Wei Li- Three Stringy Realizations of Fractional Quantum Hall Effect

8/3/2019 Wei Li- Three Stringy Realizations of Fractional Quantum Hall Effect

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Three Stringy Realizations of Fractional

Quantum Hall Effect

Wei Li

IPMU, Tokyo University, Japan

Taiwan String Theory Workshop, Jan 19, 2009


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Reference

Fractional Quantum Hall Effect via Holography:

Chern-Simons, Edge States, and Hierarchy

with Mitsutoshi Fujita (Kyoto U.)

Shinsei Ryu (Berkeley)

Tadashi Takayanagi (IPMU)

arXiv:0901.0924


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FQHE overview

Hierarchy

Edge states

FQHE from ABJM

Holographic edge states

Hall conductivity

FQHE from Deformed AdS Soliton

Holographic pure Chern-Simons and level-rank duality

Hall conductivity

Hierarchical FQHE from IIA in C2

/ZnHirzebruch-Jung Resolution

Hierarchical structure

Summary


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Outline

FQHE overview

HierarchyEdge states

FQHE from ABJM


Hall conductivity



Hall conductivity

Hierarchical FQHE from IIA in C2/Zn

Hirzebruch-Jung Resolution


Summary


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FQHE experimental setup

2DEG in xy-plane.

Low temperature Strong magnetic field Bz

Break parity symmetry.

Electric field Ey

Hall current jx = xyEy


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Hall conductivity measurement

Stormer 92

Plateaux ofxy: xy = e2

hxx = 0

Integer QHE: = 1, 2, 3, . . .

Fractional QHE: = n2m+1


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Integer quantum Hall plateaux

Figure: Bhatt

1. Conducting Landau levels (separated by gaps) Filling fraction

NeNLan

2. When Fermi level is between Landau levels:

All Landau levels below are completely filled: = N

Energy gap = xx = 0


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Fractional quantum Hall plateaux

Question: How about Fractional quantum Hall plateaux?

How to have energy gap inside partially filled Landau level?

Need to include electron-electron interaction!


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Fractional quantum Hall plateaux

Plateaux = Energy gap (Caused by e-e interaction!)

FQHE with = 1k

(k odd) can be described by Laughlins

wavefunction Laughlin 82

Quasiparticle has fractional charge ek

Quasiparticle has fractional statistics:Interchanging two quasiparticles give phase e

ik

Energy gap = BEC of composite boson (electron + k flux)


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Aside: Anyons

Figure: Shtengel

Interchanging two quasiparticles: (2, 1) = ei(2, 1)

=

0 Bos-on

Fermi-on

Any-on

Anyon only exists in 2D.


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Effective description of FQHE

What to look for?

1. (2+1)-D low energy effective field theory2. Quasiparticle with fractional charges and statistics

Answer: U(1)k Chern-Simons

Sc.s. = k4

a da + e

2

a Fext

Gives correct Hall conductivity Source term a0(x x0) creates a quasiparticle with

Charge Q =e

kand Flux =

2

k

How about FQHE with = pq

?


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Hierarchical FQHE

= 1k FQHE: Chern-Simons with single a.

Electron + k flux = Composite Boson = BEC = new

quasi-particles... Halperin 84

1. New quasiparticles are sourced by a(0)

.2. Use U(1)k1 Chern-Simons to describe the new quasiparticle

S =k04

a(0) da(0) +

1

2

a(0) Fext

+

k1

4

a

(1)

da

(1)

+

1

2

a

(1)

f

(0)

3. xy = (1

k01

k1

) e2

h


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Hierarchical FQHE

Generalize to r levels: (Hierarchy!) Quasiparticles of quasiparticles of quasiparticles of....

S =1

4

Kija

(i) da(j) +1

2

qia

i Fext

K =

k0 11 k1 1

1 k2 1

1. . .

, q = (1, 0, 0, 0, )

Filling fraction via continued fraction

=1

k0 1

k11

k2


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FQHE signature

Plateaux ofxy:

xy =

e2

h xx = 0

1. Integer quantum Hall Klitzing 80

= 1, 2, 3, . . .

2. Simplest fractional quantum HallTsui, Stormer, Laughlin 82

=1

3,

1

5,

1

7, . . .

3. Hierarchical fractional quantum Hall Haldane 83; Halperin 84

= 1(2p0 + 1) 1

2p11

2p2...

All above have odd denominators and are well understood !

4. Even denominator fractional quantum Hall

All can be realized in string theory!!


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Edge states of QHE

Question: Why isnt QHE an insulator?

Answer: Existence of edge states!! Halperin 82

Figure: Bhatt

Edge excitations are gapless!!

Edge state is a (1+1)-D chiral theory


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Why string?

FQHE is strongly coupled.

Geometrization of FQHE using AdS/CFT


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Review of stringy realization of FQHE

Earlier models1. Holographic computation of classical Hall effect

Hartnoll+Kovtun 07, OBannon 07, Myers+Wapler 08

2. Noncommutative Chern-Simons description (not holographic)

Susskind, Hellerman+Van Raamsdonk, 01...

3. Holographic construction of integer QHE

Keski-Vakkuri+Kraus, Davis+Kraus+Shah, 08

What is lacking Edge states

Hierarchical FQHE

Features of our models Based on Chern-Simons description (1, 2, 3)

Holography (1, 2)

Edge state (1, 2)

Hierarchy (3)


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FQHE overview

Hierarchy

Edge states

FQHE from ABJM


Hall conductivity



Hall conductivity

Hierarchical FQHE from IIA inC2

/ZnHirzebruch-Jung Resolution


Summary


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Outline

FQHE overview

Hierarchy

Edge states

FQHE from ABJM


Hall conductivityFQHE from Deformed AdS Soliton


Hall conductivity




Summary


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ABJM

Brane configuration: M2 probing C4/Zk singularity

AdS4/CFT3 Field theory side: N = 6 Chern-Simons with U(N)k U(N)k Gravity side: M-theory in AdS4 S

7/Zk

M-theory to IIA limit:N1/5 k


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ABJM

Brane configuration

AdS4 CP3: t x y r 1 1 2 2

N D2:

k D6-flux:

AdS4/CFT3 Field theory side: N = 6 Chern-Simons with U(N)k U(N)k Gravity side: IIA in AdS4 CP

3

Chern-Simons action:

SgaugeN=6 =k

4

Tr[a(1) da(1)]

k

4

Tr[a(2) da(2)]

Parity even!!


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How to break parity symmetry?

O


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Objects in ABJM

Flux

N D2-branes:N =

CP3

F[6]

k D6:

k =

CP1

F[2]

Particle-like Branes D0 (Di-monopole)

No tadpole

D2 wrapping CP1 (Monopole)

Tadpole k

A = has k F-strings attached D4 wrapping CP2 (Di-baryon)

No tadpole

D6 wrapping CP3 (Baryon)

Tadpole NA = has N F-strings attached

H b k i ?


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How to break parity symmetry?

Wrap M D4 on CP1

Attach to N D2-brane along (1 + 1)-dim

AdS4 CP3: t x y r 1 1 2 2

N D2:

k D6-flux:

M D4:

D4

CP1

D2

U(N)k U(N)k

U(NM)k U(N)k

1/2-BPS

M d li d


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Modeling edge states

Dibaryon (D4 on CP2) acquires tadpole F[4] A = MA= has M F-strings attached

Gauge group changes when crossing the edge!

U(N)k U(N)k = U(N M)k U(N)k

The disappeared U(M) is carried by the D4

Set M = 1 and N = 1 and treat U(1)k as spectator

U(1)k U(1)k = 1 U(1)k

Models an edge state of= 1k

FQHE!

D8 d


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D8 edge

Wrap D8 on CP3

Attach to N D2-brane along (1 + 1)-dim

AdS4 CP3: t x y r 1 1 2 2

N D2:

k D6-flux:

D8:

D8

CP3

D2

U(N)k U(N)k

U(N)kl U(N)k

1/2-BPS

M d li d t t


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Modeling edge states

Di-monopole (D0) acquires tadpole F[0]A = A= has F-strings attached

Gauge group changes when crossing the edge!

U(N)k U(N)k = U(N)k U(N)k

Set N = 1 and treat U(1)k as spectator

U(1)k U(1)k = U(1)k U(1)k

Models an interface between = 1k

FQHE and = 1k

FQHE!

A s i ABJM


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Anyons in ABJM

Dibaryon (D4 wrapping CP2) carries 1 F-string ABJM, ABJ 08

Create anyon with

Charge Q = 1 and Flux = 2k

Anyon with statistics ei/k

Rescale A ek

A s.t. Q = ek

Holographic computation of Hall conductivity


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Bending brane models the pair of edge states E



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Probe D4 action:SD4 = S

DBID4 + S

CSD4 + S

bdyD4

SCS =kD44

A dA, with kD4

e2

k

Charge density and current density j on the boundary:

=S

At

, j =S

Ax

Hall current jx = jbdy1 jbdy2

Gives correct Hall conductivity

xy = e2

h

Interface from D8


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Interface from D8

D8-edge is an interface between two FQHE with = 1k

and

= 1k

Holographic computation predicts

xy =

k2

e2

h | |

e2

h

Summary so far


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Summary so far

FQHE from ABJM

1. Supersymmetric Chern-Simons + bifundamental matter fields!!

2. Reproduce correct xy because it is topological!

Real FQHE has mass gapcan we realize pure, bosonic

Chern-Simons?

Go to Model II

Outline


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Outline

FQHE overview

Hierarchy

Edge states

FQHE from ABJM




Hall conductivity




Summary

Warm-up: AdS soliton from D3


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Warm up: AdS soliton from D3

Start from N D3-branes (N = 4 SYM IIB in AdS5)

1. Compactify D3 on space-like circle + L Geometry: AdS bubble (doubly Wick-rotated AdS black hole)

ds2 = R2dr2

f(r)r2+

r2

R2(dt2 + f(r)d2 + dx2 + dy2) + R2d25

f(r) = 1 r0

r4

cycle shrinks at r = r0.

2. Choose anti-periodic b.c. for fermions

Fermions gain mass = All scalars gain mass via quantum

correction Break SUSY completely!

At low T: (2+1)-dim pure YM AdS Soliton Witten 98

Deformed AdS soliton from D3-D7


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Deformed AdS soliton from D3 D7

Question: how to get Chern-Simons term?

Hint: SCSD3 =

14

D3 F F

Answer: Turn on axion = kL

= Add k D7 at the tip of AdS Soliton (r = r0)



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Add k D7 (wrapping S5) at the tip of AdS Soliton (r = r0):

AdS5 S5: t x y r s1 s2 s3 s4 s5

N D3:

k D7:

D3

D7

r = r0

U(N)k

U(k)N

Field theory: U(N)k Yang-Mills-Chern-Simons

Gravity: AdS soliton with k D7 at the tip



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Holography from AdS soliton + D7: Field theory: U(N)k Yang-Mills-Chern-Simons

Gravity: AdS soliton with k D7 at the tip

Question: How to get pure Chern-Simons?

Answer: Take IR limit of D3 theory: Field theory: pure U(N)k Chern-Simons

Gravity: Near r = r0 region = D7-brane theory

CS term on D7 give U(k)N Chern-Simons term

S =

1

2

C[4] F F =

N

4

Tr(A dA +

2

3 A

3

)

D7 theory is pure U(k)N Chern-Simons!!

This holographic duality in IR limit gives Level-Rank duality!



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g p p y

Choose CRR[2] = ARR[1] d

FQHE as U(1)k Chern-Simons on boundary

SD3 =k

4

A dA +

e

2

A dARR[1]

ARR[1] gives external field (B, E) for FQHE.

Charge density and current density j on the boundary:

=S

ARRt

, j =S

ARRx

Hall conductivity

xy = 1k

e2

h

Can also compute by adding edge states (D3 wrapping or D7 wrapping

S5)


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Model 1 and Model 2 : = 1k

FQHE

How about FQHE with =p

q?

Outline


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FQHE overview

Hierarchy

Edge states

FQHE from ABJM




Hall conductivity




Summary

Hierarchical FQHE


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Hierarchy of quasiparticles of quasiparticles of quasiparticlesof....

S =1

4

Kija

(i) da(j) +1

2

qia

i Fext

K =

k0 1

1 k1 1

1 k2 1

1. . .

, q = (1, 0, 0, 0, )

Signatures: Multiple U(1)

Continued fraction as filling fraction

=1

k0 1

k11

k2

Tri-diagonal matrix K

Minimal resolution ofC2/Zn(p) singularity


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/ (p) g y

Orbifold C2/Zn(p):

(z1, z2) (e

2in

z1, e

p2in

z2) where n+1 p n1 and (n, |p|) = 1

IIA on C2/Zn(p) gives Hirzebruch-Jung resolution (blowing up

minimal number of 2-cycles) Harvey+Kutasov+Martinec+Moore 01

Intersection matrix of 2-cycles:

K =

k0 1

1 k1 1

1 k2 1

1. . .

withn

p= k0

1

k1 1

k2...

Use IIA on C2/Zn(p)

to describe hierarchical FQHE!

Embedding Hierarchical FQHE in IIA


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IIA on R(1,2) S3 C2/Zn(p) (near horizon of NS5)

Resolution: r 2-cycles with harmonic 2-forms (i)

RR 3-from gives r U(1) gauge fields

CRR[3] =

i

a(i) (i)

Tri-diagonal intersection matrix

Kij =

i j

Chern-Simons from Chern-Simons term in IIA SUGRA

1

4210

HNS C dC =

1

4

ri,j=1

Kija(i) da(j)

External field from D4 wrapping 2-cycle

i qi[i]

1

2

qia

(i) dA

Embedding succeed!

Outline


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FQHE overview

Hierarchy

Edge states

FQHE from ABJM


Hall conductivity



Hall conductivity




Summary

Summary


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Embedding of FQHE in string theory

Emphasis

Based on Chern-Simons description (1, 2, 3)

Holography (1, 2)

Edge state (1, 2)

Hierarchy (3)

Summary


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FQHE from ABJM

1. Add D4 wrapping CP1

(or D8 wrapping CP3

) to ABJM to breakparity symmetry

2. Models edge states in FQHE with = 1k

3. Holographic computation of Hall conductivity along edge states

FQHE from deformed AdS soliton

1. Stringy realization of pure Chern-Simons

2. Holographic duality = Level-rank duality

3. Holographic computation of Hall conductivity without using edge

states

FQHE from Hirzebruch-Jung singularity1. First stringy realization of hierarchical FQHE

2. Continued fraction filling fraction given by intersection matrix of

resolution ofC2/Zn(p)

Wish list


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Dynamics of FQHE

Non-Abelian FQHE

Topological order of FQHE

Deeper connection between IIA on C2/Zn(p) and hierachical

FQHE


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T HAN K YOU !

Wei Li- Three Stringy Realizations of Fractional Quantum Hall Effect

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