Table Of ContentTranslation Invariant States on Twisted Algebras on a Lattice
Bernhard Baumgartner ∗, Fabio Benatti ∗∗, Heide Narnhofer ∗
∗ Fakult¨at fu¨r Physik, Universita¨t Wien
∗∗ Department of Theoretical Physics, University of Trieste
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- Abstract
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t We constructanalgebrawithtwistedcommutationrelationsandequipitwiththe shift.
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m Forappropriateirregularityofthenon-localcommutationrelationsweprovethatthetracial
state is the only translation-invariantstate.
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1 Keywords: Generalized Price-Powers-shift, generalized Jordan-Wigner- transformation,
1. invariant states
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email: [email protected]
email: [email protected]
email: [email protected]
1 Introduction
Whereas the dynamics of quantum systems with finitely many degrees of freedom is fully con-
trollable via the spectrum of the unitary operator that implements it, this is not longer true
with infinitely many degrees of freedom.While these systems can always bedescribedby opera-
tors acting on suitable Hilbert spaces, it may however happen, typically in the thermodynamic
limit of interacting quantum systems, that not all of these operators represent observable quan-
tities. In such cases, what one does is to focus not upon the dynamics, rather on its equilibrium
(time-invariant) states.
Agreatstepforwardinourunderstandingofthermodynamics,inparticularofitssecondlaw,
would follow by showing that for interacting systems the class of invariant states is drastically
reduced in comparison to those of quasifree evolutions, where e.g. every state in equilibrium
with respectto any (space-)translation invariant dynamics is also (space-) translation invariant.
For interacting systems, one hopes that invariant states might be characterized by only a few
thermodynamic quantities.
Unfortunately,scarcely anycontrolonthedynamicsofrealisticphysicalsystemsisavailable;
however, some steps forward can be made following a toy model approach and constructing
algebras and discrete dynamics on these algebras such that only one invariant state exists. A
particular instance of invariant state is the so-called tracial, or totally mixed, state ω whose
two-point correlation functions are such that ω(AB) = ω(BA) for all A,B in the algebra of
operators.
Examples of quantum dynamical systems where the tracial state is the only invariant state
are the Price-Powers shifts [1, 2],[3], [4] or the irrational rotation algebras [5],[4]. In both cases,
what forbids the existence of invariant states different from the tracial one is that operators in
the course of time anticommute infinitely often and sufficiently irregularly with one another.
Such a lack of asymptotic commutativity (abelianess) is indeed expected in real interacting
quantum systems.
In the following, we give another example of such quantum dynamical systems inspired
very much by the Price-Powers shifts. In this latter case, the algebra is created by self-adjoint
operators e = e∗, e2 = 1, k ∈ N, that commute or anticommute, e e = (−1)g(|p−k|)e e , as
k k k k p p k
prescribedbyaso-called bitstream,namelyatwo-valued functionontheintegers,g :N 7→ {0,1},
g(0) = 0. In the following, we will consider Weyl-operators in place of the e and organize them
k
in such a way that shifted Weyl-operators remain in Weyl-like relation with one another in a
so-called complementary manner [6] so that any two of them create a full matrix algebra M .
d×d
We shall show that, under the hypothesis of sufficiently irregular complementary Weyl-like
relations, the tracial state is the only translationally invariant state as for the Powers-Price
shifts. The paper is organized as follows:
• in chapter 2. we define the algebra and the complementary relations, together with some
representations either as an AF-algebra (in some cases a UHF algebra) or as a quantum-
spin chain by means of generalized Jordan-Wigner transformations;
• in chapter 3. we show that, under the assumption of sufficiently random commutation
relations, only the tracial state can be translation invariant.
2
2 The Algebra and its Automorphism
2.1 Definitions
We start with infinitely many finite-dimensional algebras A , m ∈ N, all isomorphic to d×d
m
matrix algebras, created by the operators W(m), ~k ∈ Z2 := {(k ,k ), k = 0,1,...,d−1}, that
~k d 1 2 i
satisfy the commutation relations of a discrete Weyl group
(m) (m) 2πiσ(~k1,~k2) (m) (m)
W W = e d W W , (1)
~k1 ~k2 ~k2 ~k1
with symplectic form σ(~k ,~k ) := k k −k k .
1 2 11 22 12 21
Note that W(m) = , and W(m) = (W(m))−1.
~0 1 −~k ~k
The relations between Weyl operators with different upper indices are twisted by means of
a sequence of 2×2 matrices A , n ∈ Z, with entries in {0,1,...,d−1}; explicitly,
n
W(p)W(q) = e2πiu~kp~kq(q−p)W(q)W(p) , (2)
~kp ~kq ~kq ~kp
1
u (q−p) := σ(~k ,A ~k ) . (3)
~kp~kq d p q−p q
Setting A = , the single-site relations (1) are a particular instance of (2).
0
1
The finite products define elements of an infinite discrete group. We denote them as
(1) (2) (ℓ)
W := W W ···W , (4)
I ~k1 ~k2 ~kℓ
where I denotes a sequence of vectors {~km}m∈Z with only finitely many components,
(~k ,~k ,...,~k ), possibly different from the vector ~0. We define a star operation as (W(m))∗ =
1 2 ℓ ~k
(W(m))−1, and (UV)∗ = V∗U∗ as usual. So the Weyl operators and their products are uni-
~k
tary elements of the C∗ algebra generated by the finite products W , which are assumed to be
I
linearly independent. We shall denote this algebra by A.
We shall further equip A with an automorphism α : A 7→ A such that αn W(0) = W(n).
~k ~k
Then, generic algebraic relations read (cid:16) (cid:17)
nI nJ
W αn(W ) = W(a) W(b+n) = e2πiun(I;J)αn(W )W , (5)
I J ~ka ~kb J I
(cid:16)aY=1 (cid:17)(cid:16)bY=1 (cid:17)
1 nI nJ nI nJ
u (I;J) := σ(~k , A ~k ) = u (b+n−a) . (6)
n d a b+n−a b ~ka~kb
a=1b=1 a=1b=1
XX XX
Remarks 1
1. Everysequenceof2×2matrices{An}n∈N generatesitsownalgebraA;however,forspecial
sequences {An}n∈N (as will be shown in the next section), the corresponding algebras
(m)
built by W ,m = 1,,,l will be isomorphic ∀l to the tensor product over l local sizes
~k
and therefore the total algebra can be considered as the same A equipped with the usual
shift and a different automorphism α with finite speed.
3
(m)
2. We will refer to the Weyl operators W in (1) as to the letters of the algebra A and
~k
to the products W as in (4) as to the words of A. The norm of every word equals 1 and
I
(WI)d = , as the eigenvalues of all WI are the pure phases e2πdiℓ, 0 ≤ ℓ ≤ d−1. Linear
1
combinations of words will be referred to as sentences, as in the case of the Price-Powers-
shift. A state ω : A 7→ C over the algebra amounts to a positive, normalized functional
over A: it is thus fixed by giving its values on all words.
We proceed in studying the algebras, showing the existence of a non-trivial representation.
This is not a trivial problem, see [7]. Consider the set I of multiindices I and define the
composition law of addition: I ×I 7→ I: if I = {~kmI }m∈Z and J = {~kmJ }m∈Z, then
(I,J) 7→ I +J = {~kmI +~kmJ }m∈Z ,
wherethesumofvectors~kI +~kJ isunderstoodmodulod.Then,thefamilyofoperators{W }
m m I I∈I
satisfying the multiplication law W W = W , form an Abelian group G on which the shift
I J I+J
defines an automorphism with an associated shift-invariant measure δ such that δ(Wf) = 0
I
unless I = I0 := {~0}m∈Z. Therefofre,fone cafn consider the Hilbert space ℓ2(G) spanned by the
orthonormal elements W and represent the Weyl operators W introduced before byf
I I
f Π(WI)WJ = eiπu0(I,J) WI+J . (7)
ω(I;J)
f f
| {z }
ω(I;J) is a cocycle, namely
ω(I ,I +I )ω(I ,I ) = ω(I +I ,I )ω(I ,I ) ;
1 2 3 2 3 1 2 3 1 3
In this way A is represented as a sub-algebra Π(A) ⊆ B(ℓ (G)) (it is known as the regular
2
representation)oftheboundedoperatorsonℓ (G)andthusallconsiderationsin[8]aretherefore
2
applicable to A. For instance, A has a trivial center if to any word W there exists another
I
word W such that W and W do not commute. In this case the trace on the algebra
J I J
tr(W )= 0 ∀I 6= ∅, tr(1) = 1 (8)
I
is unique and it is implemented in the regular representation by
tr(W )= hW |Π(W )|W i .
I I0 I I0
Evidently. the trace is invariant under the sfhift automorfphism α :A 7→ A
Like in [8], the main interest is in finding conditions on the cocycle (7) such that no other
invariant states exist other than the tracial state. In [8] the main tool was the high degree
of anticommutativity, a generalization of the fact, that translation invariant states over Fermi
systems have to be even. Though this criterion is sufficient only if d = 2, non-commutativity
as embodied in (2) will nevertheless turn out to be just as powerful in restricting the class of
invariant states. We will indeed give other arguments to enlarge the class of automorphisms
that allow only the tracial state as invariant state. Though not optimal, the result indicates
thatdelocalization bythedynamics as wedescribeitin(2.2)is aneffect whichis worthstudying
in more detail and in more realistic thermodynamic systems.
4
2.2 Spin-Chain Representation
We have already given a representation of the algebra over B(l (G)). However, in order to
2
bring it in closer contact with physical models we seek connections with spin chains and their
automorphisms.
This demands different representations. To demonstrate the differences we make a short
detour, considering finite algebras defined on a finite ring of N lattice points (= upper indices)
witha cyclic shift,insteadof an infiniteset of pointswith anon-recurrentshift.We getdifferent
dimensions of the Hilbert spaces. In the above mentioned representation there are (d2)N basis
vectors. For a spin-chain one would expect only dN as necessary. But this, as will turn out, is
possible for a restricted set of defining sequences only. In Section 2.3 we give then a representa-
tion with a double-spin-chain, possible for any defining sequence, employing again (d2)N basis
vectors for the finite algebra on a ring.
(m)
Moreprecisely,weshalltrytorepresenttheWeyloperatorsW ,m = 0,...,N,aselements
~k
N
of the full matrix algebra (M ) . We proceed step by step: let us define the Weyl
n=0 d×d n
operator at site 0 ≤ j ≤N to be
N
(j)
W = W ⊗W ⊗W ⊗1 ...⊗1 . (9)
~k A0,j~k A1,j~k Aj,j~k j+1 N
Theunknownsintheconstructionarethe2×2matriceswithintegerentriesfrom{0,1,...,d−1}
that we have to adjust in order to fulfil the commutation relations (2). Therefore, from (1), one
gets the condition
j
σ(A ~k,A m~)= σ(~k,m~) , (10)
ℓ,j ℓ,j
ℓ=0
X
forall ~k, m~ and 0 ≤ j ≤ N which is equivalent to j Det(A ) = 1. If d is prime, then all
ℓ=0 ℓ,n
Weyl operators are unitarily isomorphic so that the algebra created by them is M ; therefore,
d×d
P
in the rest of this section, d will be assumed to be a prime number.
We have to control whether this ansatz can really be satisfied and how far the matrices A
ℓ,k
are determined by the matrices A in (3). It turns out that A = A while A = 1. The other
n 0,j j 00
matrices A have to be calculated recursively from (2) and (3). More precisely to evaluate
ℓ,k
σ(A ~k,A ~l)+σ(A ~k,A ~l) = σ(~k,A ~l) (11)
0,1 0,2 1,1 1,2 0,1
we define the linear map A7→ A,
a a a −a
A = b 11 12 → A:= 22 12
a a a a
21 22 21 11
(cid:18) (cid:19) (cid:18) (cid:19)
b
With this map, it follows that A = Aˆ−1(A − Aˆ A ). This fixes A if we take into
1,2 1,1 0,1 0,1 0,2 1,2
account that the freedom in A is reduced to an isomorphism inside of the local algebra.
1,1
However, we can only be sure that there exists a solution A if A is invertible which surely
1,2 1,1
holds if Det(A ) 6= 0. Similarly
1,1
n−k−1
A = Aˆ−1 (A − Aˆ A ) .
n−k,n n−k,n−k 0,n−k l,n−k l,n
l=0
X
5
The equations are uniquely solvable (up to trivial local isomorphisms) under the constraint
Det(A )+...+Det(A ) 6= 1 ∀j . (12)
0,j j−1,j
This implies in addition that Det(A ) 6= 0; as a consequence the algebra generated by the
j,j
Weyl operators (9) is isomorphic to the full matrix algebra.
Only the sequence of matrices A is to our disposal, whereas the matrices A are linearly
n ℓ,k
depending on them; however, there remains enough freedom to find sequences that meet the
condition (12).Aspecialexample correspondsto choosingA = δ ;this in turncorresponds
0,j 1,j
1
to the usual shift on the lattice algebra. More generically, one may choose A to belong to a
0,j
left ideal with determinant 0 for all j 6= 0; it then follows that, for all k > 0, A also belongs
n−k,n
to this ideal and therefore Det(A ) = 1 ∀n.
n,n
As a consequence, provided Det(A ) 6= 0 ∀n we can consider the algebra to be the same
n,n
algebra i.e. the spin chain M⊗∞, but equipped with different automorphisms corresponding to
d×d
the different sequences A . Notice that in the spin-chain representation, the algebra is fairly
0,n
simple, while the automorphism (which we will again denote by α and which corresponds to
the shift in the regular representation) is complicated.
Some discussions on the spin chain representations and their importance for physics follows
in the Conclusion.
2.3 The Generalized Jordan-Wigner Transformation
We have already mentioned the representation of every algebra A as a C∗ algebra Π(A) ⊆
B(ℓ (G)). We can give another representation that can be considered as a generalization of
2
the Jordan-Wigner transformations, that relates the spin lattice with the Fermions on a lat-
tice. We represent A as a subalgebra of the doubled spin chain ∞ (M ⊗ M )
m=−∞ d×d d×d m
= ∞ (M ) , where W(0) with ~k = (k ,k ) is identified with the infinite tensor product
n=−∞ d×d n ~k 1 2 N
N +∞ +∞
(W ) ⊗(W ) ⊗ (W ) ⊗(W ) (1) , (13)
0,bn −2n 0,an −2n+1 k1,0 0 k2,k1 1 n
!
n=1 n=2
O O
where the Weyl operator W is at site n = 0, while W is at site n = 1, whereas the
k1,0 k2,k1
operators W are located at sites −2n and W at sites −2n+1. Moreover, the components
0,bn 0,an
a and b are determined by ~k and the commutation relations via
n n
k b
A 1 = n .
n
k a
2 n
(cid:18) (cid:19) (cid:18) (cid:19)
Furthermore, the action of the shift automorphism α is now represented as a 2-step translation
along the lattice.
Notice that, since the contributions from the infinite tails commute with each other by
(0) (N)
construction, finite tensor products of the form W ···W may be effectively represented
k1,k2 l1,l2
as elements of the matrix algebra 2N (M ) . It thus follows that the commutant consists
n=0 d×d n
of operators of the form
N
−∞
(1) ⊗(W ) ⊗(W ) ⊗ (W ) ⊗(W ) ,
n ℓ1,ℓ2 2j ℓ2,0 2j+1 0,bk−j(ℓ) 2k 0,ak−j(ℓ) 2k+1
n=2j−1 k=j+1
O O
6
plus operators belonging to the center. (The center becomes trivial if (12) holds. But this
condition is not needed here.)
We can consider the operators to act on the vector |Ω >= ⊗ |0 > where at each
−∞<k<∞
lattice point W |0 >=|0 > . Representing W0 by
0,k k1,k2
+∞ +∞
f
1⊗1 ⊗ (W ) ⊗(W ) (1)
k1,0 0 k2, 1 n
!
n=1 n=2
O O
and letting it acting on |Ω >= |W i we reproduce l2(G) and therefore the regular representa-
I0
tion.
f
Example As a concrete illustration of the algebraic construction of above, let us consider
the Price-Powers shift. This corresponds to d = 2, and to a generating sequence of matrices
1 0 0 1
which are either A = corresponding to g(n) = 0 or A = corresponding
n n
0 1 −1 0
(cid:18) (cid:19) (cid:18) (cid:19)
(0)
to g(n) = 1. Then, we set e = W and observe that operators at odd places have the form
0 1,0
W and therefore all commute. Consequently, we can remove them from the tensor product
0,k
(13) and represent
+∞ +∞
e = (W ) ⊗(W ) (1) .
m 0,bn k−n 1,0 k n
n=1 n=k+1
O O
2.4 Preliminary Remarks on Invariant States
We now turn to the problem of finding the invariant states under the 2-step shift. If we are
only interested in the local effects of such automorphism, we can take the periodic shift in
2N (M ), which is unitarily implemented.
n=0 d×d
The algebra created by finitely many Weyl operators is imbedded in 2N (M ), so that
N n=0 d×d
we can conclude that the automorphism α : A 7→ A is unitarily implemented. Therefore, we
N
can construct states on the local algebra defined by density operators that commute with the
unitary that implements the periodic shift. However, in general, these density operators will
not have a limit when N → ∞.
Another possibility to construct α-invariant states is to start from vectors in the infinite
tensor product ∞ |ψ i that are invariant under the shift. The simple choice where all
n=−∞ n
|ψni are identical to an eigenvector |φi of W(0,1) gives, independently of the sequence {An}n∈N,
N
already the tracial state; indeed, since for every W at some position hφ|W |φi = 0. Therefore,
I k,l
the representation is isomorphic to the regular representation.
If we choose some other vector and assume that A 6= 1 for infinitely many n, again we
n
obtain the tracial state; in fact, either |hψ|W |ψihψ|W |ψi| < 1, infinitely often so that
0,bn 0,an
+∞
|hψ|W |ψihψ|W |ψi| = 0 .
0,bn 0,an
n=−∞
Y
If for some N > 0 A = 0 for all n > N, we can choose a vector φ over 2N M(n)
n n=1 d×d
that is appropriately entangled over the lattice points to guarantee that hφ|W |φi =6 0. By
k1,0)N
averaging this vector over the period N, one gets an expectation value still 6= 0. However, in
general, one expects that it decreases with N. Therefore, if for every N we can find n >N such
7
that A 6= 0 in order to obtain another invariant state in the limit N → ∞, it is necessary to
n
have correlations between infinitely many lattice points; this can hardly be satisfied because of
monogamy of entanglement. Though we are unable to exclude that other invariant states might
be constructed, our considerations already indicate that the tracial state will turn out to be the
only invariant state under appropriate conditions on the defining sequence of matrices A .
n
3 Invariant states
Given (A,α), let ω be an invariant state such that ω ◦α = ω and consider the corresponding
GNS representation π of A as a C∗ algebra of bounded operators on the GNS Hilbert space
ω
H with cyclic vector |Ωi. Namely, ω(A) = hΩ|π (A)|Ωi; further, the shift automorphism α is
ω ω
implemented by a unitary U such that U |Ωi = |Ωi. The following simple lemmas hold.
ω ω
Lemma 1. Let Pω denote the projection onto the U -invariant subspace of H . If
0 ω ω
hΩ|π (W∗)Pωπ (W )|Ωi = 0 (14)
ω I 0 ω I
is true for all W , then ω is tracial that is ω(XY) = ω(YX) for all X,Y ∈A.
I
Proof. Since Pω ≥ |ΩihΩ|, the assumption implies
0
|ω(W )|2 = hΩ|π (W∗)|ΩihΩ|π (W )|Ωi ≤ hΩ|π (W∗)Pωπ (W )|Ωi =0 ,
I ω I ω I ω I 0 ω I
whence, by Cauchy-Schwartz, ω(W W )6= 0 only if W W = 1. Thus, ω(W W ) = ω(W W )
I J I J I J J I
for all W , whence ω(XY) = ω(YX) for A is generated by linear combinations of W s.
I,J I
Lemma 2. If ω is α-invariant, then, setting u (I) := u (I;I) in (6),
n n
N−1
1
hΩ|π (W∗)Pωπ (W )|Ωi= lim e2πiun(I)hΩ|π (W )U−nπ (W∗)|Ωi . (15)
ω I 0 ω I N→∞ N ω I ω ω I
n=0
X
Proof. The mean ergodic theorem of von Neumann ([9]) and (6) imply
N−1
1
hΩ|π (W∗)Pωπ (W )|Ωi = lim ω W∗αn W
ω I 0 ω I N→∞ N I I
nX=0 (cid:16) (cid:16) (cid:17)(cid:17)
N−1
1
= lim e2πiun(I)ω W α−n W∗ .
N→∞ N I I
nX=0 (cid:16) (cid:16) (cid:17)(cid:17)
Remarks 1
1. If W = 1, then hΩ|π (W∗)Pωπ (W )|Ωi = 1, u (I) = 0 for all n ∈ N and (15) is trivially
I ω I 0 ω I n
satisfied.
8
2. Using the spectral decomposition U = dPλe2πiλ, (15) reads
ω Sp(Uω) ω
R
N−1
1
hΩ|π (W∗)Pωπ (W )|Ωi= lim dµ (λ) e2πi(un(I)−nλ) , (16)
ω I 0 ω I N→∞ZSp(Uω) I N n=0
X
where dµ (λ) := d(hΩ|π (W )Pλπ (W∗)|Ωi.
I ω I ω ω I
We now concentrate on the sequences u(I) := {un(I)}n∈N and uλn(I) := {un(I)−λn}n∈N
and study their spectrum [9]. In order to properly introduce this notion, consider a sequence
v = {vn}n∈N taking its values vn ∈ C in a compact subset of the complex numbers. For all
k ∈ N the partial sums
N−1
1
S (k) := v∗v (17)
N N n n+k
n=0
X
arebounded,thus,thesequenceS(k) := {SN(k)}N∈N hasaccumulationpointsand,byaCantor-
like diagonalization argument (for details see the Appendix), there exists at least one subse-
quence {Nj}j∈N such that the limit
Nj−1
1
s (v) := lim v∗v
k j→∞Nj n n+k
n=0
X
exists for all k ∈ N. By setting s := s∗, one obtains a positive-definite sequence (details are
−k k
again in the Appendix), that is a sequence s(v) = {sk(v)}k∈Z such that
z∗s (v)z ≥ 0
i i−j j
i,j
X
for all sequences {zi}i∈Z such that i∈Z|zi|2 < ∞. Then, by Bochner’s theorem,
P 1
s (v) = dµ (x)e2πikx ,
k v
Z0
where dµ (x) is a positive (correlation) measure on [0,1) such that
v
1 1 Nj−1
dµ (x) = lim |v |2 .
v n
Z0 j→∞Nj n=0
X
If the correlation measure of a sequence v is the Lebesgue measure, then s (v) = 0 whenever
k
k 6= 0 and the sequence v is said to be uniformly distributed. Therefore, it makes sense to
introduce the
Definition 1 (Spectrum of a sequence). [9, 10] Given a sequence v := {vn}n∈N with values in
a compact subspace of C, its Fourier-Bohr spectrum is given by
N−1
1
Sp(v) := µ ∈ [0,1) : limsup v e−2πinµ 6= 0 .
n
( N→∞ N (cid:12) (cid:12) )
(cid:12)Xn=0 (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
9
In other words,the spectrumof a sequence v is the subsetof values µ ∈ [0,2π) such that the
sequences v(µ) = {vnexp(−2πinµ)}n∈N are not uniformly distributed. Equivalently, µ ∈/ Sp(v)
if and only if
Nj−1
1
lim v e−2πiµn = 0 (18)
n
j→∞Nj
n=0
X
for all converging subsequences of partial sums.
By means of the spectral properties of sequences, we can derive sufficient conditions that
force the invariant state ω to be tracial.
Lemma 3. Let v(I) := {e2πiun(I)} ; then, the dynamical system (A,α) has the tracial state
n≥0
as its only invariant state if for each I, Sp(v(I)) is either ∅ or {0} with
N−1 d−1
1
lim vn(I) = pj(I)e2πdij =:ν(v(I)) (∗)
N→∞N
nX=0 Xj=0 (cid:16) (cid:17)
for some p (I) ≥ 0, j ∈D, p 6= 1, d−1p (I) = 1.
j 0 j=0 j
P
Proof. If Sp(v(I)) = ∅, then, using (18) and dominated convergence, the right handside of (16)
vanishes and the result follows from Lemmas 1 and 2.
If for some I Sp(v(I)) = {0} and relation (*) holds for such I, then, using again (18) and
dominated convergence,
hΩ|π (W∗)Pωπ (W )|Ωi = ν(v(I))hΩ|π (W )Pωπ (W∗)|Ωi . (19)
ω I 0 ω I ω I 0 ω I
Since |ν(v(I))| < 1, by exchanging W and W∗, one gets
I I
hΩ|π (W∗)Pωπ (W )|Ωi< hΩ|π (W )Pωπ (W∗)|Ωi < hΩ|π (W∗)Pωπ (W )|Ωi .
ω I 0 ω I ω I 0 ω I ω I 0 ω I
Thus, hΩ|π (W∗)Pωπ (W )|Ωi = 0 and Lemmas 1 and 2 apply.
ω I 0 ω I
For some given I, for instance a singleton I = {1}, there surely exist sequences of matrices
{An}n∈N, with entries from {0,1,...,d−1}, such that Sp(v(I)) = ∅, or Sp(v(I)) = {0}.
However, in order to use the previous Lemma, we have to make sure that there exist se-
quences {An}n∈N such that conditions 1 or 2 in the previous Lemma are fulfilled for all I.
Inthefollowing,weshallconsiderthe4entriesa (n)ofthematricesA asrandomprocesses
ij n
with values from {0,1,... d − 1}. Then, we shall focus upon the space X of sequences ~x =
{~xn}n∈N, where ~xn = (a11(n),a12(n),a21(n),a22(n)) are 4-valued vectors with the entries of
the matrices A as components. If we want in addition to meet the requirements in (2.2), e.g.
n
that Det(A ) = 1 ∀n, we can restrict to 2 entries (a (n),a (n)) but still keep enough
n,n 11 12
randomness.
We equip X with the shift-automorphism (σ(~x)) = ~x and with a σ-invariant measure
n n+1
µ (defined on the σ-algebra of cylinders). Concretely, if f is a measurable function on X, then
its mean value with respect to µ is given by
µ(f)= dµ(~x)f(~x) ;
ZX
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