A supersymmetric model for quantum
diffusion in 3d
Margherita DISERTORI
joint work with T. Spencer and M. Zirnbauer
Laboratoire de Mathématiques Raphaël Salem
CNRS - University of Rouen (France)
Newton Institute, August 2008
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random matrices and SUSY approach
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a toy model for quantum diffusion
The problem
statistical properties of large matrices with random distributed
elements: (H, P (H)dH)
H (N × N matrix), P some probability distribution
limit N → ∞
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eigenvalues λ1 , ...λN
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eigenvectors ψλ
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correlation functions
largest eigenvalue...
localized: ψλ = (0, 1, 0, . . . , 0)
extended: ψλ = √1N (1, 1, 1, . . . , 1)
correlation functions hHi1 j1 . . . Hin jn i
−→ analyticity properties in some parameter of P (H)
Applications
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physics
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complex many body systems
disordered conductors
mesoscopic physics →
quantum
chaos

 quantum gravity
string theory
non commutative QFT →

quantum Hall effect...
mathematics
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zeroes of the Riemann Zeta function
random permutations
random tiling, growth processes...
Techniques
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P has rotation symmetry =⇒ P(λ1 , ...λN )
→ orthogonal polynomials (Riemann-Hilbert approach)
ex:
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H∗ = H
P(H) ∝ e−tr V(H)


 P not invariant

eigenvectors
−→


correlation functions
SUSY
moments
field theory techniques
Examples: diffusive/localized transition
a) Random Schrödinger
H = −∆ + λV,
∆ discrete Laplacian
V (j) diagonal disorder, j ∈ Zd
2) Random band matrix: H ∗ = H
Hij ind. gaussian rand. var. with i, j ∈ Λ ⊂ Zd
hHij i = 0
hHij H̄i0 j 0 i = R(i − j)δii0 δjj 0
0 ≤ R(i − j) ≤ e−|i−j|/W
band width = W ' λ−1
quantum diffusion conjecture
Green’s Function: G (E; x, y) = (H − E + iε)−1 (x, y)
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Diffusion Conjecture in 3D:
h|G (E; x, y)|2 i '
ρ(E)
(x, y)
−D∆ + ρ(E) = Im < G (E; 0, 0) >= DOS
and D = D(E) > 0 is diffusion constant.
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in 1D and 2D localization should occur with localization
length `
h|G (E; x, y)|2 i ' −1 e−|x−y|/`
Supersymmetric approach
F. Wegner, K. Efetov
1. write correlation functions in terms of ratios of
determinants
1
z∈C
z−H
d det z − H
trG(z) =
dz det w − H w=z
G(z) =
2. change of representation → new expression where saddle
analysis is possible
3. rigorous saddle analysis
1. correlation functions in terms of determinants
D(z) = det(z − H) ,
z∈C
G(z) =
1
z−H
Examples
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1 ratio
D
D(z)
D(w)
E
−→ DOS
1
hρ(E)i = − lim Im
π ε→0+
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2D ratios
D(z1 )D(z2 )
D(w1 )D(w2 )
E
−→
1
E + iε − H
00
2 points corr. funct. S2 (λ1 , λ2 )
informations on the eigenvectors
2. change of representation
algebraic operations involving fermionic and bosonic
variables
D Qn
Qnα=1 D(zα )
β=1 D(wβ )
E
SUSY
−−−−→
H
DQ
s
j∈Λ F (Qj )
E
Qs
→
DQ
E
F
(Q
)
j
j∈Λ
|Λ|2 variables
−→
n2 |Λ| variables
Hij indep. var.
−→
Qj strongly correlated
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main adv.: less variables, saddle analysis is possible, SUSY
symmetries (Q measure normalized to 1)
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main disadv: Q measure not positive
Q
2. change of representation
algebraic operations involving fermionic and bosonic
variables
D Qn
Qnα=1 D(zα )
β=1 D(wβ )
E
SUSY
−−−−→
H
DQ
s
j∈Λ F (Qj )
E
Qs
→
DQ
E
F
(Q
)
j
j∈Λ
|Λ|2 variables
−→
n2 |Λ| variables
Hij indep. var.
−→
Qj strongly correlated
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main adv.: less variables, saddle analysis is possible, SUSY
symmetries (Q measure normalized to 1)
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main disadv: Q measure not positive
Q
3. Saddle analysis
saddle approximation:
DQ
E
j∈Λ F (Qj )
Q
'
Q
j
F (Qj (s))
make this rigorous → analytical tools
small probability
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cluster expansion
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 exact integration
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convexity bounds
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Ward identities
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 saddle reduced to
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single points → “easy”

saddle is a manifold → hard
localization properties → saddle is a manifold
Measure on the saddle manifold
the case of a band matrix with gaussian measure
hQ
i
−Fβ,ε (Q,S,ψ) , β large parameter
dµs ∼
dQ
dS
dψ
=
j
j
j e
j∈Λ
1 0
Qi = Ti
Ti−1 , Ti ∈ SU(1, 1)
0 −1
Sj ∈ R3
|Sj | = 1,
ψj fermionic (couples Q and S)
Problems
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SU(1,1) symmetry ⇒ manifold is not compact
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after integrating out ψ we get a determinant in Q and S.
→ the interaction is highly non local
still a very hard problem! → try something “easier”
A nice SUSY model for quantum diffusion
vector model (no matrices), Zirnbauer (1991) → expected to
have same features of exact SUSY model for random band
matrix
main advantages
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after integrating out Grassman variables measure is
positive
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symmetries are simpler to exploit
⇒ good candidate to develop techniques to treat quantum
diffusion
The model after integrating out the Grassman variables
real positive measure, action depends only on one scalar field tj :
j ∈ Λ ⊂ Zd
Y
dµ(t) = dt e−βBε (t) = [ dtj e−tj ] e−βBε (t) det1/2 [βMε (t)]
j
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β >> 1 large parameter, ε << 1 regularization
P
P
(cosh tj − 1)
Bε (t) =
(cosh(tj − tj 0 ) − 1) + βε
<j,j 0 >
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Mε = D(t) +
j∈Λ
ε t̂
βe
> 0 positive quadratic form
features of the model
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the measure for t is positive and normalized ⇒
probability measure
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at ε = 0: B0 depends only on gradients:
B0 (tj + c) = B0 (tj ) ⇒ no mass
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Bε is non-convex and highly non local (because of the
determinant)⇒ no convexity bounds, hard to apply
renormalization
The observable:
−1 i = hetx +ty
h|Gε (E; x, y)|2 i → hetx +ty Mxy
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D(t) elliptic operator: (f, Df ) =
P
1
βD(t)+εet̂
(x, y)i
(fj − fj 0 )2 etj +tj 0 ,
<j,j 0 >
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et̂ diagonal matrix:
et̂
= δij etj
ij
random walk in a random environment: random local
conductance: Axy (jj 0 ) = etj +tj 0 −tx −ty
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tj = t constant: Axy (jj 0 ) = 1 ⇒ D = −∆
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in general A can be very small: no uniform ellipticity
Results and conjectures
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Theorem(Zirnbauer) localization in 1 dim:
hG0x i ' −1 e−|x|/β
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Conjecture in 3 dim: diffusion
hG0x i ' h[βD(t = 0) + ]−1 (x, y)i =
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1
(x, y)
−β∆ + Main Theorem (Di-Sp-Zi) In 3D, if β is large, Λ a cube
of side L containing 0 then the eigenfunctions are extended:
hG00 i = O(1)
if
εL3−α = O(1)
≡
kψk44 ≤
1
L3−α
α << 1
Proof
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1) control fluctuations: tj ' t ∀j
etx +ty M (t)−1 (x, y) ' [−β∆ + e−t ]−1 (x, y)
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2) saddle analysis to determine t:
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in d = 1, εe−t = 1/β
in d = 2, εe−t = e−β
in d = 3, t = 0.
so in 1d and 2d we get a mass as expected:
hetx +ty M (t)−1 (x, y)i '
1
(0, x)
−β∆ + mβ
Main problem: control fluctuations
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in general: hard problem
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no convexity
non locality: hard to apply renormalization techniques
D non uniformly elliptic
but we do have symmetries (inherited from the SUSY
structure)
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difficult to see them directly on Bε
easy to see them before integrating out fermions
The SUSY model
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3 bosons + 2 fermions
v = (z, x, y, χ̄, χ) z, x, y ∈ R, χ̄, χ Grassman
inner product: (v1 , v2 ) = −z1 z2 + x1 x2 + y1 y2 + χ̄1 χ2 − χ1 χ̄2
metric: (v, v) = −z 2 + x2 + y 2 + 2χχ
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sigma model constraint:
(v, v) = −1 ⇒ z 2 = 1 + x2 + y 2 + 2χχ
the submanifold is parametrized by 2 bosons + 2 fermions
horosperical coordinates: t, s, ψ̄, ψ
action is
Aε = β
X
<j,j 0 >
(vj − vj 0 , vj − vj 0 ) + ε
X
(zj − 1)
j
(vj − vj 0 , vj − vj 0 ) =
1
(cosh(tj − tj 0 ) − 1) + (sj − sj 0 )2 etj +tj0 + (ψ̄j − ψ̄j 0 )(ψj − ψj 0 )etj +tj0
2
1
(zj − 1) = (cosh tj − 1) + sj 2 etj + ψ̄j ψj etj
2
Aε invariant under symmetries that leave (vj , vj 0 ) and zj
invariant.
some SUSY identities
Z
(dte−t dsdψ̄ψ) e−Aε = 1
for all values of β, ε as long as the integral exists. β and ε may
be j dependent.
Z
m
m
=1
hSxy
i = (dte−t dsdψ̄ψ) e−Aε Sxy
where Sxy = (vx − vy , vx − vy ). In particular
1
h(cosh(tx − ty ) − 1)i = hGxy i
2
Key identity: by SUSY
m
1 = hcosh (tx − ty ) 1 − Gxy i
β
m
0 < Gxy = etx +ty
(δx − δy )M −1 (δx − δy )
Now if Gxy ≤ C uniformly in t then
hcoshm (tx − ty )i ≤ (1 − mC/β)−1
then tx − ty is small with hight probability.
Bound on Gxy : depends on the distance |x − y|
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|x − y| = 1 (n.n. case): then Gxy ≤ 1 always
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|x − y| = l > 1: false in general! True if we introduce
constraints on lower scale fluctuations (must be done
respecting SUSY)
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remove the constraints: inductive argument on scales
Bounds on Gxy
Gxy ≤ (δx − δy )
(f, D̃(t)f ) =
X
1
(δx − δy )
β D̃(t)
Axy (jj 0 )(fj −fj 0 )2
Axy (jj 0 ) = etj +tj 0 −tx −ty
<jj 0 >∈Rxy
Rxy a box containing x, y.
We need a lower bound on A inside Rxy .
in 3d it is enough to have:
Axy (jj 0 ) ≥ 1/(|j − x||j 0 − x|)α + 1/(|j − y||j 0 − y|)α
Open problems
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homogeneization in d = 3: Green function decay
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if β is small, show that in 3d, Zirnbauer model exhibits
localization
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generalize this technique to the band matrix model (the
fermionic term is more complicated, there is a another
sector)