Germán SierraInstituto de Física Teórica CSIC-UAM, Madrid

Talk at the 4Th GIQ Mini-workshop February 2011

-String theory

-Critical phenomena in 2D Statistical Mechanics

-Low D-strongly correlated systems in Condensed Matter

-Fractional quantum Hall effect

-Quantum information and entanglement

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=

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+

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+L

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s-channel

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+

t-channel u-channel

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p1 + p2 = p3 + p4

Mandelstam variables

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s = (p1 + p2)2

t = (p1 − p3)2

u = (p1 − p4 )2

Scattering amplitude

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A = A(s, t,u)

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A(s, t) =Γ(−α (s))Γ(−α (t))

Γ(−α (s) −α (t))= dx x−α (s)−1

0

1

∫ (1− x)−α ( t )−1

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=q

∑€

q

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=q

∑€

α(s) = α ′s + α (0)Regge trayectory

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q

s-t duality

String action

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Sparticle ∝ dτdx μ (τ )

dτ

⎛

⎝ ⎜

⎞

⎠ ⎟

2

∫ → Sstring ∝ dσ dτdx μ (σ ,τ )

dτ

⎛

⎝ ⎜

⎞

⎠ ⎟

2

−dx μ (σ ,τ )

dσ

⎛

⎝ ⎜

⎞

⎠ ⎟

2 ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

∫

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μ =0,1,L D −1 where D= space-time dimension

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A(s, t)∝ D x(σ ,τ )e−Sstring dσ 1∫ e i p1μ x μ (σ 1 ,−∞)L dσ 4∫ e i p4

μ x μ (σ ,∞)∫

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x μ (σ ,τ ) is a 1+1 field that satisfies the equations of motion

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d2

dσ 2−

d2

dτ 2

⎛

⎝ ⎜

⎞

⎠ ⎟x

μ (σ ,τ ) = 0 → x μ (σ ,τ ) = xRμ (τ −σ ) + xL

μ (τ + σ )

Open

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dx μ (σ ,τ )

dσ= 0,σ = 0,π → x μ (σ ,τ ) = x μ + pμτ + i

1

nα n

μ e−i n τ cosnσn=−∞

∞

∑

Closed

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x μ (0,τ ) = x μ (2π ,τ ) → x μ (σ ,τ ) = x μ + pμτ + i1

n(α n

μ e−2i n (τ −σ ) +n=−∞

∞

∑ α nμ e−2i n (τ +σ ))

Quantization

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x μ , pν[ ] = iη μ ,ν ,

α nμ ,α m

ν[ ] = nδn +m,0 η μ ,ν , α n

μ ,α mν

[ ] = nδn +m,0 η μ ,ν , α nμ ,α m

ν[ ] = 0

String=zero modes (x,p)+infinite number of harmonic oscillators

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: e i k⋅x(0,τ ) :=exp k ⋅α −n

ne i n τ

n=1

∞

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟e i k⋅(x + p τ ) exp −k ⋅

α n

ne−i n τ

n=1

∞

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟

Vertex operators: insertions of particles on the world-sheet (Fubini and Veneziano 1970)

T is a symmetric, conserved and traceless tensor

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Tab = Tba, ∂aTab = 0, η ab Tab = 0

For closed string T splits into left and right components

In light cone variables

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σ ±=τ ±σ, ∂± =1

2(∂τ + ∂σ )

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T++ =1

2(T00 + T01) = ∂+x μ ∂+x μ

T−− =1

2(T00 − T01) = ∂−x μ ∂−x μ

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Tab (σ ,τ ) (a,b = 0,1)

The energy-momentum tensor

Generator of motions on the string world-sheet

Virasoro operators

Make the Wick rotation

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σ +, σ − → z = σ + iτ , z = σ − iτ

Fourier expansion of the energy momentum tensor

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T++ → Tzz(z) = Ln z−n−2

n=−∞

∞

∑

T−− → T z z (z ) = L n z −n−2

n=−∞

∞

∑

Where are called the Virasoro operators

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Ln =1

2α n−m α m, L n =

1

2α n−m α m ,

m

∑m

∑€

Ln, L n (n ∈ Z)

Virasoro algebra

The Virasoro operators satisfy the algebra

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Ln ,Lm[ ] = (n − m) Ln +m +c

12(n3 − n)δn +m,0

where c = central charge of the Virasoro algebra

Classical version of the Virasoro algebra

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l n ,l m[ ] = (n − m)l n +m , l n = −zn +1 ∂

∂z

This contains the conformal transformations of the plane:

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l −1 = −∂z

l 0 = −z∂z

l 1 = −z2 ∂z

translations

dilatations

special conformal

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z → ′ z =a z + b

c z + d,

ad − bc =1( )

In 2D the conformal group is infinite dimensional !!

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l n (n ∈ Z)

Ln (n ∈ Z)Classical generators of conformal transformations

Quantum generators of conformal transformations

“c” represents an anomaly of conformal transformations

Physical meaning of “c”

Bosonic string: X-fields + Faddev-Popov ghost c = D - 26 Superstring: X-fields + fermionic fields + Faddev Popov ghost c = D + D/2 - 26 + 11 = 3D/2 -15

String theory does not have a conformal anomaly!! c = 0 -> D = 26 (bosonic string) and 10 (superstring)

c gives a measure of the total degrees of freedom in CFT

c= 1 (boson)

c= 1/2 (Majorana fermion/Ising model)

c= 1 (Dirac fermion/1D fermion)

c= 3/2 (boson+Majorana or 3 Majoranas)

c=….

Fractional values of c reflect highly non perturbative effects

The Belavin-Polyakov-Zamolodchikov (1984)

Infinite conformal symmetry in two-dimensional quantum field theory

Conformal transformations

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z → w = f (z), z → w = f (z )

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φh h

(z,z )(dz)h (dz )h = ′ φ h h

(w,w )(dw)h (dw )h

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Aμ (x) dx μ = ′ A μ ( ′ x ) d ′ x μ →Az(z)dz (h =1,h = 0) + A z (z )dz (h = 0,h =1)

Covariant tensors are characterized by two numbers

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h, h Conformal weights

Dilation

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z → w = λ z, z → w = λ z (λ : real)

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z → w = λ z, z → w = λ z (λ : real)

General framework of CFT

-T is a symmetric, conserved and traceless tensor with central charges (no need of an action) - There is a vacuum state |0> which satisfies

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Ln 0 = L n 0 = 0, n = −1,0,1,2,L ∞

-There is an infinite number of conformal fields in one-to-one correspondence with the states

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c = c

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Φ(z,z ) ↔ Φ ≡ limτ →−∞ Φ(σ ,τ ) 0 = limz→0 Φ(z,z ) 0

-There are special fields (and states) called primary satisfying

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L0 φh,h

= hφh,h

, L 0 φh,h

= h φh,h

Ln φh,h

= 0, L n φh,h

= 0, n > 0

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Tμν (x) dx μ dxν = ′ T μν ( ′ x ) d ′ x μ d ′ x ν →Tzz(z) (dz)2 (h = 2,h = 0) + T z (z )dz (h = 0,h = 2)

-The remaing fields form towers obtained from the primary fieldsacting with the Virasoro operators (they are called descendants)

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φh

L−1 φh

L−12 φh ,L−2 φh

L−13 φh ,L−1L−2 φh,L−3 φh

L L L L L L L L L

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L0

h

h +1

h + 2

h + 3

-The primary fields form a close operator product expansion algebra For chiral (holomorphic fields)

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φi(z)φ j (w) =Cijk

(z − w)hi +h j −hk

k

∑ φk (w) +L

T(z) T(w) =c /2

(z − w)4+

2T(w)

(z − w)2+

∂T(w)

(z − w)+L€

φi

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φ j

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φk

Vermamodule:

OPEconstants

- Fusion rules (generalized Clebsch-Gordan decomposition)

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φa × φb = Nabc

k

∑ φk, Nabc = 0,1,L

- Rational Conformal Field Theories (RCFT): finite nº primary fields - Minimal models

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c =1−6

m(m +1), m = 3,4,L

hr,s =(m +1)r − ms[ ]

2−1

4m(m +1), 1 ≤ r < m, 1 ≤ s ≤ r

A well known case is the Ising model c=1/2 (m=3)

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I ⇔ φ1,1 or φ2,3, h0 = 0

ψ ⇔ φ2,1 or φ1,3, hψ =1/2

σ ⇔ φ2,2 or φ1,2, hσ =1/16

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ψ ×ψ =I

ψ ×σ = σ

σ ×σ = I +ψ

- Conformal invariance determines uniquely the 2 and 3-point correlators

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φi(z1)φ j (z2) =δ ij

z12hi +h j

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φi(z1)φ j (z2)φk (z3) =Cijk

z12hi +h j −hk z13

hi +hk −h j z23h j +hk −hi

- Higher order chiral correlators: their number given by the fusion rules

normalization

Conformal blocks for the Ising model

Fusion rules

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σ ×σ ×σ ×σ =(I +ψ ) × (I +ψ ) = I 2( ) +ψ (2)

There are four conformal blocks:

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FI = σ (z1)L σ (z4 )I

= 2−1/ 2 zab−1/ 8 z13 z24 + z14 z23( )

1/ 2

a<b

∏

Fψ = σ (z1)L σ (z4 )ψ

= 2−1/ 2 zab−1/ 8 z13 z24 − z14 z23( )

1/ 2

a<b

∏

The non-chiral correlators (the ones in Stat Mech)

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σ(z1,z 1)L σ (z4,z 4 ) = FI z1,L z4( ) FI* z 1,L z 4( ) + Fψ z1,L z4( ) Fψ

* z 1,L z 4( )

Must be invariant underBraiding of coordinates

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z1

z2

z3

z4

Conformal blocks give a representation of the Braid group

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Fp L zi zi+1L( ) = Bp,q± Fq L zi+1 ziL( )

q

∑

Related to polynomials for knots and links, Chern-Simon theory, Anyons, Topological Quantum Computation, etc

Yang-Baxter equation

Characters and modular invariance

Conformal tower of a primary field

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φa

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χa (τ ) = TrHaqL0 −c / 24 = q−c / 24 da (n)

n≥0

∑

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da (n) : number of states at level n=0,1,2,…

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q = e iτ , τ ∈ Upper half of the complex plane

Moduli parameter of the torus

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τ

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0€

τ +1

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1 states propagation

Modular group

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T : τ → τ +1

S : τ → −1/τ€

τ → aτ + b

c τ + d,

a b

c d

⎛

⎝ ⎜

⎞

⎠ ⎟∈ Sl(2,Z) :

Generators

Fundamental region

Characters transforms under modular transformations as

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χa (τ +1) = e i(ha −c / 24 ) χ a (τ )

χ a (−1/τ ) = Sabb∑ χ b (τ )

Partition function of CFT must be modular invariant

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Z(τ ) = TrH qL0 −c / 24 q L 0 −c / 24 = Mab χ a (τ ) χ a (τ )a,b

∑

Z(τ ) = Z(τ +1) = Z(−1/τ )

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Nabc =

Sam Sbm Scm*

S1mm

∑

Verlinde formula (1988)

Fusion matrices and S-matrix and related!!

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S =1

2

1 1 2

1 1 − 2

2 − 2 0

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

Example: Ising model

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I

ψ

σ

Check

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Nσσψ =

1

42( )

2+ − 2( )

2+ 0

⎛ ⎝ ⎜ ⎞

⎠ ⎟=1

Axiomatic of CFT

Moore and Seiberg (1988-89)

- Algebra: Chiral antichiral Virasoro left right ( c ) + others

- Representation: primary fields

- Fusion rules:

- B and F matrices : BBB =BBB (Yang-Baxter) FF = FFF (pentagonal)

- Modular matrices T and S

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⊗

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⊃

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⊗

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φa, ha,h a

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Nabc

Sort of generalization of group theory-> Quantum Groups

Wess-Zumino-Witten model (1971-1984)

Field is an element of a Group manifold

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g(z,z )∈G

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SWZW =k

16πd2x Tr ∂ μ g−1∂μ g( ) −

ik

24π∫ d3y εαβγ Tr g−1∂α g−1 g−1∂ β g−1g−1∂γ g−1

( )B

∫

CFT with “colour”

Conformal invariance->

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g(z,z ) = f (z) f (z )

Currents

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J a (z) = −k∂zg g−1 = Jna z−n−1 a =1,L ,dimG

n

∑

J a (z ) = k g−1∂z g= J na z −n−1

n

∑

OPE of currents

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J a (z) J b (w) =kδab

(z − w)2+ i fabc

J c (w)

z − w+L

c

∑

Kac-Moody algebra (1967)

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Jna ,Jm

b[ ] = i fabc Jn +m

c

c

∑ + k nδab δn +m,0 k= level (entero)

Sugawara construction (1967)

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T(z) =1

2(k + g)J a (z) J a (z)

a

∑

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Ln =1

2(k + g): Jn−m

a Jma

m

∑a

∑ :

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c =k dimG

k + g

g: dual Coxeter number of G

Primary fields and fusion rules (Gepner-Witten 1986)

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φ j1⊗φ j2

= φ j

j= j1 − j2

min j1 + j2 ,k− j1 − j2( )

∑

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j = 0,1

2,L ,

k

2G=SU(2)

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(k + g)∂

∂zi

−

r S i ⋅

r S j

zi − z jj≠ i

N

∑ ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥φ j1

(z1)L φ jN(zN ) = 0

Knizhnik- Zamolodchikov equations (1984)

Heisenberg-Bethe spin 1/2 chain

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H =r S n ⋅

r S n +1

n

∑

Low energy physics is described by the WZW SU(2)@k=1

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rS i ⋅

r S j ∝ (−1)i− j

log i − j

i − j

But the spin 1 chain is not a CFT (Haldane 1983)

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rS i ⋅

r S j ∝ (−1)i− j e− i− j /ξ

-> Haldane phase and gap

FQHE/CFT correspondence

electron =

quasihole ->

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χ(z)e i 2ϕ (z)

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σ(z)ei

2 2ϕ (z )

Basis for Topological Quantum Computation (braids -> gates)

Laughlin wave function

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ψ(z1,L zN ) = (zi − z j )m e

− zk2

/ 4∑

i< j

∏

The entanglement entropy in a bipartition A U B scales as

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SA ∝ logχ

In a critical system described by a CFT (periodic BCs)

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SA =c

3logL + c1

hence one needs very large matrices to describe critical systems

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N ∝ χ κ , κ = κ (c)

Another alternative is to choose infinite dimensional matrices:

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χ =∞

(1D area law)

MPS state

auxiliary space (string like)

physical degrees

iMPS state

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χ

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χ

€

χ

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χ

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χ

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χ

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χ =∞

Example 5: level k=2, spins =1/2 and 1, D=2

SU(2)@2 = Boson + Ising c=3/2 = 1 + 1/2

spin j=1 field

spin j=1/2 field

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φ1,±1(z) = e± iϕ (z), φ1,0(z)= χ (z), h1 = hχ =1

2

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φ1/ 2,±1/ 2(z) = σ (z)e±iϕ (z ) / 2, hσ =1

16, h1/ 2 =

3

16

N spins 1

The chiral correlators can be obtained from those of the Ising model (general formula Ardonne-Sierra 2010)

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ψ s1,,K ,sN( ) = χ s zi − z j( )si s j

Pf0

1

zi − z j

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟i< j

∏

The Pfaffian comes from the correlator of Majorana fields

Similar chiral correlators have been considered inthe Fractional Quantum Hall effect at filling fraction 5/2.This is the so called Pfaffian state due to Moore and Read.

FQHE/CFT correspondence

electron = quasihole ->

€

χ(z)e i 2ϕ (z)

€

σ(z)ei

2 2ϕ (z )

Quasiholes are non abelian anyons because their wavefunctions (chiral correlators) mix under braiding of their positions.

Basis for Topological Quantum Computation (braids -> gates)

FQHE CFT Spin Models

Electron Majorana spin 1Quasihole field spin 1/2

Braid of Monodromy Adiabaticquasiholes of correlators change of H

Holonomy = Monodromy

An analogy via CFT

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σ

Then if one could get Topological Quantum Computation in the FQHE and the Spin Models.

Bibliography

Non-Abelian Anyons and Topological Quantum ComputationC. Nayak, S. H. Simon, A. Stern, M. Freedman, S. Das Sarma,arXiv:0707.1889

Applied Conformal Field TheoryPaul Ginsparg, arXiv:hep-th/9108028