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Week 10: Oct 27, 2016 Preamble to Quantum Hall Effect: Exotic Week 10: Oct 27, 2016 Preamble to Quantum Hall Effect: Exotic Phenomenon in Quantum Physics The quantum Hall effect is one of the most remarkable of all quantum-matter phenomena, quite unanticipated by the physics community at the time of its discovery in 1980. This very surprising discovery earned von Klitzing the Nobel Prize in physics in 1985. The basic experimental observation is the quantization of resistance, in two-dimensional systems, to an extreme precision, irrespective of the sample’s shape and of its degree of purity. This intriguing phenomenon is a manifestation of quantum mechanics on a macroscopic scale, and for that reason, it rivals superconductivity and Bose–Einstein condensation in its fundamental importance. As we will see later in this class, that this effect is an example of Berry phase and the “quanta” that appear in the quantization of Hall conductivity are the topological integers that emerge from quantum analog of Gauss-Bonnet theorem. This years Nobel prize to David Thouless is for explaining this exotic quantization in terms of topology. It turns out that the classical and quantum anholonomy are described by the same mathematics. To understand that, we need to introduce the concept of “curvature. As we know, it is the curved space that leads to anholonomy in classical physics. This allows us to define curvature in quantum physics. That is, using the concept of anholonomy in quantum physics, we can define the concept of curvature in quantum physics – that is, in Hilbert space. Curvature leads us to Topological Invariants that tells us that anholonomy may have its roots in topology. It will also lead to topological integers, which are the quantum numbers in quantum Hall effect. I. TOPOLOGY AND TOPOLOGICAL INVARIANTS: A. Topological Invariant for Two-Dimensional Surfaces This index, like most topological invariants in physics, arises as the integral of a geometric quantity. It is defined as follows: 1 Z χ = κ ds (1) 2π S 1 Here κ is the local curvature of the surface — a concept that we shall now define. First of all, let us consider curves in a Euclidean plane. If the curve is a circle (the simplest case), then the reciprocal of its radius, κ = 1=R, is defined to be its local curvature (and the curvature is the same at every point of the circle). Clearly, the smaller the circle, the greater the curvature, and vice versa. And a straight line, being a circle of infinite radius, has zero curvature, which makes perfect sense, since it is not curved at all. For curves that are not circular, the curvature will vary from point to point. The local curvature will of course be greater at points where the curve turns more sharply, and lower where it approaches straightness. To obtain the precise numerical value of the curvature, one finds the circle that best fits the curve at the point in question, and the inverse radius of that circle is defined to be the local curvature at that point. This notion of local curvature can be extended from two-dimensional curves to three-dimensional manifolds (surfaces in 3-space), such as a tin can, a fruit bowl, an egg, someone’s cheek, and so forth. The most na¨ıve approach to defining the local curvature of such an object (that is, the idea based on the simplest possible analogy to the two-dimensional case) would be, given a point on the object’s surface, to try to find “the best-fitting circle” at that point. Unfortunately, however, this na¨ıve approach doesn’t work, because in general, many different circles of different radii will fit snugly against the object at the chosen point, since at each point on the object’s surface, there are infinitely many different directions that one could choose. The key idea for defining the Gaussian curvature of a two-dimensional surface is one whose curvature κmax is greatest and one whose curvature κmin is least. These two circles’ curvatures are known as the manifold’s principal curvatures at that point. (By the way, some 250 or so years ago, Euler proved that for any point of any manifold, the extremal circles are always oriented at 90 degrees to each other.) We now define κ, the local Gaussian curvature at any given point, as the product of the two extremal curvatures: κ = κmaxκmin. For the pipe, since the minimal curvature is 0, it doesn’t matter what the maximal curvature is, since their product will always be 0. Thus the local curvature at every point of the pipe equals 0. 1 1 For a sphere of radius R, κmax = κmin = R , so κ = R2 . Consider next a horse’s saddle, and pick some point near its middle. At that spatial point, if you move backwards or forwards (i.e., headwards or tailwards, if the saddle is sitting on a horse), the saddle curves upwards, while if you move sideways, the saddle curves downwards. For this reason, we assign the two 2 curvatures opposite signs, so that κmax > 0 and κmin < 0. Thus the Gaussian curvature at that point, κmaxκmin, will be negative. it turns out, rather miraculously, that for any two-dimensional closed manifold, the total Gaussian curvature is always an integer. This is a consequence of a beautiful theorem, the Gauss–Bonnet theorem in differential geometry. It is named after Carl Friedrich Gauss, who was aware of a version of the theorem but never published it, and Pierre Ossian Bonnet, who published a special case of it in 1848. The Gauss–Bonnet Theorem and Topological Integers 1 Z χ = κ ds = 2(1 − g) (2) 2π S In this equation, g is the genus of the object in question — that is, the number of handles it possesses (always an integer, of course). Both g and the number χ are topological invariants. For a sphere, which clearly has no handle, g = 0 and hence χ = 2. For a torus, g = 1 and thus χ = 0. A coffee mug has one handle, and so, topologically speaking, it is a torus; that is to say, coffee mug = doughnut (at least as viewed by a topologist). The Gauss–Bonnet theorem states that the integral of the curvature over the whole surface is “quantized” (restricted to integer values, in this case), and is a topological invariant. We will next discuss anholonomy, a phenomenon that is rooted in curved space in classical systems such as Foucault pendulum. It relates directly to the notion of curvature for classical systems. This phenomena of anholonomy allows us to define the concept of curvature in quantum systems. Define Solid Angle In geometry, a solid angle symbol Ω is the two-dimensional angle in three-dimensional space that an object subtends at a point. It is a measure of how large the object appears to an observer looking from that point. A small object nearby may subtend the same solid angle as a larger object farther away. For example, although the Moon is much smaller than the Sun, it is also much closer to Earth. Indeed, as viewed from any point on Earth, both objects have approximately the same solid angle as well 3 as apparent size. A solid angle in steradians equals the area of a segment of a unit sphere in the same way a planar angle in radians equals the length of an arc of a unit circle. If the circle has a radius R, and the arc length dl, the angle is dl=R, as ld = Rdθ. Similarly for a sphere of radius R, the solid angle is dA=R2 where dA is the area of the cap, dA = R2 sin θdθdφ. Therefore, a cap of area A on the surface of a sphere of radius R will subtend a solid angle A Ω = R2 . For a triangle on a sphere as shown in Fig. (2), the solid angle is Ω = θ1 + θ2 + θ3 − π. The Foucault Pendulum: An Example of Classical Anholonomy As the earth rotates through an angle of 2π radians (360 degrees), the system’s cyclic pathway results in the plane of oscillation of the pendulum rotating through a smaller angle. An observer on Earth witnesses that the orientation of the pendulum — that is, of its plane of oscillation — slowly rotates during the course of the day, and in general does not return to its original orientation after 24 hours. II. GEOMETRIZATION OF THE FOUCAULT PENDULUM NOTE: Straight lines on a sphere are great circles When a pendulum is carried around a great circle, the angle between the pendulum’s swing and the great circle never changes. If, however, the pendulum is taken along a path that is not a great circle — for example, a path of fixed latitude other than the equator — then the angle between the pendulum’s swing and the path does change. Triangular paths on a sphere Before discussing a pendulum that moves along a circle at a fixed latitude, let us consider a triangular pathway C on the sphere — that is, a pathway consisting of three great-circle segments, as shown in Figure 2. Let θ1, θ2, and θ3 be the angles at the vertices of the triangle. We want to 4 see how the pendulum’s plane of oscillation changes as it follows the closed triangular pathway. There is no change while the pendulum moves along any one segment. There is a change, however, each time that the pendulum moves from one segment to the next. At those moments, the angle between the plane of oscillation and the path is changed by the angle between the two segments at the point where they meet.
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