Absolutely Continuous Edge Spectrum of Hall Insulators on the Lattice
aa r X i v : . [ m a t h - ph ] J a n Absolutely Continuous Edge Spectrum of Hall Insulators on theLattice
Alex Bols and Albert H. Werner
QMATH, Department of Mathematical Sciences, University of CopenhagenUniversitetsparken 5, 2100 Copenhagen, Denmarke-mail : [email protected], [email protected]
Abstract
The presence of chiral modes on the edges of quantum Hall samples is essential to our understandingof the quantum Hall effect. In particular, these edge modes should support ballistic transport andtherefore, in a single particle picture, be supported in the absolutely continuous spectrum of the single-particle Hamiltonian. We show in this note that if a free fermion system on the two-dimensional lattice isgapped in the bulk, and has a nonvanishing Hall conductance, then the same system put on a half-spacegeometry supports edge modes whose spectrum fills the entire bulk gap and is absolutely continuous.
In a finite Hall insulator, the Hall conductance is quantized to remarkable accuracy in multiples of e /h ,where e is the charge of the electron, and h is Planck’s constant. The Hall current consists of a bulkand an edge contribution which combine to give the quantized result. The edge current arises becausethe driving potential results in a skewed occupation of the edge states of the sample. These edge statesare chiral, i.e. they carry a ballistic current predominantly either clockwise or counterclockwise alongthe edge, so a different occupation of edge modes on opposite sides of the sample leads to a net currentperpendicular to the potential drop. The existence of such chiral edge modes is therefore essential to ourunderstanding of the quantum Hall effect [14, 11].The chiral edge modes must allow ballistic motion along the edge and should therefore correspondto absolutely continuous spectrum (see for example theorem 5.7 in [19]). We show in this note thatnon-trivial Hall insulators modelled by free fermions on the lattice Z inevitably yield edge modes corre-sponding to absolutely continuous spectrum, filling the entire bulk gap.The presence of edge states corresponding to (absolutely) continuous spectrum has been shown forthe Landau Hamiltonian in the continuum with a weak disorder potential, where the edge is introducedeither by a steep edge potential or appropriate half-plane boundary conditions [16, 10, 6, 12, 5]. Exceptfor the first, all these works rely on Mourre estimates to conclude purely absolutely continuous spectrumin the bulk gap.In this note we do not use Mourre estimates, but instead appeal to the bulk-edge correspondencefor Hall insulators, which has been proven in various guises for free fermions on the lattice [18, 13, 7,15, 8, 17, 9]. The topological index characterizing the edge system in this bulk-edge correspondence isa property of a unitary operator which is defined in terms of the edge states. Using a result from [2]one immediately gets that this unitary has absolutely continuous spectrum if the topological index isnonzero. We then show that the Hamiltonian of the half-plane system inherits this absolutely contiuousspectrum.A possible concrete advantage of our method is the following. The techniques using Mourre estimateslead to purely continuous spectrum in the gap. That method therefore seems to exclude from the startthe possibility of dealing with a mobility gap. Our method may be more promising in this regard. We consider free electrons moving on the lattice Z modelled by a local bulk Hamiltonian H on l ( Z ).Associated to this bulk Hamiltonian there is an edge Hamiltonian ˆ H acting on l ( Z × N ) which is obtained y restricting the bulk Hamiltonian to the half-space. We spell out detailed assumptions on the bulkand edge Hamiltonians at the end of this section.We assume that for some open interval ∆∆ ∩ σ ( H ) = ∅ . Such an interval ∆ is called a bulk gap . We let P F = χ ≤ µ ( H ) for any µ ∈ ∆ denote the Fermi projection. Theorem 2.1.
Let H be a bulk Hamiltonian with bulk gap ∆ and let Ind B ( H, ∆) := − Ind( P F , U ) ∈ Z be the bulk index , where U := e i arg ~X with ~X = ( X , X ) the vector of position operators.If Ind B = 0 then ∆ ⊂ σ ac ( ˆ H ) . i.e. the edge Hamiltonian ˆ H has absolutely continuous spectrum everywhere in the bulk gap ∆ . The bulk index Ind(
U, P F ) is defined as the index of the pair of projections UP F U † and P F [3]: Definition 2.2. If U is unitary and P is a projection such that the difference U † P U − P is compact, then we define Ind(
P, U ) := dim ker( P − UP U † − ) − dim ker( P − UP U † + ) ∈ Z . This index is stable under norm-continuous changes of both U and P as long as it remains welldefined, see [3]. Remarks.
1. It is believed that the bulk index equals the Hall conductance of the material as obtainedfrom the Green-Kubo formula, up to a factor e /h with e the charge of the electron, and h Planck’sconstant. Under very natural homogeneity assumptions on the Hamiltonian (in particular, if theHamiltonian is translation invariant) this has been proven, see [4, 1].2. It is shown in [1] that the difference P F − UP F U † with U as in the statement of theorem 2.1 isindeed compact, so the bulk index is well defined. We now spell out the assumtions on the Hamiltonians H and ˆ H .Lattice points are labelled by ~x = ( x , x ) ∈ Z and we have corresponding states | ~x i : Z → C : ~y δ ~x,~y . These states form an orthonormal basis of the Hilbert space.The bulk Hamiltonian H is a self-adjoint operator on l ( Z ) which is exponentially local: |h ~x, H ~y i| ≤ C e −k ~x − ~y k /ξ for some C < ∞ , ξ > ~x, ~y ∈ Z .The Hilbert space for the edge system is ˆ H = l ( Z × N ). The natural inclusion of the half-space lattice Z × N in the bulk lattice Z × Z induces an injectionˆ H ι −→ H . The half-space Hamiltonian ˆ H is a self-adjoint operator on l ( Z × N ) that agrees with the bulkHamiltonian in the bulk: (cid:12)(cid:12)(cid:12) h ~x, ( ι † Hι − ˆ H ) ~y i (cid:12)(cid:12)(cid:12) ≤ C e −k ~x − ~y k /ξ − y /ξ ′ for some C < ∞ , ξ, ξ ′ > ~x, ~y ∈ Z × N . i.e. up to a boundary condition which is exponentiallylocalized near the edge of the system, the edge Hamiltonian equals the restriction of the bulk Hamiltonianto the half-space Z × N . In particular, the half-space Hamiltonian is itself exponentially local. Edge index and bulk-edge correspondence
Let g : R → [0 ,
1] be a smooth function interpolating from 1 to 0 and such that g ′ is supported in thebulk gap ∆. We have P F = g ( H ) . Consider now the unitary W g ( ˆ H ) where W g is the function W g : R → C : x e π i x . (1)This unitary is local and supported near the edge of the half-space. In particular: Lemma 3.1 ([7]) . Let ˆΠ denote the projection on { ~x ∈ Z × N | x ≥ } , then the commutator [ W g ( ˆ H ) , ˆΠ ] is trace class. This lemma follows immediately from lemmas A.2. and A.3. in [7].It follows that W g ( ˆ H ) † ˆΠ W g ( ˆ H ) − ˆΠ = W g ( ˆ H ) † [ ˆΠ , W g ( ˆ H )] is also trace class and in particularcompact so the index Ind E ( H, ∆) := Ind(Π , W g ( ˆ H )) ∈ Z is well defined. We call this integer the edge index . Remark.
The edge index may a priori depend on the boundary conditions defining the edge Hamiltonian ˆ H . The following theorem implies that this is not the case, justifying our notation Ind E ( H, ∆) . Theorem 3.2 (Theorem 2.11 of [9]) . Under the above assumptions on H , Ind B ( H, ∆) = Ind E ( H, ∆) for any bulk gap ∆ . Remark.
In [9] the bulk and edge indices are given als Fredholm indices. The equivalence to the definition2.2 is established in theorem 5.2. in [3].
Theorem 2.1 is an almost direct consequence of the following proposition from [2].
Proposition 4.1 (Theorem 2.1. of [2]) . Let U be a unitary operator on a Hilbert space and P anorthogonal projection such that [ U, P ] is trace class, then the index Ind(
U, P ) is a well-defined finiteinteger. If Ind(
U, P ) = 0 , then the absolutely continuous spectrum of U is the entire unit circle. The unitary W g ( ˆ H ) and the projecion ˆΠ satisfy the assumptios of this proposition due to lemma3.1.We are now ready to give the proof of the main theorem. Proof of Theorem 2.1 :
Using theorem 3.2, if Ind B ( H, ∆) = 0 also Ind E ( H, ∆) = Ind( W g ( ˆ H ) , ˆΠ ) =0. It follows then from lemma 3.1 and proposition 4.1 that the absolutely continuous spectrum of theedge unitary W g ( ˆ H ) is the whole unit circle.For any ǫ < | ∆ | / ǫ = { x | dist( x, ∆ c ) > ǫ } be the gap ∆ shrunk by ǫ . Similarly, for any δ < π let∆ ′ δ = { e i θ | θ ∈ ( δ, π − δ ) } be the unit circle with a closed δ -neighbourhood around 1 excluded. We canchoose g in such a way that x W g ( x ) := e π i g ( x ) is a smooth function that satisfies ∆ ǫ = g − (cid:0) ∆ ′ δ ( ǫ ) (cid:1) for all such ǫ , and such that W g is invertible on ∆ ǫ . Then lemma A.2 applies to give W g (cid:0) σ ac ( ˆ H ) ∩ ∆ ǫ (cid:1) = σ ac (cid:0) W g ( ˆ H ) (cid:1) ∩ ∆ ′ δ = ∆ ′ δ . where we used that the absolutely continuous spectrum of W g ( ˆ H ) is the whole unit circle.Since W g is invertible on ∆ ǫ = W − g (∆ ′ δ ) it follows that ∆ ǫ is contained in the absolutely continuousspectrum of ˆ H for any ǫ . Since the absolutely continuous spectrum is a closed set, it follows that thegap ∆ is contained in the absolutely continuous spectrum of ˆ H , as required. (cid:3) Spectral mapping for absolutely continuous spectrum
Let A be a bounded self-adjoint operator on a separable Hilbert space H and Let g : R → U (1) be asmooth function. By the spectral calculus this defines a unitary operator B = g ( A ). We investigate therelation between the absolutely continuous spectra of A and B .Let O be A or B and let X be R if O is self-adjoint, and U (1) if O is unitary. Take a continuousfunction f ∈ C ( X ) from X to C and a state ψ ∈ H . Through the continuous functional calculus wedefine a bounded linear functional on C ( X ) by f
7→ h ψ, f ( O ) ψ i By Riesz-Markov there is associated to this functional a unique measure µ ( O ) ψ on X such that µ ( O ) ψ ( X \ σ ( O )) = 0 and h ψ, f ( O ) ψ i = Z X d µ ( O ) ψ ( λ ) f ( λ ) . Let H ( O ) ac = { ψ | µ ( O ) ψ is absolutely continuous w.r.t. Lebesgue } and P ( O ) ac the orthogonal projectiononto this subspace. Then O ac = P ( O ) ac OP ( O ) ac is the absolutely continuous part of O and σ ac ( O ) = σ ( O ac ).We have that x ∈ σ ac ( O ) if and only if x ∈ supp µ ( O ) ψ for some ψ ∈ H ( O ) ac . Lemma A.1. If g : R → U (1) is continuous and invertible on σ ( A ) , then σ ac ( g ( A )) = g (cid:0) σ ac ( A ) (cid:1) . Proof :
We have x ∈ σ ac ( A ) if and only if there is a ψ ∈ H ( A ) ac such that x ∈ supp µ ( A ) ψ . To the samestate ψ is associated the linear functional f
7→ h ψ, f ( g ( A )) ψ i and the unique measure µ ( g ( A )) ψ on U (1) such that h ψ, f ( g ( A )) ψ i = Z U (1) d µ ( g ( A )) ψ ( λ ) f ( λ ) . At the same time, we have h ψ, f ( g ( A )) ψ i = Z R d µ ( A ) ψ ( λ ) f ( g ( λ )) . The spectral measures µ ( A ) ψ and µ ( g ( A )) ψ are therefore related by a change of variable: µ ( g ( A )) ψ (∆) = µ ( A ) ψ ( g − (∆))for any Borel set ∆. Since g is continuous and invertible, we have then that the measure µ ( A ) ψ is absolutelycontinuous w.r.t. Lebesgue on R if and only if µ ( g ( A )) is absolutely continuous w.r.t. Lebesgue on U (1),and their supports are related by supp µ ( g ( A )) ψ = g (cid:16) supp µ ( A ) ψ (cid:17) i.e. x ∈ supp µ ( A ) ψ if and only if g ( x ) ∈ supp µ ( g ( A )) ψ . From this we conclude the proof of the lemma. (cid:3) We now extend this result a bit as follows,
Lemma A.2. If g : R → U (1) is differentiable on σ ( A ) and invertible on ∆ A = g − (∆ B ) for some opensubset ∆ B ⊂ U (1) , then if B = g ( A ) we have g (cid:0) σ ac ( A ) ∩ ∆ A (cid:1) = σ ac ( B ) ∩ ∆ B Proof :
Let P ( A )∆ A be the spectral projection for A on ∆ A and P ( B )∆ B the spectral projection of B on∆ B . Then letting A ′ = P ( A )∆ A AP ( A )∆ A and B ′ = P ( B )∆ B BP ( B )∆ B we have g ( A ′ ) = B ′ and g : ∆ A = σ ( A ′ ) → ∆ B is differentiable and invertible. Therefore, lemma A.1 applies and gives σ ac ( B ′ ) = g ( σ ac ( A ′ )) . ut σ ac ( A ′ ) = σ ac ( A ) ∩ ∆ A and σ ac ( B ′ ) = σ ac ( B ) ∩ ∆ B so σ ac ( B ) ∩ ∆ B = g (cid:0) σ ac ( A ) ∩ ∆ A (cid:1) as required. (cid:3) Acknowledgements
A.B. and A.H.W. were supported by VILLIUM FONDEN through the QMATH Centre of Excellence(grant no. 10059). A.H.W. thanks the VILLIUM FONDEN for its support with a Villium YoungInvestigator Grant (grant no. 25452).
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