Tribe 3.

Emergence of

Cleon - Simons theory

in CUT .

C and " statistical transmutation"

).

Gil Young do.

In this part of the lecture , we will

derive the Chern - Simons theory from

a microscopic theory :.

t '

&electrons in a fractional

quantum Hall effects. )

the effective theorythat we have

extensively studied.

&he 'll provide heuristic pictures

of " fractional gators"

.

( includingother examples ?

Again , we focus on V -- You

+ ,,

NEZ.

in which. . .

I" T

¥4' "

electron %uti ) - magnetic fluxes

Then,

the theory for"

electrons "is

. . .

f-frtotciae-A.IT→ mffY

^- sink- iota?-1051.4 -

iEtA%To}

& ur

.in#Tt1cn.CTutToIcrsIelectron 's r.ir ' i

kinetic Iinteraction btwn

energy he electrons.

We will apply slightly more electric

field & magnetic field a measure

the conductivities

,fgenerating magnetic

flux.

i. e .

, f- = Go t ftur

I small external field.

fullfor " probing " the system .

electromagneticfield

.

Thus :

f.fr#Ci2e-AolF-zILC-iEtAiF5.lC-iotATF3

}.

+ interactions.

Tlet 's ignorethis for a mowed

C for the clarity ?.

from the physical Picture . . -

it is essentiallymagnetic fields←T I which make the

¥4T problem difficult

electron uti ) - magnetic fluxes

⇒ its tempting co Noite the theoryof

'

¥7,4 =as a

"

composite"

CW/ internal structure !

However.

"

magnetic flux " is"

external "

which is"

given"

,& so we can not

detach it simply so electrons.

We by - pass this problem by writing2h t I

④ = ⑨ ④HITTITE'sYan .ca ,

T in the opposite

electronI

some particle dir, as

the backgroundthen

A T A T~ ← moving

µ ⑤ = ¥01 = ⑥ in He absence

yaT I tf T y

'

of backgroundfields

.

spread"

d"

attached

To ②①

I :what is this new ⑧ particle ?

⇒ we will see this is a boson

② how do we introduce the"

artificial"

magnetic flux"

t"

attached co

" statist

]de ② ?

caltransmutation

"

let's stare from ②,

which we know

how co do :

→ we can introduce the fluxes

& attach teen w the particle

by Clem - Sinuous term.

Consider : fnew particle

A -- f Et Cide

tao- So ) §

r

- sink-iota- A-IET .Kioto- A'583

- at en "anna .

.

ten.

8- f -- o

Sao

⇒ 8+8=2*1 . ¥ Eiirdiaj.

i. e., E Jia ;

= 21T k NE

.

-

I' I

# of

artificial "

magnetic" flux

→ which is what we hoped for !

Now,

we confirm To n boson. ,

for this, we show instead . .

{ ¥'g-

boson

= Fenner

Remember that,

{ exchange } = I { fullbraiding }

( "

monodromy"

phase ).

i. e.

Fermion:

⑥ & = e" "

.

on ① = eZiti

.

i. e., Boson 8

② I = e" °

on ① = e

"°

.

Now,

Fhuxo),8xCk - fluxes

at ik← conventional

= e£O&Em. e

AB - phase .

+self - statistics

⇒ implying ,

{exchange } = @

i Off eitk .

→ We was"

Ferri or"

,kE2utl .

i. e. , O EE = O.

Thus ,we see that Ogg =o

,

implying that we hare a boson &.

* Statistics is"

Trans mutated "

from " fermion "co

"boson

"

.

Next, we want no have

du Approximation such that

the attached fluxes are speaded .

co cancel the background fluxes.

i. e, an = a-

µ+ fan

?"

fluctuations.

"

such that - a- at Ao,

- o

I '

generatingthe background

magneticfields

.

beading co . . .

A -- fr Iot Cite

tfao- FA . ) §

- sink- if

+Ea - SEIFTIC - if

HE- SATE ]

-1- Em

"

Sandu fax.

41T . k

Here weperformed. .

4¥ Em 'C Int fan ) do ( Eat fax )

= 4¥ Em '

TenguI,

← just a number.

ignored .

+ ←¥ E" "

Ink fax

/ + ←÷ E" "

Sanatan.

only a average particle+ Sao . 2¥ E "

"

Lia,

- number.

me g we keep this.

Fer total :

f.fr#Ci2e-AolF-zILC-iEtAiF5.lC-iotATF3 }.

+ interactions.

"

Fermion"

under the uniformmagnetic

Iffields

"

Boson " w/o background field

but interacting w/ artificial

£-1 E' Cite+fa .- FA . , §

Gage field.

- sink- if+

ta - SEETIC -

iota- FATE ]

-I Em

"

Sandu fax.

41T . k

What do we know of bosons

at the verylow - T ?

C w/ high densities ).

⇒ We know that Hey undergothe Bose-Einstein condensation

.

e.g , superconductorsuperfluid . . .

' -

etc .

What is the BEC ?

⇒ Bosons become coherent .

€g eIO ← ahead phase .

* > = Ffaaverage densitymodulator

in density

Plug . no guediert ( hyguediero )

£ ( g- t Sf ) C

tsao- SAO ) + C . - . + f Left . . . )

.

+£ g- CEO- taita )'

+ C. tfosgosatjfinding

- 1- Em "

Sanada , off41T . k

" Not important"

- fa.

. g- ?from the expansive.

= -

g- FA o t

Gao- f Ao ) If .

-

e-

I + ICHI - taboo

ifyou look at

this. . . 1- Eh "

said . fax

JFC If ,

-

£00)

-

44kand

=TMCfq- fan )+ C i - . Higher gnedieos .

. ).

rewriting

A = - FA .

. I

+ Ju fan - if En "Sandu fax

.

- gu fan C←

n

Emporia.

next, using Ju = 2¥ Et "

tuba.

= - SA of - 2¥ Em " SAU dub ,

+ I EM 'bar do fax

21T

- 4¥ E" "

fan to fan..

and then solve the equ . of . notion

for fam ,we obtain . .

A -- - SAof + 4¥ E " "

bid ba

a"

voltage drop"

- 2¥ EMA but .FA

?

⇒ completing the derivation !

Next, we can ask what's the physical

meaning of j"

bn← carvers of bar .

imagine that

< Es = Fg

eio.J-hisjbneco.msO = O t Own T

"a came

in space ,

then,

t.

Iif ex

← anti - cortex

Remarks

I.

This approach is known as the composite

boson theory

⇒ we can also use He so-called

composite fermion theory .

Here we attach only the

even number of fluxes a

the electron

Remarkably ,whatever the approach

is used ,the result lanais the same

.

2. Appearance of Clem - Simons theory

is somewhat universal.

→ When you see a particle ,

which is neither boson nor

fermion ,it is very likely

you can describe them by

Chem - Simons theory.

e. g ,I spin liquid .

Additional Reference:

Aspects of Chern-Simons theory, GV Dunnes, arxiv: 9902115

Shou-Cheng Zhang, The Chern-Simons-Landau-Ginzburg theory for the fractional quantum Hall effect, Int. J. Mod. Phys. B 1992