Spectral statistics of random band matrices: old
and new results
László Erdős
Institute for Science and Technology, Austria
Stochastic and Interactions
(Bálint is 60 )
Rényi Institute, Budapest
July 24, 2015
1
INTRODUCTION
Universality conjecture for disordered quantum systems:
A disordered quantum systems with sufficient complexity exhibits
one of the following two behaviors:
A) Localized evectors, lack of transport, and Poisson local spectral
statistics (strong disorder)
B) Delocalization, quantum diffusion and random matrix (RMT)
local statistics (weak disorder).
At first sight, localization is surprising (Anderson). Still, mathematically it is much more accessible (Fröhlich-Spencer, AizenmanMolchanov, Minami, ...).
2
Two popular models to study the dichotomy
(1) Random Schrödinger operators: in lattice box Λ := [1, L]d ∩ Zd
In d = 1 it corresponds to a narrow band matrix with i.i.d. diagonal:

v
1
 1
 1 v2
X

H = −∆ +
vx = 
1


x


1
...




1

1 vL−1 1 

1
vL
Follows behavior (A) [Localization, Poisson]
3
(2) Wigner random matrices:
H = (Hxy ),
H = H∗
EHxy = 0.
entries are identically distributed and independent up to symmetry.
H models a mean-field hopping mechanism with random quantum
transition rates. No spatial structure (dim is irrelevant or d = 0).
Follows behavior (B) [Delocalization, RMT]
Random band matrices: intermediate model that interpolates between (1) and (2).
They can be used to probe the transition between (A) and (B).
They also model quantum diffusion
4
Random band matrices (RBM)
Λ := [1, L]d ∩ Zd lattice box represents the configuration space.
H = (Hxy )x,y∈Λ,
H = H∗
EHxy = 0.
Entries are independent but no longer identically distributed. VariR
ance is given by a band profile f (even function, Rd f = 1)
|x − y|
1
f
sxy := E|Hxy |2 =
Wd
W
Key parameter: Band width W ∈ [1, L] (range of interaction).
Nontrivial spatial structure like RS, but technically more accessible
[Disertori, Pinson, Spencer, Zirnbauer, Shcherbina, Schenker...]
Normalization: Level spacing around energy E is
1
∆= d ,
L %
1
%=
4 − E2
2π
q
5
ANDERSON TRANSITION FOR BAND MATRICES
W = O(1) [∼ Random Schrödinger] ←→ W = L [Wigner ensemble]
Varying W can probe the transition between (A) and (B).
Facts: Transition occurs at
W ∼ L1/2
(d = 1)
SUSY [Fyodorov-Mirlin, 1991]
p
log L
(d = 2)
RG scaling [Abrahams, 1979]
W ∼ W0(d)
(d > 3)
extended states conjecture
W ∼
In terms of the localization length of (typical) eigenvectors, `:
` ∼ W2
(d = 1)
` ∼ exp(W 2)
(d = 2)
`=L
(d > 3)
6
Rigorous mathematical results on ` ∼ W 2 (d = 1)
Upper bound: ` 6 W 8+ε [Schenker, 2008] — localization methods.
Lower bound: ` > W [E-Yau-Yin, 2010] — RM methods.
Improving this to ` W is hard: it requires to prove some version
of quantum diffusion. We have two different methods:
• Time evolution with rescaling converges to the heat equation
Result: ` > W 1+1/6 [E-Knowles]
Method: Chebyshev expansion, Feynman graph expansion
• Resolvent profile is given by that of the diffusion.
Result: ` > W 1+1/4 (roughly) [E-Knowles-Yau-Yin]
Method: Matrix self-consistent eqn. Fluctuation averaging thm.
7
Quantum diffusion for band matrices I: Time evolution
Expected quantum transition probability from 0 to x in time t:
2
−itH/2
%(t, x) = Ehx|e
|0i
Diffusive rescaling: t = ξT , x =
√
ξW X, ξ 1.
Expect: diffusion holds up to time scale t ∼ W 2, then it stops.
(eventually it is a 1d disordered system)
Choose ξ = W κ, ideally one should take κ = 2; we can do κ = 1/3.
This scale is enough to detect diffusion but not enough to see how
it stops.
8
Theorem [E-Knowles, 2011]. Fix 0 < κ < 1/3. Then for all T > 0,
% W κT, W 1+κ/2X
λ2
4
G(λT, X)
→
dλ q
π 1 − λ2
0
Z 1
weakly in X (tested against a smooth function), where
1 X2
1
1
−
e 2T D
,
D :=
X 2f (X)dX.
G(T, X) := √
2
2πT D
I.e. the limit behavior is given by a superposition of heat kernels.
Similar result holds in higher dimensions (κ = d/3).
Z
λ ∈ [0, 1] represents the fraction of macroscopic time T that the
particle spends moving effectively; the remaining fraction 1 − λ of
T represents the time the particle “wastes” in backtracking. Backtracking is due to a self-energy renormalization.
Corollary: eigenvector localization length ` > W 1+1/6.
(Not optimal power)
9
Quantum diffusion for band matrices II: Mesoscopic statistics
X
η
φη (λj − E),
Yφ (E) :=
j
with
φη (e) := η −1φ(e/η),
Eigenvalue density at energy E on scale η (smoothed by testfn φ)
η
η
Question: Joint statistics of Yφ (E1), Yφ (E2), . . .
Microscopic scale: η ∼ ∆: Poisson vs. RMT (GUE, GOE)
Macroscopic scale: η ∼ 1: No universality, model dependent (DOS)
Mesoscopic scale: ∆ η 1: Universalities with a phase transition.
Special physical motivation: fluctuation of conductance comes (partly)
from the fluctuation of the number of states in a mesoscopic window
around the Fermi level E [Thouless]
11
η
η
η
The correlations of Yφ (E1), Yφ (E2), . . . , Yφ (Ek ) are equivalent to the
truncated correlation functions smoothed on scale η, e.g.
ω
ω
η
η
Yφ (E− ) ; Yφ (E+ ) =
2
2
ZZ
ω η
ω (2)
η
φ (x−E+ )φ (y−E− )p (x, y)dxdy
2
2
E.g. for GUE, if the sine-kernel held on any scale, we had
1
sin e/∆ 2
de ∼ 2
∼
e/∆
ω
|e−ω|≤η
Z
if
∆ηω1
(∗)
Looks easy, by extrapolation: For GUE, (*) was indeed proved by
Boutet de Monvel and Khorunzhy [1999]. However:
• Asymptotics (∗) does not hold for general delocalized systems: the
sine-kernel fails on mesoscopic scale. Instead: Altshuler-Shklovskii
formulas (1986 in physics, now we proved it at least for RBM).
• Sine kernel in (∗) may fail to predict the subleading term – New
observation, contradicting to several physics predictions
12
Mesoscopic phase transition occurs at the Thouless energy
−1
η0 = (time for diffusion to reach the boundary)
diff coeff
=
L2
For RBM: η0 ∼ W 2/L2 [E-Knowles, 2011]
Altshuler-Shklovskii (AS) formulas
(1) In the diffusion regime, η η0
VarY η ∼ (η/η0)d/2η −2
ω
ω E
η
η
Yφ (E − ) ; Yφ (E + ) ∼ ω d/2−2
2
2
D
(d = 1, 2, 3)
(d = 1, 3)
(2) In the mean field regime, η η0 , the same holds with d = 0.
• Compare with Poisson: Var [ ηY η ] ∼ [ηY η ]. Predicts the Anderson transition in d = 1 at ` ∼ W 2 via the crossover at η ∼ W −2.
• d = 2: critical case, leading term vanishes. Subleading terms?
13
Theorem (Mesoscopic Universality for RBM) [E-Knowles, 2013].
W 2 , and assume η W −d/3 .
L
Suppose diffusive regime η η0 =
Away from the spectral edges, we have for the density correlator
E
η
η
Yφ (E1) ; Yφ (E2)
1
η
−ε ) ,
D 1
ED 2
E =
Θ
(E
,
E
)
1
+
O(W
1
2
η
η
(LW )d φ1,φ2
Yφ (E1) Yφ (E2)
1
2
D
where Θ(E1, E2) is an explicit formula, depending only on the band
profile f but independent of the distribution of the matrix elements.
• The technical bound η W −d/3 is needed for the box band
profile. For general f we need η > W −%d with some % > 0.
• Θ = “one-loop diagrams after self-energy renormalization”.
• Leading term is a higher order effect (cancellation in correlation)
14
Related result [Shcherbina, 2013] with SUSY
The 2-point correlation function of the determinant of 1d RBM is
y sin(x − y)
x
; Det H − E −
∼
Det H − E −
L%
L%
x−y
if W 2 L (optimal power) + Technical condition: Gaussian with a
very special covariance.
The only rigorous result where the threshold W 2 L shows up.
Det is easier via SUSY because only the fermionic sector is present.
Challenge No. 1:
Demonstrate the transition at W 2 ∼ L for other observables (localization length, local eigenvalue statistics etc.) and for general RBM.
Wide open
15
LOCAL SEMICIRCLE LAW
ρ (x) =
−2
1
2π
4 − x2
2
1
Limiting density of the eigenvalues is %sc(x) = 2π
1
1
1
m(z) = d Tr
= d Tr G(z),
L
H −z
L
q
(4 − x2)+
msc(z) =
Z
%sc(x)
dx
x−z
FACT: Suppose for some fixed η > 0 and any E we have
|m(z) − msc(z)| 6 ε,
z = E + iη
then the local density in spectral windows of size η about E is given
by %sc(E) up to a precision ε.
Similar control on Gxx(z) implies efn localization length ` η −1.
16
Theorem [E-Yau-Yin, 2011]. Local semicircle law holds if η W −1:
1
,
|m(z) − msc(z)| .
Wη
|Gxy (z) − δxy msc(z)| .
1
(W η)1/2
(in d = 1, with very high probability and modulo log corrections).
Consequently, eigenfunction localization length is ` ≥ W .
Related results
• Local semicircle law for the expectation Em, uniform in η,
error W −2, (d = 3, Gaussian, with a special covariance).
[Disertori-Pinsker-Spencer, 2002] SUSY
• Local semicircle law for the expectation Em at η = W −0.99
with error W −1 (in d = 1, Bernoulli distr) [Sodin, 2011]
For Em one needs to compute ETrG and not ETrGTrG∗ or EGxx
17
Challenge No. 2: Control the regime L−1 η W −1
We expect for any η L−1 that
|m(z) − msc(z)| .
1
,
Lη
|Gxy (z) − δxy msc(z)| .
1
(Lη)1/2
(i.e. L instead of W ) in the (presumed) delocalized phase, i.e. for
W 2 L. This would prove complete eigenfunction delocalization:
1
√
kuk2
kuk∞ .
L
i.e. ` ∼ L; localization length is comparable with the sample size.
There is one special RBM where we could prove this: Block band
matrices.
18
BLOCK BAND MATRICES
Let H be N × N hermitian matrix, consisting of W × W blocks. The
variance of all entries in the (j, k) block is identical:
1
δjk + sjk δjk0 δj 0k δα,β 0 δα0,β
Ehjk,αβ hj 0k0,α0β 0 =
W
(here j, k label the block, α, β label the entries within a box).

·

·

·
H=
·


·
·
·
·
·
·
∗
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·

·
·


·

·


·
·
∗ = h31,12
Thus the bandwidth is at least W .
N)
The block-variances are given by the matrix I + S (of size W
19
Theorem [E-Bao, 2015] Suppose that W N 6/7 and I +S is strictly
diagonally dominant and stochastic matrix (e.g. S = a∆ for some
small a < 1/4d). Moreover, the third and fourth moments of h’s
√
match with those of a Gaussian. Let |E| < 2. Then
|Gxy (z) − δxy msc(z)| .
1
(N η)1/2
,
z = E + iη
and we have complete eigenfunction delocalization:
1
kuαk∞ . √ kuαk2
N
for any eigenfunction, Huα = λαuα, with energy |λα| <
√
2.
Main Lemma: For the Gaussian case we have
1
E|Gab(z)|2n ≤ N C0 δab +
(N η)n
Lemma ⇒ Theorem Perturbation argument (Green function comparison theorem)
20
Proof of the main lemma uses SUSY representation (superbosonization formula) partially motivated by [Shcherbina 2014].
Our observables are different and we allow narrow band (W N ),
while Shcherbina has W ∼ N , so factors like eN/W are not affordable
for us.
Supersymmetric formalism is a major challenge for mathematicians.
It is widely and successively used by physicists, but most of their
arguments do not make sense mathematically. Still, clearly correct.
Part of the formalism is rigorous and we can use it. But the really
elegant part (saddle point approximation in the superspace) is nonrigorous.
21
SUPERSYMMETRIC FORMALISM
Complex (”bosonic”) Gaussian integrals
Let A be a k × k matrix with ReA > 0, then
Z
Ck
dφdφ̄ e
−φ∗ Aφ
1
=
detA
Z
C
∗
dφdφ̄ e−φ Aφ φiφj =
k
1
(A−1)ij
detA
Grassmann (”fermionic”) Gaussian integrals
Let B be a k × k matrix, then
Z
dψdψ̄ e
ψ ∗ Bψ
= detB
Z
∗ Bψ
ψ
dψdψ̄ e
ψ̄iψj = (−1)i+j detB (i|j)
where ψ = (ψ1, ψ2, . . . , ψk )0 and ψ ∗ = (ψ̄1, ψ̄2, . . . , ψ̄k ) are 2k independent anticommuting variables.
B (i|j) is the minor with the i-th row and j-th column removed.
22
Functions of ψ are defined by (finite) Taylor expansions, e.g.
1
ψ̄ψ
e
= 1 + ψ̄ψ + ψ̄ψ ψ̄ψ + . . . = 1 + ψ̄ψ
2
since ψ 2 = 0, so everything is a polynomial (of degree at most 2k).
Integrals are defined by the iteration of the simple rules and linearity
Z
dψa = 0,
Z
ψbdψa = δab.
For example the k = 2 Gaussian integral above
Z
=
ψ ∗ Bψ
dψdψ̄ e
Z
=
Z
dψ1dψ2dψ̄1dψ̄2 eψ̄1b11ψ1+ψ̄1b12ψ2+ψ̄2b21ψ1+ψ̄2b22ψ2
dψ1dψ2dψ̄1dψ̄2
1 + ψ̄1b11ψ1 + . . . + ψ̄1b11ψ1
ψ̄2b22ψ2 + . . .
= b11b22 − b12b21
23
Why is it useful? Compensates for the bosonic determinant!
E

1
= (−i)N Edet(H − z)
H − z aa

Z
dφdφ̄ e−iφ̄(H−z)φφ̄aφa
for Imz > 0. Determinant makes taking the expectation hard.
Use fermions to express the determinant:
1
E
= E
H − z aa
=
Z
dφdφ̄
Z


−
dψdψ̄ e
Z
dφdφ̄
P
ij
Z
dψdψ̄ e−iφ̄(H−z)φ−iψ̄(H−z)ψ φ̄aφa
E|hij |2 (φ̄i φj +ψ̄i ψj )(φ̄j φi +ψ̄j ψi )+iz(φ̄·φ+ψ̄·ψ)
φ̄aφa
1,
For mean field model (Wigner matrices), E|hij |2 = N
=
Z
dφdφ̄
Z
dψdψ̄
1
−N
e
h
(φ̄·φ)2 +2(φ̄·ψ)(ψ̄·φ)+(ψ̄·ψ)2
i
+iz(φ̄·φ+ψ̄·ψ)
φ̄aφa
Use Hubbard-Stratonovich decoupling:
−a2
e
=
Z
2
eiaxe−x dx
to reduce quartic to quadratic. We need a 2 × 2 matrix version:
24
1
−N
e
h
(φ̄·φ)2 +2(φ̄·ψ)(ψ̄·φ)+(ψ̄·ψ)2
!
i
=
Z
dR e
2
−N 1
2 StrR −i
P
j
Φ∗j RΦj
!
φj
x ξ
, R := ∗
is a 2 × 2 supermatrix with x, y ∈ R
ψj
ξ iy
and ξ, ξ ∗ (new) Grassmann variables and Str is the supertrace
where Φj =
Str
x ξ
ξ ∗ iy
!
:= x − iy,
i.e.
StrR2 = x2 + y 2
(convergence!)
The result is quadratic in Φ, so it can be integrated out:
1
E Tr
=N
H −z
Z
1
dR e−N S[R]
R − z 11
1
StrR2 + Str log(R − z)
2
saddle point anal. in 2+2 dim superspace (dimension reduction!)
with the action
S[R] :=
Rigorously: first integrate out ξ, ξ ∗, then saddle point in x, y.
Key technical issue: appropriate contour deformation.
25
The above presentation is the simplest version of the technique: for
the trace and for mean field models.
Instead of H-S, we use (a version of the) superbosonisation formula
Z
F
ψ ∗ψ
φ∗ψ
ψ ∗φ
φ∗ φ
!
dφdψ
=
Z
dµ(x)dydω dξ F
x ω
ξ y
detW y
!
detW
x − ωy−1ξ
,
φ, ψ are W × r complex and Grassmann matrices, resp. (W r)
x is a r × r unitary matrix; y ≥ 0 is r × r hermitian matrix, ω and ξ
are two r × r Grassmann matrices, µ is Haar measure.
For block matrices, each block is replaced by a 2 × 2 (actually 4 × 4)
supermatrix (use the formula for each block, i.e. k = N/W times).
Dimension reduction: from N supervectors we go down to k = N/W
finite supermatrices (effective sigma-model)
26
Roughly speaking, we have
E|Gpq,11|2n ∼
Z
dXdBdΩdΞ e
h
i
−W K(X)+L(B)
P (Ω, Ξ, X, B)Qpq,n(...)
where X = (X1, . . . Xk ), etc. with Xj ∈ U (2), Bj ≥ 0 positive definite,
Ωj , Ξj are 2 × 2 Grassmann matrices.
K(X) = −
X
s̃ij TrXiXj + iE
ij
X
j
TrXj +
X
log detXj ,
j
L(B) is similar,
−W
1
− ij s̃ij TrΩi Ξj
Xj−1Ωj Bj−1Ξj )
P (Ω, Ξ, X, B) = e
det(1 +
W
j
and Qpq,n contains the observable, independent of W .
P
Y
The main contribution is determined by the saddle point of the
sigma-model part K(X) + L(B) which is a point where all Xj ’s and
all Bj ’s are aligned.
Analysis is more complicated if k = N/W is large, this is where the
technical limitation W N 6/7 comes from.
27
SUMMARY
• Proof of quantum diffusion for RBM to time scale T W 1/3
(dream: T ∼ W 2)
• Localization length ` ≥ W 1+1/4 (dream: ` ∼ W 2).
• Proof of the AS formulas for RBM: mesoscopic universality
• Local semicircle law down to optimal scale η ∼ N −1 for special
block band matrices via rigorous SUSY.
• All results are beyond the Gaussian case.
• Open problem: extensions towards random Schrödinger?
28
HAPPY BIRTHDAY, BÁLINT!
29