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A New Paradigm in Quantum Chaos: Aubry's Theory of Equilibrium States for the Adiabatic Holstein Model
A New Paradigm in Quantum Chaos: Aubry's Theory of Equilibrium States for the Adiabatic Holstein Model. R . S. MACKAY
Nonlinear Systems Laboratory, Mathematics Institute, University of Warwick Coventry CVJ> 7AL, U.K. C . BAESENS
Laboratoire Léon Brillouin, CEΝ Saclay - 91191 Gif-sur-Yvette, France
1. - Introduction. AUBRY and co-workers have introduced a new direction in the subject of quantum chaos: they show existence of spatially chaotic equilibrium states for some models of electron-phonon systems [1]. This is in our opinion a most interesting theoretical development, and furthermore it sheds light on many experimental results on charge-density waves. In this lecture, we give a pedagogical treatment of Aubry's theory, with some simplifications and improvements in the proofs. The approach is closely analogous to the derivation of chaotic orbits for symplectic twist maps near an anti-integrable limit, described in sect. 3 of the preceding lecture.
2. - The model. The adiabatic Holstein model consists of i) classical variables un (which can be thought of, for example, as the deviation of the length of a molecular bond from a reference value) at sites η in some lattice L, which we shall take initially to be finite with Ν sites, and ii) 2M electrons which are supposed noninteracting (apart from Pauli's exclusion principle), and which fill up the energy levels of a tight-binding Hamiltonian (2.1)
H =
u-tA, 51
52
R.
s.
M A C K A Y and
c.
BAESENS
in pairs (spin up and down) from the lowest level until the electrons are all used up. The operator Δ is assumed to connect each site to only finitely many sites, e.g. symmetric nearest-neighbour coupling on a one-dimensional lattice (2.2)
(Δψ)η = ψ η_ι + ψ η +ι ,
or possibly to have exponential decaying interaction between distant sites. Here ψ is the wave function for up-down pairs of electrons, and {ψη } are its components with respect to the basis of on-site wave functions {\n)} at sites n. The Schrödinger equation for (2.1) can be written as (Ηψ)η = ηηφη-ίΣ
(2.3)
= Εφη .
\imΦπι m
The term u in the Hamiltonian represents an effect of the {un } on the electrons, and the term tA represents hopping of electrons between sites. A configuration u = {un } is assigned an energy (2.4)
W(u, t) = ±
Σ ni + Σ Ε η
Δ
Ν
(Μ,
ί),
veO
where ν is a label for the eigenstates and Ο denotes the occupied eigenstates of H. The aim is to find the equilibrium states of the model, that is, the configurations u of stationary W. The condition of stationarity is (2.5)
v
- Σ
un=
dE /dun.
veO
Evaluating the right-hand side leads to (2.6)
UN=
-
Σ
2
\Φη\ =
- P n ,
veO
ν
where the {ψ } are an orthonormal set of eigenvectors corresponding to energies E\ A proof is given in appendix D. The quantity pn can be interpreted as the probability of finding an up-down electron pair on site n.
3. - The anti-integrable limit t = 0. In the case t = 0, the sites are uncoupled and it is easy to solve the model exν plicitly. The eigenstates can be chosen to be ψ = | v), the wave function on site v v, with energies E = uv. Thus the occupied states are the M states for which uv is smallest, and the energy of a configuration u is (3.1) Δ
n
v e O
A
N E W
P A R A D I G M
IN
Q U A N T U M
C H A O S :
A U B R Y ' S
T H E O R Y
53
E T C .
The stationary states are seen to be those for which (3.2)
Un --
1 0
if if
weO, ntO.
This means we can arbitrarily choose M of the sites to have un = - 1 and the remaining Ν - M sites to have un = 0. Since p n = - un for equilibrium states, the sites with un = - 1 have an electron pair and those with un = 0 do not. The combination of un = - 1 and p n = 1 is called a bipolaron. Thus the equilibrium states correspond to all the ways of putting M bipolarons on a lattice of Ν sites. Note that all the equilibrium states have the same energy, namely W = = - (1/2) M. Furthermore, the electronic spectrum consists of an M-fold degeneracy at Ε = - 1 , which are all occupied, and an (N - M)-fold degeneracy at Ε = 0, which are unoccupied, so there is a gap between the occupied and unoccupied levels. The above discussion shows that we have a form of spatial chaos at t = 0. This result is very natural in the case t = 0 because then the sites are independent. What is significant is that these states can be continued for t nonzero provided it is small enough, as we now explain.
4. - Small hopping. Suppose the hopping amplitude t is small but nonzero. The main result of [1] is that for each choice of equilibrium configuration u at t = 0, with un= - ση, ση = 0 or 1, there is a unique continuation u(t) as an equilibrium state of Wt, for t small enough (where Wt(u) means W(u, t)). Thus the chaos persists for small t. The {ση } are called pseudospins. In principle, there might also be other equilibrium states, but it is shown in[l] that they must have significantly higher energy. Now we give a proof of this continuation result. We restrict attention to the one-dimensional case with nearest-neighbour coupling only (2.2). Our notation and basic mathematical tools are explained in appendix A. We compare our proof with that of [1] in appendix B. From (2.6), the equilibrium states are the fixed points of the operator (4.1)
St(u) = u',
where
(4.2)
<=-Σ ι « ι . 2
veO
54
R.
s.
MACKAY
and
c.
BAESENS
Note that it follows that (4.3)
-l^un^0,
for all equilibrium states. Given a configuration ιι(0) = - σ, we define a space U of perturbations u with the norm (4.4)
τ = ||u|| = sup \un + ση | .
The space U is a Banach space (see appendix A) with origin shifted to - σ. The 1 operator S can be thought of as mapping U χ Τ —>U where 7 = R represents the set of possible We will show that S is differentiate with respect to (if, t) e U x Γ for τ and £ small enough, and that u(O) is a nondegenerate fixed point of S0 (i.e. 1 g specDf//S0(ii(0))). Hence by the implicit-function theorem (appendix A) there is a locally unique fixed point u(t) of St, for t small enough, and it depends differentiably on t It can be continued as long as / - Ou S remains invertible, and it satisfies the differential equation (4.5)
du/dt = (I-
1
OuSy
dS/dt
(u(t)).
We will use this in sect. 5 to obtain good estimates of how far u(t) can be continued. Step 1: Electronic gap. The first step is to prove that, for t and τ small enough, the spectrum of Η continues to consist of M eigenvalues near - 1 and Ν -M eigenvalues near 0. More precisely, it is shown in[l] that for (4.6)
τ + 2t <
I
the spectrum of Η divides into M eigenvalues in an interval (4.6α)
7_! = [ - 1 - 2 ί , - 1 + τ + 2ί],
and Ν — M eigenvalues in an interval (4.66)
/0 = [ - τ - 2 ί , 2 ί ] ,
with / _ ! and I0 disjoint. So we have an electronic gap (4.7)
AE^s
= l-
2r-4t.
We give a slightly different proof of this result, which serves to introduce some of the ingredients we will use later. For t = 0, the spectrum of Η consists precisely of the values un, and hence for τ < 1/2 we have the desired result. For Ε φ spec Η', define the resolvent operator (4.8)
G(E) =
(E-Hy
1
on wave functions {e.g., [2]). We will often drop the dependence on Ε from the
A NEW PARADIGM IN QUANTUM CHAOS: AUBRY'S THEORY ETC.
55
notation. Now H depends differentiably on t, and computing the derivative of G with respect to t gives (4.9)
- GAG,
DtG=
as long as G remains defined. Taking / 2 norms (see appendix A), we obtain (4.10)
d||G||2/d^||G||I||/l||2.
The operator G has / 2-norm (4.11)
l|G||2 = < R \
where S is the shortest distance from Ε to the spectrum of H, and, in the 1-D case (2.2) we are considering, (4.12)
114 = 2 .
Solving the differential inequality (4.10) yields (4.13)
*(0)-<*(ί)^2ί.
Hence the spectrum of Η moves by at most 2t from its position at t = 0. Since Η is Hermitian, the spectrum is real, proving the required result. Step 2: Reformulation. Our next step is to rewrite (4.2) in terms of the spectral projection operator P. Let Γ be a contour in the complex 2?-plane which surrounds the interval 7_χ once but excludes the interval I0. Then define (4.14)
P=ih\ 2πί
J
G(E)dE,
which is the projection onto the space spanned by the eigenvectors corresponding to spectrum inside Γ (e.g., [2]). It follows that (4.2) can be rewritten as (4.15)
u;=-(n\P\n).
Step 3: Differentiability. To carry out the proof via the implicit-function theorem, we need to show that S is jointly differentiate with respect to (u, t) and that 1 is not in the spectrum of OuS0(u(0)). Joint differentiability is automatic from (4.14), (4.15) as long as there remains an electronic gap so that the contour Γ can be chosen to avoid the spectrum of H. Since we will need to refer to it often, we will write (4.16)
J = OuS.
56
R.
s.
MACKAY
and
c.
BAESENS
Then J is given by (4.17)
Jmn = - (m I dP/dun \m) = -
\{m\G(E) \ n)(n \ G(E)
\m)dE.
r Writing (4.18)
Gmn =
(m\G\n),
then (4.19)
=
Jmn
-
J Gmn Gnm
dE.
Γ
Note that this reformulation has the additional advantage that it allows one to take an infinite lattice from the start. Step 4: Nondegeneracy. Finally, we show that 1 φ s p e c / at ii(0), t = 0. This is easy because / = 0 at t = 0. To see this, note that at t = 0 (4.20)
for m * η,
Gmn = 0
and (4.21)
Gnn = (E-uny
l
.
Hence the off-diagonal terms of J in (4.19) are automatically zero and the diagonal terms are given by integrals of double poles, which are also zero. This completes the proof of the continuation theorem. In the next section, we find explicit estimates for how large a coupling t these states can be continued to. 5. - Explicit estimates. We will show that u(t) can be continued for at least t < tQ = 0.089939 (eq. (5.46)). Our strategy of proof is that as long as / - / remains invertible, u(t) can be continued. We use the differential equation (4.5) to derive a differential inequality for r(t), which we solve. For this we need estimates on J and OTS. We derive them via estimates on the resolvent operator G. Step 1: Exponential decay of Gmn. For S a positive lower bound to the distance from Ε to the spectrum of H, the elements of G(E) are all bounded by (5.1)
1
IG™ I s M l G H ^ - .
We will show that the off-diagonal elements are exponentially small:
(5.2)
\Gmn\
2
ssal™-*!/ ,
57
A NEW PARADIGM IN QUANTUM CHAOS: AUBRY'S THEORY ETC.
with 2
2t
(5.3)
2 S
+ V^ + * ' 4
4
4
and (5.4)
λ
9
where 2
(5.5)
2
ζ = π ώ ι ( ί / * » 1).
To prove (5.2), since ||G||2 is finite, the equation (5.6)
|0)
(Ε-Η)ξ=
has a unique solution ξ in / 2 . Given Ν > 0, let yj be the vector with components (5.7)
for
η < N,
for
Ν.
Then η' = (E - H)r) has components (5.8) Y]n = 0 otherwise. Using (4.11), we obtain (5.9) for (5.10)
Ρ =Σ Ν
2
2
2
2
2
| s , | ^ t * - ( | f o l + I&-1I )
^ 1. This can be written as Ρ * ^ Χ(Ρ^_! - PN+1 ),
£ 1,
with (5.11)
K =
2
t
2
/ S
.
This recurrence inequality can be proved to yield (see appendix C) (5.12)
Ν 1
ΡΝ^ζλ ~ Ρ0,
N&l,
with ζ and λ as above. Since |0) has norm 1, (5.13)
2
PQ ^ S~ .
58
R . s.
M A C K A Y and
c.
BAESENS
Hence it follows that 1
2
Ι^Ι^Ρ^ζλ"- *" ,
(5.14) and so
ιfo|
(5.15)
2
^ Α " /
,
for Ν ^ 1. But ξΝ is just Gm. The same technique can be used for Ν < 0 and for the other columns of G. This completes the proof of (5.2). Step 2: Exponential decay of Jmn. Using the above estimates (5.1), (5.2) on Gmn, formula (4.19) and the contour Γ which passes up the vertical line Re Ε = = - 1 / 2 and is closed by a large semicircle in the left half-plane, we can obtain good estimates on Since the spectrum of Η is contained in the intervals (4.6a), (4.66), for (5.16)
- i +%, 2
E=
y real, we can choose (5.17)
= ^ l + y \S
where ε is defined in (4.7). Hence, for the off-diagonal elements we find 00
(5.18)
\Jmn\
2
J C A'»-»" dy,
si ±-
— 00
because the contribution from the large semicircle goes to zero as its radius goes to infinity. The integral can be written as twice that from 0 to infinity. Changing variable from y to 2
and using
2
2
= 1 + 4ΐ/ /ε = 4£ /ε
(5.18a) 2
2
ζ^ί /8 ,
2
we obtain 00
(5.19)
I»--I f
ξ ^ - ' ά η
8
7τε
where (5.20) and
s=
2
2
4t /e ,
59
A NEW PARADIGM IN QUANTUM CHAOS: AUBRY'S THEORY ETC.
Since ξ ^ 1 for η ^ 1, the integral in (5.19) is bounded by the case \m - n\ = 1, which can easily be calculated to be π/2. Hence 1
\Jmn \ ^ l e - « ! " - * ! .
(5.22) ι
Note that, using ξ ^ η~ for η ^ 1, one can obtain the slightly better estimate (5.23)
N
^fs ,
\Jnm\<
where (5.24)
Ν =
\m-n\,
and 1 · 3 · 5 · . . . · ( 2 Λ Τ - 1) <
5 2
·
ι
5)
i 1
Step 3: Estimates of\\j\\œ and - J ) " \\M . To obtain good estimates for the diagonal elements of J, we need to subtract out the double pole that we noticed in the case t = 0. It is easy to check from the definition (4.8) that the resolvent operator G satisfies the identity (5.26)
G = G° - tG° AG l
where G° is the case t = 0 (so its matrix Gmn is diagonal with G n n = (E Hence (5.27)
Gnn = (E- uJ-Hl
- t(n|AG|n))
l
= (E - un)~ {l
- t[Gn+hn
un)~ ).
+ G n_ l f J ) .
It follows from (4.19) that (5.28) =
Jnn = " ~h
\ Γ
{ E
~
U
n 2) 1(
~
"
2 t [ G n ++ 1
J+2 G
^
>
' t »+i.n +
G n-
l
t
fn) d E .
The first term gives zero, being a double pole. We estimate the remaining terms using (5.29)
\Gn+hn\
^ ψ
2
,
which comes from (5.2), and \E - un\ ^8. Hence 00
(5.30)
\Jnn\^
^ J ί- (4< /* 2
2
2
4
+ 4*7* )%·
60
R. s. MACKAY and c. BAESENS
Using the change of variables (5.18a), we obtain
: , - i 4 f7jL+ · ΐ \ _ Ε 2 _
(5.31)
(5.32)
l « / r a| = f ( 2 + | s ) .
The estimates (5.23) and (5.32) lead to an estimate for \\J\\„, by summing the estimates for a row of J. Alternatively, one can sum up the integrands in (5.19) rather than the estimates (5.23), and then bound the resulting integral. Following this route, the sum of the off-diagonal elements of a row of J is bounded by (5.33)
2s
J' =
Using ξ =£ η
1
- 1(1
-
SO
for η ^ 1, we obtain 2s
(5.34)
-K>?-*) (5.35)
1
ε
- A .
Vi-
by the calculus of residues. Hence, adding in the estimate (5.32) for the diagonal element, we obtain
^1
(5.36)
2
M
s
+
|
s
^ -2 ν τ ^
2
+
2
=
Thus, if (5.36) is less than 1, then (5.37)
M =
I-J
is invertible, and we obtain the estimate (5.38)
||Af-l. M l
) - ^
Step 4: Bound on OTS. spect to t can be written (5.39)
at
—
!_I
2
e
+
i
e
«
+
_ J
-1 =
From (4.9) and (4.15), the derivative of S with re-
J Σ Gnm(Gm+ijn
+
Gm-if7l)dE.
61
A NEW PARADIGM IN QUANTUM CHAOS: AUBRY'S THEORY ETC.
Using the estimates (5.1) and (5.2) on G, we obtain 00
(5.40)
4ί
2π Jί
di
2
VÂ 1-λ
4ί 4 ί
+
— 00
d#.
Changing variables as in (5.18a) gives d< di
(5.41) Using f ^ ^
1
j3/2 1f / 2
.1/2
d^
+
J/2
for η ^ 1, we can estimate this by
V2 d^
f
(5.42)
dt
(i?-e)V^-l " 2
This integral can be evaluated by the changes of variable η = cosh ζ followed by χ = exp [2z], and gives dun dt
(5.43)
8 πε V T
tg-
Vi
1
Step 5: Differential inequality. The results (4.5), (5.38) and (5.43) lead to the following differential inequality: 8 dr di
(5.44)
tg-
1
Vi VT
valid as long as i) ε > 0, ii) s < 1, and iii) the denominator of (5.44) is positive. We recall that the definitions of ε and s are given in eqs. (4.7) and (5.20), respectively. We proceed now to deduce a bound on r(t) and hence deduce a range 0 ^ t < t0 for which u(t) continues. If zh(t) is the solution of (5.44) with the inequality replaced by equality and initial condition z h (0) = 0, then it follows that (5.45)
r(t)^rb(t),
up to the first t for which one of the above three conditions fails. It is
62
R.
s.
MACKAY
and
c.
BAESENS
easy to see that, if conditions i) or ii) fail, then condition iii) will already have failed, so (5.45) holds up to the first t = t0 for which condition iii) fails. We calculated rb (t) numerically until the denominator passed through 0. Up to numerical errors, the denominator remained positive for t
io = 0.089 939.
Hence u(t) can be continued for at least t
(5.47)
tg~ x
for χ ^ 0.
^x
Secondly, we use (5.48)
2
2s + 3 s / 2 ^ 4((1 - s)~
1/2
- 1)
for s < 1,
as can be seen by Taylor-expanding the right-hand side. Hence we obtain 16* πε (1 - s) 2
dr
(5.49)
^d t
^
~ 1-
1
§((l-SR
/
2
-D
Next, for 0 ^ s ^ s0 < 1, we have by convexity (5.50)
1 2
(1 - s ) " / - 1 ^ ((1 - s o ) "
1 /2
- l)s/so •
Hence, for t ^ t0 and τ ^ τ 0 , we have (5.51)
2
dr/di ^ At/(1
-Bt ),
with (5.52)
A = 16/^(1 - s 0) ,
and (5.53)
5 = 2 4 ( ( l - s 0) -
1 / 2
-l)/so4,
where (5.54)
ε 0 = 1 - 2τ 0 - 4*0 ,
and (5.55)
s0 =
2
4t$/e 0,
A NEW PARADIGM IN QUANTUM CHAOS: AUBRY'S THEORY ETC.
63
provided (5.56)
ε 0 > 0,
s 0 < 1 and
1 - Bt$ > 0.
Equation (5.51) can easily be integrated, and gives (5.57)
2
-|^log(l-^ ),
r(t)^
as long as ί ^ t0 and τ ^ τ 0 . By trial and error, the following values were found to work: (5.58)
«o = 0.0767116,
(5.59)
τ 0 = 0.0705115.
They are close to optimal values obtainable from (5.52)-(5.57). Hence the continuation theorem holds at least for t ^ t0 given by (5.58). 6. - Properties. The bipolaronic states found above have a number of interesting properties, which are discussed in[l]. We recall first of all that they have an electronic gap (4.7). This suggests that the material will be an insulator. Secondly, the bipolaronic states are all nondegenerate local minima of W. 2 The second variation D W at an equilibrium is just M (5.37). So for | | / | | ο ο < 1 we have (6.1)
2
||D W|U^l-||Jl|oo>0.
In fact, the first inequality is an equality because the diagonal elements of J are all nonnegative, but this is not important. Thirdly, they have a phonon gap. Small vibrations ξη of the classical variables un about an equilibrium are governed by (6.2)
ξ=
2
-Ό Ψξ,
where we have scaled the masses associated to the classical variables to unity. So the frequencies ω of normal modes of vibration are the square roots of the eigenvalues μ of M. The latter satisfy (6.3)
0<1-||/||2^.
The first inequality is a consequence of the inequality \\J\\2 ^ \\J\\œ (A.9) for symmetric matrices. In fact, one can also show that μ ^ 1, because J can be shown to be nonnegative definite. Fourthly, the response to certain forms of external field can be shown to de-
64
R . s.
M A C K A Y and
c.
BAESENS
cay exponentially. Suppose we modify the energy functional W by adding a term
Σ^Λ,
(6.4)
η
representing the effect of an external field h which interacts with the variables it. Then it is easy to see that the first-order change in an equilibrium state is given by 1
3if/3A = - M " .
(6.5)
We showed in (5.22) that the elements of / decay exponentially from the diagol nal. We now show that the elements of M~ do also. We recall (5.23) (6.6) for N=
N
\Jmn \ ^ \m-n\^l,
?fs ,
and (5.32)
(6.7)
\ J n \n
^
±(2+
Let C be the matrix whose elements are given by these bounds. Then the analysis of sect. 5 showed that ||C||oo < 1 for t < t0, with the explicit bound (5.36). Supl pose we can show that the elements of (I - C)~ decay exponentially from the 1 diagonal. Then the same estimates hold for M " . This is because, for ||C||oo < 1, we have | | / | | ο ο ^ ||0|| 00 < 1 and hence the absolutely convergent series expansion (6.8)
M~
l
l
= (I-
J)~ =
k^O So (6.9)
Mmn
— Σ
Σ
Jm,
—'Jmk-Amk
>
k^o r where γ ranges over all paths of length k from m0 = m to mk = n. This can be bounded in absolute value by the corresponding sum for C, which sums to 1
To show exponential decay for the elements of (/ - Ο " , we adopt a slightly more general setting. Let (6.10)
A =
I-C.
It is symmetric, positive definite, of bounded / 2-norm and commutes with the Z Z shift Θ: R — > R , defined by (Θύ)η = u n + 1. Denote its elements by (6.11)
Amn = a | m_ w( .
A NEW PARADIGM IN QUANTUM CHAOS: AUBRY'S THEORY ETC.
65
Its spectrum is the set of values Α(θ) = a0 + 2 Σ an cos ηθ,
(6.12)
for real Θ, with eigenvectors ξ(θ) whose components are (6.13)
ξη = exp [ίηθ].
Denote its inverse by B. It has the same eigenvectors ξ(θ) as A, with eigenvalues (6.14)
Β(θ) = 1/Α(θ),
so it has the same form Bmn = 6 | m_ w| ,
(6.15)
with {bn} the Fourier coefficients of Β(θ), i.e. Β(θ) = b0 + 2 Σ Κ cos
(6.16) Now, suppose the {an} (6.17)
.
decay exponentially: 1/n
α = lim sup \an\
< 1.
From (6.6), this can be achieved, with α = s. Then we claim that the {bn } also decay exponentially, though possibly with a different exponent (6.18)
l/n
ß = lim sup
\bn\
To prove this, notice that A(6) is analytic in the strip | I m 0 | < log a " . By positive definiteness, Α(θ) ^ A m ni > 0 on the real axis, where A m ni is the smallest eigenvalue of A, which can be estimated by (6.19)
Am
ni
= 1 - lldk ^ 1 - ||C||. .
It is also periodic in θ with period 2π. Hence there exists γ > 0 such that Α(θ) is analytic and nonzero for | Im θ \ < γ. Hence Β(θ) is analytic in | Im 01 < γ, and so the {bn} decay exponentially with (6.20)
/2 ^ exp [ - 7].
To obtain an explicit estimate of β, we need to estimate how close the nearest zero of A(0) can be to the real axis. Now the derivative (6.21)
A ' (θ) =
-
2
Σ
η
α
η
sin ηθ.
So (6.22) 5 - Rendiconti S.I.F. - CXIX
|A'(0)| ^ 2 Σ n\an \ coshny,
66
R.
s.
MACKAY
and
c.
BAESENS
for I Im θ I ^ y. Using the bound (5.22) Κ I^
(6.23)
which, though weaker than (6.6), is easier to work with, we obtain (6.24)
|A'(fl|
S ]e X p [ j /
2
\(l-sexp[y])
.+
2
S e X p [] _ 2 /
^ 2 (1 - sexpf-*/]) J
Hence, integrating along vertical lines in the complex plane, we deduce that (6.25)
\Α(Θ) I >
λ η ώ ι
- ε"
1
- . 1 - 2scosh?/ 4- s
So if (6.26)
κ = ελ π
s sinh y + s 2 1 - 2scosh?/
then I A(0) | > 0. Let γ be the first y for which this is violated, which one can calculate as a function of s and κ, by solving a quadratic equation in exp[y]. We obtain 1
(6.27)
( s " + s)/c + V γ = log -
1
+
(s
_ 1
-s)V
, 1 < log s - .
Then we deduce (6.20). Note that from (6.19) and (5.36) κ can be bounded by (6.28)
2
κ ^ ε - (2s + 3 s / 2 + 2(1 - s ) "
1 /2
- 2).
With a little more work one could no doubt deduce explicit estimates of the form n
(6.29)
bn**C'ß' ,
though one may have to take ß' slightly larger than β. Fifthly, the above result has a noninfinitesimal version. Suppose one changes the initial configuration at t = 0 by adding or removing one bipolaron. What is the change in the configuration at nonzero ϊ! It is shown in[l] that the change decays exponentially from the site where the alteration took place. The l exponent can be taken to be the same as for the decay of the elements of M~ . By adding pseudospins am one at a time, this result allows one to represent the solution M(0 which continues u(0) = - σ as a sum (6.30)
un=
Σ m: am = l
m
(b )n
A NEW PARADIGM IN QUANTUM CHAOS: AUBRY'S THEORY ETC. m
with b
67
a configuration exponentially localized around site m, and
(6.31)
m
E(6 )™=-1. η
m
Thus the solution can still be thought of as consisting of localized distortions b which we continue to call bipolarons, but now they are slightly extended. ν Note that the eigenfunctions ψ may or may not be localized. In particular, if σ is periodic, then they can be chosen to be Bloch waves which are not localized. But this is irrelevant, both physically and mathematically, because all that matters is the span Ο of the occupied eigenstates. Other properties are discussed in[l]. 7. - Extensions.
These results can be extended in many directions. We mention a few. Firstly, one might ask what are the limits to the continuation theorem for From the implicit-function theorem, u(t) has a locally unique continuation _ 1 if it has electronic and phonon gaps (strictly speaking, | | Μ | | ο ο < o o ) . The former guarantees that S is differentiate, the second is the required nondegeneracy condition. Numerical results of Le Daeron [3] and Raimbault [4] indicate in the ID case (2.2) that electronic and phonon gaps are preserved up to t — 0.40 for a Fibonacci choice of σ (that is, the sequence in which a straight line of slope golden ratio cuts the horizontal (ση = 0) and vertical (ση = 1) lines of the integer lattice), and that it is the phonon gap that is lost there. Generically for a finite system, crossings of energy levels are avoided in oneparameter families, so in this case the first condition to fail must be phonon gap. Beware, however, that degeneracies of energy levels can occur robustly in twoparameter families (for real symmetric H) and three-parameter families (for general Hermitian H); also, even for one-parameter families there may be very close energy levels sometimes. Furthermore, for an infinite lattice, the Hamiltonian might not diagonalize over bound states. We conjecture that G, J and l M~ have exponential decay as long as there are electronic and phonon gaps. Secondly, one might ask what happens when the electronic gap or the phonon gap is lost. Generically, if M loses invertibility, but S remains different i a t e , we expect some sort of bifurcation, that is, annihilation by or merging with another equilibrium state. This is a theorem in the finite case, giving fold bifurcations or (if there is some symmetry) pitchfork or transcritical bifurcations. This applies also to periodic configurations of an infinite system. The equilibrium states with which our local minima collide must generically be of 2 Morse index 1 (i.e. one negative eigenvalue for D W ) . As these do not exist at the anti-integrable limit, it is an interesting question where they come from. Maybe they are present as nondifferentiable critical points of W. One should be
68
R.
s.
M A C K A Y and
c.
BAESENS
able to use the minimax principle (mountain pass lemma) to prove existence of many minimax points, which are generically index-1 equilibria for smooth systems. We do not know, however, if there is a general analogue of the bifurcation theorem for infinite systems. Numerical results of [3] for quasi-periodic configurations σ with ratio ω of number of electron pairs to sites (generalizations of the Fibonacci chain to other slopes) suggest that u(f) makes a transition to a state where there exists an analytic function g of period 1 such that (7.1)
u
n =
g ( n a j )
and the transition greatly resembles that from a cantorus to an invariant circle for area-preserving twist maps. So the state merges into a continuum of states related by phase shift. This has not yet been explained. Thirdly, one might ask what is the ground state. Unlike for area-preserving twist maps, there is not yet a general theory of the minimum-energy states for the adiabatic Holstein model, but it is proved in [1] that the ground state must be one of these bipolaronic states for t < t0. It is straightforward to extend the theory to other lattices and other operators A. We believe that the operator A need not even be of finite range; exponentially decaying interaction should suffice. Polaronic and mixed polaronic-bipolaronic states can also be considered. If instead of putting the electrons in the lowest levels one allows any choices (subject to Pauli's exclusion principle), then the electronic density at t = 0 can take the values 0, 1 or 2 at each site, instead of just the values 0 and 2. Such states can also be continued for t small. In a magnetic field the spin degeneracy is lifted and the ground state may even be one of these states if the field is strong enough. Finally, it has been suggested by AUBRY that it may be possible to include temperature in the model, via an electronic filling function (7.2)
1
σ(Ε) = (1 + exp [(E - EF )/kT\) " ,
instead of σ(Ε) = 1 for E G Ι_λ, 0 for Ε G 7 0· There are other models of charge density wave systems which may also be amenable to similar analysis [1]. It is clear that this is a very interesting direction for further research.
Note added in proofs. We have recently simplified the proof of the continuation result still further, and extended it to many generalizations of the adiabatic Holstein model (Warwick preprint, to appear).
A N E W PARADIGM
IN QUANTUM
CHAOS:
AUBRY'S
THEORY
69
ETC.
** *
We thank S . AUBRY for introducing us to this problem, and S . AUBRY, G. and J . - L . RAIMBAULT for answering many questions and providing early versions of the paper [1]. C.B. thanks the Equipe Turbulence Plasma at the Université de Provence and the Institut Méditerranéen de Technologie, Marseille, for their hospitality while these notes were written up.
ABRAMOVICI
APPENDIX
A
Notation and basic mathematical results. In this appendix, we summarize our notation and recall the basic mathematical results we use. For further background on functional analysis, see, e.g. [5,6]. 1. A Banach space is a complete normed linear space U. We denote the norm of u e U by ||%||. Examples: Denote the set of doubly infinite sequences u = {un} of real Z numbers by R . Then the following are Banach spaces: Z
(A.1)
/„
=
{a
G R
(A.2)
/ = {if G R : \\u|| = ^ Σ K |
(A.3)
Λ =
: LLICLLOO =
sup
\un
\ <
2
Z
2
2
<*>},
< <
Z
u
G R
: ||M||I =
Σ
\un
I
<
2. The norm of a linear operator A:U —> V from a Banach space U to a Banach space V is defined by (A.4)
For e x a m p l e , for A :/„—»/» (A.5)
sup{W|/|k||:|kHO}.
|[A|| =
, ||A||„ = s u p S \Amn\ m
,
η
for A : / 2 ^ / 2 , (A.6)
||A||2 = sup { I λ I : λ G spec A } ,
and, for A : /x —>/Xj (A.7)
ΙΐΑίΙ^βιιρΣίΑ™!. n
m
The linear operator A is called bounded if ||A|| < oo. It is saidl to be invertible if _1 1 there exists a bounded linear map A such that AA = A~ A = I. Note the general relation (A.8)
||Α||<||Α|Μ|Α||.,
70
R. Ύ
proved by considering Α Α,
s.
MACKAY
and
c.
BAESENS
and its special case for a symmetric matrix A
(A.9)
IIaIU ^ M U .
3. A map F: U-+V from a Banach space U to a Banach space V is called 1 C if for each ueU there exists a bounded linear map, denoted by OF : U —> V, such that (A.10)
\\F(u + ξ) - F(u) - DF?|| = o(||?||)
as ||f||
0,
and OF depends continuously on u G U. 7
4. The product U x Τ of two Banach spaces t/ and T is a Banach space with the norm (A.11)
\\(u, i)|| = max (||u||, ||t||),
ueU,
teT.
The1 partial derivatives of a function1 F : U x T-^V with respect to w G U and k î are denoted by T>VF and D^i , respectively. 5. Implicit-function1 theorem. Let t/, Τ and y be Banach spaces and F: U x T^>V be a C map; suppose that ^(0) e Î7 satisfies F(u(0), 0) = 0 and OuF is invertible at (u(0), 0). Then there exists Ε > 0 and neighbourhood U' of u(0) such that for ||f|| < Ε there is a unique u(t) e U' such that (A.12)
F(u(t),t)
= 0.
1
Furthermore u(t) depends C on t, with 1
(A.13)
OTu = - (DuF)- OTF(u(t),
t).
The implicit-function theorem is proved using the contraction mapping principle. 6. Application to 1 fixed-point problems. Let U, Τ be Banach spaces and S : U x Τ —> U be a C map; suppose that ^(0) G U satisfies S(u(0), 0) = ^(0) and that / - OuS is invertible at (u(0), 0). Then, by applying the implicit-function theorem to the map F: U x Τ —> £/ defined by (A.14)
F(u,t) =
S(u,t)-u,
we find there is a locally unique fixed point u(t) (i.e. S(u(t), i) = u(t)) and (A.15)
APPENDIX
OTu = (I - OuSy'OrSiuit),
t).
Β
Comments on the proof of [1]. In this appendix we summarize the proof given in[l] for continuation of bipolaronic states and comment on it. Rather than using the implicit-function theorem, they use the contraction
A
N E W
P A R A D I G M
IN
Q U A N T U M
C H A O S :
A U B R Y ' S
T H E O R Y
71
E T C .
mapping principle directly. They search for a fixed point of St in a set of configurations of the form (B.l)
ί/(σ, r) = {u:une[-l,
- l + τ] if ση = 1, un e [ - τ , 0] if ση = 0}
for some 0 < τ < 1/2. They prove that St is a contraction on this set for t < some t0, when τ(ί) is chosen suitably, and hence there is a unique fixed point u(t) of St in υ(σ, τ) and it depends continuously on t. They restrict attention to finite lattices, as we shall do also, and at the end they prove a result which allows them to take the infinite limit. For simplicity of exposition, we will again consider only the one-dimensional case (2.2). To prove St is a contraction on ί/(σ,τ) one has to show that St maps ί/(σ, τ) into itself and contracts distances by at least some λ < 1, in some metric. They show that St maps ί/(σ, τ) into itself if (B.2)
2
2
8t + τ ^ 4τ(1 - τ)(1 - 2τ - 4t).
Thus, for £ ^ ti, the maximum of t in this region, £/(σ, τ) is mapped into itself for any τ in the interval τλ (t) ^ τ ^ τ2 (t), where τλ (t), τ 2 (t) are the lower and upper boundaries of this region. Since £/(σ, τ) is compact and convex and St is continuous in a, it follows by a theorem of Brouwer that ί/(σ, τ) contains a fixed point. Note that it is not necessary here to state which norm we are using, as all norms are equivalent for a finite system. To show that the fixed point is unique and depends continuously on t, it suffices to show that St is a contraction in some metric. We will choose / 2 (see appendix A), as this is the simplest to get results for. To prove contraction in / 2 , since £/(σ, τ) is convex it suffices to show that the derivative J = OuS has / 2 norm less than 1 in [/(σ, τ), for t and τ small enough. This was done in earlier reports by AUBRY and ABRAMOVICI, referred to in[l], though not in[l] itself. The operator / is represented by a matrix (see[l]) (B.3)
Jmn = ^—
=
OUm
Ζ
^ —
tiv>
v e O , v ' e O '
+C.C,
— tiv
where Ο ' denotes the unoccupied states and c.c. denotes complex conjugate. To estimate ||/|| 2 we first use a trick given in [1] to make clear νthat / is small. By ν eliminating un between the eigenequations for ψ and ψ one obtains the identity (B.4)
ν
tW? = (Εν.-Εν)ψηψ η*
,
where ν
(B.5)
ν
= ψ; * (Δψ ) η - φΐ (Δψ ' )* .
Hence (B.6)
Jmn = t
2
Σ
m
\
+C.C,
72
R . s.
M A C K A Y and
c.
BAESENS
2
which has an explicit factor t in front. Now, since / is symmetric, T
(B.7)
Ύ
\\J\\2 = sup {X JX
:ΧΧ
= 1}.
Using Ev> - Ev ^ ΔΕ, and then extending the sum to all v, v', we obtain 2
(B.8)
\\j\\2^16t /àE*.
Hence, for 2 1/3
(16£ ) < 1 - 2 τ - 4 ί ,
(B.9)
2
we have \\j\\2 < 1. We see that there is a region A G R such that, for (t, τ) e A, St is a contraction of ί/(σ, τ). Let £0 be the supremum of t over the region A. Then for ί < i 0 we have a unique fixed point in [7(σ, τ) for suitable τ and it depends continuously on t. To deduce uniform continuity of u(t) (i.e. independent of the size of the lat1 tice), one has to find /«, estimates on M " and OTS, and then use (4.5). In[l], / o o estimates on J and DTS are found by using a polynomial approximation for the filling fraction σ(2?) = 1 for E G 7_ χ , 0 for Ε e / 0> and then estimates similar to the one we made above for ||/|| 2. This works, but is tedious and also gives poorer estimates 1than we found in sect. 5. This allows them to deduce exponential decay of M " and to take the limit of an infinite lattice.
APPENDIX
C
Solution of a recurrence inequality. In this appendix, we prove that, if (C.I)
P
N
^ K ( P
N
.
X
- P
N
+
) L,
for Ν ^ 1, some Κ ^ 0, and PN are nonnegative and decreasing, then (C.2)
Ν
Ρ Ν ^ ζ λ
1
foriV^l,
Ρ 0
where (C.3)
ζ = min(l, K),
and (C.4)
λ = 2K/{1
2 1/2
+ (1 + 4K ) }
.
Let (C.5)
ΡΝ =
ΡΝ+Ι/ΡΝ·
Then (C.6)
1
O^p^p^-tf- ,
N&l.
A
N E W
P A R A D I G M
IN
Q U A N T U M
C H A O S :
A U B R Y ' S
T H E O R Y
E T C .
73
The quantity λ in (C.4) is one root of the equation 1
(C.7)
X = X~ -K~
1
.
It follows that, if pN > λ, then pN+1 < λ and furthermore (C.8)
l
1
^ 1 - K~ PN
<1-Κ~ λ
N P N + l P
2
= λ.
Hence 2
(C.9)
ΡΝ+2<λ ΡΝ.
If ρ ^ λ, then by definition (CIO)
ΡΝ+1^λΡΝ.
Hence Ν 1
(C.ll)
ΡΝ^λ ~ Ρ0.
In the case Κ < 1 this result can be improved by noting that pN ^ Κ for all Ν ^ 0, else pN+1 < 0, which is not permitted. Hence in this case we obtain (C.12)
Ρχ^Κχν-'Ρο.
This completes the proof.
APPENDIX D
Variation of the electronic energy with configuration u. In this appendix we prove that (D.l)
^
2
= Σ\Φΐ\
=
(η\Ρ\η),
where (D.2)
Ρ
=
Σ
Ε
ν
veO
is the total electronic energy and the other notation is as in (2.6) and (4.14), (4.15). If there is no degeneracy, this can be proved by standard perturbation theory. One differentiates ( 2 . 3 ) with respect to un and takes the inner product with ψ, assumed normalized, leading to
which when summed yields (D.l). If there is degeneracy, however, it is ν not always possible to choose eigenfunctions ψ to depend differentiably on parameters. The spectral projection operator P, nonetheless, remains
74
R . s.
M A C K A Y and
c.
BAESENS
differentiable provided there is an electronic gap, and this allows a proof (in fact we will give two proofs) of (D.l) as follows. The energy F can be written as (D.4)
F = Tr
HP.
For the first proof, differentiating (D.4) gives ÖUN
\ ÖUN
ÖUN
Since H = u - tA, we obtain
(D.6)
Ψ^= η> π
the projection onto the state \n). Taking the basis {|m)} shows that the first term of (D.5) reduces to (D.7)
TrKNP
=
(n\P\n).
Using the decomposition of the space of wave functions into Ο Θ Ο', which are the ranges of the projection operators Ρ and (D.8)
Q = I - P ,
write the second term of (D.5) as (D.9)
Tr ( Ρ Η ψ - Ρ + QH^Q)
.
2
Now Ρ = Ρ, and differentiating this gives (D.10)
FQ
= ^
and
aun
oun
aun
oun
Using HP = PH, (D.9) becomes (D.ll)
TY(HPQ^-
+
HQP^—
dun
\
dun
But QP = PQ = 0. Hence this is zero and the result (D.l) follows. The second proof uses formula (4.14) for P . From (D.4) (D.12)
F = Tr - ^ r ί EG(E) dE. 2 τα J r
Using (D.6), (D.13)
^ L
=
T
A _ ( r
EG
q M
A N E W PARADIGM
IN QUANTUM CHAOS: AUBRY'S THEORY ETC. ν
Now use a basis of eigenstates {ψ }
75
to find
2
By the calculus of residues this reduces to Σ | ft | , as required. veO
REFERENCES [1] S. A U B R Y , G. A B R A M O V I C I and J.-L. R A I M B A U L T : J. Stat. Phys., 67, 675 (1992). [2] A. M E S S I A H : Quantum Mechanics, Vol. II, Chap. XVI, para. 15 (Wiley, New York, N.Y., 1962). [3] P. Y. L E D A E R O N and S. A U B R Y : / . Phys. C, 16, 4827 (1983); /. Phys. (Paris), 44, C3, 1573 (1983). [4] J.-L. R A I M B A U L T : Configurations bipolaroniques et fluctuations quantiques du reseau dans les chaînes de Peierls, thèse de doctorat (Nantes, 1990). [5] M. R E E D and B. S I M O N : Methods of Modem Mathematical Physics: I. Functional Analysis (Academic Press, New York, N.Y., 1972). [6] V. T R É N O G U I N E : Analyse fonctionelle (Mir, Moscow, 1985).
Nonlinear Systems Laboratory, Mathematics Institute, University of Warwick Coventry CVJ> 7AL, U.K. C . BAESENS
Laboratoire Léon Brillouin, CEΝ Saclay - 91191 Gif-sur-Yvette, France
1. - Introduction. AUBRY and co-workers have introduced a new direction in the subject of quantum chaos: they show existence of spatially chaotic equilibrium states for some models of electron-phonon systems [1]. This is in our opinion a most interesting theoretical development, and furthermore it sheds light on many experimental results on charge-density waves. In this lecture, we give a pedagogical treatment of Aubry's theory, with some simplifications and improvements in the proofs. The approach is closely analogous to the derivation of chaotic orbits for symplectic twist maps near an anti-integrable limit, described in sect. 3 of the preceding lecture.
2. - The model. The adiabatic Holstein model consists of i) classical variables un (which can be thought of, for example, as the deviation of the length of a molecular bond from a reference value) at sites η in some lattice L, which we shall take initially to be finite with Ν sites, and ii) 2M electrons which are supposed noninteracting (apart from Pauli's exclusion principle), and which fill up the energy levels of a tight-binding Hamiltonian (2.1)
H =
u-tA, 51
52
R.
s.
M A C K A Y and
c.
BAESENS
in pairs (spin up and down) from the lowest level until the electrons are all used up. The operator Δ is assumed to connect each site to only finitely many sites, e.g. symmetric nearest-neighbour coupling on a one-dimensional lattice (2.2)
(Δψ)η = ψ η_ι + ψ η +ι ,
or possibly to have exponential decaying interaction between distant sites. Here ψ is the wave function for up-down pairs of electrons, and {ψη } are its components with respect to the basis of on-site wave functions {\n)} at sites n. The Schrödinger equation for (2.1) can be written as (Ηψ)η = ηηφη-ίΣ
(2.3)
= Εφη .
\imΦπι m
The term u in the Hamiltonian represents an effect of the {un } on the electrons, and the term tA represents hopping of electrons between sites. A configuration u = {un } is assigned an energy (2.4)
W(u, t) = ±
Σ ni + Σ Ε η
Δ
Ν
(Μ,
ί),
veO
where ν is a label for the eigenstates and Ο denotes the occupied eigenstates of H. The aim is to find the equilibrium states of the model, that is, the configurations u of stationary W. The condition of stationarity is (2.5)
v
- Σ
un=
dE /dun.
veO
Evaluating the right-hand side leads to (2.6)
UN=
-
Σ
2
\Φη\ =
- P n ,
veO
ν
where the {ψ } are an orthonormal set of eigenvectors corresponding to energies E\ A proof is given in appendix D. The quantity pn can be interpreted as the probability of finding an up-down electron pair on site n.
3. - The anti-integrable limit t = 0. In the case t = 0, the sites are uncoupled and it is easy to solve the model exν plicitly. The eigenstates can be chosen to be ψ = | v), the wave function on site v v, with energies E = uv. Thus the occupied states are the M states for which uv is smallest, and the energy of a configuration u is (3.1) Δ
n
v e O
A
N E W
P A R A D I G M
IN
Q U A N T U M
C H A O S :
A U B R Y ' S
T H E O R Y
53
E T C .
The stationary states are seen to be those for which (3.2)
Un --
1 0
if if
weO, ntO.
This means we can arbitrarily choose M of the sites to have un = - 1 and the remaining Ν - M sites to have un = 0. Since p n = - un for equilibrium states, the sites with un = - 1 have an electron pair and those with un = 0 do not. The combination of un = - 1 and p n = 1 is called a bipolaron. Thus the equilibrium states correspond to all the ways of putting M bipolarons on a lattice of Ν sites. Note that all the equilibrium states have the same energy, namely W = = - (1/2) M. Furthermore, the electronic spectrum consists of an M-fold degeneracy at Ε = - 1 , which are all occupied, and an (N - M)-fold degeneracy at Ε = 0, which are unoccupied, so there is a gap between the occupied and unoccupied levels. The above discussion shows that we have a form of spatial chaos at t = 0. This result is very natural in the case t = 0 because then the sites are independent. What is significant is that these states can be continued for t nonzero provided it is small enough, as we now explain.
4. - Small hopping. Suppose the hopping amplitude t is small but nonzero. The main result of [1] is that for each choice of equilibrium configuration u at t = 0, with un= - ση, ση = 0 or 1, there is a unique continuation u(t) as an equilibrium state of Wt, for t small enough (where Wt(u) means W(u, t)). Thus the chaos persists for small t. The {ση } are called pseudospins. In principle, there might also be other equilibrium states, but it is shown in[l] that they must have significantly higher energy. Now we give a proof of this continuation result. We restrict attention to the one-dimensional case with nearest-neighbour coupling only (2.2). Our notation and basic mathematical tools are explained in appendix A. We compare our proof with that of [1] in appendix B. From (2.6), the equilibrium states are the fixed points of the operator (4.1)
St(u) = u',
where
(4.2)
<=-Σ ι « ι . 2
veO
54
R.
s.
MACKAY
and
c.
BAESENS
Note that it follows that (4.3)
-l^un^0,
for all equilibrium states. Given a configuration ιι(0) = - σ, we define a space U of perturbations u with the norm (4.4)
τ = ||u|| = sup \un + ση | .
The space U is a Banach space (see appendix A) with origin shifted to - σ. The 1 operator S can be thought of as mapping U χ Τ —>U where 7 = R represents the set of possible We will show that S is differentiate with respect to (if, t) e U x Γ for τ and £ small enough, and that u(O) is a nondegenerate fixed point of S0 (i.e. 1 g specDf//S0(ii(0))). Hence by the implicit-function theorem (appendix A) there is a locally unique fixed point u(t) of St, for t small enough, and it depends differentiably on t It can be continued as long as / - Ou S remains invertible, and it satisfies the differential equation (4.5)
du/dt = (I-
1
OuSy
dS/dt
(u(t)).
We will use this in sect. 5 to obtain good estimates of how far u(t) can be continued. Step 1: Electronic gap. The first step is to prove that, for t and τ small enough, the spectrum of Η continues to consist of M eigenvalues near - 1 and Ν -M eigenvalues near 0. More precisely, it is shown in[l] that for (4.6)
τ + 2t <
I
the spectrum of Η divides into M eigenvalues in an interval (4.6α)
7_! = [ - 1 - 2 ί , - 1 + τ + 2ί],
and Ν — M eigenvalues in an interval (4.66)
/0 = [ - τ - 2 ί , 2 ί ] ,
with / _ ! and I0 disjoint. So we have an electronic gap (4.7)
AE^s
= l-
2r-4t.
We give a slightly different proof of this result, which serves to introduce some of the ingredients we will use later. For t = 0, the spectrum of Η consists precisely of the values un, and hence for τ < 1/2 we have the desired result. For Ε φ spec Η', define the resolvent operator (4.8)
G(E) =
(E-Hy
1
on wave functions {e.g., [2]). We will often drop the dependence on Ε from the
A NEW PARADIGM IN QUANTUM CHAOS: AUBRY'S THEORY ETC.
55
notation. Now H depends differentiably on t, and computing the derivative of G with respect to t gives (4.9)
- GAG,
DtG=
as long as G remains defined. Taking / 2 norms (see appendix A), we obtain (4.10)
d||G||2/d^||G||I||/l||2.
The operator G has / 2-norm (4.11)
l|G||2 = < R \
where S is the shortest distance from Ε to the spectrum of H, and, in the 1-D case (2.2) we are considering, (4.12)
114 = 2 .
Solving the differential inequality (4.10) yields (4.13)
*(0)-<*(ί)^2ί.
Hence the spectrum of Η moves by at most 2t from its position at t = 0. Since Η is Hermitian, the spectrum is real, proving the required result. Step 2: Reformulation. Our next step is to rewrite (4.2) in terms of the spectral projection operator P. Let Γ be a contour in the complex 2?-plane which surrounds the interval 7_χ once but excludes the interval I0. Then define (4.14)
P=ih\ 2πί
J
G(E)dE,
which is the projection onto the space spanned by the eigenvectors corresponding to spectrum inside Γ (e.g., [2]). It follows that (4.2) can be rewritten as (4.15)
u;=-(n\P\n).
Step 3: Differentiability. To carry out the proof via the implicit-function theorem, we need to show that S is jointly differentiate with respect to (u, t) and that 1 is not in the spectrum of OuS0(u(0)). Joint differentiability is automatic from (4.14), (4.15) as long as there remains an electronic gap so that the contour Γ can be chosen to avoid the spectrum of H. Since we will need to refer to it often, we will write (4.16)
J = OuS.
56
R.
s.
MACKAY
and
c.
BAESENS
Then J is given by (4.17)
Jmn = - (m I dP/dun \m) = -
\{m\G(E) \ n)(n \ G(E)
\m)dE.
r Writing (4.18)
Gmn =
(m\G\n),
then (4.19)
=
Jmn
-
J Gmn Gnm
dE.
Γ
Note that this reformulation has the additional advantage that it allows one to take an infinite lattice from the start. Step 4: Nondegeneracy. Finally, we show that 1 φ s p e c / at ii(0), t = 0. This is easy because / = 0 at t = 0. To see this, note that at t = 0 (4.20)
for m * η,
Gmn = 0
and (4.21)
Gnn = (E-uny
l
.
Hence the off-diagonal terms of J in (4.19) are automatically zero and the diagonal terms are given by integrals of double poles, which are also zero. This completes the proof of the continuation theorem. In the next section, we find explicit estimates for how large a coupling t these states can be continued to. 5. - Explicit estimates. We will show that u(t) can be continued for at least t < tQ = 0.089939 (eq. (5.46)). Our strategy of proof is that as long as / - / remains invertible, u(t) can be continued. We use the differential equation (4.5) to derive a differential inequality for r(t), which we solve. For this we need estimates on J and OTS. We derive them via estimates on the resolvent operator G. Step 1: Exponential decay of Gmn. For S a positive lower bound to the distance from Ε to the spectrum of H, the elements of G(E) are all bounded by (5.1)
1
IG™ I s M l G H ^ - .
We will show that the off-diagonal elements are exponentially small:
(5.2)
\Gmn\
2
ssal™-*!/ ,
57
A NEW PARADIGM IN QUANTUM CHAOS: AUBRY'S THEORY ETC.
with 2
2t
(5.3)
2 S
+ V^ + * ' 4
4
4
and (5.4)
λ
9
where 2
(5.5)
2
ζ = π ώ ι ( ί / * » 1).
To prove (5.2), since ||G||2 is finite, the equation (5.6)
|0)
(Ε-Η)ξ=
has a unique solution ξ in / 2 . Given Ν > 0, let yj be the vector with components (5.7)
for
η < N,
for
Ν.
Then η' = (E - H)r) has components (5.8) Y]n = 0 otherwise. Using (4.11), we obtain (5.9) for (5.10)
Ρ =Σ Ν
2
2
2
2
2
| s , | ^ t * - ( | f o l + I&-1I )
^ 1. This can be written as Ρ * ^ Χ(Ρ^_! - PN+1 ),
£ 1,
with (5.11)
K =
2
t
2
/ S
.
This recurrence inequality can be proved to yield (see appendix C) (5.12)
Ν 1
ΡΝ^ζλ ~ Ρ0,
N&l,
with ζ and λ as above. Since |0) has norm 1, (5.13)
2
PQ ^ S~ .
58
R . s.
M A C K A Y and
c.
BAESENS
Hence it follows that 1
2
Ι^Ι^Ρ^ζλ"- *" ,
(5.14) and so
ιfo|
(5.15)
2
^ Α " /
,
for Ν ^ 1. But ξΝ is just Gm. The same technique can be used for Ν < 0 and for the other columns of G. This completes the proof of (5.2). Step 2: Exponential decay of Jmn. Using the above estimates (5.1), (5.2) on Gmn, formula (4.19) and the contour Γ which passes up the vertical line Re Ε = = - 1 / 2 and is closed by a large semicircle in the left half-plane, we can obtain good estimates on Since the spectrum of Η is contained in the intervals (4.6a), (4.66), for (5.16)
- i +%, 2
E=
y real, we can choose (5.17)
= ^ l + y \S
where ε is defined in (4.7). Hence, for the off-diagonal elements we find 00
(5.18)
\Jmn\
2
J C A'»-»" dy,
si ±-
— 00
because the contribution from the large semicircle goes to zero as its radius goes to infinity. The integral can be written as twice that from 0 to infinity. Changing variable from y to 2
and using
2
2
= 1 + 4ΐ/ /ε = 4£ /ε
(5.18a) 2
2
ζ^ί /8 ,
2
we obtain 00
(5.19)
I»--I f
ξ ^ - ' ά η
8
7τε
where (5.20) and
s=
2
2
4t /e ,
59
A NEW PARADIGM IN QUANTUM CHAOS: AUBRY'S THEORY ETC.
Since ξ ^ 1 for η ^ 1, the integral in (5.19) is bounded by the case \m - n\ = 1, which can easily be calculated to be π/2. Hence 1
\Jmn \ ^ l e - « ! " - * ! .
(5.22) ι
Note that, using ξ ^ η~ for η ^ 1, one can obtain the slightly better estimate (5.23)
N
^fs ,
\Jnm\<
where (5.24)
Ν =
\m-n\,
and 1 · 3 · 5 · . . . · ( 2 Λ Τ - 1) <
5 2
·
ι
5)
i 1
Step 3: Estimates of\\j\\œ and - J ) " \\M . To obtain good estimates for the diagonal elements of J, we need to subtract out the double pole that we noticed in the case t = 0. It is easy to check from the definition (4.8) that the resolvent operator G satisfies the identity (5.26)
G = G° - tG° AG l
where G° is the case t = 0 (so its matrix Gmn is diagonal with G n n = (E Hence (5.27)
Gnn = (E- uJ-Hl
- t(n|AG|n))
l
= (E - un)~ {l
- t[Gn+hn
un)~ ).
+ G n_ l f J ) .
It follows from (4.19) that (5.28) =
Jnn = " ~h
\ Γ
{ E
~
U
n 2) 1(
~
"
2 t [ G n ++ 1
J+2 G
^
>
' t »+i.n +
G n-
l
t
fn) d E .
The first term gives zero, being a double pole. We estimate the remaining terms using (5.29)
\Gn+hn\
^ ψ
2
,
which comes from (5.2), and \E - un\ ^8. Hence 00
(5.30)
\Jnn\^
^ J ί- (4< /* 2
2
2
4
+ 4*7* )%·
60
R. s. MACKAY and c. BAESENS
Using the change of variables (5.18a), we obtain
: , - i 4 f7jL+ · ΐ \ _ Ε 2 _
(5.31)
(5.32)
l « / r a| = f ( 2 + | s ) .
The estimates (5.23) and (5.32) lead to an estimate for \\J\\„, by summing the estimates for a row of J. Alternatively, one can sum up the integrands in (5.19) rather than the estimates (5.23), and then bound the resulting integral. Following this route, the sum of the off-diagonal elements of a row of J is bounded by (5.33)
2s
J' =
Using ξ =£ η
1
- 1(1
-
SO
for η ^ 1, we obtain 2s
(5.34)
-K>?-*) (5.35)
1
ε
- A .
Vi-
by the calculus of residues. Hence, adding in the estimate (5.32) for the diagonal element, we obtain
^1
(5.36)
2
M
s
+
|
s
^ -2 ν τ ^
2
+
2
=
Thus, if (5.36) is less than 1, then (5.37)
M =
I-J
is invertible, and we obtain the estimate (5.38)
||Af-l. M l
) - ^
Step 4: Bound on OTS. spect to t can be written (5.39)
at
—
!_I
2
e
+
i
e
«
+
_ J
-1 =
From (4.9) and (4.15), the derivative of S with re-
J Σ Gnm(Gm+ijn
+
Gm-if7l)dE.
61
A NEW PARADIGM IN QUANTUM CHAOS: AUBRY'S THEORY ETC.
Using the estimates (5.1) and (5.2) on G, we obtain 00
(5.40)
4ί
2π Jί
di
2
VÂ 1-λ
4ί 4 ί
+
— 00
d#.
Changing variables as in (5.18a) gives d< di
(5.41) Using f ^ ^
1
j3/2 1f / 2
.1/2
d^
+
J/2
for η ^ 1, we can estimate this by
V2 d^
f
(5.42)
dt
(i?-e)V^-l " 2
This integral can be evaluated by the changes of variable η = cosh ζ followed by χ = exp [2z], and gives dun dt
(5.43)
8 πε V T
tg-
Vi
1
Step 5: Differential inequality. The results (4.5), (5.38) and (5.43) lead to the following differential inequality: 8 dr di
(5.44)
tg-
1
Vi VT
valid as long as i) ε > 0, ii) s < 1, and iii) the denominator of (5.44) is positive. We recall that the definitions of ε and s are given in eqs. (4.7) and (5.20), respectively. We proceed now to deduce a bound on r(t) and hence deduce a range 0 ^ t < t0 for which u(t) continues. If zh(t) is the solution of (5.44) with the inequality replaced by equality and initial condition z h (0) = 0, then it follows that (5.45)
r(t)^rb(t),
up to the first t for which one of the above three conditions fails. It is
62
R.
s.
MACKAY
and
c.
BAESENS
easy to see that, if conditions i) or ii) fail, then condition iii) will already have failed, so (5.45) holds up to the first t = t0 for which condition iii) fails. We calculated rb (t) numerically until the denominator passed through 0. Up to numerical errors, the denominator remained positive for t
io = 0.089 939.
Hence u(t) can be continued for at least t
(5.47)
tg~ x
for χ ^ 0.
^x
Secondly, we use (5.48)
2
2s + 3 s / 2 ^ 4((1 - s)~
1/2
- 1)
for s < 1,
as can be seen by Taylor-expanding the right-hand side. Hence we obtain 16* πε (1 - s) 2
dr
(5.49)
^d t
^
~ 1-
1
§((l-SR
/
2
-D
Next, for 0 ^ s ^ s0 < 1, we have by convexity (5.50)
1 2
(1 - s ) " / - 1 ^ ((1 - s o ) "
1 /2
- l)s/so •
Hence, for t ^ t0 and τ ^ τ 0 , we have (5.51)
2
dr/di ^ At/(1
-Bt ),
with (5.52)
A = 16/^(1 - s 0) ,
and (5.53)
5 = 2 4 ( ( l - s 0) -
1 / 2
-l)/so4,
where (5.54)
ε 0 = 1 - 2τ 0 - 4*0 ,
and (5.55)
s0 =
2
4t$/e 0,
A NEW PARADIGM IN QUANTUM CHAOS: AUBRY'S THEORY ETC.
63
provided (5.56)
ε 0 > 0,
s 0 < 1 and
1 - Bt$ > 0.
Equation (5.51) can easily be integrated, and gives (5.57)
2
-|^log(l-^ ),
r(t)^
as long as ί ^ t0 and τ ^ τ 0 . By trial and error, the following values were found to work: (5.58)
«o = 0.0767116,
(5.59)
τ 0 = 0.0705115.
They are close to optimal values obtainable from (5.52)-(5.57). Hence the continuation theorem holds at least for t ^ t0 given by (5.58). 6. - Properties. The bipolaronic states found above have a number of interesting properties, which are discussed in[l]. We recall first of all that they have an electronic gap (4.7). This suggests that the material will be an insulator. Secondly, the bipolaronic states are all nondegenerate local minima of W. 2 The second variation D W at an equilibrium is just M (5.37). So for | | / | | ο ο < 1 we have (6.1)
2
||D W|U^l-||Jl|oo>0.
In fact, the first inequality is an equality because the diagonal elements of J are all nonnegative, but this is not important. Thirdly, they have a phonon gap. Small vibrations ξη of the classical variables un about an equilibrium are governed by (6.2)
ξ=
2
-Ό Ψξ,
where we have scaled the masses associated to the classical variables to unity. So the frequencies ω of normal modes of vibration are the square roots of the eigenvalues μ of M. The latter satisfy (6.3)
0<1-||/||2^.
The first inequality is a consequence of the inequality \\J\\2 ^ \\J\\œ (A.9) for symmetric matrices. In fact, one can also show that μ ^ 1, because J can be shown to be nonnegative definite. Fourthly, the response to certain forms of external field can be shown to de-
64
R . s.
M A C K A Y and
c.
BAESENS
cay exponentially. Suppose we modify the energy functional W by adding a term
Σ^Λ,
(6.4)
η
representing the effect of an external field h which interacts with the variables it. Then it is easy to see that the first-order change in an equilibrium state is given by 1
3if/3A = - M " .
(6.5)
We showed in (5.22) that the elements of / decay exponentially from the diagol nal. We now show that the elements of M~ do also. We recall (5.23) (6.6) for N=
N
\Jmn \ ^ \m-n\^l,
?fs ,
and (5.32)
(6.7)
\ J n \n
^
±(2+
Let C be the matrix whose elements are given by these bounds. Then the analysis of sect. 5 showed that ||C||oo < 1 for t < t0, with the explicit bound (5.36). Supl pose we can show that the elements of (I - C)~ decay exponentially from the 1 diagonal. Then the same estimates hold for M " . This is because, for ||C||oo < 1, we have | | / | | ο ο ^ ||0|| 00 < 1 and hence the absolutely convergent series expansion (6.8)
M~
l
l
= (I-
J)~ =
k^O So (6.9)
Mmn
— Σ
Σ
Jm,
—'Jmk-Amk
>
k^o r where γ ranges over all paths of length k from m0 = m to mk = n. This can be bounded in absolute value by the corresponding sum for C, which sums to 1
To show exponential decay for the elements of (/ - Ο " , we adopt a slightly more general setting. Let (6.10)
A =
I-C.
It is symmetric, positive definite, of bounded / 2-norm and commutes with the Z Z shift Θ: R — > R , defined by (Θύ)η = u n + 1. Denote its elements by (6.11)
Amn = a | m_ w( .
A NEW PARADIGM IN QUANTUM CHAOS: AUBRY'S THEORY ETC.
65
Its spectrum is the set of values Α(θ) = a0 + 2 Σ an cos ηθ,
(6.12)
for real Θ, with eigenvectors ξ(θ) whose components are (6.13)
ξη = exp [ίηθ].
Denote its inverse by B. It has the same eigenvectors ξ(θ) as A, with eigenvalues (6.14)
Β(θ) = 1/Α(θ),
so it has the same form Bmn = 6 | m_ w| ,
(6.15)
with {bn} the Fourier coefficients of Β(θ), i.e. Β(θ) = b0 + 2 Σ Κ cos
(6.16) Now, suppose the {an} (6.17)
.
decay exponentially: 1/n
α = lim sup \an\
< 1.
From (6.6), this can be achieved, with α = s. Then we claim that the {bn } also decay exponentially, though possibly with a different exponent (6.18)
l/n
ß = lim sup
\bn\
To prove this, notice that A(6) is analytic in the strip | I m 0 | < log a " . By positive definiteness, Α(θ) ^ A m ni > 0 on the real axis, where A m ni is the smallest eigenvalue of A, which can be estimated by (6.19)
Am
ni
= 1 - lldk ^ 1 - ||C||. .
It is also periodic in θ with period 2π. Hence there exists γ > 0 such that Α(θ) is analytic and nonzero for | Im θ \ < γ. Hence Β(θ) is analytic in | Im 01 < γ, and so the {bn} decay exponentially with (6.20)
/2 ^ exp [ - 7].
To obtain an explicit estimate of β, we need to estimate how close the nearest zero of A(0) can be to the real axis. Now the derivative (6.21)
A ' (θ) =
-
2
Σ
η
α
η
sin ηθ.
So (6.22) 5 - Rendiconti S.I.F. - CXIX
|A'(0)| ^ 2 Σ n\an \ coshny,
66
R.
s.
MACKAY
and
c.
BAESENS
for I Im θ I ^ y. Using the bound (5.22) Κ I^
(6.23)
which, though weaker than (6.6), is easier to work with, we obtain (6.24)
|A'(fl|
S ]e X p [ j /
2
\(l-sexp[y])
.+
2
S e X p [] _ 2 /
^ 2 (1 - sexpf-*/]) J
Hence, integrating along vertical lines in the complex plane, we deduce that (6.25)
\Α(Θ) I >
λ η ώ ι
- ε"
1
- . 1 - 2scosh?/ 4- s
So if (6.26)
κ = ελ π
s sinh y + s 2 1 - 2scosh?/
then I A(0) | > 0. Let γ be the first y for which this is violated, which one can calculate as a function of s and κ, by solving a quadratic equation in exp[y]. We obtain 1
(6.27)
( s " + s)/c + V γ = log -
1
+
(s
_ 1
-s)V
, 1 < log s - .
Then we deduce (6.20). Note that from (6.19) and (5.36) κ can be bounded by (6.28)
2
κ ^ ε - (2s + 3 s / 2 + 2(1 - s ) "
1 /2
- 2).
With a little more work one could no doubt deduce explicit estimates of the form n
(6.29)
bn**C'ß' ,
though one may have to take ß' slightly larger than β. Fifthly, the above result has a noninfinitesimal version. Suppose one changes the initial configuration at t = 0 by adding or removing one bipolaron. What is the change in the configuration at nonzero ϊ! It is shown in[l] that the change decays exponentially from the site where the alteration took place. The l exponent can be taken to be the same as for the decay of the elements of M~ . By adding pseudospins am one at a time, this result allows one to represent the solution M(0 which continues u(0) = - σ as a sum (6.30)
un=
Σ m: am = l
m
(b )n
A NEW PARADIGM IN QUANTUM CHAOS: AUBRY'S THEORY ETC. m
with b
67
a configuration exponentially localized around site m, and
(6.31)
m
E(6 )™=-1. η
m
Thus the solution can still be thought of as consisting of localized distortions b which we continue to call bipolarons, but now they are slightly extended. ν Note that the eigenfunctions ψ may or may not be localized. In particular, if σ is periodic, then they can be chosen to be Bloch waves which are not localized. But this is irrelevant, both physically and mathematically, because all that matters is the span Ο of the occupied eigenstates. Other properties are discussed in[l]. 7. - Extensions.
These results can be extended in many directions. We mention a few. Firstly, one might ask what are the limits to the continuation theorem for From the implicit-function theorem, u(t) has a locally unique continuation _ 1 if it has electronic and phonon gaps (strictly speaking, | | Μ | | ο ο < o o ) . The former guarantees that S is differentiate, the second is the required nondegeneracy condition. Numerical results of Le Daeron [3] and Raimbault [4] indicate in the ID case (2.2) that electronic and phonon gaps are preserved up to t — 0.40 for a Fibonacci choice of σ (that is, the sequence in which a straight line of slope golden ratio cuts the horizontal (ση = 0) and vertical (ση = 1) lines of the integer lattice), and that it is the phonon gap that is lost there. Generically for a finite system, crossings of energy levels are avoided in oneparameter families, so in this case the first condition to fail must be phonon gap. Beware, however, that degeneracies of energy levels can occur robustly in twoparameter families (for real symmetric H) and three-parameter families (for general Hermitian H); also, even for one-parameter families there may be very close energy levels sometimes. Furthermore, for an infinite lattice, the Hamiltonian might not diagonalize over bound states. We conjecture that G, J and l M~ have exponential decay as long as there are electronic and phonon gaps. Secondly, one might ask what happens when the electronic gap or the phonon gap is lost. Generically, if M loses invertibility, but S remains different i a t e , we expect some sort of bifurcation, that is, annihilation by or merging with another equilibrium state. This is a theorem in the finite case, giving fold bifurcations or (if there is some symmetry) pitchfork or transcritical bifurcations. This applies also to periodic configurations of an infinite system. The equilibrium states with which our local minima collide must generically be of 2 Morse index 1 (i.e. one negative eigenvalue for D W ) . As these do not exist at the anti-integrable limit, it is an interesting question where they come from. Maybe they are present as nondifferentiable critical points of W. One should be
68
R.
s.
M A C K A Y and
c.
BAESENS
able to use the minimax principle (mountain pass lemma) to prove existence of many minimax points, which are generically index-1 equilibria for smooth systems. We do not know, however, if there is a general analogue of the bifurcation theorem for infinite systems. Numerical results of [3] for quasi-periodic configurations σ with ratio ω of number of electron pairs to sites (generalizations of the Fibonacci chain to other slopes) suggest that u(f) makes a transition to a state where there exists an analytic function g of period 1 such that (7.1)
u
n =
g ( n a j )
and the transition greatly resembles that from a cantorus to an invariant circle for area-preserving twist maps. So the state merges into a continuum of states related by phase shift. This has not yet been explained. Thirdly, one might ask what is the ground state. Unlike for area-preserving twist maps, there is not yet a general theory of the minimum-energy states for the adiabatic Holstein model, but it is proved in [1] that the ground state must be one of these bipolaronic states for t < t0. It is straightforward to extend the theory to other lattices and other operators A. We believe that the operator A need not even be of finite range; exponentially decaying interaction should suffice. Polaronic and mixed polaronic-bipolaronic states can also be considered. If instead of putting the electrons in the lowest levels one allows any choices (subject to Pauli's exclusion principle), then the electronic density at t = 0 can take the values 0, 1 or 2 at each site, instead of just the values 0 and 2. Such states can also be continued for t small. In a magnetic field the spin degeneracy is lifted and the ground state may even be one of these states if the field is strong enough. Finally, it has been suggested by AUBRY that it may be possible to include temperature in the model, via an electronic filling function (7.2)
1
σ(Ε) = (1 + exp [(E - EF )/kT\) " ,
instead of σ(Ε) = 1 for E G Ι_λ, 0 for Ε G 7 0· There are other models of charge density wave systems which may also be amenable to similar analysis [1]. It is clear that this is a very interesting direction for further research.
Note added in proofs. We have recently simplified the proof of the continuation result still further, and extended it to many generalizations of the adiabatic Holstein model (Warwick preprint, to appear).
A N E W PARADIGM
IN QUANTUM
CHAOS:
AUBRY'S
THEORY
69
ETC.
** *
We thank S . AUBRY for introducing us to this problem, and S . AUBRY, G. and J . - L . RAIMBAULT for answering many questions and providing early versions of the paper [1]. C.B. thanks the Equipe Turbulence Plasma at the Université de Provence and the Institut Méditerranéen de Technologie, Marseille, for their hospitality while these notes were written up.
ABRAMOVICI
APPENDIX
A
Notation and basic mathematical results. In this appendix, we summarize our notation and recall the basic mathematical results we use. For further background on functional analysis, see, e.g. [5,6]. 1. A Banach space is a complete normed linear space U. We denote the norm of u e U by ||%||. Examples: Denote the set of doubly infinite sequences u = {un} of real Z numbers by R . Then the following are Banach spaces: Z
(A.1)
/„
=
{a
G R
(A.2)
/ = {if G R : \\u|| = ^ Σ K |
(A.3)
Λ =
: LLICLLOO =
sup
\un
\ <
2
Z
2
2
<*>},
< <
Z
u
G R
: ||M||I =
Σ
\un
I
<
2. The norm of a linear operator A:U —> V from a Banach space U to a Banach space V is defined by (A.4)
For e x a m p l e , for A :/„—»/» (A.5)
sup{W|/|k||:|kHO}.
|[A|| =
, ||A||„ = s u p S \Amn\ m
,
η
for A : / 2 ^ / 2 , (A.6)
||A||2 = sup { I λ I : λ G spec A } ,
and, for A : /x —>/Xj (A.7)
ΙΐΑίΙ^βιιρΣίΑ™!. n
m
The linear operator A is called bounded if ||A|| < oo. It is saidl to be invertible if _1 1 there exists a bounded linear map A such that AA = A~ A = I. Note the general relation (A.8)
||Α||<||Α|Μ|Α||.,
70
R. Ύ
proved by considering Α Α,
s.
MACKAY
and
c.
BAESENS
and its special case for a symmetric matrix A
(A.9)
IIaIU ^ M U .
3. A map F: U-+V from a Banach space U to a Banach space V is called 1 C if for each ueU there exists a bounded linear map, denoted by OF : U —> V, such that (A.10)
\\F(u + ξ) - F(u) - DF?|| = o(||?||)
as ||f||
0,
and OF depends continuously on u G U. 7
4. The product U x Τ of two Banach spaces t/ and T is a Banach space with the norm (A.11)
\\(u, i)|| = max (||u||, ||t||),
ueU,
teT.
The1 partial derivatives of a function1 F : U x T-^V with respect to w G U and k î are denoted by T>VF and D^i , respectively. 5. Implicit-function1 theorem. Let t/, Τ and y be Banach spaces and F: U x T^>V be a C map; suppose that ^(0) e Î7 satisfies F(u(0), 0) = 0 and OuF is invertible at (u(0), 0). Then there exists Ε > 0 and neighbourhood U' of u(0) such that for ||f|| < Ε there is a unique u(t) e U' such that (A.12)
F(u(t),t)
= 0.
1
Furthermore u(t) depends C on t, with 1
(A.13)
OTu = - (DuF)- OTF(u(t),
t).
The implicit-function theorem is proved using the contraction mapping principle. 6. Application to 1 fixed-point problems. Let U, Τ be Banach spaces and S : U x Τ —> U be a C map; suppose that ^(0) G U satisfies S(u(0), 0) = ^(0) and that / - OuS is invertible at (u(0), 0). Then, by applying the implicit-function theorem to the map F: U x Τ —> £/ defined by (A.14)
F(u,t) =
S(u,t)-u,
we find there is a locally unique fixed point u(t) (i.e. S(u(t), i) = u(t)) and (A.15)
APPENDIX
OTu = (I - OuSy'OrSiuit),
t).
Β
Comments on the proof of [1]. In this appendix we summarize the proof given in[l] for continuation of bipolaronic states and comment on it. Rather than using the implicit-function theorem, they use the contraction
A
N E W
P A R A D I G M
IN
Q U A N T U M
C H A O S :
A U B R Y ' S
T H E O R Y
71
E T C .
mapping principle directly. They search for a fixed point of St in a set of configurations of the form (B.l)
ί/(σ, r) = {u:une[-l,
- l + τ] if ση = 1, un e [ - τ , 0] if ση = 0}
for some 0 < τ < 1/2. They prove that St is a contraction on this set for t < some t0, when τ(ί) is chosen suitably, and hence there is a unique fixed point u(t) of St in υ(σ, τ) and it depends continuously on t. They restrict attention to finite lattices, as we shall do also, and at the end they prove a result which allows them to take the infinite limit. For simplicity of exposition, we will again consider only the one-dimensional case (2.2). To prove St is a contraction on ί/(σ,τ) one has to show that St maps ί/(σ, τ) into itself and contracts distances by at least some λ < 1, in some metric. They show that St maps ί/(σ, τ) into itself if (B.2)
2
2
8t + τ ^ 4τ(1 - τ)(1 - 2τ - 4t).
Thus, for £ ^ ti, the maximum of t in this region, £/(σ, τ) is mapped into itself for any τ in the interval τλ (t) ^ τ ^ τ2 (t), where τλ (t), τ 2 (t) are the lower and upper boundaries of this region. Since £/(σ, τ) is compact and convex and St is continuous in a, it follows by a theorem of Brouwer that ί/(σ, τ) contains a fixed point. Note that it is not necessary here to state which norm we are using, as all norms are equivalent for a finite system. To show that the fixed point is unique and depends continuously on t, it suffices to show that St is a contraction in some metric. We will choose / 2 (see appendix A), as this is the simplest to get results for. To prove contraction in / 2 , since £/(σ, τ) is convex it suffices to show that the derivative J = OuS has / 2 norm less than 1 in [/(σ, τ), for t and τ small enough. This was done in earlier reports by AUBRY and ABRAMOVICI, referred to in[l], though not in[l] itself. The operator / is represented by a matrix (see[l]) (B.3)
Jmn = ^—
=
OUm
Ζ
^ —
tiv>
v e O , v ' e O '
+C.C,
— tiv
where Ο ' denotes the unoccupied states and c.c. denotes complex conjugate. To estimate ||/|| 2 we first use a trick given in [1] to make clear νthat / is small. By ν eliminating un between the eigenequations for ψ and ψ one obtains the identity (B.4)
ν
tW? = (Εν.-Εν)ψηψ η*
,
where ν
(B.5)
ν
= ψ; * (Δψ ) η - φΐ (Δψ ' )* .
Hence (B.6)
Jmn = t
2
Σ
m
\
+C.C,
72
R . s.
M A C K A Y and
c.
BAESENS
2
which has an explicit factor t in front. Now, since / is symmetric, T
(B.7)
Ύ
\\J\\2 = sup {X JX
:ΧΧ
= 1}.
Using Ev> - Ev ^ ΔΕ, and then extending the sum to all v, v', we obtain 2
(B.8)
\\j\\2^16t /àE*.
Hence, for 2 1/3
(16£ ) < 1 - 2 τ - 4 ί ,
(B.9)
2
we have \\j\\2 < 1. We see that there is a region A G R such that, for (t, τ) e A, St is a contraction of ί/(σ, τ). Let £0 be the supremum of t over the region A. Then for ί < i 0 we have a unique fixed point in [7(σ, τ) for suitable τ and it depends continuously on t. To deduce uniform continuity of u(t) (i.e. independent of the size of the lat1 tice), one has to find /«, estimates on M " and OTS, and then use (4.5). In[l], / o o estimates on J and DTS are found by using a polynomial approximation for the filling fraction σ(2?) = 1 for E G 7_ χ , 0 for Ε e / 0> and then estimates similar to the one we made above for ||/|| 2. This works, but is tedious and also gives poorer estimates 1than we found in sect. 5. This allows them to deduce exponential decay of M " and to take the limit of an infinite lattice.
APPENDIX
C
Solution of a recurrence inequality. In this appendix, we prove that, if (C.I)
P
N
^ K ( P
N
.
X
- P
N
+
) L,
for Ν ^ 1, some Κ ^ 0, and PN are nonnegative and decreasing, then (C.2)
Ν
Ρ Ν ^ ζ λ
1
foriV^l,
Ρ 0
where (C.3)
ζ = min(l, K),
and (C.4)
λ = 2K/{1
2 1/2
+ (1 + 4K ) }
.
Let (C.5)
ΡΝ =
ΡΝ+Ι/ΡΝ·
Then (C.6)
1
O^p^p^-tf- ,
N&l.
A
N E W
P A R A D I G M
IN
Q U A N T U M
C H A O S :
A U B R Y ' S
T H E O R Y
E T C .
73
The quantity λ in (C.4) is one root of the equation 1
(C.7)
X = X~ -K~
1
.
It follows that, if pN > λ, then pN+1 < λ and furthermore (C.8)
l
1
^ 1 - K~ PN
<1-Κ~ λ
N P N + l P
2
= λ.
Hence 2
(C.9)
ΡΝ+2<λ ΡΝ.
If ρ ^ λ, then by definition (CIO)
ΡΝ+1^λΡΝ.
Hence Ν 1
(C.ll)
ΡΝ^λ ~ Ρ0.
In the case Κ < 1 this result can be improved by noting that pN ^ Κ for all Ν ^ 0, else pN+1 < 0, which is not permitted. Hence in this case we obtain (C.12)
Ρχ^Κχν-'Ρο.
This completes the proof.
APPENDIX D
Variation of the electronic energy with configuration u. In this appendix we prove that (D.l)
^
2
= Σ\Φΐ\
=
(η\Ρ\η),
where (D.2)
Ρ
=
Σ
Ε
ν
veO
is the total electronic energy and the other notation is as in (2.6) and (4.14), (4.15). If there is no degeneracy, this can be proved by standard perturbation theory. One differentiates ( 2 . 3 ) with respect to un and takes the inner product with ψ, assumed normalized, leading to
which when summed yields (D.l). If there is degeneracy, however, it is ν not always possible to choose eigenfunctions ψ to depend differentiably on parameters. The spectral projection operator P, nonetheless, remains
74
R . s.
M A C K A Y and
c.
BAESENS
differentiable provided there is an electronic gap, and this allows a proof (in fact we will give two proofs) of (D.l) as follows. The energy F can be written as (D.4)
F = Tr
HP.
For the first proof, differentiating (D.4) gives ÖUN
\ ÖUN
ÖUN
Since H = u - tA, we obtain
(D.6)
Ψ^= η> π
the projection onto the state \n). Taking the basis {|m)} shows that the first term of (D.5) reduces to (D.7)
TrKNP
=
(n\P\n).
Using the decomposition of the space of wave functions into Ο Θ Ο', which are the ranges of the projection operators Ρ and (D.8)
Q = I - P ,
write the second term of (D.5) as (D.9)
Tr ( Ρ Η ψ - Ρ + QH^Q)
.
2
Now Ρ = Ρ, and differentiating this gives (D.10)
FQ
= ^
and
aun
oun
aun
oun
Using HP = PH, (D.9) becomes (D.ll)
TY(HPQ^-
+
HQP^—
dun
\
dun
But QP = PQ = 0. Hence this is zero and the result (D.l) follows. The second proof uses formula (4.14) for P . From (D.4) (D.12)
F = Tr - ^ r ί EG(E) dE. 2 τα J r
Using (D.6), (D.13)
^ L
=
T
A _ ( r
EG
q M
A N E W PARADIGM
IN QUANTUM CHAOS: AUBRY'S THEORY ETC. ν
Now use a basis of eigenstates {ψ }
75
to find
2
By the calculus of residues this reduces to Σ | ft | , as required. veO
REFERENCES [1] S. A U B R Y , G. A B R A M O V I C I and J.-L. R A I M B A U L T : J. Stat. Phys., 67, 675 (1992). [2] A. M E S S I A H : Quantum Mechanics, Vol. II, Chap. XVI, para. 15 (Wiley, New York, N.Y., 1962). [3] P. Y. L E D A E R O N and S. A U B R Y : / . Phys. C, 16, 4827 (1983); /. Phys. (Paris), 44, C3, 1573 (1983). [4] J.-L. R A I M B A U L T : Configurations bipolaroniques et fluctuations quantiques du reseau dans les chaînes de Peierls, thèse de doctorat (Nantes, 1990). [5] M. R E E D and B. S I M O N : Methods of Modem Mathematical Physics: I. Functional Analysis (Academic Press, New York, N.Y., 1972). [6] V. T R É N O G U I N E : Analyse fonctionelle (Mir, Moscow, 1985).