REFi CEIC/90/206

ATIONAL CENTRE FORTHEORETICAL PHYSICS

COMPOSITE CHERN-SIMONS GAUGE BOSONIN ANYON GAS

Nguyen Van Hieu

and

Nguyen Hung Son

INTERNATIONAL

ATOMIC ENERGYAGENCY

UNITED NATIONSEDUCATIONAL,

SCIENTIFICAND CULTURALORGANIZATION

1990 MIRAMARE-TRIESTE

IC/90/206

International Atomic Energy Agency

and

United. Nations Educational Scientific and Cultural Organization

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

COMPOSITE CHERN-SIMONS GAUGE BOSON IN ANYON GAS *

Nguyen Van Hieu** and Nguyen Hung Son **International Centre for Theoretical Physics, Trieste, Italy.

ABSTRACT

It was shown that in a free anyon gas there exists a composite vector gauge field withthe effective action containing a Chern-Simons term. The momentum dependence of the energy ofthe composite boson was found. The mixing between Chern-Simons boson and photon gives riseto the appearance of new quasiparticles - Chern-Simons polaritons. The dispersion equations ofChern-Simons polaritons were derived.

MIRAMARE - TRIESTE

August 1990

* To be submitted for publication.** Permanent address: Institute of Theoretical Physics, Academy of Science of "Vietnam,

P.O. Box 429, Bo Ho, Hanoi 10000, Vietnam.

I- Introduction.

One of the characteristic features of the gauge fields in 2+1

dimensions Is the possible existence of the topological Chern-

Simons (CS) term In the Lagranglan / 1 r J / In connection with the

study of two-dimensional systems, in condensed matter physics

(quantum Hall-effect, high-T, superconductivity etc..) there arises

a great Interest to the study of the .gauge fields with the CS term

In many-body systems. / 4 7 g / in particular, It was shown that the CS

gauge fields might play a crucial role in the theory of high-Tc/10-1 7

superconductivity / . °

In the precedent work of Randjbar-Daemi, Salam and Strath-

dee /1^/ a simple model of the GS superconductivity at finite

temperature has been investigated. From the expression of the two-

point vertex In the one-loop effective action one obtains the dis-

persions of the massive vector particles which arise due to the

mixing of the photon with CS gauge boson and derives then the

penetration deepth of the Meisner effect. In this paper we

continue the work in Ref.12 and study the GS gauge field In the

system of free anyons - the quasiparticles obeying some fractional/-) A/

statistics. It is known ' that a system of free anyons Is equi-

valent to a two-dimensional system of interacting fermions with

the minimal interaction through some vector field:

p a — p a + B(ra)(ra - r?)

B,(ra) = (1-v)e., Y — ^ ^ (1)

Here rf are the components I = 1,2 of the radius vector ra of the

a-th fermion,v is the fractional number characterising the sta-

tistics of anyons, £_,., is the antisymmetric tensor with e12= -e21

= 1. PIrst we prove that the fermion system with the minimal

•jn + ovontinn thrritii-ph tho vQrtnr f lair ! ! A \ -f c unidiralont uH+h

the fermion system Interacting with some OS gauge field both at

classical and quantum levels. Then in the path-Integral expression

of the generating functional (grand partition functional) we

Integrate over the fennion fields and obtain the one-loop

effective action of the GS gauge field which Is interpreted as the

niiqj-i + Tim f -f pi rt nf nnmo r*/~\mr\r\a1 +o ni ioai-novi+inlp In fha am/Tin a\ra —

+ cirn fVio rn i in i -Intr hcitmDQn f h i o nomrv-icf+ci ntjiirro h/~i"nn nan/H -t-ViCk

!l iTiifp^ "pi Ou +r* tho DV! c! + ortriiii n f tian now n i i a c ^ i Q f t i n l u n _ r» i -

rr\rsn TVio H-f ononoi nn n f tl-ici T-.rii civt tnno l o ^ Cl• -• T o ^ TViiii

"I q + -f nn 1c Hnno hrith at tho tomTiarta + nvci fp — O anrf ra+ f i n i t o

2. Composite GYiern-Slmons gauge field.

Consider some system of free anyone which Is equivalent to a

system of self-interacting fermions In 2+1 dimensions with their

i i l l U i U ' . ' t i U l l L-L1X '-'i-i^il »1IW I t V 1,^1 XJ-1—-J.U \ I / m lUi Hit b tT L-111 1 J .inimJ.liJ.m

V L l i X L J J V 1 ^ TT'_ iJ.»_*U L.UJ1*_: i t l i U X i_jX.LX Li X V X k-1 ^ X W X WX 111 U i V J 1 1 '_ • LJ. tXU'll

X X ' _ '_- X ' _ X lliX.1—'! It—>

Trh 'tT\ 1 - I ( T ^ " T W * V ^H* _i. 'j.-j 'f' J — r f i i , u t i n •LJ

h/' T 1 l a a -)-uiri~.r>i"M7mrirt£in+ oninnr f- iaiH THnan fho af+ inn rif fho/ i iX> . / X.k_> LX V* VI i_' L-WUlpW^UIlb L'L'J.UL'l XXUi.Ul l i l ' ^ l l L, 1 1W Ldl W l ,X> . ' i l \J X L l I U

system of Interacting ferraions equivalent to the anyon system mayIn

^' '" J ~ IT1 ^ / i L J n1-"

--' — x1 ^ — n sf x n ''if x! ~ '••( r* t) i? ' x) — B ' " * X X ' X X* _ / • _ ;

B,(r,t) = 2%(J\-v)e,,d,h(T-Tt)^(r,t)ilKr\t)d2v (4)

where D(r-r') is the Green function of the Laplace equation in two

dimensions

- 72D(r) = 5(r) (5)

and equals

P(r) = - d/43C)ln r 2 (6)

Introduce now another system consisting of a two-component

fermion field ty(x) interacting with CS gauge field $"(z) whose

action equals

{7)

with some constant, ae. Here e ^ is the antisymmetric tensor with

= s^^,.Since the gauge field ^ is non-dynamical,the solu-

tion of the equation of motion for this field means the expres-

sions of their components in terms of the fermion field

hfr(R"*

i°(r,t) = (g/ae.) e1 j aj [orr-r*.)(|)+ (r',t)x±<\>(T'9 t)d2r'

Substituting the solution (8) into the r.h.s. of Eq.(7) we obtain

the expression which is identical with the action (3) of the

system of interacting fermions equivalent to the anyon system.

so,[fl>,$,2 i - s a n y o r , [ ^ ] (9)

if and only if the coupling constant g,m and the fractional number

v satisfy the condition

2%(1-v) = fg2/(£j (10)

This well-known property of the. classical CS gauge field, however,

does not mean automatically that two system,?, with the action (3)

and (7) are equivalent at the quantum level. In order to check

this equivalence we now prove that the Green functions of fermion

fields in two systems are identical.

In the path-integral formalism the Green function are the

functional derivatives of the quantities

W [T),T| ] = in Z Cn.fj ]any on " l any on •' ; , . ^

W [T],7] ] - In ZreCT),T] ]

where Z onC rj, 7] ] and Zcs[7]trj ] are the generating functionals

for the systems with the action (3) and (7)

3 = |ciXl)]CD(|)]CD4]eip|l[scs[(|),i]),i£] + [(fj ) + d V j j (13)

Here [Ml denotes briefly the measure of the path-integral over

the gauge field with the gauge fixing condition and the Fadeev-

Popov ghosts. In this path-integral let us make a shift of the

field #

\lDdl exp U Scs[tj>,(p,jJ ]}• = nDstt exp h Scsti\>,§,4+2 lj (14)

It is easy to check that

where S°q[iJ is the action of the free CS gauge field

We have then the equality

Zro[Ti,fj 3 = Z o Z _ [iq.T] ] (17)where ZQ is some constant independent of the sources T| and fj

ZQ = FtD l exp jtS°sU ]l (18)

From Eq.(17) it follows that the functional derivatives of

WCC.[TJ,r\ 1 and W a n nti?,Tj ] are identical, and two systems with the

actions (3) and (7) are equivalent at the quantum level. This

means that the system of self-interacting fermlons (3) can be re-

placed by the system (7) of fermions interacting with the CS gauge

field. This CS gauge field may be interpreted as the quantum field

5

3. Effective action and dispersion of gauge boson.

The results of the precedent section show that in the

V. w _k_ --L_i_L !_' L L.41_iji U i . U W I U * - ^ CJ.J. J.J _'i. i t_tj LJ L-»_Ui TT »_ LLJ.it _L \_Lj_l_LJ, Lv»—• -A, V *JJ

another system of fermions interacting with a OS .gauge field. Con-

sider now the generating functional (grand partition functional)

of the latter in the absence of the sources n and fj

7 - Ir riruir rtRir r u i n - m i * [c r<\\ 7K > i ] lWlD J I L w " jj

^ ^ r '

qrsi-i Ah+o in fhci o f f c i c f •? T/CI ^3*- t oT^ C f j f l n f +hci P.P. rrraiirrci -PI cil HS l l _ .-

S..^[,41 = S^Lil + SJ43 (21)

S.U] = - iln dent? id* - led*) ~ ml (22)

•Tha riLj-n+nt'*ha + 'T on rci l ni t l ot"T r n a t f-Tni+o iomrvdTo+11 rCi T pan

i l l A'L 1—-!- * 1 £ _ LU.1JV_A _L->_' t_F l r X L J . a . ^ 1 1 ( r ± U i i ?l l_*_L UA • -i.il

3xd2y (23)

aJ +h como nnl i3T'Hi7a + i inn f i inr>t1nn IT ('rr— 11 ) inT* -T + o T?miT>1cir' t yonotf4- I'll l_"_'ili'_ L ' U X U J . J . A J U U J . U U J_ U 1 J W i? _ i - _/A 1 I A, ( . - . , i O-- & • ' * -L %—'X -t v 1-' i ' - 1 WU. -t. *— A 1 -A- U- i A*-'

f rir»m 17 f ,-T 1

n^Ci-y; = f7/t?TL)3|ei^""^in)il,(qJci3q (24)

I n +ho nnn_-PQin + 1u1o + 1j-- l i m i t we hairo f n i inrnHnn1 ovrrpQaa-f nrja

T7 in i - -In^ -f>f n i

Q ffrf ) f'25)

..,4i*(<«J*«--.i-- . - - (

rr n n p_ 4p n Q 1 f ^

ffq) = fV2;Q|3cH ^ o (26)

00

KfTJz = 5 (28)o / + exp i-(um)(^ En

where k is the Boltzman constant and S p is the Fermi level. At

zero temperature

K(0) = far, E p // 2 (29)

It is straightforward to verify the gauge invariance of the action

(23) with the polarization function of the form determined by

Sqs.(25).

Let us fix the gauge by the condition 4 = 0. In this gauge

the total effective action of the GS gauge field equals

(30)

In order to find the dispersion of the composite gauge boson it is

necessary to diagonalize the effective action (30). This operation

depends on the sign of ae. Consider the case of positive ae and try

Az(q) =

Then we obtain

= -f-J - Q + ^ ] ^ q ) d3 q (32)

q0 + (q

From the field equation in momentum space we derive the dispersion

law

qrt = 0 - -SJsr- (33)

Since the energy q0 must be non-negative, Eqs.£31) are valid only

In the range

£ <- " (34)For large values of q2

> Q (35)

instead of Eqs.(31) we use the ansatz:

^(Q) = -2-t$(<l) + $*(-q)l

djq) = - -i>-i<p(q) ~ $+(-q)l

and obtain

The field equation now gives the disperssion law

% ' 4r - a (38>Thus in the. case of positive x the energy of the composite gauge

boson with momentum q equals

- Qi (39)

In the case of negative ae we use the ansatz (36) and obtain

q (40)

The. energy of the composite gauge boson with momentum q now equals

4. Chern-SImons polaritons.

Due to the minimal coupling of the charged fermions with the

electromagnetic field &Jx) there arises the transition between

two vector gauge Held a (x) and d (x). In the lowest order of the

electromagnetic coupling constant e the photon - CS boson transi-

tion term in the effective action equals

33y (42)

Tfdn I n Thta TTV>eir>o_V L-iJ . H I OltU' £.'4. ' _ ' _ • ' _

U7T-tar>Q t h o TVI 1 ar>"f 17a + "I n n "P1 inr> + 1 nn IT 1 > - i i i UIQ a rriftLi'_X w l/H'*- LJI_'.l.' 1 i J-iJLJ. b l U l l J. LUIt- U j_i_'H J-J-j ( ^, , \ •-<-• iy y TH.il . . . ' ,«•, X

rio>-it o a r t i Ir.n UIc. wrir»V airq-ln In +ho crcmrro w!+h /, — A Tn',*'„-4 4 t l-"- -* t ± ± ' J U f tl ' _ . Tf-JXAi. i-ij^O. J-1 i X H l^liL.- ^V-X L ~ " ^ | ' — - 1TX u!4 U* ^J . i l l

~ ^~ UJ

mrimpntnm oriano TWO hGn;Q t h e n

\f(q)d3q (43)

In order to simplify the expression of the action we must intro-

duce for both vector fields AA(x) and a.(z) the transformations as

•lr\ TTnc /Q-1\ r\-n f '~kt-,\ Tn f h o n a a o r,f n n a i + ' l u o » onri I n + h a -p

-1 x-/ t /',-v ) _ L r f / / i l JU P ' i _ . n 1 1

V " + (44)&Jq) = -^r-tirq/1 - t r - g n

•fc

the case of negative as toother with Eqs.(36) we replacei

&JQ) = —h-it(q) + £ ' ( - q . ) ]{ A&\

ajq) = -

r\

^D - I n _ ?n ' - /Pn i

ho nhAtnn ._ f!Q >*x.T3i~irj + v a n a i + i ri-n fri uuo T»"T OCS +I~I +htj o v ? o t o n n c i n f4 t»_ |_*44L.' U L ' l l \_rl_r LJ'_'ii_ll_*i 1 :J X '„£! 1L1 A t J- *—'4 4 f'-j -4- V UK 1 X J-t-'1.^ t 1 . 1 L- 4 i *~.- '..JV J-U ^ U l l ^ ' L ' *—' i

j- - xi tat ions called poiaritons which are the

of the polaritons it is convenient to apply the Green function

technique. Denote $'"(q) and $°(q) the two-point Green function of

PQ hnarin '.mri nhn+riri roari i r< tTP .•qh^pnre fif +lie1?'t>-'i-; U'-' l^'Jii '^ l i lU L--4i'_/ Li_'Uj X >_-l_<^_'« » J-ii LilL UULf^liL'. ^_.»X Lr4i>_^-L.X

LiiX4iLJi tXU'il, • / 1 t- J L i l l ' J II \ -_J / L-44'~

presence, of this transition. They satisfy the Dyson equation

f(q) ™q' V 9'$,(q)2fj°(q)

(48)

where R(q) is the matrix element of the transition. From Eq.(45)

it follows that for positive ae and in the range (34)

%(q) = - -?§- Q 5° (49)q0

while for positive se and in the range (35) as well as for negative

ae Eq. (47) gives

% n \ ( 5 0 )

From Dyson equation (48) with the transition matrix element %(q)

given in Eq.(49) or (50) and the well-known expression for g°(qj

we obtain the dispersion equations for the polariton in different

2

Case 1: ae = |ae|, -^- < Q :

Q)2qo = ° (52)

Cane ;?: ae = |ae|, 4 E - >

Case 3: ae = -|ac|:

Q ' -tr] + -f Q)\ = ° <54)

Each of Eqs.(52)-(54) has foiu1 solutions : two physical and two

non-physical. The physical solutions are those which tend to the

dispersion laws of the GS boson

and the bare photon

10

HO = S

Their analytical expressions q0 = qo(q) are complicated, but in

each range of q2 we can find the corresponding approximation ones.

5. Diesussions.

The exslstence of CS gauge fields in the condensed matter and

their role In physical processes of different natures are widely

discussed, in this work we shown that in the system of self-

interacting fermions equivalent to a free anyon gas there exists

indeed a composite boson described by a CS gauge field. The

dispersion equations of this CS boson depend on the sign of the

coupling constant ae which is related to the fractional numbers v

determining the statistics of anyons

sign ae = slgn(1-v)

They are2

% = I-Sir " Ql- * >0

2

% = -fsr + Q • * <0

Constant Q determining the energy range of the CS boson is of the

order of the Fermi level Ep of the fermion system. The mixing bet-

ween GS boson and photon gives rise to the appearence of two

polariton branches in the same energy range. The polaritons might

be observed either In the Haman scattering experiments or in the

luminescence spectrum of the . system. In particular, the Raman

scattering experiments would check the disper sion equations of

two polariton branches.

li

ACKNOWLEDGMENTS

The authors would like to thank Professor Abdus Salam,the

IAEA and UNESCO for the hospitality at the International Centre

for Theoretical Physics, Trieste, Italy. We express our deep

gratitude to Profs. D.Amati, Yu Lu, S.RandJbar-Daemi, Abdus Salam,

J.Strathdee, E.Tosatti and M.Tosi for the fruitful discussions and

encouragements.

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