Localization with Less Larmes: Simply MSA

0.22 MB·English

by  Victor Chulaevsky

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Localization with Less Larmes: Simply MSA
2

c
Victor Chulaevsky1
e
D
D´epartementdeMath´ematiques etd’Informatique,
Universit´edeReims,MoulindelaHousse,B.P.1039,
1
51687ReimsCedex2,France
3 E-mail:[email protected]

]
h December2008
p
-
h Abstract: We give a short summary of the fixed-energy Multi-Scale Analysis
t (MSA) of the Anderson tight binding model in dimension d≥1 and show that
a
this technique admits a straightforwardextension to multi-particle systems. We
m
hope that this shortnote may serve as an elementary introduction to the MSA.
[

2
1. Introduction
v
4
In this paper, we study spectral properties of random lattice Schr¨odinger oper-
3
ators at a fixed, but arbitrary, energy E ∈ R, in the framework of the MSA.
6
The idea of the fixed-energy scale induction goes back to [FS83], [Dr87] 1 and
2
. [S88]. While the fixed-energy analysis alone does not allow to prove spectral lo-
2
calization, it provides a valuable information. Besides, from the physical point
1
ofview,asufficiently rapiddecay,withprobabilityone,ofGreenfunctions infi-
8
nitevolumes,combinedwiththecelebratedKuboformulaforthezero-frequency
0
: conductivity σ(E), shows that σ(E)=0 for the disorderedsystems in question.
v
The main motivation for studying in this paper only the fixed-energy prop-
i
X erties of randomHamiltonians came froman observationthat such analysiscan
be made very elementary, even for multi-particle systems considered as difficult
r
a since quite a long time (cf. recent works [CS08], [CS09A], [AW08], [CS09B],
[BCSS08], [BCS08]).
An earlier version of this manuscript represented an extended variant of the
talk given by the author of these lines in the framework of the program MPA
run by the Isaac Newton Institute in 2008. Questions and remarks made by
participantstotheprogrammadeitclearthatthebrevityofthefirstversionwas
sometimesexcessive,andthatadditionalillustrationsandexplanationswouldbe
useful. Suchmodifications arepartially implementedin the presentversion,and
1 IthankA.KleinandT.SpencerforpointingoutthePhDthesisbyH.vonDreifus[Dr87]
(scientificadvisorT.Spencer)wherethefixed-energyapproachhadbeenused.
2 VictorChulaevsky
it will probably evolve further in order to make this (very) short introduction
to the MSA fairly clear not only for mathematicians, but also for researchers
with different backgrounds, interested more by results and basic ideas than by
formal constructions. Producing clear illustrations is a time-consuming process,
but live discussions evidence that they can be more useful than equations.
2. The models and some geometric definitions
Apopularformofasingle-particleHamiltonianinpresenceofarandomexternal
potential gV(x;ω) is as follows:
H(ω)=−∆+gV(x;ω), (2.1)
where the parameter g ∈ R is often called the coupling constant and ∆ is the
nearest-neighbor lattice Laplacian:
(∆f)(x)= f(y), x,y ∈Zd,
X
y:ky−xk=1
and V(x;ω) acts as a multiplication operator on Zd. For the sake of simplicity,
the random field {V(x;ω),x ∈ Zd} will be assumed IID, although a large class
of correlated random fields can also be considered. In this paper, we consider
only the case of ”large disorder”, i.e., we assume that |g| is sufficiently large,
althoughthe caseof”lowenergies”canalsobe considered.An IID randomfield
on Zd is completely determined by its marginal probability distribution at any
sitex∈Zd,e.g.,forx=0.Weassumethatthemarginaldistributionfunctionof
potential V (defined by F (t)=P{V(0;ω)≤t}, t∈R) is Ho¨lder-continuous:
V
∀t∈R, ∀ǫ∈(0,1]F (t+ǫ)−F (t)≤Constǫb, (2.2)
V V
for some b>0.
For N >1 particles, positions of which will be denoted by x ,...,x , or, in
1 N
vector notations, x = (x ,...,x ) ∈ ZNd, we introduce an interaction energy
1 N
U(x). Again, for the sake of simplicity of presentation, we assume that
U(x)=U(x ,...,x )= U (kx −x k), (2.3)
1 N X 2 j k
1≤j<k≤N
whereU (r),r ≥0,isaboundedtwo-bodyinteractionpotential.TheN-particle
2
Hamiltonian considered below will have the following form:
N
H(N)(ω)= −∆(j)+gV(x ;ω) +U(x). (2.4)
X(cid:16) j (cid:17)
j=1
GivenalatticesubsetΛ⊂Zd,wewillworkwithsubsetsthereofcalledboxes.
It is convenient to allow boxes Λ (u)⊂Λ of the following form:
ℓ
Λ (u)={x: kx−uk≤ℓ−1},
ℓ
LocalizationwithLessLarmes:SimplyMSA 3
wherek·kisthesup-norm:kxk=max |x |.Further,weintroduceanotion
1≤j≤d j
of internal and external ”boundaries” relative to Λ:
∂−Λ (u)={x∈Λ: kx−uk≤ℓ−1}
ℓ
∂+Λ (u)={x∈Λ: dist(x,Λ (u))=1}.
ℓ ℓ
We also define the boundary ∂Λ (u)∈Λ by
ℓ
∂Λ (u)= (x,x′): x∈∂−Λ (u),x′ ∈∂+Λ (u), kx−x′k=1 .
ℓ ℓ ℓ
(cid:8) (cid:9)
By resolvent identity, if operators A and A+B are invertible, then
(A+B)−1 =A−1−A−1B(A+B)−1.
Observe now that the the second-order lattice Laplacian has the form
∆= X Γx,x′, Γx,x′ =|δxihδx′|,
(x,x′):kx−x′k=1
so that (Γx,x′f)(y)=δx,yf(x′). Given a box Λℓ(u)⊂Λ, the Laplacian ∆Λ in Λ
with Dirichlet boundary conditions on ∂+Λ reads as follows:
∆Λ =(cid:16)∆Λℓ(u)⊕∆Λcℓ(u)(cid:17)+ X (Γx,x′ +Γx′,x).
(x,x′)∈∂Λℓ(u)
Similarly, for the Hamiltonian H (with Dirichlet boundary conditions) we can
Λ
write:
HΛ =(cid:2)(cid:0)∆Λℓ(u)+VΛℓ(u)(cid:1)⊕(cid:16)∆Λcℓ(u)+VΛcℓ(u)(cid:17)(cid:3)+ X (Γx,x′ +Γx′,x),
(x,x′)∈∂Λℓ(u)
with Λc(u)=Λ (u)\Λ (u). Therefore,by the resolventidentity combinedwith
ℓ L ℓ
the above decomposition, we have:
G(u,y;E)= X G(u,x;E)Γx,x′G(x′,y;E),
(x,x′)∈∂Λℓ(u)
usually called the Geometric Resolvent Identity, yielding immediately the Geo-
metric Resolvent Inequality (referred to as GRI in what follows):
|G(u,y;E)|≤ max |G(u,x;E)|·|∂+Λ (u)| · max |G(x′,y;E)|.
(cid:16) ℓ (cid:17)
x∈∂−Λℓ(u) x′∈∂+Λℓ(u)
(2.5)
Throughout this paper, we use a standard notation [[a,b]]:=[a,b]∩Z.
4 VictorChulaevsky
3. Green functions in a finite volume
Definition 3.1.A box Λ (u) is called (E,m)-non-singular ((E,m)-NS) if
L
max |G(u,y;E)|≤e−γ(m,L),
y∈∂ΛL(u)
where
γ(m,L)=m(L+L3/4)=m 1+L−1/4 L. (3.1)
(cid:16) (cid:17)
Otherwise, it is called (E,m)-singular ((E,m)-S).
Remark. The function γ(m,L) defined in Eqn (3.1) will be often used below.
It allows us to avoida ”massive rescaling of the mass”,which would, otherwise,
inevitablymakenotationsandassertionsmorecumbersome.Obviously,forlarge
values of L, γ(m,L)/(mL)≈1.
Itisconvenienttointroducethefollowingproperty(orassertion),thevalidity
ofwhichdependsuponparametersu∈Zd,L∈N∗,m>0,p>0,aswellasupon
the probability distribution of the random potential {V(x;ω),x∈Zd}:
(S.L,m) P{Λ (u) is (E,m)-S}≤L−p.
L
In order to distinguish between single- and multi-particle models, we will often
write (S.L,k,N) where N ≥1 is the number of particles.
Definition 3.2.A lattice subset Λ⊂Zd of diameter L is called E-non-resonant
(E-NR) if kG (E)k≤e−Lβ, β ∈(0,1), and E-resonant ( E-R), otherwise. It is
Λ
called (E,ℓ)-completely non-resonant ((E,ℓ)-CNR) if it is E-NR and does not
contain any E-R cube Λ′ of diameter ≥3ℓ.
Lemma 3.1.If the marginal CDF F(s)=F (s) is Ho¨lder-continuous, then
V
′
∃L∗ >0,β′ ∈(0,1): ∀L≥L∗ P Λ (u) is not (E,L2/3)-CNR ≤e−Lβ .
n L o
Consider a pair of boxes Λ (u) ⊂ Λ (x). If Λ (u) is (E,m)-NS, then GRI
ℓ L ℓ
implies that function f(x)=f (x):=G(x,y;E) satisfies
y
|f(u)|≤q· max |f(v)|, (3.2)
v:kv−uk=ℓ
e
with
q =q(d,ℓ;E)=2dℓd−1e−γ(m,ℓ). (3.3)
e e
Now suppose that Λ (u) is (E,m)-S, but Λ (x) is E-CNR and, in addition, for
ℓ L
someA>0andforanywwith dist(w,Λ (u))=ℓtheboxΛ (w)is(E,m)-NS.
Aℓ ℓ
Then, by GRI applied twice,
|G(u,y)|≤2d(6ℓ)d−1eLβ max |G(w,y;E)|
w:kw−uk=(A+1)ℓ
≤2d(6ℓ)d−1eLβ 2d(2ℓ)d−1e−γ(m,ℓ) max |G(v,y;E)|.
v:kv−uk∈[Aℓ,(A+2)ℓ]
Therefore, with
q(d,ℓ,E):=4d2(12ℓ2)d−1eLβe−γ(m,ℓ) >q(d,ℓ,E), (3.4)
e
LocalizationwithLessLarmes:SimplyMSA 5
we obtain
|G(u,y)|≤q(d,ℓ,E) max |G(v,y;E)|. (3.5)
v:kv−uk∈[Aℓ,(A+2)ℓ]
e
With these observations in mind, we study in the next section decay properties
of functions f : Λ (x) → C obeying, for any point u ∈ Λ (x), one of the Eqns
L L
(3.2),(3.5).Observethat we do notrequire thatq andq be smaller than1,but,
ofcourse,theabovebounds(andthoseobtainedinSection4)areusefulonlyfor
e
q,q <1.Finally, note that,sinceq <q,it willbe convenientto replaceq byq in
thebound(3.2).Withthismodification,weseethattheonlydifferencebetween
e e e
bounds (3.2) and (3.5) is in the distance kv−uk figuring in these inequalities.
4. Radial descent: A few simple lemmas
Definition 4.1.Consider a set Λ⊂Zd (not necessarily finite), a subset S ⊂Λ
and a bounded function f : Λ→C. Let L≥0 be an integer and q >0. Function
f will be called (ℓ,q)-subharmonic in Λ and regular on Λ\S (or, equivalently,
(ℓ,q,S)-subharmonic), if for any u6∈S with dist(u,∂Λ)≥ℓ, we have
|f(u)|≤q max |f(y)| (4.1)
y:ky−uk=ℓ
and for any u∈S
|f(u)|≤q max |f(y)|. (4.2)
y:dist(y,S)∈[1,2ℓ−1]
On Fig.1, the pink square (labeled by the letter S) is a singular set, and the
greenbelt-shaped areais formed by non-singular(NS) neighboring squares(the
four white squares are examples of neighboring NS-squares). A thin white layer
between the pink singular square and its NS-neighborhood has the width = 1
(the minimal distance on the lattice between two disjoint sets).
For our purposes,it suffices to consider a particular case where Λ=Λ (x) is
L
a cube of side L with center x and S =Λe(v)∩Λ, ℓ=Aℓ−1. In this particular
ℓ
case Eqn (4.2) becomes e
∀u∈S |f(u)|≤q max |f(y)|. (4.2A)
y:ℓ≤ky−uk≤Aℓ−1
We will use the notation M(f,Λ) := max |f(x)|. Our goal is to obtain
x∈Λ
an upper bound on the value f(x) exponential in L/ℓ. To this end, we study
separately several cases.
Lemma 4.1.Let Λ=Λ (x) and consider an (ℓ,q,∅)-subharmonic function f :
L
Λ→C. Then
|f(x)|≤q[L/ℓ]−1M(f,Λ). (4.3)
Proof. Let n ≥ 2 and consider a point u ∈ Λ (x). Since Λ (u) ⊂
L−nℓ ℓ
Λ (x), the subharmonicity condition implies that
L−(n−1)ℓ
|f(u)|≤q max ≤qM(f,Λ (x)). (4.4)
L−(n−1)ℓ
y∈ΛL(x):ky−uk=ℓ
In other words, we have
M(f,Λ (x))≤qM(f,Λ (x)). (4.5)
L−nℓ L−(n−1)ℓ
6 VictorChulaevsky
Fig. 1. An example of a singular subset (pink). All squares inside the green area are non-
singular.Fourwhitesquaresareexamplesofnon-singularsquares
Eqn(4.5)canbeiterated,aslongasthesetΛ (x)isnon-empty,soweobtain
L−nℓ
by induction
M(f,Λ (x))≤qn−1M(f,Λ (x))≤qn−1M(f,Λ (x)). (4.6)
L−nℓ L−ℓ L
Now the assertionof the lemma follows fromthe inclusion x∈Λ (x). ⊓⊔
L−[L/ℓ]ℓ
Lemma 4.2.Let Λ = Λ (x) and consider an (ℓ,q,S)-subharmonic function
L
f :Λ→C. Suppose that diam(S)≤Aℓ−1 and dist(S,∂Λ (x))<ℓ. Then
L
|f(x)|≤q[(L−A)/ℓ]−2M(f,Λ). (4.7)
Proof. ObservethatS ⊂(Λ (x)\Λ (x).Therefore,functionf is(L,q,∅)-
L L−(A+1)ℓ
subharmonic in Λ (x), and it suffices to apply Lemma 4.1. ⊓⊔
L−(A+1)ℓ
Lemma 4.3.Let f bean (ℓ,q,S)-subharmonic function on Λ=Λ (x). Suppose
L
that diam(S)=Aℓ−1 and
(r+2)ℓ≤ dist(S,∂Λ (x))<(r+3)ℓ,
L
so that
S ⊂Λ (x)\Λ .
L−(r+2)ℓ L−(r+A+3)ℓ
Then
|f(x)|≤q[(L−A)/ℓ]−3M(f,Λ). (4.8)
LocalizationwithLessLarmes:SimplyMSA 7
Fig. 2. ”Radial” induction in Lemma 4.2. Here, a singular subset (pink) is at distance < ℓ
fromtheboundary, anditsufficestostartthe inductionfromtheinnersquare(its boundary
isindicatedbythedoubleline).Thesingularsetcanbehereevenan”incomplete” square.
Proof. We start as in Lemma 4.1. For points u∈Λ (x)\S we use Eqn (4.1):
L−ℓ
|f(u)|≤q max |f(y)|≤q M(f,Λ (x)),
L
y:|y−u|=L
and for points u∈S ⊂Λ (x) Eqn (4.2) leads to a similar upper bound:
L−ℓ
|f(u)|≤q max |f(y)|≤qM(f,Λ (x)).
L
y:|y−u|∈[Aℓ,(A+2)ℓ−1]
Therefore,
M(f,Λ (x))≤q M(f,Λ (x)). (4.9)
L−ℓ L
We can iterate this argument and obtain, for n=1,...,r
M(f,Λ (x))≤q M(f,Λ (x)). (4.10)
L−nℓ L−(n−1)ℓ
Indeed, if 1≤n≤r, then
{u: dist(S,u)∈[1,2ℓ−1]}⊂Λ (x)⊆Λ (x).
L−nℓ L−rℓ
This leads to the following upper bound:
M(f,Λ (x))≤qr M(f,Λ (x)). (4.11)
L−rℓ L
8 VictorChulaevsky
Fig.3. Two-fold”radial”inductioninLemma4.3.Here,asingularsubset(pink)isatdistance
r′∈[(r+2)ℓ,(r+3)ℓ)fromtheboundary.
If L−(A+r+3)ℓ < 2ℓ, we stop the induction and obtain for x ∈ Λ(x) the
requited upper bound.
Otherwise, we make at least one inductive step (or more) inside the box
Λ (x). Since Λ (x)⊂(Λ (x)\S), Eqn (4.11) implies
L−(r+A+3)ℓ L−(r+A+3)ℓ L−rℓ
M(f,Λ (x))≤qr M(f,Λ (x)). (4.12)
L−(r+A+3)ℓ L
Observe that function f is (ℓ,q,∅)-subharmonic on Λ (x). Applying
L−(r+A+3)ℓ
Lemma 4.1 to this cube, we conclude that
|f(x)|≤q[(L−r−A)/ℓ]−3qrM(f,Λ (x))≤q[(L−A)/ℓ]−3M(f,Λ (x)). ⊓⊔
L L
5. Inductive bounds of Green functions
Taking into account observations made at the end of Section 3, namely, Eqns
(3.2) and (3.5), we come immediately to the following
Lemma 5.1.Suppose that a box Λ (u) is E-NR and does not contain any
L
(E,m)-S boxe of size ℓ. Then, with q =q(d,ℓ,E) defined in Eqn (3.12),
max |G (u,y;E)|≤q[L/ℓ])kG (E)k. (5.2)
Λ Λ
y∈∂−ΛL(u)
LocalizationwithLessLarmes:SimplyMSA 9
In particular, if ℓ−1 and ℓ/L are sufficiently small, then ∀y ∈∂Λ (u), we have
L
|G (u,y;E)|≤e−γ(m,L)kG (E)k. (5.3)
Λ Λ
Proof. Apply Lemma 4.3. ⊓⊔
Lemma 5.2.SupposethataboxΛ (u)is E-CNRanddoes notcontain anypair
L
of non-overlapping (E,m)-S boxes of size ℓ. Then
max |G (u,y)|≤e−γ(m,L). (5.4)
y∈∂ΛL(u) ΛL
Proof. If Λ (u) contains no (E,m)-S box Λ (x), the assertion follows from
L ℓ
Lemma 5.1. Suppose there exists a box Λ (x)⊂Λ (u) which is (E,m)-S. Since
ℓ L
all boxes Λ (w) with kw −xk = 2ℓ are disjoint with Λ (x), none of them is
ℓ ℓ
(E,m)-S, by hypotheses of the lemma. Therefore, either Lemma 4.2 or Lemma
4.3 applies, with A=4, giving the required upper bound. ⊓⊔
Lemma 5.3.Suppose that the finite-volume approximations H (ω), Λ ⊂ Zd,
Λ
|Λ|<∞, of random LSO H(ω) in ℓ2(Zd) satisfy the following hypotheses:
(W)P dist[E,Σ(H (ω))]≤e−|Λ|−β ≤e−|Λ|−β′, for some β,β′ >0;
n Λ o
(DS.ℓ) ∀v such that Λ (v)⊂Λ (u), P{Λ (u) is (E,m)-S}≤ℓ−p.
ℓ L ℓ
Set L=[ℓ3/2]. If p>6d, then H satisfies (S.L,m,1).
ΛL
Proof. Consider a box Λ (u). By Lemma 5.2, it must be (E,m)-NS, unless one
L
of the following events occurs:
– Λ (u) is not E-CNR
L
– Λ (u) contains at least two non-overlapping (E,m )-S boxes Λ (x),Λ (y).
L k ℓ ℓ
The probability of the former event is bounded by (W). Further, by virtue of
(DS.ℓ) the probability of the latter event is bounded by
12L2dℓ−2p ≤ 21L−p(3/22−2pd) ≤L−p. ⊓⊔
Corollary 5.1.(S.Lk,m,1) implies (S.Lk+1,m,1) (provided that Wegner-
type bound (W) holds true).
Theorem 5.1.If an IID random potential V(x;ω) satisfies the assumption
(2.2), then (S.Lk,1) holds true for all k ≥0.
In the single-particle case, treated in this section, for Hamiltonians with an
IID potential, it is well-known that the exponential decay of Green functions
implies that the respective Hamiltonian has pure point spectrum, and that
all its eigenfunctions decay exponentially (with probability one). This follows
fromSimon–Wolff criterion(cf. [SW86]), which aplies also to a largeclass of so-
called non-deterministic potentials. However, the method of [SW86] does apply
to multi-particle Hamiltonians. For this reason, the results of the next section
do not lead directly to spectral localization for multi-particle Hamiltonians.
10 VictorChulaevsky
6. Simplified multi-particle MSA
In this section, we study decay properties of Green functions for an N-particle
HamiltonianH(N)(ω)definedinEqn(2.4).Westressthatestimatesgivenbelow
are far from optimal; they only show that for any given number of particles
N ≥ 1 and any m > 0, there exists a threshold g = g (m) > 0 for the
N N
disorderparametergsuchthatif|g|≥g ,thenGreenfunctionsintheN-particle
N
model(withashort-rangeinteraction)decayexponentiallywithratem.tisalso
importanttorealizethatusingtheMSA,initstraditionalform,foranN-particle
system with large N inevitably requires using large values of g . Indeed, this
N
phenomenon occurs even in the single-particle MSA in high dimension d ≫ 1:
the respective threshold g = g (m,d) → ∞ as d → ∞. A direct inspection of
1 1
the conventional MSA show that, typically, V(·;ω, g (m,d)=O(d).
1
Given a number m > 0 and an integer L > 0, we will define a decreasing
0
sequence of decay exponents m(n), n≥1 as follows:
m(1) =m; m(N) =m(N−1)−L−1/8, N >1.
0
It is clear that in order to have m(N) > 0, we have to assume that m = m(1)
is sufficiently large (depending on N). In turn, this requires the thresholds
g ,...,g to be large enough, as explained above.
1 N−1
Compared to the single-particle MSA scheme presented in previous sections,
wehavetoreplacetheproperty(S.L,I)byadifferentone,(MS.L,I,N)given
below,andtoincludein(MS.L,I,N)therequirementthatthedecayofGreen
functions holds for all n′ <N:
(MS.Lk,I,N): The property (S.Lk,I,n) holds for 1≤n≤N −1,
P Λ(n)(u) is (E,m(n))-S ≤L−p(n,g),
n L o
wherep(n,g)→∞as|g|→∞,n=1,...,N−1,andE ∈I.Inthecaseoflarge
disorder,one can set I =R, and in the case of ”low energies”,I is a sufficiently
small interval near the bottom of the spectrum.
Remark. For any k = 0 and any n > 1, the validity of the above statement is
proved exactly in the same way as for n = 1. Then for k > 0 it is reproduced
inductively.DenotingP(N−1,g)=min p(n,g),weseethat,under the
1≤n≤N−1
hypothesis (MS.Lk,I,N), we have P(N −1,g)→∞ as |g|→∞.
For the sake of notational simplicity, we consider in Lemma 6.1 below only
partitions[[1,N]]=J Jc oftheinterval[[1,N]]intotwo(non-empty)consec-
utivesub-intervals,J =`[[1,n′]],Jc =[[n′+1,N]].Inotherwords,weconsidera
union of two subsystems with particles 1,...,n′ and n′+1,...,N, respectively.
We introduce the ”diagonal” subset of ZNd
D:= x=(x ,...,x ), x ∈Zd .
1 1 1
(cid:8) (cid:9)
Lemma 6.1.Let [[1,N]] = J ∪ Jc, J = [[1,n′]], Jc = [[n′ + 1,N]] and
consider a cube Λ(LN)(u) = ΛL(n′)(u′) × ΛL(n′′)(u′′), with u′ = (u1,...,un′),
u′′ = (un′+1,...,uN). Let Σ′ = {λa,a = 1,...,|Λ(n′)(u′)|} be the spectrum
of H(n′) , and Σ′′ ={µ ,b= 1,...,|Λ(n′′)(u′′)|} the spectrum of H(n′′) .
Λ(n′)(u′) b Λ(n′′)(u′′)
Suppose that