2.5 The free Green function
2.5.3 Properties of the free Green function
Figure 2.1 displays the gauge-independent, regularized part of the exact and semiclas- sical Green function. As one expects, the exact Green function decays exponentially once the points are separated by a distance,|r−r0|>2ρ, (iez >4ν) which cannot
be traversed classically.∗ Asr → r0, it has a logarithmic singularity (cf (2.71)) like
the (complex valued) field-free Green function [45]. Our method to evaluate the free Green function numerically with high precision and efficiency is discussed in Appen- dix E. There,Gb0
ν is displayed atν = 57.75, as a function of|r−r0|/ρ.
∗The abovementioned independent solution of (4.2) grows exponentially beyond the classically al- lowed region. Its derivation is sketched in Appendix B.
0 1 2 3 4 5 6 7 8 10−6 10−5 10−4 10−3 10−2 10−1 −0.1 0 0.1 PSfrag replacements √ z bG 0 ,ν bG 0(sc ) ν | bG 0 −ν bG 0(sc ) ν |
Figure 2.1: Regularized gauge-independent part of the free Green function. Top: Exact (solid line) and semiclassical (dashed line) functionsGb0
ν atν = 10.1. Bottom: Error of the
semiclassical approximation. Even at this moderate value ofνstrong deviations occur only at the classical turning point√z = 2√ν ≈ 6.36and at small distances. (The deviations arise since the semiclassical approximation does not account for the logarithmic singularity atz= 0
and the tunnelling into distances larger than the cyclotron diameter.)
Differential expressions
The gauge invariant part of the Green function has the remarkable property that its derivatives can be expressed by the function itself, at a different energy. For the regu- larized version one finds
z d dzGb 0 ν(z) =−(12 −ν) Gb0ν+Gb0ν−1 −z 2Gb 0 ν (2.68) z2 d 2 dz2Gb 0 ν(z) =(32 −ν)(12−ν) Gb0ν+ 2Gb0ν−1+Gb0ν−2 +z(12 −ν) Gbν0+Gb0ν−1+z 2 4Gb 0 ν . (2.69)
These formulas were obtained by employing the differential properties of the confluent hypergeometric function [43].
21
Asymptotic behaviour
Finally, we note the behaviour of the Green function at small distances. It has a loga- rithmic singularity, similar to the Green function of the field-free case [45].
b G0ν(z) = Lν(z) + O(zlogz) asz→ 0, (2.70) where Lν(z) := cos(πν) 4π log(z) + Ψ( 1 2 +ν)−2Ψ(1) −sin(4πν) . (2.71) Here,Ψis the Digamma function [43]. As for the derivatives of the gauge independent part of the Green function, one finds the asymptotic expressions
z d dzGb 0 ν(z) = cos(πν) 4π 1−z ν log(z) + Ψ(12 −ν)−2Ψ(1)−1+ O(z2logz), (2.72) z2 d 2 dz2Gb 0 ν(z) =− cos(πν) 4π + O(zlogz), asz→0. (2.73)
They were deduced from the logarithmic representation of U in terms of the regular Kummer function [43; eq. (13.6.1)]. These formulas will be needed in Section 4.4.
Notes
1. The motion on magnetic surfaces of finite curvature received some attention in recent years, both, in the classical [46–49] and the quantum treatment [46,47,50– 52]. One motivation for introducing a non-vanishing curvature is the possibility to study the quantum spectrum of the free magnetic motion on a compact domain (a modular domain in the case of constant negative curvature). This has consid- erable mathematical advantages, since the spectrum remains discrete in the limit of vanishing field.
2. After (lengthy) deliberations, the author decided to avert from Landau’s defini- tion of the magnetic length`B =b/
√
2. The latter is appropriate (only) for the Landau gauge (2.12). The lengthb, which is the appropriate scale of the symmet- ric gauge, proves more convenient, since it avoids the appearance of the factor
2and√2at various places. It gives the radius of a disk, the areab2πof which assumes the role of Planck’s quantum, cf Eq. (3.11a). Moreover, the flux through the disk equates the “flux quantum”Φ0=h/q=B b2π.
3. Scaled spectroscopy is applied successfully in atomic physics, eg [53]. The mea- surement of absorption spectra in the semiclassical direction allowed in particular to extract the actions of classical periodic orbits from Rydberg spectra [54]. It should be noted that scaled spectra have some unusual mathematical features since they do not belong to one self-adjoint operator. Rather, they stem from a family of Hamilton operators parametrized by an effective value of which de- pends on the spectral variable. As a result, the eigenfunctions are not orthogonal, although they are proper solutions of the Schr¨odinger equation with a real en- ergy. More severely, the stability of spectral points with respect to changes in an external parameter known from self-adjoint operators does not hold in general.
Chapter 3
Introducing a boundary
The motion in the magnetic plane turns into a non-trivial problem, once the particle is restricted to a bounded domain.
3.1
Motion in a restricted domain
Let us assume that the particle is confined to move in a domainD ⊂ 2, which is com- pact and singly connected. The classical equation of motion (2.4) still applies in the interior of the domain (ie, inD◦). Here, the particle moves on arcs of constant curva- ture, which may at some point impinge on the boundaryΓ =∂D. At these instances, the trajectories must obey the law of specular reflection to qualify as a classical solu- tion. This follows directly from Hamilton’s principle of requiring an extremal action, as will be shown in Sect. 6.3.1. Clearly, any trajectory which was reflected once must run into the boundary again. It follows, that the phase space is in general split up in two disjunct parts. One part consists of skipping orbits. Their classical motion is no longer described by a continuous Hamiltonian flow (but by a discrete map), and may range from regular (integrable) to completely chaotic (hyperbolic). It is characterized completely by the shape of the billiard and the size of the cyclotron radius. Below, in Section 3.2, we will briefly review this classical billiard problem. The remaining part of phase space describes the trivial motion on closed cyclotron orbits. It has a finite volume whenever the cyclotron radius is small enough to allow for a disk of radiusρto fit into the domain. We will call the magnetic field strong, accordingly, if the cyclotron radius is comparable to or smaller than the size of the billiard – a criterion which is purely classical.
In the corresponding quantum problem, the eigenfunctions are required to satisfy the Schr¨odinger equation in the open domainD◦, together with a boundary condition on the border lineΓ(as discussed in Sect. 3.3). One observes that, at strong fields, the spectrum reflects the partitioning of the classical phase space. There are eigenstates which hardly touch the boundary, and have energies very close to the Landau levels.
They are called bulk states, because in the limit of strong fields they constitute the major part of the spectrum. We will see that these states are based on that part of phase space which is given by unperturbed cyclotron motion. At the same time, one finds eigenstates which are localized at the boundary. These edge states correspond to the skipping trajectories, and are expected to reflect the underlying billiard motion. Albeit being an effect of the boundary they may be quite significant. For instance, they typically exhibit a directed probability flux causing a large magnetic moment. This way they balance the magnetic moments of the bulk, leading to a zero mean magnetization, as discussed in Section 3.4.
The separation into edge and bulk states is intuitively clear and often used. Early studies were concerned with the surface electron states inside metals [55, 56], and af- ter the discovery of the Quantum Hall Effect [57, 58] the notion of edge states was extensively employed to explain this phenomenon [59–64]. (In the latter problem the Hamiltonian must include an additional impurity potential.) However, the above char- acterization of edge states is rather vague and we are not aware of a general quantitative definition in the literature. In due course, we will propose a spectral measure, which allows to quantify the edge character of a state. Having a meaningful spectral density of edge states at our disposal, it will be worthwhile to consider the quantum problem also in the exterior.
Motion in the exterior
The exterior billiard problem is obtained by restricting the particle to the domain 2\D – henceforth called the exterior domain. From the classical point of view, there is little difference between the interior and the exterior dynamics. A particle impinging on the boundary from outside is reflected specularly and performs a skipping motion around the billiard. Like in the interior, the skipping trajectories cover a finite volume in phase space and are described by a discrete billiard bounce map. Complete cyclotron orbits, on the other hand, now exist for any ρ. The corresponding phase space volume is unbounded, because the cyclotron center may be located at an arbitrarily large distance from the billiard.
The fact that a “free particle” may not escape to infinity but is trapped on a cy- clotron orbit is reflected by the exterior quantum spectrum. It is discrete, in marked contrast to the field-free scattering situation. The exterior quantum problem requires the stationary wave function to satisfy the Schr¨odinger equation in 2\D◦, again with a boundary condition onΓ. In addition, the normalization condition (2.16) implies that the wave functions must vanish at infinity. In the absence of a boundary, the spectrum would be given by a discrete set of Landau energies, each infinitely degenerate, as shown in the preceeding chapter. The presence of a billiard lifts this degeneracy, turn- ing each Landau level into a spectral accumulation point. This means, that there are infinitely many discrete eigen-energies in any finite vicinity of each Landau energy.
We shall address the general quantum problem in Section 3.3. There, the main con- cern will be on the boundary conditions and the average spectral behaviour, whereas the actual quantization is performed in Chapter 4. However, to prepare for the semi- classical quantization in Chapter 6, it is first necessary to take a closer look at the classical problem.