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![Page 1: Germán Sierra IFT-CSIC/UAM, Madrid in collaboration with Ignacio Cirac and Anne Nielsen MPQ,Garching Workshop: ”New quantum states of matter in and out.](https://reader034.fdocuments.in/reader034/viewer/2022051621/56649ea85503460f94bac1cf/html5/thumbnails/1.jpg)
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
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A bit of history (Gebhard-Vollhardt 1987)
€
ψG ∝PG FS = Π i (1− ni↑ ni↓ ) FS
Fermi state of a chain with N sites at half filling
€
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)
€
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
€
Sna S0
b ≈ δa b (−1)n c log n
n−
1
4π 2 n2
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥, (n → ∞)
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The Gutzwiller state has only spin degrees of freedomand can be mapped into a hardcore boson state
€
↑ ↔ 0 empty
↓ ↔ a* 0 occupied
€
ψG = ψ (n1,L ,nN / 2)n1 ,K ,nN / 2
∑ an1
* K anN / 2
* 0
€
ni : position of the i-boson (i.e. spin down)
€
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
€
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)
€
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
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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,…
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- Overlap between the HS state and the Bethe state
€
HS Bethe N ≈ 0.99, N = 20 sites
- Truncate the HS Hamiltonian to NN and NNN couplings
€
H = J1n
∑r S n⋅
r S n +1 + J2
r S n⋅
r S n +2
with
€
J2 /J1 =1/4 = 0.25
The J1-J2 model is critical with no log corrections at
€
J2 /J1 ≅ 0.2411
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€
za = na + i ma
€
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
€
↑ ↔ 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
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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:
€
φ1/ 2 × φ1/ 2 =φ0Fusion rule:
Unique conformal block
(N even)
The WZW model
CFT with c =1 and two primary fiels
€
φ0, φ1/ 2, h0 = 0, h1/ 2 =1/4
€
zn = e2π i n / N
€
z = n + i mThe Kalmeyer-Laughlin state
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€
φsn(zn ) = χn e i sϕ (z) / 2 , sn = ±1
€
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
€
χn =1 (n : even), χn = e iπ (sn −1)/ 2 (n : odd) Marshall sign rule
€
sn =1: ↑ ↔ 0 qn = 0
sn = −1: ↓ ↔ an* 0 qn =1
€
si =1− 2qi
€
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
€
zn = e2π i n / N
Map: Spin-> hard boson
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The KL state is approached asymptotically
€
ψWZW (z1,L zN / 2) ∝ χ (zn ) (zn − zm )2 fM (zn )n =1
M
∏n< m
∏n
∏
where
In 2D
€
fM (zn ) = zn 1 −zn
zm
⎛
⎝ ⎜
⎞
⎠ ⎟
−1
→e− zn2
/ 4 (M = N /2 →∞)m(≠n )
N
∏
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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?
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Hamiltonians from null vectors
€
χ3 / 2, 3 / 2 = J−1+ φ1/ 2, 1/ 2
€
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
€
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
€
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)
€
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
€
Ci, a z1,K zn( ) = wijj(≠ i)
n
∑ ( t ja + iε abc ti
b t jc ), wij =
zi + z j
zi − z j
€
H i = Ci,a∗
a
∑ Ci,a, H iψ = 0, i =1,K ,N
€
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
€
zn = e2π i n / N →cn = 0
€
HHS ψ = E0ψ
€
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
€
zn =1
recover the HS Hamiltonian
1D-model
€
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
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Uniform
Dimer Random
€
zn = e2π i n / N
€
zn = e2π i(n +δ (−1)n ) / N
€
zn = e2π i(n +φ n ) / N
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Decoupling equations -> eqs. for spin correlators
€
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)
€
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
€
tna t0
b = (−1)nδ a b
Si(π n)
4π n, Si(z) = dt
sin t
t0
z
∫
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Four point spin correlator
€
t10 t50
0 t1250 tn
0 n =1,L ,200
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Excited states (uniform model)
They are known exactly in terms of quasimomenta
€
S z =N
2− M, {m j} j =1
M = {1,2,L ,N −1}, m j +1 > m j +1
Energy and momenta
€
E = m j (m j − N),j
∑ P = 2π m jj
∑ /N
The wave functions of excited states are also chiral correlators
€
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)
∑ ,
€
zi = ni + i mi
Take the z’s generic in the complex,
€
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)
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Area law on the sphere
€
−γ=0.341 ± 0.057≈ −1
2log2 = −0.347Topological entropy:
Same as the bosonic Laughlin states at
€
ν =1
2
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Two leg ladder and entanglement Hamiltonian
€
zn.A = e(2 i n + χ )π /Q, zn.B = e(2 i n − χ )π /Q , n =1,L Q
€
ρA = TrB ψ ψ = e− HA
€
HA = c0 + c2 H2 + c3 H3 + cr Hr
Density matrix for leg A
Entanglement Hamiltonian
€
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 )
€
χ= Inter legdistance
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Primary fields:
€
φ0, φ1/ 2, φ1
€
φ1/ 2 × φ1/ 2 =φ0 + φ1, φ1 × φ1 =φ0, φ1 × φ1/ 2 =φ1/ 2Fusion rules
€
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
€
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)
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Spectrum in the uniform case
There are not accidental degeneracies-> No Yangian symmetry
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SU(2)@k=2 = Boson + Ising (c= 3/2 = 1+ 1/2)
Primary spin 1 fields (h=1/2)
€
φ±1(z j ) = e±iϕ (z j ), φ0(z j ) = (−1) j χ (z j )
€
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
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Renyi-2 entropy
€
SL = −logTr ρ L2
€
SU(2)@k = 2 →c =3
2
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Spin-spin correlator
CFT prediction
€
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
€
φ1/ 2,m1(z1)L φ1/ 2,mN
(zN )p
p =1, 2L , 2N / 2−1
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Mixing spin 1/2 and spin 1 for SU(2)@k=2
€
φ1/ 2K N1/2 K φ1/ 2 φ1 K N1 K φ1 p
€
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
€
φ1,±1(z) = e± iϕ (z), φ1,0(z)= χ (z)Spin 1 field
€
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
€
χ(z)
€
σ(z)
Ising fusion rules
€
χ ×χ =id, χ ×σ = σ , σ ×σ = id + χ
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Moore-Read wave function for FQHE @5/2 (1992)
€
ψe (z) = χ (z)e i 2ϕ (z)
CFT = boson (c=1) + Ising (c=1/2)
Electron operator
Ground state wave function
€
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
€
ψqh (z) = σ (z)ei
2 2ϕ (z )
Quasihole operator
€
ψqh K Nqh K ψ qh ψ e K Ne K ψ ep
€
21
2Nqh −1
-> Degeneracy
Fusion rules of Ising
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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
€
σ
€
χ
In the FQHE braiding is possible because electrons live effectively in 2 dimensions
To have “braiding” for the spin systems we needs 2D
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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)
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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
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THANK YOU
Grazie Mille