Edge currents for magnetic barriers

Edge currents and eigenvalue estimates for magnetic barrier Schrödinger operators

Nicolas Dombrowski Department of Mathematics, University of Helsinki, FI-00014 Helsinki, Finland Peter D. Hislop Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506-0027, USA  and  Eric Soccorsi CPT, CNRS UMR 7332, Aix Marseille Université, 13288 Marseille, France & Université du Sud Toulon-Var, 83957 La Garde, France
Abstract.

We study two-dimensional magnetic Schrödinger operators with a magnetic field that is equal to for and for . This magnetic Schrödinger operator exhibits a magnetic barrier at . The unperturbed system is invariant with respect to translations in the -direction. As a result, the Schrödinger operator admits a direct integral decomposition. We analyze the band functions of the fiber operators as functions of the wave number and establish their asymptotic behavior. Because the fiber operators are reflection symmetric, the band functions may be classified as odd or even. The odd band functions have a unique absolute minimum. We calculate the effective mass at the minimum and prove that it is positive. The even band functions are monotone decreasing. We prove that the eigenvalues of an Airy operator, respectively, harmonic oscillator operator, describe the asymptotic behavior of the band functions for large negative, respectively positive, wave numbers. We prove a Mourre estimate for a family of magnetic and electric perturbations of the magnetic Schrödinger operator and establish the existence of absolutely continuous spectrum in certain energy intervals. We prove lower bounds on magnetic edge currents for states with energies in the same intervals. For a different class of perturbations, we also prove that these lower bounds imply stable lower bounds for the asymptotic edge currents. We study the perturbation by slowly decaying negative potentials. Using the positivity of the effective mass, we establish the asymptotic behavior of the eigenvalue counting function for the infinitely-many eigenvalues below the bottom of the essential spectrum.

AMS 2000 Mathematics Subject Classification: 35J10, 81Q10, 35P20
Keywords: magnetic Schrödinger operators, snake orbits, magnetic field, magnetic edge states, edge conductance

1. Statement of the problem and results

We continue our analysis of the spectral and transport properties of perturbed magnetic Schrödinger operators describing electrons in the plane moving under the influence of a transverse magnetic field. In [13], two of us studied the original Iwatsuka model for which . The basic model treated in this paper consists of a transverse magnetic field that is constant in each half plane so that it is equal to for and for . We choose a gauge so that the corresponding vector potential has the form . The second component of the vector potential is obtained by integrating the magnetic field so that , independent of . The fundamental magnetic Schrödinger operator is:

 H0:=p2x+(py−b|x|)2,  px:=−i∂/∂x,  py:=−i∂/∂y, (1.1)

defined on the dense domain . This operator extends to a nonnegative self-adjoint operator in .

The magnetic field is piecewise constant and equals on the half-planes , where . The discontinuity in the magnetic field at is called a magnetic edge. Classically, a particle moving within a distance of of the edge moves in a snake orbit [19]. Half of a snake orbit lies in the half-plane , and the other half of the orbit lies in . We prove that the quantum model has current flowing along the magnetic edge at and that the current is localized in a small neighborhood of size of .

1.1. Fiber operators and reflection symmetry

Due to the translational invariance in the -direction, the operator on is unitarily equivalent to the direct integral of operators , , acting on . This reduction is obtained using the partial Fourier transform with respect to the -coordinate and defined as

 (Fu)(x,k)=^u(x,k):=1√2π∫Re−iyku(x,y)dy,   (x,k)∈R2.

Then we have where

 H0:=∫⊕Rh(k)dk,

and the fiber operator acting in is

 h(k):=p2x+(k−b|x|)2,   k∈R.

Since the effective potential is unbounded as , the self-adjoint fiber operators have compact resolvent. Consequently, the spectrum of is discrete. We write for the eigenvalues listed in increasing order. They are all simple (see [12, Appendix: Proposition A.2]) and depend analytically on . As functions of , these functions are called the band functions or dispersion curves and their properties play an important role. For fixed , we denote by the -normalized eigenfunctions of with eigenvalue . These satisfy the eigenvalue equation:

 h(k)ψj(x,k)=ωj(k)ψj(x,k),  ψj(x,k)∈L2(R),  ∥ψj(⋅,k)∥=1. (1.2)

We choose all to be real, and for and . The rank-one orthogonal projections , , depend analytically on by standard arguments.

The full operator exhibits reflection symmetry with respect to . Let be the parity operator:

 (IPf)(x,y):=f(−x,y), (1.3)

so that . The Hilbert space has an orthogonal decomposition corresponding to the eigenspaces of with eigenvalue . The Hamiltonian commutes with so each eigenspace of is an -invariant subspace.

This symmetry passes to the fiber decomposition. For each we have , where is the restriction to of the operator defined in (1.3). Since the eigenvalues of are simple, for each , there is a map so that

 (IPψj)(x,k)=θj(k)ψj(x,k),   k∈R,   j∈N∗,

as is -normalized and real-valued. We show that is independent of . Since the mapping , the orthogonal projector onto , is analytic, it follows that for every . Consequently, each eigenfunction is either even or odd in .

We have an -invariant decomposition , according to the eigenvalues of the projection . From this then follows that , where

 h±(k):=h(k)|H±, H±:={f∈H, IPf=±f}.

We analyze the spectrum of by studying the spectrum of the restricted operators letting . Bearing in mind that and for every , we have

 ω+j(k)=ω2j−1(k), ω−j(k)=ω2j(k), j∈N∗.

1.2. Effective potential

The fiber operator has an effective potential:

 Veff(x,k):=(k−b|x|)2,  x,k∈R.

The properties of this potential determine those of the band functions.

Positive . There are two minima of at . The potential consists of two parabolic potential wells centered at and has value . As , the potential wells separate and the barrier between the two minima grows to infinity.

Negative . The effective potential is a parabola centered at and is the minimum. Consequently, as , the minimum of this potential well goes to plus infinity.

1.3. Band functions

The behavior of the effective potential determines the band functions. For , the symmetric double wells of indicate that there are two eigenvalues near each level of a harmonic oscillator Hamiltonian. The splitting of these eigenvalues is exponentially small in the tunneling distance in the Agmon metric between . As , this tunneling effect is suppressed and these two eigenvalues approach the harmonic oscillator eigenvalue exponentially fast. For , there is a single potential well with a minimum that goes to infinity as . Hence, the band functions diverge to plus infinity in this limit. Several band functions along with the parabola are shown in Figure 1.

1.4. Relation to edge conductance

Dombrowski, Germinet, and Raikov [10] studied the quantization of the Hall edge conductance for a generalized family of Iwatsuka models including the model discussed here. Let us recall that the Hall edge conductance is defined as follows. We consider the situation where the edge lies along the -axis as discussed above. Let be a compact energy interval. We choose a smooth decreasing function so that . Let be an -translation invariant smooth function with . The edge Hall conductance is defined by

 σIe(H):=−2πtr (g′(H)i[H,χ]),

whenever it exists. The edge conductance measures the current across the axis with energies below the energy interval .

Theorem 2.2 of [10] presents the quantization of edge currents for the generalized Iwastuka model. For this model, the magnetic field is simply assumed to be monotone and to have values at . The energy interval is assumed to satisfy the following condition. There are two nonnegative integers for which

 I⊂((2n−−1)|b−|,(2n−+1)|b−|)∩((2n+−1)|b+|,(2n++1)|b+|),  n±≠0. (1.4)

If , the corresponding interval should be taken to be . Under condition (1.4), Dombrowski, Germinet, and Raikov [10] proved

 σIe(H)=(sign b−)n−−(sign b+)n+.

Applied to the model studied here where and , and under condition (1.4), we have

 σIe(H)=−(n−+n+).

In particular, if , and , we have .

We complement this result by proving in sections 3 and 4 the existence and localization of edge currents for and its perturbations. Following the notation of those sections, we prove, roughly speaking, that there is a nonempty interval between the Landau levels and and a finite constant , so that for any state , where is the spectral projector for and the interval , we have

 ⟨ψ,vyψ⟩⩾cn2b1/2∥ψ∥2>0,  vy:=−(py−b|x|).

This lower bound indicates that such a state carries a nontrivial edge current for . We prove that this estimate is stable for a family of magnetic and electric perturbations of .

1.5. Contents

We present the properties of the band functions for the unperturbed fiber operator in section 2. The emphasis is on the behavior of the band functions as . The basic Mourre estimate for the unperturbed operator is derived in section 3 and its stability under perturbations is proven. As a consequence, this shows that there is absolutely continuous spectrum in certain energy intervals. Existence, localization, and stability of edge currents for a family of electric and magnetic perturbations is established in section 4. These edge currents and their lower bounds are valid for all times. We also prove a lower bound on the asymptotic velocity for a different class of perturbations in Theorem 4.2. In section 5, we study perturbations by negative potentials decaying at infinity. We demonstrate that such potentials create infinitely-many eigenvalues that accumulate at the bottom of the essential spectrum from below. We establish the asymptotic behavior of the eigenvalue counting function for these eigenvalues accumulating at the bottom of the essential spectrum.

1.6. Notation

We write and for the inner product and norm on . The functions are written with coordinates , or, after a partial Fourier transform with respect to , we work with functions . We often view these functions on as parameterized by . In this case, we also write and for the inner product and related norm on . So whenever an explicit dependance on the parameter appears, the functions should be considered on . We indicate explicitly in the notation, such as , for , when we work on those spaces. We write for for . For a subset , we denote by the set . Finally for all we put .

1.7. Acknowledgements

ND is supported by the Center of Excellence in Analysis and Dynamics Research of the Finnish Academy. PDH thanks the Université de Cergy-Pontoise and the Centre de Physique Théorique, CNRS, Luminy, Marseille, France, for its hospitality. PDH was partially supported by the Université du Sud Toulon-Var, La Garde, France, and National Science Foundation grant 11-03104 during the time part of the work was done. ES thanks the University of Kentucky for hospitality.

Remark 1.

After completion of this work, we learned of a similar analysis of the band structure by Nicolas Popoff [16] in his 2012 thesis at the Université Rennes I. We thank Nicolas for many discussions and for letting us use his graph in Figure 1.

Remark 2.

After completing this paper, we discovered the paper “Dirichlet and Neumann eigenvalues for half-plane magnetic Hamiltonians,” by V. Bruneau, P. Miranda, and G. Raikov [6]. Their Corollary 2.4, part (i), is similar to our Theorem 5.1.

2. Properties of the band functions

In this section, we prove the basic properties of the band functions . We have the basic identity:

 ωj(k)=⟨ψj(⋅,k),h(k)ψj(⋅,k)⟩.

According to section 1.1, the eigenfunctions of are either even and lie in , or odd and lie in , with respect to the reflection . We label the states so that the eigenfunctions and , for . The restrictions of to are denoted by , with eigenvalues and , respectively.

Following the qualitative description in section 1.2, we have the following asymptotics for the band functions. When , the band function satisfies , whereas as , we have .

Proposition 2.1.

The band functions are differentiable and the derivative satisfies

 ω′j(k)=−2b[(ωj(k)−k2)ψj(0,k)2+ψ′j(0,k)2]. (2.1)

As a consequence, we have a classification of states:

1. Odd states: . The band functions satisfy:

 (ω−j)′(k)=ω′2j(k)=−2bψ′2j(0,k)2<0. (2.2)
2. Even states: . The band functions satisfy:

 (ω+j)′(k)=ω′2j−1(k)=−2b(ω2j−1(k)−k2)ψ2j−1(0,k)2. (2.3)
Proof.

The Feynman-Hellmann Theorem gives us

 ω′j(k) = ∫R 2(k−b|x|)ψj(x,k)2 dx = −1b∫∞0ψj(x,k)2ddx(k−bx)2 dx +1b∫0−∞ψj(x,k)2ddx(k+bx)2 dx.

Integrating by parts, and using the ordinary differential equation (1.2), we obtain (2.1). Note that since is in the domain of (see [14, Lemma 3.5]). ∎

Let us note that we cannot have both and . As consequences, the band functions for odd states are strictly monotone decreasing . For even states, there is a minimum at satisfying

 ω2j−1(κj)=κ2j.

We will prove in Proposition 2.4 that this is the unique critical point of these band functions and that it is a non-degenerate minimum. This shows that there is an effective mass at this point. This is essential for the discussion in section 5.

2.1. Absolutely continuous spectrum for H0

The spectrum of is the union of the ranges of the band functions . The band functions are analytic and nonconstant by Propositions 2.1 and 2.4. Consequently, from [18, Theorem XIII.86], the spectrum of is purely absolutely continuous.

2.2. Band function asymptotics k→−∞.

As , we will prove that the fiber Hamiltonian is well approximated by an Airy operator

 hAi(k):=p2x+2b|k||x|+k2, (2.4)

in the sense that the band functions of are close to the band functions of the Airy operator . In order to establish this, let be the standard Airy function whose zeros are located on the negative real axis. The Airy function satisfies the Airy ordinary differential equation:

 Ai′′(x)=xAi(x).

By scaling and translations, it follows that the Airy function satisfies

 (p2x+γ3x)Ai(γx+σ)=−γ2σAi(γx+σ),  γ,σ∈R. (2.5)

The model Airy Hamiltonian in (2.4) has discrete spectrum and eigenfunctions satisfying

 hAi(k)~ΨAij(x,k)=~ωj(k)~ΨAij(x,k). (2.6)

It follows from (2.5) that the eigenfunction in (2.6) is a multiple of the scaled and translated Airy function. The non-normalized solution for the eigenvalue is

 ~ΨAij(x,k)=Ai((2b|k|)1/3|x|+k2−~ωj(k)(2b|k|)2/3), k<0, x∈R,

with an eigenvalue given by

 ~ωj(k)=k2−(2b|k|)2/3σ.

We determine as follows. The operator commutes with the parity operator so its states are even or odd. The odd eigenfunctions of must satisfy . Consequently, the -normalized odd eigenfunctions are given by

 ΨAi,oj(x,k)=CAi,j(b,k)(sign x)Ai((2b|k|)1/3|x|+zAi,j),  ΨAi,oj(0,k)=0, (2.7)

where is the zero of and the corresponding eigenvalue is

 ~ω2j(k)=k2−(2b|k|)2/3zAi,j.

The even eigenfunctions of must have a vanishing derivative at and are given by

 ΨAi,ej(x,k)=CAi′,j(b,k)Ai((2b|k|)1/3|x|+zAi′,j),  (ΨAi,ej)′(0,k)=0, (2.8)

and the corresponding eigenvalue is

 ~ω2j−1(k)=k2−(2b|k|)2/3zAi′,j,

where is the zero of . The normalization constant , for is given by

 CX,j(b,k):=((2b|k|)1/32cX,j)1/2,where  cX,j:=∫∞0Ai(v+zX,j)2 dv. (2.9)

We now obtain estimates on the band functions as .

Proposition 2.2.

For each , as , we have

 ∥(h(k)−[k2−(2b|k|)2/3zX,j])ΨAi,uj(⋅,k)∥⩽b4/3(2|k|)2/3DX,j, (2.10)

where the constant , given in (2.11), is independent of the parameters , and or , for even or odd states, respectively. This immediately implies the eigenvalue estimate

 |ωj(k)−[k2−(2b|k|)2/3zX,j]|⩽b4/3(2|k|)2/3DX,j,  k→−∞.
Proof.

In order to prove (2.10), we note that

 h(k)−hAi(k)=b2x2,

so that with the definition of in (2.7) for and (2.8) for , and the normalization constant in (2.9), we have

 ∥[h(k)−hAi(k)]ΨAi,uj(⋅,k)∥2 = 2b4(2|k|b)5/3C2X,j∫∞0v4Ai(v+zX,j)2 dv = b8/3(2|k|)4/3D2X,j,

where the constant , given by

 DX,j:=(∫∞0v4Ai(v+zX,j)2 dvcX,j)1/2, (2.11)

is finite since as . ∎

2.3. Band functions asymptotics k→+∞

For , the effective potential consists of two double wells that separate as . Consequently approaches as . The eigenvalues of the double well potential consists of pairs of eigenvalues whose differences are exponentially small as . Thus, the effective Hamiltonian for is the harmonic oscillator Hamiltonian:

 hHO(k):=−d2dx2+(bx−k)2.

We let and , for every , denote the energy levels of the harmonic oscillator. Let denote the normalized eigenfunction of the harmonic oscillator so that . It can be explicitly expressed as

 ΨHOj(x,k):=1(2jj!)1/2(bπ)1/4e−b/2(x−k/b)2Hj(b1/2(x−k/b)), (2.12)

where is the Hermite polynomial.

Proposition 2.3.

For each , there exists a constant , depending only on , so that for , we have,

 ∥(h(k)−ej(b))ΨHOj(±x;k)∥⩽Cjbe−k2/(4b). (2.13)

This immediately implies the eigenvalue estimate

 0<∓(ω±j(k)−ej(b))⩽Cjbe−k24b, k⩾κj, (2.14)

and the difference of the two eigenvalues is bounded as

 0⩽ω−j(k)−ω+j(k)⩽2Cjbe−k24b, k⩾κj. (2.15)
Proof.

1. Since is the eigenfunction of the harmonic oscillator Hamiltonian, we have for all ,

 (h(k)−ej(b))ΨHOj(±x,k)=((bx±k)2−(bx∓k)2)χR∓(x)ΨHOj(±x,k),

so that for any , we have

 ∥(h(k)−ej(b))ΨHOj(±x,k)∥⩽∥(bx∓k)2ΨHOj(±x,k)∥L2(R∓). (2.16)

Here stands for the characteristic function of . From (2.16), the identity

 ∥(bx∓k)2ΨHOj(±x,k)∥L2(R∓)=∥(bx−k)2ΨHOj(x,k)∥L2(R−),

and (2.12), it follows that

 ∥(h(k)−ej(b))ΨHOj(±x,k)∥⩽cjbe−k2/(4b), k⩾0, (2.17)

for some constant depending only on .

2. Let . In light of (2.17) we have

 ∥(h(k)−ej(b))(ΨHOj)±(k)∥H⩽cjbe−k2/(4b), k⩾0. (2.18)

Further since

 ∥(ΨHOj)±(k)∥2=(1±∫RΨHOj(x,k)ΨHOj(−x,k)dx)/2,

with

for some constant depending only on , we deduce from (2.18) that

 dist(σ(h±(k)),ej(b))⩽Cjbe−k2/(4b), k⩾0, (2.19)

where depends only on .

3. As for each , from the minimax principle, the result (2.14) for follows readily from (2.19). The case of is more complicated. In the section 2.4, we prove in the derivation of Proposition 2.4 that the band function has a unique absolute minimum at a value . Furthermore, . We also prove that for and for . The facts that the analytic band function is monotone increasing for and converges to as due to (2.19) imply the result (2.14) for . ∎

2.4. Even band functions ω+j(k): the effective mass

We prove that the even states in , with band functions , have a unique positive minimum at . We prove that the even band function is concave at . This convexity means that there is a positive effective mass. This positive effective mass plays an important role in the perturbation theory and creation of the discrete spectrum discussed in section 5.

Proposition 2.4.

The band functions , corresponding to the even states of , each have a unique extremum that is a strict minimum. The minimum is attained at a single point . This point is the unique real solution of , and . The concavity of the band function at is strictly positive and given by:

 (ω+j)′′(κj)=ω′′2j−1(κj)=4κjbψ2j−1(0,κj)2>0. (2.20)

We also have for .

Proof.

1. We first prove that there exists a unique minimum for the band function. The Feynman-Hellmann formula yields

 (ω+j)′(k)=−2∫R(b|x|−k)ψ+j(x,k)2dx, k∈R. (2.21)

Next, recalling (2.3), we get that

 (ω+j)′(k)=2bf+j(k)ψ+j(0,k)2, f+j(k):=k2−ω+j(k), (2.22)

since and . Moreover, taking into account that we see that

 ω+j(k)⩽ω+j(0)=e2j−1(b), k∈R+, (2.23)

as in this case. Therefore we have and for all from (2.23). The function is continuous in hence there exists such that . Moreover, being real analytic, the set is at most discrete so we may assume without loss of generality that is its smallest element.

2. We next prove that is decreasing for and increasing for . It follows from (2.21) that . Integrating this inequality over the interval , we obtain

 ω+j(k)<ω+j(κj)+∫kκj2tdt=ω+j(κj)+(k2−κ2j),  k>κj,

and hence for all . This result with the fact that and (2.22) imply that for .

3. To study the concavity of the band function and establish (2.20), we differentiate (2.22) with respect to and obtain

 (ω+j)′′(k)=−2b([(ω+j)′(k)−2k]ψ+j(0,k)2−2f+j(k)ψ+j(0,k)∂kψ+j(0,k)),  k∈R