germán sierra instituto de física teórica csic-uam, madrid talk at the 4th giq mini-workshop...
TRANSCRIPT
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
€
=
€
+
€
+L
€
s-channel
€
+
t-channel u-channel
€
p1 + p2 = p3 + p4
Mandelstam variables
€
s = (p1 + p2)2
t = (p1 − p3)2
u = (p1 − p4 )2
Scattering amplitude
€
A = A(s, t,u)
€
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
€
=q
∑€
q
€
=q
∑€
α(s) = α ′s + α (0)Regge trayectory
€
q
s-t duality
String action
€
Sparticle ∝ dτdx μ (τ )
dτ
⎛
⎝ ⎜
⎞
⎠ ⎟
2
∫ → Sstring ∝ dσ dτdx μ (σ ,τ )
dτ
⎛
⎝ ⎜
⎞
⎠ ⎟
2
−dx μ (σ ,τ )
dσ
⎛
⎝ ⎜
⎞
⎠ ⎟
2 ⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
∫
€
μ =0,1,L D −1 where D= space-time dimension
€
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 ,∞)∫
€
x μ (σ ,τ ) is a 1+1 field that satisfies the equations of motion
€
d2
dσ 2−
d2
dτ 2
⎛
⎝ ⎜
⎞
⎠ ⎟x
μ (σ ,τ ) = 0 → x μ (σ ,τ ) = xRμ (τ −σ ) + xL
μ (τ + σ )
Open
€
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
€
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
€
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
€
: 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
€
Tab = Tba, ∂aTab = 0, η ab Tab = 0
For closed string T splits into left and right components
In light cone variables
€
σ ±=τ ±σ, ∂± =1
2(∂τ + ∂σ )
€
T++ =1
2(T00 + T01) = ∂+x μ ∂+x μ
T−− =1
2(T00 − T01) = ∂−x μ ∂−x μ
€
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
€
σ +, σ − → z = σ + iτ , z = σ − iτ
Fourier expansion of the energy momentum tensor
€
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
€
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
€
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
€
l n ,l m[ ] = (n − m)l n +m , l n = −zn +1 ∂
∂z
This contains the conformal transformations of the plane:
€
l −1 = −∂z
l 0 = −z∂z
l 1 = −z2 ∂z
translations
dilatations
special conformal
€
z → ′ z =a z + b
c z + d,
ad − bc =1( )
In 2D the conformal group is infinite dimensional !!
€
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
€
z → w = f (z), z → w = f (z )
€
φh h
(z,z )(dz)h (dz )h = ′ φ h h
(w,w )(dw)h (dw )h
€
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
€
h, h Conformal weights
Dilation
€
z → w = λ z, z → w = λ z (λ : real)
€
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
€
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
€
c = c
€
Φ(z,z ) ↔ Φ ≡ limτ →−∞ Φ(σ ,τ ) 0 = limz→0 Φ(z,z ) 0
-There are special fields (and states) called primary satisfying
€
L0 φh,h
= hφh,h
, L 0 φh,h
= h φh,h
Ln φh,h
= 0, L n φh,h
= 0, n > 0
€
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)
€
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
€
L0
h
h +1
h + 2
h + 3
-The primary fields form a close operator product expansion algebra For chiral (holomorphic fields)
€
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
€
φ j
€
φk
Vermamodule:
OPEconstants
- Fusion rules (generalized Clebsch-Gordan decomposition)
€
φa × φb = Nabc
k
∑ φk, Nabc = 0,1,L
- Rational Conformal Field Theories (RCFT): finite nº primary fields - Minimal models
€
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)
€
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
€
ψ ×ψ =I
ψ ×σ = σ
σ ×σ = I +ψ
- Conformal invariance determines uniquely the 2 and 3-point correlators
€
φi(z1)φ j (z2) =δ ij
z12hi +h j
€
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
€
σ ×σ ×σ ×σ =(I +ψ ) × (I +ψ ) = I 2( ) +ψ (2)
There are four conformal blocks:
€
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)
€
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
€
z1
z2
z3
z4
Conformal blocks give a representation of the Braid group
€
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
€
φa
€
χa (τ ) = TrHaqL0 −c / 24 = q−c / 24 da (n)
n≥0
∑
€
da (n) : number of states at level n=0,1,2,…
€
q = e iτ , τ ∈ Upper half of the complex plane
Moduli parameter of the torus
€
τ
€
0€
τ +1
€
1 states propagation
Modular group
€
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
€
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
€
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 )
€
Nabc =
Sam Sbm Scm*
S1mm
∑
Verlinde formula (1988)
Fusion matrices and S-matrix and related!!
€
S =1
2
1 1 2
1 1 − 2
2 − 2 0
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
Example: Ising model
€
I
ψ
σ
Check
€
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
€
⊗
€
⊃
€
⊗
€
φa, ha,h a
€
Nabc
Sort of generalization of group theory-> Quantum Groups
Wess-Zumino-Witten model (1971-1984)
Field is an element of a Group manifold
€
g(z,z )∈G
€
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->
€
g(z,z ) = f (z) f (z )
Currents
€
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
€
J a (z) J b (w) =kδab
(z − w)2+ i fabc
J c (w)
z − w+L
c
∑
Kac-Moody algebra (1967)
€
Jna ,Jm
b[ ] = i fabc Jn +m
c
c
∑ + k nδab δn +m,0 k= level (entero)
Sugawara construction (1967)
€
T(z) =1
2(k + g)J a (z) J a (z)
a
∑
€
Ln =1
2(k + g): Jn−m
a Jma
m
∑a
∑ :
€
c =k dimG
k + g
g: dual Coxeter number of G
Primary fields and fusion rules (Gepner-Witten 1986)
€
φ j1⊗φ j2
= φ j
j= j1 − j2
min j1 + j2 ,k− j1 − j2( )
∑
€
j = 0,1
2,L ,
k
2G=SU(2)
€
(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
€
H =r S n ⋅
r S n +1
n
∑
Low energy physics is described by the WZW SU(2)@k=1
€
rS i ⋅
r S j ∝ (−1)i− j
log i − j
i − j
But the spin 1 chain is not a CFT (Haldane 1983)
€
rS i ⋅
r S j ∝ (−1)i− j e− i− j /ξ
-> Haldane phase and gap
FQHE/CFT correspondence
electron =
quasihole ->
€
χ(z)e i 2ϕ (z)
€
σ(z)ei
2 2ϕ (z )
Basis for Topological Quantum Computation (braids -> gates)
Laughlin wave function
€
ψ(z1,L zN ) = (zi − z j )m e
− zk2
/ 4∑
i< j
∏
The entanglement entropy in a bipartition A U B scales as
€
SA ∝ logχ
In a critical system described by a CFT (periodic BCs)
€
SA =c
3logL + c1
hence one needs very large matrices to describe critical systems
€
N ∝ χ κ , κ = κ (c)
Another alternative is to choose infinite dimensional matrices:
€
χ =∞
(1D area law)
MPS state
auxiliary space (string like)
physical degrees
iMPS state
€
χ
€
χ
€
χ
€
χ
€
χ
€
χ
€
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
€
φ1,±1(z) = e± iϕ (z), φ1,0(z)= χ (z), h1 = hχ =1
2
€
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)
€
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
€
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