Germán Sierra IFT-CSIC/UAM, Madridin collaboration with Ignacio Cirac and Anne Nielsen MPQ,Garching

Workshop:

”New quantum states of matter in and out equilibrium”

The Galileo Galilei Institute of Theoretical Physics,Firenze, 18 April 2012

A bit of history (Gebhard-Vollhardt 1987)

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ψG ∝PG FS = Π i (1− ni↑ ni↓ ) FS

Fermi state of a chain with N sites at half filling

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FS =Π k <kFck↑∗ ck↓

∗ 0 kF =π

2

Spin-spin correlator in the limit N -> infty (Gebhard-Vollhardt)

Eliminated the states doubly occupied (Gutzwiller projection)

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Sna S0

b = (−1)nδ a b

Si(π n)

4π n≈ δ a b (−1)n 1

8 n−

1

4π 2n2

⎡

⎣ ⎢

⎤

⎦ ⎥ (n >>1)

Spin-spin correlator in the AF Heisenberg model

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Sna S0

b ≈ δa b (−1)n c log n

n−

1

4π 2 n2

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥, (n → ∞)

The Gutzwiller state has only spin degrees of freedomand can be mapped into a hardcore boson state

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↑ ↔ 0 empty

↓ ↔ a* 0 occupied

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ψG = ψ (n1,L ,nN / 2)n1 ,K ,nN / 2

∑ an1

* K anN / 2

* 0

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ni : position of the i-boson (i.e. spin down)

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ψ(n1,K ,nN / 2) = zn i

i

∏ (zn i− zn j

)2

i< j

∏ , zn = e2π i n / N

1D version of a bosonic Laughlin state at

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ν =1

2This state is a spin singlet

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H =1

2

r S n⋅

r S m

sin2(π (n − m) /N)n<m

∑

Haldane-Shastry Hamiltonian (1988)

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H = −2zn zm

(zn − zm )2n<m

∑r S n⋅

r S m€

ψG Is the exact ground state of the Hamiltonian

AFH model with exchange couplings inversely proportionally to the chord distance

Long range version of the AFH with NN couplings

Properties of the HS model

- Elementary excitations: spinons (spin 1/2 with fractional statistics)

- Degenerate spectrum described by the Yangian symmetry

- Closely related to the Calogero-Sutherland model - Low energy physics described by the WZW SU(2)@k=1

- The HS model is at the fixed point of the RG while the AFH is a marginal irrelevant perturbation which gives rise to log corrections

Haldane, Bernard, Pasquier, Talstra, Schoutens, Ludwig,…

- Overlap between the HS state and the Bethe state

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HS Bethe N ≈ 0.99, N = 20 sites

- Truncate the HS Hamiltonian to NN and NNN couplings

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H = J1n

∑r S n⋅

r S n +1 + J2

r S n⋅

r S n +2

with

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J2 /J1 =1/4 = 0.25

The J1-J2 model is critical with no log corrections at

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J2 /J1 ≅ 0.2411

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za = na + i ma

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ψKL (z1,L zN / 2) = χ (zn ) (zn − zm )2 e− |zq |2 / 4

q∑

n< m

∏n

∏

Kalmeyer-Laughlin wave function (1987)

Bosonic Laughlin wave function on the square lattice at

Where the z’s are the position of hard core bosons€

ν =1

2

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↑ ↔ 0 empty

↓ ↔ a* 0 occupied

The KL state is the wave function of a spin chiral liquid

Is there a H for which the KL state is the GS? -> Parent Hamiltonian

The Haldane-Shastry (1D) and the Kalmeyer-Laughlin (2D) states have a common origin

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SU(2)k=1

The Haldane-Shastry state:

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φ1/ 2 × φ1/ 2 =φ0Fusion rule:

Unique conformal block

(N even)

The WZW model

CFT with c =1 and two primary fiels

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φ0, φ1/ 2, h0 = 0, h1/ 2 =1/4

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zn = e2π i n / N

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z = n + i mThe Kalmeyer-Laughlin state

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φsn(zn ) = χn e i sϕ (z) / 2 , sn = ±1

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φs1(z1)L φsN

(zN ) = δsi ,0

i∑ e iπ (sn −1)/ 2

n:odd

∏ (zi − z j )si s j / 2

i< j

∏

Bosonization

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χn =1 (n : even), χn = e iπ (sn −1)/ 2 (n : odd) Marshall sign rule

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sn =1: ↑ ↔ 0 qn = 0

sn = −1: ↓ ↔ an* 0 qn =1

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si =1− 2qi

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φs1(z1)K φsN

(zN ) = zn i

i

∏ (zn i− zn j

)2

i< j

∏

Spin ½ primary field

Choosing we recover the HS state

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zn = e2π i n / N

Map: Spin-> hard boson

The KL state is approached asymptotically

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ψWZW (z1,L zN / 2) ∝ χ (zn ) (zn − zm )2 fM (zn )n =1

M

∏n< m

∏n

∏

where

In 2D

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fM (zn ) = zn 1 −zn

zm

⎛

⎝ ⎜

⎞

⎠ ⎟

−1

→e− zn2

/ 4 (M = N /2 →∞)m(≠n )

N

∏

Questions:

- can one derive the parent Hamiltonians for the HS and KL states using purely CFT methods?

- can one generalize these states to higher spin?

Hamiltonians from null vectors

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χ3 / 2, 3 / 2 = J−1+ φ1/ 2, 1/ 2

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χa, m (z) = K b,m ′a,m J−1

b φ1/ 2, m ′∑ (z), a =1,2,3, m = ±1/2

In the spin ½ module at k=1 there is the null vector

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Kb,m′a,m =

2

3δa,b δm,m′ − i εabc tm,m′

c

c

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟, t a =

σ a

2

This is the heighest weight vector of a spin 3/2 multiplet. The whole multiplet is given by

(Kac, Gepner, Witten)

Clebsh-Gordan coefficients for

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1

2⊗1 →

3

2

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ψm1 ,L ,mN(z1L zN ) = φm1

(z1)L φmN(zN )

Satisfy the algebraic equations (use Ward identities)

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Ci, a z1,K zN( ) ψ (z1L zN ) = 0, i =1,K N, a =1,2,3

Decoupling eqs for these null vectors imply that the conformal blocks

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Ci, a z1,K zn( ) = wijj(≠ i)

n

∑ ( t ja + iε abc ti

b t jc ), wij =

zi + z j

zi − z j

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H i = Ci,a∗

a

∑ Ci,a, H iψ = 0, i =1,K ,N

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H = H ii

∑ , H ψ = 0Parent Hamiltonian

This construction works for generic z’s

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HHS =−zn zm

(zn − zm )2 +1

12wn,m (cn − cm )

⎛

⎝ ⎜

⎞

⎠ ⎟

n ≠m

∑r S n⋅

r S m

Uniform case

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zn = e2π i n / N →cn = 0

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HHS ψ = E0ψ

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wn,m =zn + zm

zn − zm

, cn = wn,m, E0 =m(≠n )

∑ 1

16 n ≠m

∑ wn,m2 −

N(N +1)

16

Non uniform case -> Yangian is broken-> less degeneracy

If

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zn =1

recover the HS Hamiltonian

1D-model

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H,r S [ ] = H,

r Λ [ ] = 0,

r S ,

r Λ [ ] ≠ 0,

r S = S i,

i

∑r Λ = wij

r S i ×

i, j

∑r S j

Yangian symmetry

Uniform

Dimer Random

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zn = e2π i n / N

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zn = e2π i(n +δ (−1)n ) / N

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zn = e2π i(n +φ n ) / N

Decoupling equations -> eqs. for spin correlators

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wi j tia t j

a + wi k t ja tk

a +3

4wi j

k(≠ i, j )

∑ = 0, i ≠ j

In the uniform case, N -> infinite (Gebhard-Vollhardt 1987)

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tna t0

b = (−1)nδa b

(−1)n

4 N sin(πn /N)

sin(2π n(m −1/2) /N)

m −1/2m=1

N / 2

∑

Exact formula for finite N

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tna t0

b = (−1)nδ a b

Si(π n)

4π n, Si(z) = dt

sin t

t0

z

∫

Four point spin correlator

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t10 t50

0 t1250 tn

0 n =1,L ,200

Excited states (uniform model)

They are known exactly in terms of quasimomenta

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S z =N

2− M, {m j} j =1

M = {1,2,L ,N −1}, m j +1 > m j +1

Energy and momenta

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E = m j (m j − N),j

∑ P = 2π m jj

∑ /N

The wave functions of excited states are also chiral correlators

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φm1(z1)L φmN

(zN ) → E0 = −(N 3 + 2N) /12 → S = 0

φm∞(∞)φm1

(z1)L φmN(zN )φm0

(0) → Eexc = N /2 → S =1

2⊗

1

2= 0⊕1

φm1(z1)L φmN

(zN ) J a (0) → Eexc = N −1 → S =1

J a (∞)φm1(z1)L φmN

(zN ) → Eexc = N −1 → S =1

J a (∞)φm1(z1)L φmN

(zN ) J b (0) → Eexc = 2(N −1) → S =1⊗1 = 0⊕1⊕2

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H i =1

2| wij |2 +

j (≠ i)

∑ 2

3| wij |2 ti

a t ja +

j (≠ i)

∑ 2

3wij

* wik t ja tk

a

j ≠k(≠ i)

∑

−2 i

3wij

* wik ε abc tia t j

b tkc

j ≠k(≠ i)

∑ ,

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zi = ni + i mi

Take the z’s generic in the complex,

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wij =g(zi)

zi − z j

+ h(zi) g(z), h(z): generic

Generalization to 2D

If this is the Parent H for the “KL state”

Greiter et al. have constructed H’s for the large N limit

The three body term breaks time reversal (in 1D it was absent)

Area law on the sphere

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−γ=0.341 ± 0.057≈ −1

2log2 = −0.347Topological entropy:

Same as the bosonic Laughlin states at

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ν =1

2

Two leg ladder and entanglement Hamiltonian

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zn.A = e(2 i n + χ )π /Q, zn.B = e(2 i n − χ )π /Q , n =1,L Q

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ρA = TrB ψ ψ = e− HA

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HA = c0 + c2 H2 + c3 H3 + cr Hr

Density matrix for leg A

Entanglement Hamiltonian

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H2 = −2zn zm

znm2

n<m

∑r S n⋅

r S m , H3 = −i

zn zm zp

znm znp zmpn<m

∑r S n⋅(

r S m ×

r S p )

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χ= Inter legdistance

Primary fields:

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φ0, φ1/ 2, φ1

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φ1/ 2 × φ1/ 2 =φ0 + φ1, φ1 × φ1 =φ0, φ1 × φ1/ 2 =φ1/ 2Fusion rules

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ψs1K sN= φs1

(z1)L φsN(zN ) , si = 0,±1

For spin 1 there is only one conformal block

Using the null vectors method one finds the parent Hamiltonian

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H = −4

3wi j

2

i≠ j

∑ −1

3wi j

2 + 2 wk i wk j

k(≠ i, j )

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

i≠ j

∑ tia t j

a +1

6wi j

2 tia t j

a( )

i≠ j

∑2

+1

6wi j wi kti

a t ja ti

b tkb

i≠ j≠k

∑

(See also Greiter et al and Paredes)

Spectrum in the uniform case

There are not accidental degeneracies-> No Yangian symmetry

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

Primary spin 1 fields (h=1/2)

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φ±1(z j ) = e±iϕ (z j ), φ0(z j ) = (−1) j χ (z j )

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ψs1L sN= (−1)

sii: odd∑zi − z j( )

si s j

Pf0

1

zi0 − z j

0 , si = 0, N : eveni

∑i< j

∏

Majorana fermion

In the uniform case we expect the low energy spectrum of this model to be described by SU(2)@k=2 model

Renyi-2 entropy

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SL = −logTr ρ L2

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SU(2)@k = 2 →c =3

2

Spin-spin correlator

CFT prediction

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tna t0

a ≈(−1)n sinπ n

N

⎛

⎝ ⎜

⎞

⎠ ⎟b

, b = −3

4

Suggest existence of log corrections (Narajan and Shastry 2004)

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φ1/ 2 × φ1/ 2 = φ0 + φ1

Take again SU(2)@k=2

Fusion rule of spin 1/2 field

Number of chiral correlators of N spin 1/2 fields

Now the GS is NOT unique but degenerate !!

Example N = 6 -> 4 GS

The spin Hamiltonian contains 4 body terms

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φ1/ 2,m1(z1)L φ1/ 2,mN

(zN )p

p =1, 2L , 2N / 2−1

Mixing spin 1/2 and spin 1 for SU(2)@k=2

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φ1/ 2K N1/2 K φ1/ 2 φ1 K N1 K φ1 p

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21

2N1/2 −1

->

The degeneracy only depends on the number of spin 1/2 fields

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

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φ1,±1(z) = e± iϕ (z), φ1,0(z)= χ (z)Spin 1 field

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

is the Majorana field and is the spin field of the Ising model

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χ(z)

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σ(z)

Ising fusion rules

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χ ×χ =id, χ ×σ = σ , σ ×σ = id + χ

Moore-Read wave function for FQHE @5/2 (1992)

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ψe (z) = χ (z)e i 2ϕ (z)

CFT = boson (c=1) + Ising (c=1/2)

Electron operator

Ground state wave function

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ψe (z1)K ψ e (zN ) = (zi − z j )2

i< j

∏ χ (z1)K χ (zN )

χ (z1)K χ (zN ) = Pfaffian1

zi − z j

= det1

zi − z j

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ψqh (z) = σ (z)ei

2 2ϕ (z )

Quasihole operator

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ψqh K Nqh K ψ qh ψ e K Ne K ψ ep

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21

2Nqh −1

-> Degeneracy

Fusion rules of Ising

FQHE CFT Spin Models

Electron field spin 1Quasihole field spin 1/2

Braiding of Monodromy Adiabaticquasiholes of correlators change of H

An analogy via CFT

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σ

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χ

In the FQHE braiding is possible because electrons live effectively in 2 dimensions

To have “braiding” for the spin systems we needs 2D

The SU(2)@k=2 in 2D is the spin analogue of the Moore-Read state

In the FQHE the z’s are the positions of the electrons or quasiholes

In the spin models the z’s parametrize the couplings of the Hamiltonian. They are not real positions of the spins.

Braiding amounts to change these couplings is a certain way.

One can in principle do topological quantum computation in these spin systems.

But one has first to show that Holonomy = Monodromy

This problem has been solved for the Moore-Read state(Bonderson, Gurarie, Nayak, 2010)

Conclusions

- Using WZW we have proposed wave functions for spin systems which are analogue of FQH wave functions

- Generalization of the Haldane-Shastry model in several directions 1) non uniform 2) higher spin 3) degenerate ground states 4) 1D -> 2D

Prospects

- Physics of the generalized HS models- WZW’s with other Lie groups and supergroups, other chiral algebras- Relation with the CFT approach to the Calogero-Sutherland model- Excited states - Topological Quantum Computation with HS models

THANK YOU

Grazie Mille