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On the low lying spectrum of the magnetic Schrödinger operator with kagome periodicity
Philippe Kerdelhué, Jimena Royo-Letelier
To cite this version:
Philippe Kerdelhué, Jimena Royo-Letelier. On the low lying spectrum of the magnetic Schrödinger
operator with kagome periodicity. 2014. �hal-00969569�
On the low lying spectrum of the magnetic Schr¨ odinger operator with kagome periodicity
Philippe Kerdelhu´e
1& Jimena Royo-Letelier
21 D´epartement de Math´ematiques, CNRS UMR 8628, F-91405 Orsay Cedex, France
Philippe.Kerdelhue@math.u-psud.fr
2jimena.royo-letelier@m4x.org
Abstract
We study in a semi-classical regime a periodic magnetic Schr¨ odinger operator in R
2. This is inspired by recent experiments on artificial magnetism with ultra cold atoms in optical lattices, and by the new interest for the operator on the hexagonal lattice describing the behavior of an electron in a graphene sheet. We first review some results for the square (Harper), triangular and hexagonal lattices. Then we study the case when the periodicity is given by the kagome lattice considered by Hou. Following the techniques introduced by Helffer-Sj¨ostrand and Carlsson, we reduce this problem to the study of a discrete operator on ℓ
2(Z
2; C
3) and a pseudo-differential operator on L
2(R; C
3), which keep the symmetries of the kagome lattice. We estimate the coefficients of these operators in the case of a weak constant magnetic field. Plotting the spectrum for rational values of the magnetic flux divided by 2πh where h is the semi-classical parameter, we obtain a picture similar to Hofstadter’s butterfly. We study the properties of this picture and prove the symmetries of the spectrum and the existence of flat bands, which do not occur in the case of the three previous models.
1 Introduction
We consider in a semi-classical regime the Schr¨ odinger magnetic operator P
h,A,V, defined as the self-adjoint extension in L
2(R
2) of the operator given in C
0∞(R
2) by
P
h,A,V0= (hD
x1− A
1(x))
2+ (hD
x2− A
2(x))
2+ V (x) , (1.1) where D
xj=
1i∂
xj. Our goal is to study the spectrum of P
h,A,Vas a function of A and the semi-classical parameter h > 0, when V has its minima in the kagome lattice and both V and B = ∇ ∧ A are invariant by the symmetries of the kagome lattice.
Our interest in this mathematical problem is motivated by recent experiments on artifi- cial magnetism with ultra cold atoms ([DGJO11, JZ03]), that lead to new geometries for this problem. To our knowledge, the Hamiltonian in (1.1) has not been obtained in a laboratory with ultra cold atoms, but we mention that a two-dimensional kagome lattice for ultra cold atoms has been recently achieved ([JGT
+12]) using optical potentials. Our main motivation is to understand and analyze mathematically various considerations of Hou in [Hou09].
Let us explain the setting of our problem. A n-dimensional Bravais lattice is the set of
points spanned over Z by the vectors of a basis { ν
1, · · · , ν
n} of R
n. A fundamental domain
of the Bravais lattice is a domain of the form V =
t
1ν
1+ · · · + t
nν
n; t
1, · · · , t
n∈ [0, 1] .
The kagome lattice is not a Bravais lattice, but is a discrete subset of R
2invariant un- der translations along a triangular lattice and containing three points per fundamental domain of this lattice (see Figures 1 and 6d). Each point of the lattice has four nearest neighbours for the Euclidean distance. The word kagome means a bamboo-basket (kago) woven pattern (me) and it seems that the lattice was named by the Japanese physicist K. Husimi in the 50’s ([Mek03]).
Let Γ
△be the triangular lattice spanned by B = { 2ν
1, 2ν
2} , where
ν
ℓ= r
ℓ−1(1, 0) (1.2)
and r is the rotation of angle
π/
3and center the origin. The kagome lattice can be seen as the union of three conveniently translated copies of Γ
△:
Γ = n
m
α,ℓ= 2α
1ν
1+ 2α
2ν
2+ ν
ℓ; (α
1, α
2) ∈ Z
2, ℓ = 1, 3, 5 o
. (1.3)
Figure 1: The kagome lattice and its labelling.
We label the points of Γ by their coordinates in B : Γ = ˜ n
˜
m
α,ℓ= (α
1, α
2) + ˜ ν
ℓ; (α
1, α
2) ∈ Z
2, ℓ = 1, 3, 5 o ,
where
˜ ν
ℓ= 1
2 κ
ℓ−1(1, 0) (1.4)
are the coordinates of ν
jin the basis B . The map κ : Z
2→ Z
2here before is given by
κ(α
1, α
2) = ( − α
2, α
1+ α
2) (1.5)
and represents the rotation r in the basis B , that is, r(m ^
α,ℓ) = κ( ˜ m
α,ℓ).
We will often consider j = 1, . . . , 6 as an element of
Z/
6Z. Depending on the situation, we will give the properties of the kagome lattice in terms of the points m
α,ℓor in terms of their coordinates ˜ m
α,ℓ.
The symmetries of Γ are given by those of Γ
△. For j ∈
Z/
6Zconsider the translations t
j(x) = x + 2ν
jand define
G = the subgroup of the affine group of the plane generated by r, t
1and t
2. (1.6) Setting (gu)(x) = u(g
−1(x)) for g ∈ G , we define a group action of G on C
∞(R
2) which can be extended as an unitary action on L
2(R
2).
Hypothesis 1.1. The electric potential V is a real nonnegative C
∞function such that
gV = V for all g ∈ G , (1.7)
V ≥ 0 and V (x) = 0 if and only if x ∈ Γ , (1.8)
Hess V (x) > 0 ∀ x ∈ Γ . (1.9)
We associated with the magnetic vector potential A = (A
1, A
2) the 1-form ω
A= A
1dx
1+ A
2dx
2.
The magnetic field B is then associated with the 2-form obtained by taking the exterior derivative of ω
A:
dω
A= B (x)dx
1∧ dx
2.
In the case of R
2, we identify this 2-form with B. The renormalized flux of B through a fundamental domain V of Γ
△is by definition
γ = 1 h
Z
V
dω
A.
Hypothesis 1.2. The magnetic potential A is a C
∞vector field such that the correspond- ing magnetic 2 form satisfies
gB = B for all g ∈ G . (1.10)
In the case when A = 0 (see for example Chapter XIII.16 in [RS80]), the spectrum of P
h,A,Vis continuous and composed of bands. The general case, even when the magnetic field is constant, is very delicate. The spectrum of P
h,A,Vcan indeed become very singular (Cantor structure) and depends crucially on the arithmetic properties of γ/(2π).
To approach this problem, we are often led to the study of limiting models in different asymptotic regimes, such as discrete operators defined over ℓ
2(Z
2; C
n), or equivalently, γ - pseudo-differential operators defined on L
2(R; C
n) and associated with periodical symbols.
The discrete operators considered are polynomials in τ
1, τ
2, τ
1∗and τ
2∗with coefficients in M
n(C), where τ
1and τ
2are the discrete magnetic translations on ℓ
2(Z
2; C
n) given by
(τ
1v)
α= v
α1−1,α2, (τ
2v)
α= e
iγα1v
α1,α2−1. (1.11) We also recall that the γ-quantization of a symbol p(x, ξ, γ ) with values in M
n(C) is the pseudo-differential operator defined over L
2(R; C
n) by
(Op
Wγp)u
(x) = 1 2πγ
Z Z
R2
e
i(x−y)ξγp
x + y 2 , ξ, γ
u(y) dy dξ . (1.12)
In this article, following the ideas in [HS88a], § 9, we first analyze the restriction of P
h,A,Vto a spectral space associated with the bottom of its spectrum, and we show the existence of a basis of this space such that the matrix of this operator keeps the symmetries of Γ.
In order to state our first theorem, let us explain more in detail this procedure. First of all, the harmonic approximation together with Agmon estimates shows the existence of an exponentially small (with respect to h) band in which one part of the spectrum (including the bottom) of P
h,A,Vis confined. We name this part the low lying spectrum. The rest of the spectrum is separated by a gap of size h/C .
Consider δ ∈ (0, 1/8) and a non negative radial smooth function χ, such that χ = 1 in B(0, δ/2) and supp χ ⊂ B(0, δ). For any m ∈ Γ define
V
m( · ) = X
n∈Γ\{m}
χ( · − n) and
P
m= P
h,A,V+ V
m. (1.13)
All the P
mare unitary equivalent and
b = lim inf
|x|→∞
V
m(x) (1.14)
is positive and does not depend on m. The spectrum of P
mis discrete in the interval [0, b].
The first eigenvalue of P
mis simple and we note it λ(h). We can prove that there exists then ǫ
0> 0 such that σ(P
m) ∩ I (h) = { λ(h) } , where I (h) = [0, h(λ
har,1+ ǫ
0)] and λ
har,1is the first eigenvalue of the operator associated with P
mby the harmonic approximation when h = 1 (see Section 5.3 for more details). We define
Σ = the spectral space associated with I (h) . (1.15) We denote by d
Vthe Agmon distance associated with the metric V dx
2(see [DS99], § 6) and
S = min { d
V(n, m); n, m ∈ Γ, n 6 = m } . (1.16) We then have:
Theorem 1.3. Under Hypotheses 1.1 and 1.2, there exists h
0> 0 such that for h ∈ (0, h
0) there exists a basis of Σ in which P
h,A,VΣ has the matrix λ(h)I + W
γ, where for all n, ˜ m ˜ ∈ Γ ˜ and α ∈ Z
2, W
γsatisfies
(W
γ)
n,˜m˜= (W
γ)
m,˜˜ n, (1.17)
(W
γ)
n,˜m˜= e
−iγ2( ˜m−˜n)∧α(W
γ)
(˜n+α),( ˜m+α), (1.18)
(W
γ)
n,˜m˜= (W
γ)
κ(˜n),κ( ˜m). (1.19)
Moreover, there exists C > 0 such that for every ǫ > 0 there exists h
ǫ> 0, such that for h ∈ (0, h
ǫ)
| (W
γ)
n,˜m˜| ≤ C exp
− (1 − ǫ)d
V(m, n) h
, (1.20)
| (W
γ)
n,˜˜n| ≤ C exp
− (2S − ǫ) h
. (1.21)
The coefficients of W
γare related to the interaction between different sites of the kagome lattice. Our next result concerns the study of this matrix, when we only keep the main terms for the Agmon distance. In order to estimate these terms, we need additional hypothesis. Here we assume (see [HS84] for more details):
Hypothesis 1.4. A. The nearest neighbors for the Agmon distance are the same of those for the Euclidean distance, i.e. S = d
Vm
(1,0),3, m
(0,0),1.
B. Between two nearest neighbors m, n ∈ Γ there exists an unique minimal geodesic ζ
m,nfor the Agmon metric.
C. This geodesic ζ
m,ncoincides with the Euclidean one that is the segment between m and n.
D. The geodesic ζ
m,nin non degenerate in the sense that there is a point
1x
0∈ ζ
m,n\ { m, n } such that the function x 7→ d
V(x, m) + d
V(x, n) − d
V(m, n) restricted to a transverse line to ζ
m,nat x
0has a non degenerate local minimum at x
0.
Under this hypothesis, we will estimate the main terms in the case of a weak and constant magnetic field B = hB
0, given by the gauge
A(x
1, x
2) = hB
02 ( − x
2, x
1) , B
0> 0 . (1.22) The discrete model associated with the kagome lattice is
Q
γ,ω=
0 e
i(ω+γ8)τ
1∗+ e
−iγ2τ
1∗τ
2e
−i(ω+γ8)(τ
1∗+ τ
2∗) e
−i(ω+γ8)τ
1+ e
−iγ2τ
1τ
2∗0 e
i(ω+γ8)e
−iγ2τ
1τ
2∗+ τ
2∗e
i(ω+γ8)(τ
1+ τ
2) e
−i(ω+γ8)e
−iγ2τ
1∗τ
2+ τ
20
(1.23) acting on ℓ
2(Z
2; C
3).
We also introduce the symbol p
kag(x, ξ, γ, ω) =
0 e
i(ω+γ8)e
−ix+ e
−i(x−ξ)e
−i(ω+γ8)e
−ix+ e
−iξe
−i(ω+iγ8)e
ix+ e
i(x−ξ)0 e
i(ω+γ8)e
i(x−ξ)+ e
−iξe
i(ω+γ8)e
ix+ e
iξe
−i(ω+γ8)e
−i(x−ξ)+ e
iξ0
(1.24)
and its Weyl-quantization P
γ,ωkag= Op
Wγp
kag(x, ξ, γ, ω) acting on L
2(R; C
3).
We now state two theorems linking the Schr¨odinger operator and these two models.
Theorem 1.5. Let V satisfies Hypothesis 1.4. There exists b
0> 0, h
0> 0, C > 0 and R ∈ L (ℓ
2(Z
2; C
3)) such that for h ∈ (0, h
0), P
h,A,V| Σ is unitary equivalent with
Q
γ= λ(h) I − ρ (Q
γ,ω+ R
γ) , (1.25) where
ρ = h
1/2b
0e
−Sh(1 + O (h)) , (1.26)
ω = O (h) (1.27)
and
k R
γk
L(ℓ2(Z2;C3))≤ C exp
− 1 Ch
. (1.28)
1Actually this condition does not depend on the choice of the pointx0 (see [HS84]).
Theorem 1.6. Under the same Hypothesis of Theorem 1.5, there exists a symbol r(x, ξ) 2π-periodic in x and ξ such that P
h,A,V| Σ has the same spectrum than
λ(h) I − ρ
P
γ,ωkag+ Op
Wγr(x, ξ)
, (1.29)
where ρ and ω are given by (1.26) and (1.27), and k Op
Wγr(x, ξ) k
L(L2(R;C3))≤ C exp
− 1 Ch
. (1.30)
Remark 1.7. In the case of the square, triangular and hexagonal lattices, using Hypothesis 1.1 and 1.2, it is possible to prove that the terms corresponding to the interaction between nearest neighbours of the lattices are equal, so ω = 0. The situation is more complex for the kagome lattice, and we are only able to prove equality for half of these terms. We point out that we do not see any a priori reason for equality between all terms, although this is assumed in some articles ([Hou09, HA10]).
We then study the dependence on ω of the spectra.
Proposition 1.8. Let σ
γ,ωbe the spectrum of Q
γ,ω. We have σ
γ,ω+π4
= σ
γ−6π,ω, (1.31)
σ
γ,−ω= σ
−γ,ω. (1.32)
Thus it in enough to consider ω ∈ [0, π/8] to obtain all the spectra.
In order to compute the spectrum of Q
γ,ω, we give a last representation in the case when γ/(2π) is a rational number.
For p, q ∈ N
∗we define the matrices J
p,q, K
q∈ M
q(C) by J
p,q= diag(exp (2iπ(j − 1)p/q)) and (K
q)
ij=
1 if j = i + 1 (mod q)
0 if not .
(1.33) Theorem 1.9. Let γ = 2πp/q with p, q ∈ N
∗relatively primes and denote by σ
γ,ωthe spectrum of Q
γ,ω. We have
σ
γ,ω= [
θ1,θ2∈[0,1]
σ(M
p,q,ω,θ1,θ2) , (1.34)
where M
p,q,ω,θ1,θ2∈ M
3q(C) is given by
M
p,q,ω,θ1,θ2=
0
qM
p,q,ω,θ13 1,θ2M
p,q,ω,θ15 1,θ2h M
p,q,ω,θ131,θ2
i
∗0
qM
p,q,ω,θ351,θ2
h
M
p,q,ω,θ15 1,θ2i
∗h
M
p,q,ω,θ35 1,θ2i
∗0
q
(1.35)
with
M
p,q,ω,θ13 1,θ2= e
i(ω+π4pq)e
i2πθ1K
q+ e
−iπpqe
i2π(θ1+θ2)K
qJ
p,qM
p,q,ω,θ15 1,θ2= e
−i(ω+π4pq)e
i2πθ1K
q+ e
−i2πθ2J
p,q∗(1.36) M
p,q,ω,θ35 1,θ2= e
i(ω+π4)pqe
−iπpqe
−2iπ(θ1+θ2)K
q∗J
p,q∗+ e
−i2πθ2J
p∗q.
Remark 1.10. Formally we obtain (1.24) and (1.35) by replacing the pair of operators (τ
1, τ
2) in (1.23) by (op
Wγ(e
ix), op
Wγ(e
iξ)) and (e
−i2πθ1K
q∗, e
i2πθ2J
p,q). Note that these pairs of operators have the same commutation relation
τ
2τ
1= e
iγτ
1τ
2,
op
Wγ(e
iξ) op
Wγ(e
ix) = e
iγop
Wγ(e
ix) op
Wγ(e
iξ) , (e
i2πθ2J
p,q)(e
−i2πθ1K
q∗) = e
iγ(e
−i2πθ1K
q∗)(e
i2πθ2J
p,q) ,
and we obtain three isospectral operators Q
γ,ω, P
γ,ωkagand M
p,q,ωwhere M
p,q,ωacts on L
2([0, 1]
2; C
3q) by
( M
p,q,ωu)(θ
1, θ
2) = M
p,q,ω,θ1,θ2u(θ
1, θ
2) .
In the formalism of rotational algebras, it is said that these three isospectral operators are representations of the same Hamiltonian in different rotation algebras (see [BKS91]).
In Figures 2 and 3 we present the equivalent of Hofstadter’s butterfly for the kagome lattice in the case when ω = 0 and ω = π/8, obtained by numerically diagonalizing the matrices M
p,q,0,θ1,θ2and M
p,q,π8,θ1,θ2
. In the first case we recover that one obtained by Hou in [Hou09].
Figure 2: Hofstadter’s butterfly for the kagome lattice when ω = 0.
Figure 3: Hofstadter’s butterfly for the kagome lattice when ω = π/8.
We notice that for fixed γ = 2πp/q the spectrum is composed of 3q (possibly not disjoint) bands, which are the images of
[0, 1] × [0, 1] ∋ (θ
1, θ
2) 7→ λ
kp,q,ω,θ1,θ2, 1 ≤ k ≤ 3q (1.37) where λ
kp,q,ω,θ1,θ2is the kth eigenvalue of M
p,q,ω,θ1,θ2.
Since the smallest positive integer for which the operator Q
γ,ωis invariant by the trans- formation γ 7→ γ + 2πk is k = 8, we plot in the vertical axis of Figures 2 and 3 the bands of the spectrum for
γ 2π = p
q , p, q relatively prime and 0 ≤ p < 8q ≤ 400 .
We first observe some symmetries in these butterflies and prove the proposition Proposition 1.11. Let σ
γ, ω be the spectrum of Q
γ,ω. We have
σ
γ,ω⊂ [ − 4, 4] , (1.38)
σ
γ+16π,ω= σ
γ,ω(translation invariance), (1.39)
e ∈ σ
γ+8π,ω⇔ − e ∈ σ
γ,ω(translation anti-invariance). (1.40) In the case when ω = 0, we have
σ
−γ,0= σ
γ,0(reflexion with respect to the axis γ = 0), (1.41) e ∈ σ
8π−γ,0⇔ − e ∈ σ
γ,0(reflexion with respect to the point (4π, 0)). (1.42) In the case when ω = π/8, we have
σ
6π−γ,π8
= σ
γ,π8
(reflexion with respect to the axis γ = 3π), (1.43) e ∈ σ
−2π−γ,π8
⇔ − e ∈ σ
γ,π8
(reflexion with respect to the point ( − π, 0)). (1.44)
Second we note the presence of isolated points in σ
4π 3 ,0, σ
8π3 ,0
, σ
4π,0, σ
π,π8
, σ
3π,π8
, σ
7π 3 ,π8and σ
−π3,π8
.
To see more precisely the last phenomenon, we plot in Figures 4b and 5 the bands of the spectra σ
4π,0, σ
4π3 ,0
and σ
8π 3 ,0.
(a) (b)
Figure 4: Spectrum bands of Q
γ,ωfor (a) (γ, ω) = (0, 0) and (b) (γ, ω) = (4π, 0).
(a) (b)
Figure 5: Spectrum bands of Q
γ,ωfor (a) (γ, ω) = (4π/3, 0) and (b) (γ, ω) = (8π/3, 0).
Numerically it seems that the second, third and fourth bands of σ
4π3 ,0
are reduced to {− √
3 } , and that the third, fourth and fifth band of σ
8π3 ,0
are reduced to {− 1 } . This leads to the definition
Definition 1.12. Let λ
0be a reel number and n a positive integer. { λ
0} is called a flat band of multiplicity n of σ
γ,ωif the k
thband of σ
γ,ωis reduced to { λ
0} for exactly n values of k.
One can easily compute the characteristic polynomials of the 3 × 3 matrices M
0,1,0,θ1,θ2and M
2,1,0,θ1,θ2. For the other cases, we use the symbolic computation software Mathematica and obtain
Proposition 1.13. 1. (a) {− 2 } and { 0 } are flat bands of multiplicity 1 of σ
0,0and σ
4π,0respectively. σ
0,0is composed of the three touching bands {− 2 } , [ − 2, 1]
and [1, 4]. σ
4π,0is composed of the three disjoint bands [ − 2 √ 3, − √
3], { 0 } and [ √
3, 2 √
3].
(b) {− √
3 } and {− 1 } are flat bands of multiplicity 3 of σ
4π3,0
and σ
8π3 ,0
respectively.
2. (a) {− √
2 } and {− 2 } are flat bands of multiplicity 2 of σ
π,π8
and σ
3π,π8
respectively.
σ
3π,π8
is composed of the flat band {− 2 } and the four touching bands [1 − √
6, 1 − √
3], [1 − √
3, 1], [1, 1 + √
3], and [1 + √
3, 1 + √ 6].
(b) {−
√6+2√2} and {−
√6−2√2} are flat bands of multiplicity 6 of σ
7π3,π8
and σ
−π3,π8
respectively.
Remark 1.14. This phenomenon does not occur for the square, triangular and hexagonal models.
Remark 1.15. 1. Proposition 1.13 ensures the existence of eigenvalues of infinite multi- plicity for Q
γ,ωand P
γ,ωkagfor several values of (γ, ω).
2. Since the models Q
γ,ωand P
γ,ωkagonly take into account the interactions beetween nearest wells, and ω = O (h) does not a priori vanish, the existence of eigenvalues for Q
γ,0when γ equals to 4π/3, 8π/3 or 4π does not imply the existence of eigenvalues for the correspond- ing initial Schr¨ odinger operator P
h,A,V. However, Proposition 1.13 together with Theorem 1.5 ensure that, when the values of A and h lead to one of these values of γ, there exists C > 0 suth that a part of the low lying spectrum of P
h,A,Vis included in an interval of length at most C h
3/2exp( − S/h) and separated from the rest of the spectrum by intervals of lengh at least C
−1h
1/2exp( − S/h).
Remark 1.16. In the light of Proposition 1.13 we can state the following conjecture : if σ
2πp/q,ωcontains a flat band for a real number ω and two relatively prime integers p and q with q > 0, then its multiplicity is q.
An interesting question is to see how the invariances of the initial problem are conserved in the reduced model p
kag. The invariance by rotation of angle π/3 gave the application κ on the indices α, so the transpose application
tκ(x, ξ) = (ξ, − x + ξ) is seen as the rotation of angle − π/3 on the phase space R
x× R
ξ. We introduce the translations ˜ t
1(x, ξ) = (x + 2π) and ˜ t
2(x, ξ) = (x, ξ + 2π), and the symmetry s(x, ξ) = (ξ, x). We then have
Proposition 1.17.
p
kag◦ ˜ t
1= p
kag, (1.45) p
kag◦ ˜ t
2= p
kag, (1.46)
0 1 0 0 0 1 1 0 0
−1
p
kag◦
tκ
2
0 1 0 0 0 1 1 0 0
= p
kag, (1.47)
0 0 1 0 1 0 1 0 0
(p
kag◦ s)
0 0 1 0 1 0 1 0 0
= p
kag. (1.48)
Remark 1.18. 1. The invariance by the rotation of angle π/3 seems lost, but fortunately the action of the group generated by ˜ t
1, ˜ t
2, s and
tκ
2on the set of the microlocal wells, which are at energy λ the connected components of { (x, ξ); det(λ I
3− p
kag(x, ξ, 0, ω)) = 0 } , is transitive.
2. As shown in [Ker95], the invariances of p
kaggive operators commuting with P
γ,ωkag.
We will develop these points in a further work joint with B.Helffer and devoted to the
microlocal study of P
γ,ωkag.
Outline of the article
This article is organized as follows:
• In Section 2 we present a general theorem on the Weyl quantization of periodical symbols.
• In Section 3 we review the cases of the square, triangular and hexagonal lattices.
We describe and prove the symmetries of the corresponding spectra.
• In Section 4 we study the properties of the kagome lattice and we construct a family of potentials invariant by the symmetries of Γ
△, whose minima are located in Γ.
• Section 5 is devoted to the semi-classical analysis of the low lying spectrum of P
h,A,Vfor h small. We derive the discrete operator W
γand prove Theorem 1.3.
• In Section 6, we study the properties of W
γand prove Theorem 1.5. We give the representation using the pseudo-differential operator acting on L
2(R; C
3) and prove Theorem 1.6. We then study the case when γ/(2π) is rational and prove Theorem 1.9. We end this article by proving Propositions 1.8, 1.11 and 1.13.
Ackowledgements : This article is a revisited version of the second part of J. Royo- Letelier’s PhD thesis (defended in June 2013 at the Universit´e de Versailles Saint-Quentin- en-Yvelines) with B. Helffer as advisor and written with the help of P. Kerdelhu´e. We warmly thank B. Helffer for suggesting us this problem and for his precious help with the realization of this article. The second author thanks the Institute of Science and Technology Austria (IST Austria) in which she was staying as a post-doc while this article was finalized. The first author thanks P. Gamblin for useful conversations.
2 Quantization of periodical symbols
We first give a general theorem on the γ-quantization of a periodic symbol, which will be used to study the symmetries of the butterflies associated to the square, triangular, hexag- onal and kagome models, and in the proofs of Theorems 1.5 and 1.6. This theorem was first established in [HS88a] and [Ker92] for Harper’s and triangular models, and under the restriction 0 < γ < 2π. We present here a slightly different proof to avoid this restriction.
Let n ∈ N
∗and (β, γ) 7→ p
β,γbe a function on Z
2× R
∗with values in M
n(C) such that :
∀ N ∈ N, ∃ C
N> 0, ∀ (β, γ) ∈ Z
2× R
∗, | p
β,γ| ≤ C
N(1 + β
1+ β
2)
−N, (2.1)
∀ (β, γ) ∈ Z
2× R
∗, p
−β,γ= p
∗β,γ. (2.2) We define the symbol
p(x, ξ, γ ) = X
β∈Z2
p
β,γe
i(β1x+β2ξ)(2.3) and its Weyl quantization P
γintroduced in (1.12). A straightforward computation gives that P
γacts on L
2(R; C
n) by
P
γu(x) = X
β∈Z2
p
β,γe
iγ2β1β2e
iβ1xu(x + β
2γ) . (2.4)
We also consider the discrete operator Q
γ= X
β∈Z2
p
β,γe
iγ2β1β2τ
1β1τ
2β2where τ
1and τ
2are the discrete magnetic translations defined in (1.11), and A
γthe infinite matrix defined by
(A
γ)
α,β= e
−iγ2α∧βp
α−β,γand acting on ℓ
2(Z
2; C
n) by
(A
γv)
α= X
β∈Z2
(A
γ)
α,βv
β.
Theorem 2.1. A
γand Q
γare unitary equivalent. P
γ, A
γand Q
γhave the same spectrum.
Proof. The first hypothesis anables to prove the convergence of the series defining p, A
γand Q
γ, and the second one gives the self-adjointness of A
γ, Q
γand P
γ.
Q
γacts on ℓ
2(Z : C
n) by
Q
γu(α) = X
β∈Z2
p
β,γe
−iγ2β1β2e
iγα1β2u
α−β= X
β∈Z2
p
α−β,γe
iγ2(α1α2−β1β2−α∧β)u
βso A
γand Q
γare unitary equivalent.
The operator P
γcommutes with the translation u( · ) 7→ u( ·− 2π), so Floquet theory applies and the spectrum of P
γis the union over θ ∈ R of the spectra of the operators P
γθacting on the space n
u ∈ L
2loc(R; C
n) ; u( · + 2π) = e
i2πθu( · ) a.e. o by P
γθu(x) = X
β∈Z2
p
β,γe
iγ2β1β2e
iβ1xu(x + β
2γ ) .
We notice that P
γθhas the same spectrum than its conjugate ˜ P
γθ= e
−iθxP
γθe
iθxacting on L
2(R/2πZ; C
n) by
P ˜
γθu(x) = X
β∈Z2
p
β,γe
iγ2β1β2e
iβ1xe
iγβ2θu(x + β
2γ) .
The union over θ ∈ R of the spectra of the operators ˜ P
γθis the union over θ ∈ [0, 2π/γ]
(or θ ∈ [2π/γ, 0] in the case when γ < 0) of these spectra. Hence the spectrum of P
γis the spectrum of the operator ˇ P
γacting on L
2(R/2πZ × R/
2πγZ; C
n) by
P ˇ
γu(x, θ) = X
β∈Z2
p
β,γe
iγ2β1β2e
iβ1xe
iγβ2θu(x + β
2γ, θ) .
We define the unitary Fourier transform F mapping L
2((R/2πZ) × (R/
2πγZ); C
n) on ℓ
2(Z
2; C
n) by
( F u)
α= γ
1/22π
Z Z
e
−i(α1x+γα2θ)u(x, θ) dx dθ
and a straightforward computation gives
F P ˇ
γ= Q
γF .
So ˇ P
γand Q
γare unitary equivalent. Hence P
γand Q
γhave the same spectrum.
3 The square, triangular and hexagonal lattices
3.1 Presentation of the models
(a) (b)
(c) (d)
Figure 6: (a) square, (b) triangular, (c) hexagonal and (d) kagome lattices. In each case we had drawn a fundamental domain of the Bravais lattice. The points of the lattice correspond to the minima of the electric potential.
The spectral properties of P
h,A,Vhave been studied for the square, triangular and hexag- onal lattices. When plotting the spectrum as a function of γ , we obtain a picture with several symmetries, which are determined by the symmetries of the lattice. In the case of the square lattice, we get the famous Hofstadter butterfly. In this section we review and prove the symmetries of these spectra using the pseudo-differential operators associated with these lattices. We recall that the symbols corresponding to the square, triangular and hexagonal lattices are respectively
p
(x, ξ) = cos x + cos ξ , (3.1)
p
△(x, ξ) = cos x + cos ξ + cos(x − ξ) , (3.2) p
hex(x, ξ) =
0 1 + e
ix+ e
iξ1 + e
−ix+ e
−iξ0
. (3.3)
The size of the matrix is the number of points of the lattice in each fundamental domain.
The number of terms of the form e
i(ax+bξ)is the product of this number by the number of nearest neighbours of each point of the lattice.
To study the symmetries of the Hofstadter’s butterflies associated with each model, we will use the following direct consequence of Theorem 2.1.
Proposition 3.1. Let n ∈ N
∗and β 7→ p
βbe a function on Z
2(here p
βdoes not depend on γ ) with values in M
n(C) satisfying (2.1) and (2.2). Consider the symbol p(x, ξ) defined in (2.3), its Weyl quantization P
γand denote by σ(P
γ) the spectrum of P
γ. We have:
If p
β= 0 for β
1and β
2odd, then ∀ γ ∈ R , σ(P
γ+2π) = σ(P
γ) . (3.4) If p
β= 0 for β
1and β
2even, then ∀ γ ∈ R , σ(P
γ+2π) = − σ(P
γ) . (3.5) Remark 3.2. Property (3.4) applies to the Harper model p
and the hexagonal model p
hex. Property (3.5) applies to the triangular model p
△.
Proof. First we notice that the magnetic translations τ
1and τ
2defined in (1.11) don’t change when we replace γ by γ + 2π. Hence Theorem 2.1 gives
σ(P
γ+2π) = σ
Op
Wγ
X
β∈Z2
( − 1)
β1β2p
βe
i(β1x+β2ξ)
,
so (3.4) is proved.
Since the application (x, ξ) 7→ (x + π, ξ + π) is affine and symplectic the operators Op
Wγ
X
β∈Z2
( − 1)
β1β2p
βe
i(β1x+β2ξ)
and Op
Wγ
X
β∈Z2
( − 1)
β1β2p
βe
i(β1(x+π)+β2(ξ+π))
are unitary equivalent. Then, σ
X
β∈Z2
( − 1)
β1β2p
βe
i(β1x+β2ξ)
= σ
X
β∈Z2
( − 1)
β1β2p
βe
i(β1(x+π)+β2(ξ+π))
= − σ
X
β∈Z2
( − 1)
(β1+1)(β2+1)p
βe
i(β1x+β2ξ)
,
which yields (3.5).
3.2 The square lattice
The square lattice is the Bravais lattice associated with the basis { (1, 0), (0, 1) } of R
2. Each point of the lattice has 4 nearest neighbours for the Euclidean distance. One of the models used in this case is the discrete operator L
γdefined on ℓ
2(Z
2, C) by
L
γ= 1
2 (τ
1+ τ
1∗+ τ
2+ τ
2∗) , (3.6)
where τ
1, τ
2are the discrete magnetic translations defined in (1.11).
Using a partial Floquet theory
2, we are led to the study of the spectrum of a family (parametrized by θ
2) of discrete Schr¨odinger operators L
γ,θ2acting over ℓ
2(Z) by
(L
γ,θ2v)
n= v
n+1+ v
n−12 + V
θ2(n)v
n, (3.7)
where V
θ2(n) = cos (γn + θ
2) is the discrete potential.
Notice that L
γ,θ2+γ
is unitary equivalent with L
γ,θ2
. When γ/(2π) is irrational, the spec- trum of L
γ,θ2does not depend on θ
2(see [HS88a], § 1). This is no longer the case when γ/(2π) is rational. In 1976 Hofstadter performed a formal study of the spectrum of L
γ,θ2as a function of γ/(2π) ∈ Q ([Hof76]). His approach suggests a fractal structure for the spectrum and leads to Hofstadter’s butterfly. The method consists in studying numerically the case γ = 2πp/q, with p, q ∈ N relative primes. Hofstadter observed that in this case, the spectrum is formed of q bands which can only touch at their boundary. Hofstadter’s butterfly is obtained by placing in the y-axis of a graph the bands of the spectrum (see Figure 7a). Moreover, Hofstadter derived rules for the configuration of the bands related to the expansion of p/q as continued fraction. This configuration strongly suggests the Cantor structure of the spectrum of L
γ,θ2when γ/(2π) is irrational. A longtime open problem, proposed by Kac and Simon in the 80’s and called the “Ten Martinis problem”
([Sim], Problem 4), was to prove that for irrational γ/(2π), the spectrum of L
γis a Cantor set. After many efforts starting with the article of Bellissard and Simon in 1982 ([BS82]), the problem was finally solved in 2009 by Avila and Jitomirskaya ([AJ09]).
(a) (b)
Figure 7: (a) The spectrum of P
γcorresponding to the square lattice (Hofstadter’s but- terfly). (b) Energy bands for γ = 2π/3.
In order to compute the spectrum of L
γfor γ = 2πp/q, we may use again the Floquet theory. Introducing the Floquet condition v
n+q= e
i2πθ1qv
n, we are led to the computation of the eigenvalues of a family (parametrized by θ
1and θ
2). Denoting σ
γ= σ(L
γ) we obtain
σ
γ= [
θ1,θ2∈[0,1]
σ(M
p,q,θ 1,θ2) ,
2The classical reference for Floquet theory is [RS80],§XII.16. We also refer to the review about periodic operators in Subsections 2.1 and 2.2 of [PST06].
where
M
p,q,θ 1,θ2= 1 2
e
i2πθ1K
q+ e
−i2πθ1K
q∗+ e
i2πθ2J
p,q+ e
−i2πθ2J
p,q∗with J
p,q, K
qdefined in (1.33).
For 1 ≤ k ≤ q the kth band of σ
γis given by the image of
E
kp,q: [0, 1] × [0, 1] → R , (θ
1, θ
2) 7→ λ
kp,q,θ1,θ2, (3.8) where λ
kp,q,θ1,θ2
is the kth eigenvalue of M
p,q,θ1,θ2
(see Figure 7b).
In [HS88a], § 1, it was proved that L
γis unitary equivalent with the pseudo-differential operator P
γdefined in (2.4) for p
given in (3.1). Helffer and Sj¨ ostrand developed in [HS88a, HS89, HS90] sophisticated techniques (inspired by the work of the physicist Wilkinson ([Wil84]) to study the operator P
γ. In particular, they justified in various regimes the approximation for the low spectrum of P
h,A,Vby the spectrum of P
γ. When plotting σ
γas a function of γ (see Figure 7a), we observe the following properties.
Proposition 3.3.
σ
γ⊂ [ − 2, 2] , (3.9)
σ
γ+2π= σ
γ(translation invariance), (3.10) σ
−γ= σ
γ(reflexion with respect to the axis γ = 0), (3.11) σ
2π−γ= σ
γ(reflexion with respect to the axis γ = π), (3.12)
− e ∈ σ
γ⇔ e ∈ σ
γ(reflexion with respect to the axis e = 0) . (3.13) Proof. The definition of p
(x, ξ) together with the fact that the Weyl quantizations of e
ix, e
−ix, e
iξ, e
−iξare unitary operators yield (3.9). Property (3.4) gives (3.10). We obtain (3.11) noticing that
P
−γ= Op
W−γp
(x, ξ)
= Op
Wγp
(x, − ξ)
= Op
Wγp
(x, ξ)
= P
γ. Properties (3.10) and (3.11) imply (3.12). Finally, we have that
p
(x + π, ξ + π) = − p
(x, ξ)
so P
γand − P
γare conjugate by the unitary operator u 7→ e
γiπ·u( · − π). This yields (3.13).
3.3 The triangular lattice
The triangular lattice
3is the Bravais lattice associated with the basis { (1, 0), (
1/
2, −
√3/
2) } . Each point of the lattice has 6 nearest neighbors for the Euclidean distance. This case was studied by Claro and Wannier in [CW79]. These authors exhibit an analogous structure to the case of the square lattice. In the case γ = 2πp/q, with p, q ∈ N relative primes, the spectrum is formed of q bands which can only touch at their boundary (see Figure 8a). In [Ker92], the first author studied rigorously the operator P
h,A,Vin this case. He justified
3We note that the triangular and hexagonal lattices are sometimes respectively called hexagonal and honeycomb lattices.
the reduction to the pseudo-differential operator P
γ△defined in (2.4) with p
△given in (3.2).
As in the case of the square lattice discussed before, when γ = 2πp/q the spectrum can be computed by considering the family of matrices in M
q(C) defined by
M
p,q,θ△1,θ2
= 1 2
e
i2πθ1K
q+ e
−i2πθ1K
q∗+ e
i2πθ2J
p,q+ e
−i2πθ2J
p,q∗+e
−iπp/qe
i2π(θ1+θ2)J
p,qK
q+ e
−iπp/qe
−i2π(θ1+θ2)J
p,q∗K
q∗.
(a) (b)
