en tica l p a rticles is d escrib ed b y a w a ve fu n ctio n ª (r 1 ;r 2 ;:::r n ) w h ere r 1 ;r 2 ; : ::r n sy m b o lica lly d en o te p o sition an d o th er q u a n tu m n u m b ers o f p a rticles 1 ; 2 :::n , resp e ctiv ely. In sim p le term s, a w a ve fu n ctio n is a co m p lex fu n ction w h ose m o d u lu s sq u a red jª (r 1 ;r 2 ;:::r n )j2 g iv es th e p ro b a b ility d en sity of ¯ n d in g th ese n p a rticles w ith th e g iv en p o sitio n a n d o th er q u a n tu m n u m b ers.
whose modulus
a complex function squared |ψ (r1, r2, ... rn )|2 gives the probability density of finding these n particles with the given position and other quantum numbers.
W h a t h a p p en s w h en w e ex ch an g e p a rticle 1 w ith p a rticle 2 ? T h e n ew con ¯ g u ra tio n w ill b e d escrib ed b y th e w av e fu n ctio n ª (r 2 ;r 1 ;:::r n ). S in c e th e p a rticles a re id en tic al, th e p h y sica l sta te o f th e sy stem g iven b y m o d u lu s sq u a red ª ª ¤ sh o u ld b e u n ch an g ed . H en ce, th e q u a n tu m sta te ª ca n a t m o st p ick u p a p h a se u n d er ex ch an g e. T h a t is., ª (r 2 ;r 1 ;:::r n ) = e iµ ª (r 1 ;r 2 ;::: r n ) :
(1 )
R ep ea tin g th is ex ch a n ge , w e g et b ack th e o rig in a l q u a n tu m sta te: ª (r 1 ;r 2 ;:::r n ) = e iµ ª ( r 2 ;r 1 ;:::r n ) = e 2 iµ ª (r 1 ;r 2 ;:::r n ) (2 ) w h ich im p lies th a t e 2 iµ = 1 giv in g tw o p ossib ilities; µ = 0 o r µ = ¼ . T h ese tw o p o ssib ilities rep resen t th e tw o c la sses of p a rticles. H en ce th e q u an tu m sta te fo r th ese tw o cla sses u n d er ex ch a n g e o f p article 1 a n d p a rticle 2 sa tisfy ª (r 2 ;r 1 ;:::r n ) = ª (r 2 ;r 1 ;:::r n ) =
ª (r 1 ;r 2 ;:::r n ): ¡ ª (r 1 ;r 2 ;:::r n ):
(3 ) (4 )
S u p p ose p a rtic le 1 a n d p article 2 h av e th e sa m e set of q u a n tu m n u m b ers r 1 = r 2 . C learly ª (r 1 ;r 1 ; r 3 ;:::r n ) 6= 0 fo r th e w av e fu n ctio n o b ey in g (3 ) w h ereas ª (r 1 ;r 1 ;
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r 3 ;:::r n ) = 0 fo r th e w av e fu n ctio n o b ey in g (4 ). R ecallin g P auli's exclu sion prin ciple, w hich states that n o tw o identical ferm ion s can have the sam e set of qu an tum n u m bers, w e dedu ce that (4 ) den otes the property of ferm ions un der exchan ge an d (3) represen ts the property of boson s u nder exchan ge. T h u s th e tw o u n iv ersa l categ o ries o f boson s an d ferm ion s em erg e n atu ra lly fro m th e sim p le ex ch a n g e p ro cess in th ree d im en sio n s. W e w ill see in th e n ex t se ction th a t th e restriction to tw o d im en sion s resu lts in p ecu lia r fea tu res. 3 . E x ch a n g e o f P a rtic le s in T w o D im e n sio n s In th e p rev io u s sectio n , w e d id n o t sp ecify w h eth er th e e x ch a n g e is clo ck w ise or a n ticlo ck w ise. H ow ev er in a tw o d im en sion a l p la n e, w e w ill see th at clockw ise exchan ge is di® eren t from an ticlockw ise exchan ge. S u ch a d istin ctio n d isa p p ea rs o n ce w e h av e an ex tra th ird d im en sio n . C o n sid er n id en tical p a rticles w h ich ca n on ly m ov e in tw o d im en sion s, say th e x ¡ y p lan e a s sh ow n in F igure 1 . Figure 1.
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L et u s d en ote th e clo ck w ise ex ch a n ge of p a rticle 1 w ith (+ ) p a rticle 2 b y P 12 (a s sh ow n b y d a sh ed lin e in F igu re 1 ).
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A s m en tion ed in th e p rev io u s sectio n , th e p h y sica l sta te rem a in s u n ch a n g ed u n d er su ch a n o p eratio n . T h ere fo re th e w a ve fu n ctio n ca n at m ost ch a n g e b y a p h a se: ª (r 1 ;r 2 ;:::r n ) ! P e
iº 1 2
(+ ) 12 ª
(¡ ) 12 ª
particles followed exchange of the
(5 )
w h ere º 1 2 en co d es th e n a tu re o f th e p articles. In stea d of clo ck w ise ex ch a n g e, w e p erfo rm an ticlo ck w ise ex ch a n g e ( ) P 12¡ (a s sh o w n b y d o tted lin e in F igure 1 ). L et ¯ 1 2 b e th e p h a se p icked u p b y th e w a v e fu n ctio n : ª (r 1 ;r 2 ;:::r n ) ! P
exchange of two by anticlockwise
(r 1 ;r 2 ;:::r n ) =
ª (r 1 ;r 2 ;:::r n ) ;
A clockwise
same particles is no exchange at all.
( r 1 ;r 2 ;:::r n ) =
e i¯ 1 2 ª (r 1 ;r 2 ;:::r n ) :
(6 )
It is a n o b v io u s fa ct th a t a clockw ise exchan ge of tw o particles follow ed by an ticlockw ise exchan ge of the sam e particles is n o exchan ge at all { th a t is, P
( ¡ ) (+ ) 1 2 P 12 ª
(r 1 ;r 2 ;:::r n ) = e i¯ 1 2 e iº 1 2 ª (r 1 ;r 2 ;:::r n ) ´ ª (r 1 ;r 2 ;:::r n ) ;
(7 )
¯ 12 = ¡ º 1 2 :
(8 )
w h ich im p lies T h u s, w e see th at th e p h a se u n d er a n ticlo ck w ise ex ch a n g e is in v erse o f th e p h a se fo r th e clo ck w ise ex ch an g e. A s lo n g a s w e p reten d to liv e in th e tw o d im en sion a l sp a ce (h ere it is th e x ¡ y p lan e), th e clo ck w ise ex ch a n g e is d istin ct fro m a n ticlo ck w ise ex ch a n ge a n d th ere is n o restriction o n º 1 2 . It ca n ta k e a n y va lu e b etw ee n 0, a n d ¼ . T h e tw o ex trem e va lu es º 12 = 0 an d º 1 2 = ¼ corre sp o n d to b oso n s a n d ferm ion s, resp ectiv ely. T h e p a rticles w ith in term ed ia te va lu es º 12 2 (0 , ¼ ) rep resen t anyon s. N o w w e a llo w th e p ossib ility of m o v em en t in th e th ird d im e n sion a lso . T h en th e p a rticle w ill h av e th e freed o m
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Suggested Reading [1] F Wilczek, Anyons, Scientifc American, 58, May 1991. [2] G S Canright and S M Girvin, Quantum Possibilities in Two Dimensions, Science, 247, 1197, 1990. [3] Halperin, The Quantized Hall Effect, Scientific American, 52, April 1986. [4] Laughlin, The relationship between high-temperature superconductivity and fractional quantum Hall effect, Science, 242, 525, 1988.
o f m ov in g in th e z -d irectio n a s w ell. C lea rly, the clockw ise exchan ge can be easily m ade in to an anticlockw ise exchan ge b y ro tatin g th e d o tted cu rv e in F igu re 1 a b o u t th e x -a x is in th e th ree-d im en sio n al sp a ce. T h is o b serva tio n a u tom a tica lly restricts th e clo ck w ise p h a se to b e eq u a l to a n ticlo ck w ise p h a se e iº 1 2 = e ¡ iº 12 :
(9 )
T h e ab o v e eq u atio n a lso im p lies th at th e ex ch a n g e o f p article 1 a n d p a rticle 2 tw ice is n o ex ch a n ge a t a ll in th ree d im en sio n s g iv in g rise to tw o p o ssib le va lu es º 12 = 0 or ¼ a s o b tain ed in th e p rev io u s sectio n . T h u s, w e see th a t th e p resen ce o f th e th ird d im en sio n p lay s a sig n i¯ ca n t ro le in rem o v in g th e d istin c tio n b etw een clo ck w ise an d a n ticlo ck w ise ex ch a n g es o f id en tica l p a rticles. In th is sectio n , w e h av e d em o n stra ted th a t th e tw o d im en sio n al p la n e is very sp ecia l w ith tw o ty p es of ex ch a n g es { clo ck w ise a n d a n ticlo ck w ise giv in g rise to a n ew set o f p a rticles ca lled anyon s b esid es th e w ell-k n ow n u n iv ersa l classes o f b o so n s a n d ferm io n s. 4 . C o n c lu sio n s In th is a rticle, u sin g th e tech n iq u e of ex ch a n g e, w e h av e sh ow n th a t o n ly tw o ca teg ories o f p articles ( boson s an d ferm ion s) a re a llow ed in o u r th ree d im en sio n al w o rld w h erea s a n ew set o f p a rticle s ca lled anyon s a re a lso a llo w ed in a tw o d im en sio n a l p la n e. T h is cru cia l d i® eren ce stem s fro m th e fa ct th a t th e clo ck w ise ex ch a n g es a n d a n ticlo ck w ise ex ch a n ges a re d istin ct in tw o d im en sio n s b u t n o t in th ree d im en sio n s.
Address for Correspondence P Ramadevi Physics Department Indian Institute of Technology Mumbai 400 076, India. E-mail:
[email protected]
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A c k n o w le d g m e n ts I w ou ld lik e to th a n k A m eey a , M o h a n , S u rya N a ya k , U m a san ka r, T R G o v in d a ra ja n fo r th eir co m m en ts a n d su g gestio n s.
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