* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Integer Quantum Hall Effect - (Dawn of topology in
Survey
Document related concepts
Scalar field theory wikipedia , lookup
History of quantum field theory wikipedia , lookup
Atomic orbital wikipedia , lookup
Molecular Hamiltonian wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Hydrogen atom wikipedia , lookup
Quantum state wikipedia , lookup
Quantum group wikipedia , lookup
Electron configuration wikipedia , lookup
Bra–ket notation wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Ferromagnetism wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Topological quantum field theory wikipedia , lookup
Canonical quantization wikipedia , lookup
Transcript
Integer Quantum Hall Effect (Dawn of topology in electronic structure) Raffaele Resta Dipartimento di Fisica, Università di Trieste Trieste, May 2016 . . . . . . Outline 1 What topology is about 2 Topology shows up in electronic structure 3 Classical Hall effect 4 2d noninteracting electrons in a magnetic field 5 Quantum Hall Effect . . . . . . Topology Branch of mathematics that describes properties which remain unchanged under smooth deformations Such properties are often labeled by integer numbers: topological invariants Founding concepts: continuity and connectivity, open & closed sets, neighborhood...... Differentiability or even a metric not needed (although most welcome!) . . . . . . Topology Branch of mathematics that describes properties which remain unchanged under smooth deformations Such properties are often labeled by integer numbers: topological invariants Founding concepts: continuity and connectivity, open & closed sets, neighborhood...... Differentiability or even a metric not needed (although most welcome!) . . . . . . thors Pictures onardo Da Vinci Shop mposers Posters mposers T shirts mous Physicists Math Mug (set theory): If you consider the set of all sets that have never been considered .... A coffee cup and a doughnut are the same The above shapes are topologically equivalent and are of Genus 0 Math Mug - Topology To a Topologist This is a Doughnut http://cosmology.uwinnipeg.ca/Cosmology/Properties-of-Space.htm (7 of 11) [03/01/2002 7:23:17 PM] Topological invariant: genus (=1 here) Math Mug: Real Life is a Special Case (black background) . . . . . . Gaussian curvature: sphere In a local set of coordinates in the tangent plane z =R− ( Hessian H= Gaussian curvature √ R2 − x 2 − y 2 ≃ 1/R 0 0 1/R ) K = det H = . x2 + y2 2R . 1 R2 . . . . Gaussian curvature: sphere In a local set of coordinates in the tangent plane z =R− ( Hessian H= Gaussian curvature √ R2 − x 2 − y 2 ≃ 1/R 0 0 1/R ) K = det H = . x2 + y2 2R . 1 R2 . . . . Positive and negative curvature Bob Gardner's "Relativity and Black Holes" Special Relativity > For example, a plane tangent to a 2/3/12 sphere lies the sphere, and and so Black a sphere is of Relativity positive Bob Gardner's "Relativity Holes" Special 4:27 PM entirely on one side of curvature. In fact, a sphere of radius r is of curvature 1/r2. 2/3/12 4:27 PM > A cylinder is of zero curvature since a tangent plane lies on one side of the cylinder and the points of saddle shaped (or more precisely, a hyperbolic paraboloid) is of negative curvature. A tangent For example, a plane tangent to a sphere lies entirely on one side ofAthe sphere, and surface so a sphere is of positive inof blue) are a lineare containing the point of tangency. Also, of course, a plane is of zero plane lies on both sides of the surface. Here, the point of tangency isintersection red and the(here points intersection curvature. In fact, a sphere of radius r is of curvature 1/r2. curvature.paraboloid. blue. Pringles potato chips are familiar examples of sections of a hyperbolic Smooth surface, local set of coordinates on the tangent plane ) ( 2 2 K = det http://www.etsu.edu/physics/plntrm/relat/curv.htm A saddle shaped surface (or more precisely, a hyperbolic paraboloid) is of negative curvature. A tangent plane lies on both sides of the surface. Here, the point of tangency is red and the points of intersection are blue. Pringles potato chips are familiar examples of sections of a hyperbolic paraboloid. ∂ z ∂x 2 ∂2z ∂y ∂x ∂ z ∂x∂y ∂2z ∂y 2 Page 6 of 9 If two surfaces have the same curvature, we can smoothly transform one into the other without changing distances (the transformation is called an isometry). For example, a sheet of paper (used here to represent a curvature zero plane) can be rolled up to form a cylinder (which also has zero curvature). However, we cannot role the paper smoothly into a sphere (which is of positive curvature). For example, if we try to giftwrap a basketball, then the paper will overlap itself and have to be crumpled. We also cannot role the . . . . . . paper smoothly over a saddle shaped surface (which is of negative curvature) since this would require ripping a curved space but not in a flat space may lead to the idea of using it as a way to measure curvature. We cannot prove it mathematically here, but it turns out that if we choose a loop that is small enough around a point in a curved space, the amount of change in the direction of a vector that is parallel transported along it is proportional to the area enclosed by the loop. So, the ratio between the area of the loop and the amount of change in the direction of the vector (whatever way we chose to measure it) can be used as a measure to the curvature of the surface that includes the loop. Actually we define curvature by the value of this ratio. Twist angle vs. Gaussian curvature Gaussian curvature of the spherical surface Ω = 1/R 2 Angular mismatch for parallel transport: ∫ Riemann Curvature Tensor γ = dσ K Riemann tensor is a rank (1,3) tensor that describes the curvature in all directions at a given point in space. It takes 3 vectors and returns a single vector. The vectors that are fed to the tensor should be very small and have a length ε. If we use the first two vectors to form a tiny parallelogram and we parallel transport the third vector around this parallelogram, the vector that the function returns is approximately the vector difference between the original vector and the vector after the parallel transport. Mathematically it can be written like this: Equivalently: sum of the three angles: ∫ α1 + α2 + α3 = π + γ = π + . dσ K . . . . . a curved space but not in a flat space may lead to the idea of using it as a way to measure curvature. We cannot prove it mathematically here, but it turns out that if we choose a loop that is small enough around a point in a curved space, the amount of change in the direction of a vector that is parallel transported along it is proportional to the area enclosed by the loop. So, the ratio between the area of the loop and the amount of change in the direction of the vector (whatever way we chose to measure it) can be used as a measure to the curvature of the surface that includes the loop. Actually we define curvature by the value of this ratio. Twist angle vs. Gaussian curvature Gaussian curvature of the spherical surface Ω = 1/R 2 Angular mismatch for parallel transport: ∫ Riemann Curvature Tensor γ = dσ K Riemann tensor is a rank (1,3) tensor that describes the curvature in all directions at a given point in space. It takes 3 vectors and returns a single vector. The vectors that are fed to the tensor should be very small and have a length ε. If we use the first two vectors to form a tiny parallelogram and we parallel transport the third vector around this parallelogram, the vector that the function returns is approximately the vector difference between the original vector and the vector after the parallel transport. Mathematically it can be written like this: Equivalently: sum of the three angles: ∫ α1 + α2 + α3 = π + γ = π + . dσ K . . . . . choose a loop that is small enough around a point in a curved space, the amount of change in the direction of a vector that is parallel transported along it is proportional to the area enclosed by the loop. So, the ratio between the area of the loop and the amount of change in the direction of the vector (whatever way we chose to measure it) can be used as a measure to the curvature of the surface that includes the loop. Actually we define curvature by the value of this ratio. What about integrating over a closed surface? Figure 4. The Gauss–Bonnet formula trated here by a toroidal surface with local curvature K is positive on those face that resemble a sphere and nega the hole, that resemble a saddle. Bec of handles, equals one, the integral o the entire surface vanishes. One can Shiing-shen Chern’s quantum genera Gauss–Bonnet formula plausible by c angular mismatch of parallel transpo around the small red patch in the fig allel transport around the small loop enclosing the area dA. The left side of the Gauss–Bonnet equation is geometric and not quantized a priori. But the right side is manifestly quantized; the integer g is the number of handles characterizing the topology of S. (For the torus in figure 4, g = 1.) So, if we change K arbitrarily by denting the Riemann Curvature Tensor surface—so long as we do not punch through new handles—the quantized right side of equation 3 does not Riemann tensor is a rank (1,3) tensor that describes the curvature in all directions at a change. The Gauss–Bonnet given point in space. It takes 3 vectors and returns a single vector. The vectors that areformula has a well-known modern generalization, due to Shiing-shen Chern. The Gaussfed to the tensor should be very small and have a length ε. If we use the first twoapplies not only to the geometry of Bonnet-Chern formula surfaces but also to the geometry of the eigenstates paravectors to form a tiny parallelogram and we parallel transport the meterized third vector around by F and q. It looks exactly like equation 3, exthis parallelogram, the vector that the function returns is approximately vector cept that Kthe is now the adiabatic curvature of equation 1, the surface S is a torus parameterized by the two difference between the original vector and the vector after theand parallel transport. fluxes F and q. The right-hand side of the Chern’s generMathematically it can be written like this: alization of equation 3 is still an integer. Indeed, it is the so-called Chern number. But there is a difference: It is not necessarily an even integer, and g can no longer be interpreted as a tally of handles. But if the Chern number doesn’t count handles, why does it have to be an integer? One can see why by considering the of parallel transport around the little . . . loop Where U and V are the vectors of length ε that form the parallelogram, R failure is Riemann in figure 4. The loop can be thought of as the boundary of Sphere: K = Torus: ∫ 1 , R2 dσ K = 4πR 2 ∫ dσ K = ??? 1 = 4π R2 Returning to Laughlin’s argume fundamental units, the Hall conduct average charge transport in Laugh ment are Chern numbers. That ex ported charge, averaged over many tized. And thus it explains the surpr structure discovered in 1980 by von The Hofstadter model The glory of Chern numbers in th from a theoretical model investigat Hofstadter,9 three years before t widely read book Gödel, Escher, Ba Braid. The model, which was first by Rudolph Peierls and his student independent electrons on a 2D latt mogeneous magnetic field. The mo several reasons. First, its rich conje perimentally realized in 2001 by C Klitzing, and coworkers in a 2D el lattice potential.10 The experiment v detailed Hofstadter-model predictio and coworkers.4 Second, the only kn the Hall conductance in Hofstadter numbers. Third, the model provide built of Chern numbers. The thermodynamic properties in Hofstadter’s model are determine the . magnetic. flux, the temperature . choose a loop that is small enough around a point in a curved space, the amount of change in the direction of a vector that is parallel transported along it is proportional to the area enclosed by the loop. So, the ratio between the area of the loop and the amount of change in the direction of the vector (whatever way we chose to measure it) can be used as a measure to the curvature of the surface that includes the loop. Actually we define curvature by the value of this ratio. What about integrating over a closed surface? Figure 4. The Gauss–Bonnet formula trated here by a toroidal surface with local curvature K is positive on those face that resemble a sphere and nega the hole, that resemble a saddle. Bec of handles, equals one, the integral o the entire surface vanishes. One can Shiing-shen Chern’s quantum genera Gauss–Bonnet formula plausible by c angular mismatch of parallel transpo around the small red patch in the fig allel transport around the small loop enclosing the area dA. The left side of the Gauss–Bonnet equation is geometric and not quantized a priori. But the right side is manifestly quantized; the integer g is the number of handles characterizing the topology of S. (For the torus in figure 4, g = 1.) So, if we change K arbitrarily by denting the Riemann Curvature Tensor surface—so long as we do not punch through new handles—the quantized right side of equation 3 does not Riemann tensor is a rank (1,3) tensor that describes the curvature in all directions at a change. The Gauss–Bonnet given point in space. It takes 3 vectors and returns a single vector. The vectors that areformula has a well-known modern generalization, due to Shiing-shen Chern. The Gaussfed to the tensor should be very small and have a length ε. If we use the first twoapplies not only to the geometry of Bonnet-Chern formula surfaces but also to the geometry of the eigenstates paravectors to form a tiny parallelogram and we parallel transport the meterized third vector around by F and q. It looks exactly like equation 3, exthis parallelogram, the vector that the function returns is approximately vector cept that Kthe is now the adiabatic curvature of equation 1, the surface S is a torus parameterized by the two difference between the original vector and the vector after theand parallel transport. fluxes F and q. The right-hand side of the Chern’s generMathematically it can be written like this: alization of equation 3 is still an integer. Indeed, it is the so-called Chern number. But there is a difference: It is not necessarily an even integer, and g can no longer be interpreted as a tally of handles. But if the Chern number doesn’t count handles, why does it have to be an integer? One can see why by considering the of parallel transport around the little . . . loop Where U and V are the vectors of length ε that form the parallelogram, R failure is Riemann in figure 4. The loop can be thought of as the boundary of Sphere: K = Torus: ∫ 1 , R2 dσ K = 4πR 2 ∫ dσ K = ??? 1 = 4π R2 Returning to Laughlin’s argume fundamental units, the Hall conduct average charge transport in Laugh ment are Chern numbers. That ex ported charge, averaged over many tized. And thus it explains the surpr structure discovered in 1980 by von The Hofstadter model The glory of Chern numbers in th from a theoretical model investigat Hofstadter,9 three years before t widely read book Gödel, Escher, Ba Braid. The model, which was first by Rudolph Peierls and his student independent electrons on a 2D latt mogeneous magnetic field. The mo several reasons. First, its rich conje perimentally realized in 2001 by C Klitzing, and coworkers in a 2D el lattice potential.10 The experiment v detailed Hofstadter-model predictio and coworkers.4 Second, the only kn the Hall conductance in Hofstadter numbers. Third, the model provide built of Chern numbers. The thermodynamic properties in Hofstadter’s model are determine the . magnetic. flux, the temperature . Grisha doesn't want it. Emotiona 1 week ago Gauss-Bonnet theorem (1848) Separate from that cash, Grisha has been awarded the Fields Medal, equivalent beginnin SEOW! to the Nobel Prize in math, which also comes with a nice clump of cash. (There 2 weeks ag is no Nobel Prize in math.) Over a smooth closed surface: ∫ Grisha doesn't want it. 1 dσ K = 2(1 − g) 2π S Geometr Geometric Mounted S 4 weeks ag The Gho Tuesday F start Maybe his mama could talk some sense into this boy. But taking a look at this Genus g integer: counts theinto number ofprobably “handles” Rasputin lookin' mofo, if she could talk sense him, she'd start by 4 weeks ag g for homeomorphic notSame dressin' him funny anymore. surfaces (continuous stretching and bending into a new shape) Differentiability not needed ...other d Court oka spying 4 weeks ag It's News Shutting d 1 month ag The Midd Party (M 2 months a Matty Boy, can you 'splain the Poincaré Conjecture to your gentle readers, g=0 g=1 trinket99 World Off With H some of whom have serious issues with the math? . . . . . . Grisha doesn't want it. Emotiona 1 week ago Gauss-Bonnet theorem (1848) Separate from that cash, Grisha has been awarded the Fields Medal, equivalent cée gaz (8) beginnin SEOW! to the Nobel Prize in math, which also comes with a nice clump of cash. (There Math mugs Math humor Mathematician jokes Math mug Math mugs Math gift Math gifts is no Nobel Prize in math.) 2 weeks ag 2/3/12 12:05 PM Archimedes Over a smooth closed surface: ∫ Grisha doesn't want it. 1 dσ K = 2(1 − g) 2π S Geometr Geometric Mounted S Eukleides (Euclid) Descartes Fermat Pascal Newton Leibniz Math Mug: Certified Math Geek. Euler Lagrange 4 weeks ag Laplace Gauss Ada Byron, Lady Lovelace The Gho Tuesday F start Riemann Maybe his mama could talk some sense into this boy. But taking a look at this Cantor Genus g integer: counts theinto number ofprobably “handles” Rasputin lookin' mofo, if she could talk sense him, she'd start by surfaces Qu'y-a-t-il dans mon panier ...other d Authors Pictures Leonardo Da Vinci Shop Composers Posters Composers T shirts Famous Physicists g for homeomorphic notSame dressin' him funny anymore. 4 weeks ag Math Mug (set theory): If you consider the set of all sets that have never been considered .... (continuous stretching and bending into a new shape) Produit(s) Differentiability not Enlever needed Court oka spying 4Lubian weeks ag cessoires (10) potage It's News blanche Shutting d Math Mug - Topology To a Topologist This is a Doughnut 1 month ag The Midd Party (M 2 months a divers (19)Matty Boy, can you 'splain the Poincaré Conjecture to your gentle readers, g=0 g=1 some of whom have serious issues with the math? g=1 g=2 trinket99 World Off With H Math Mug: Real Life is a Special Case (black background) http://mathematicianspictures.com/Math_Mugs_p01.htm Page 4 of 6 . . . . . . Nonsmooth surfaces: Polyhedra Euler characteristic Euler characteristic - Wikipedia, the free encyclopedia χ=V −E +F 2/3/12 5:27 PM This result is known as Euler's polyhedron formula or theorem. It corresponds to the Euler characteristic of the sphere (i.e. χ = 2), and applies identically to spherical polyhedra. An illustration of the formula on some polyhedra is given below. Name Image Toroidal Euler polyhedroncharacteristic: - Wikipedia, the free encyclopedia Vertices Edges Faces V E F V−E+F Tetrahedron 4 6 Hexahedron or cube 8 12 4 Wikipedia, the 2 free encyclopedia From In geometry, a toroidal polyhedron is a polyhedron with a genus of 1 or greater, representing topological torus surfaces. 6 2 Octahedron 6 12 Non-self-intersecting toroidal polyhedra are embedded tori, while self-intersecting toroidal polyhedra are toroidal as abstract polyhedra, which can be verified 8 2 by their Euler characteristic (0 or less) and orientability (orientable), and their self-intersecting realization in Euclidean 3-space is a polyhedral immersion. Dodecahedron 20 30 12 2 12 30 20 2 1 Stewart toroids 2 Császár and Szilassi polyhedra 3 Self-intersecting tori The surfaces of nonconvex polyhedra can have various Euler4 characteristics; See also 5 References 6 External Vertices Edges Faces Eulerlinks characteristic: Name Image V E F A polyhedral torus can be constructed to approximate a torus surface, from a net of quadrilateral faces, like this 6x4 example. χ=0 Contents Icosahedron 2/3/12 5:34 PM Toroidal polyhedron V−E+F χ = 2(1 − g) Stewart toroids Tetrahemihexahedron 6 12 7 1 A special category of toroidal polyhedra are constructed exclusively by regular polygon faces, no intersections, and a further restriction that adjacent . toroids, . faces may not exist in the same plane. These are called Stewart This toroidal polyhedron . constructed from. . . Nonsmooth surfaces: Polyhedra Euler characteristic Euler characteristic - Wikipedia, the free encyclopedia χ=V −E +F 2/3/12 5:27 PM This result is known as Euler's polyhedron formula or theorem. It corresponds to the Euler characteristic of the sphere (i.e. χ = 2), and applies identically to spherical polyhedra. An illustration of the formula on some polyhedra is given below. Name Image Toroidal Euler polyhedroncharacteristic: - Wikipedia, the free encyclopedia Vertices Edges Faces V E F V−E+F Tetrahedron 4 6 Hexahedron or cube 8 12 4 Wikipedia, the 2 free encyclopedia From In geometry, a toroidal polyhedron is a polyhedron with a genus of 1 or greater, representing topological torus surfaces. 6 2 Octahedron 6 12 Non-self-intersecting toroidal polyhedra are embedded tori, while self-intersecting toroidal polyhedra are toroidal as abstract polyhedra, which can be verified 8 2 by their Euler characteristic (0 or less) and orientability (orientable), and their self-intersecting realization in Euclidean 3-space is a polyhedral immersion. Dodecahedron 20 30 12 2 12 30 20 2 1 Stewart toroids 2 Császár and Szilassi polyhedra 3 Self-intersecting tori The surfaces of nonconvex polyhedra can have various Euler4 characteristics; See also 5 References 6 External Vertices Edges Faces Eulerlinks characteristic: Name Image V E F A polyhedral torus can be constructed to approximate a torus surface, from a net of quadrilateral faces, like this 6x4 example. χ=0 Contents Icosahedron 2/3/12 5:34 PM Toroidal polyhedron V−E+F χ = 2(1 − g) Stewart toroids Tetrahemihexahedron 6 12 7 1 A special category of toroidal polyhedra are constructed exclusively by regular polygon faces, no intersections, and a further restriction that adjacent . toroids, . faces may not exist in the same plane. These are called Stewart This toroidal polyhedron . constructed from. . . Outline 1 What topology is about 2 Topology shows up in electronic structure 3 Classical Hall effect 4 2d noninteracting electrons in a magnetic field 5 Quantum Hall Effect . . . . . . whereas the electrons in the Landau level give rise to the same Hall current as that obtained when all the electrons are in the level and can move freely. Clearly this process must be occuring but its range of validity must be carefully examined as an accompaniment to highly accurate measurements of Hall resistance. For high-precision measurements we used a normal resistance R, in series with the device. The voltage drop, U„across R„and the voltages UH and Upp across and along the device was measured with a high impedance voltmeter (R &2 x10'0 corresponds to the minimum at V~ =8.7 V in Fig. 1, because the thicknesses of the gate oxides of these two samples differ by a factor of 3.6. Our experimental arrangement was not sensitive enough to measure a value of R» of less than 0.1 0 which was found in the gate-voltage region 23.40 V& V &23.80 V. The Hall resistance in this gate voltage region had a value of 6453.3+ 0.1 Q. This inaccuracy of + 0.1 0 was due to the limited sensitivity of the voltmeter. We would like to mention that most of the samples, especially devices with a small length-to-width ratio, showed a minimum in the Hall voltage as a function of V at gate voltage close to the left side of the plateau. In Fig. 2, this minimum is relatively shallow and has a value of 6452. 87 0 at V~ =23.30 V. In order to demonstrate the insensitivity of the Hall resistance on the geometry of the device, measurements on two samples with a length-towidth ratio of /W=0. 65 and I/W=25, respectively, are plotted in Fig. 3. The gate-voltage scale Debut of topology in electronic structure (IQHE) 400 200. I 23.0 23.5 24.5 24.0 =Vg /V Figure from von Klitzing et al. (1980). 6454. + ~ ow z. 6453. I I .0 — +~)~+K «~~+ p h/4e2 + / Gate voltage Vg was supposed to control the carrier density. II 6400- 1 I 8=13.0 T 1 I T =1.8 K (9 I I 6452. 1 I I I d 6300- I I I l I I B=13.9 T I I I I 1 T I I I I I 23.5 24.5 24.0 =Vg /V FIG. 2. Hall resistance RH, and device resistance, Rpp, between the potential probes as a function of the gate voltage ~~ in a region of gate voltage corresponding to a fully occupied, lowest (n =0) Landau level. The plateau in RH has a value of 6453.3+ 0.1 Q. The geometry of the device was =400 pm, 8'=50 pm, and L» I I I -- ------L=260 I 23.0 0I K I 6451 6200 =1.8 pm, W=400pm L=1000pm W=40 pm 0 Plateau flat to five decimal figures I 6450 0.98 0.99 1.00 = gate FIG. 3. Hall resistance RH 101 voltage 102 Vg/r'el. units for two samples with dif- Natural resistance unit: I B=13 T. &. 2 1 klitzing = h/e = 25812.807557(18) ohm. This experiment: RH = klitzing / 4 =130 pm; ferent geometry in a gate-voltage region V~ where the n =0 Landau level is fully occupied. The recommended value h/4e' is given as 6453.204 496 . . . . . . whereas the electrons in the Landau level give rise to the same Hall current as that obtained when all the electrons are in the level and can move freely. Clearly this process must be occuring but its range of validity must be carefully examined as an accompaniment to highly accurate measurements of Hall resistance. For high-precision measurements we used a normal resistance R, in series with the device. The voltage drop, U„across R„and the voltages UH and Upp across and along the device was measured with a high impedance voltmeter (R &2 x10'0 corresponds to the minimum at V~ =8.7 V in Fig. 1, because the thicknesses of the gate oxides of these two samples differ by a factor of 3.6. Our experimental arrangement was not sensitive enough to measure a value of R» of less than 0.1 0 which was found in the gate-voltage region 23.40 V& V &23.80 V. The Hall resistance in this gate voltage region had a value of 6453.3+ 0.1 Q. This inaccuracy of + 0.1 0 was due to the limited sensitivity of the voltmeter. We would like to mention that most of the samples, especially devices with a small length-to-width ratio, showed a minimum in the Hall voltage as a function of V at gate voltage close to the left side of the plateau. In Fig. 2, this minimum is relatively shallow and has a value of 6452. 87 0 at V~ =23.30 V. In order to demonstrate the insensitivity of the Hall resistance on the geometry of the device, measurements on two samples with a length-towidth ratio of /W=0. 65 and I/W=25, respectively, are plotted in Fig. 3. The gate-voltage scale Debut of topology in electronic structure (IQHE) 400 200. I 23.0 23.5 24.5 24.0 =Vg /V Figure from von Klitzing et al. (1980). 6454. + ~ ow z. 6453. I I .0 — +~)~+K «~~+ p h/4e2 + / Gate voltage Vg was supposed to control the carrier density. II 6400- 1 I 8=13.0 T 1 I T =1.8 K (9 I I 6452. 1 I I I d 6300- I I I l I I B=13.9 T I I I I 1 T I I I I I 23.5 24.5 24.0 =Vg /V FIG. 2. Hall resistance RH, and device resistance, Rpp, between the potential probes as a function of the gate voltage ~~ in a region of gate voltage corresponding to a fully occupied, lowest (n =0) Landau level. The plateau in RH has a value of 6453.3+ 0.1 Q. The geometry of the device was =400 pm, 8'=50 pm, and L» I I I -- ------L=260 I 23.0 0I K I 6451 6200 =1.8 pm, W=400pm L=1000pm W=40 pm 0 Plateau flat to five decimal figures I 6450 0.98 0.99 1.00 = gate FIG. 3. Hall resistance RH 101 voltage 102 Vg/r'el. units for two samples with dif- Natural resistance unit: I B=13 T. &. 2 1 klitzing = h/e = 25812.807557(18) ohm. This experiment: RH = klitzing / 4 =130 pm; ferent geometry in a gate-voltage region V~ where the n =0 Landau level is fully occupied. The recommended value h/4e' is given as 6453.204 496 . . . . . . More recent experiments Plateaus accurate to nine decimal figures In the plateau regions ρxx = 0 and σxx = 0: “quantum Hall insulator” . . . . . . Outline 1 What topology is about 2 Topology shows up in electronic structure 3 Classical Hall effect 4 2d noninteracting electrons in a magnetic field 5 Quantum Hall Effect . . . . . . Gaussian (a.k.a. CGS) units Permittivity of free space ε0 = 1 4π Permeability of free space µ0 = 4π In vacuo D ≡ E and H ≡ B All fields have the same dimensions Newtonian & Hamiltonian mechanics: ( ) 1 dv =f=Q E+ v×B M dt c H= 1 2M ( p− )2 Q A(r) + QΦ(r) c . . . . . . Atomic Gaussian units 1 H= 2M ( )2 1 p − A(r) + QΦ(r) c Schrödinger Hamiltonian for the electron H= )2 1 ( e −i~∇ + A(r) − eΦ(r) 2me c me = 1, ~ = 1, e = 1, (c = 137) 1 a.u. of energy = 1 hartree = 2 rydberg = 27.21 eV 1 H= 2 ( )2 1 −i∇ + A(r) − Φ(r) c Warning: Other “atomic units” with e = . √ 2 . . . . . Atomic Gaussian units 1 H= 2M ( )2 1 p − A(r) + QΦ(r) c Schrödinger Hamiltonian for the electron H= )2 1 ( e −i~∇ + A(r) − eΦ(r) 2me c me = 1, ~ = 1, e = 1, (c = 137) 1 a.u. of energy = 1 hartree = 2 rydberg = 27.21 eV 1 H= 2 ( )2 1 −i∇ + A(r) − Φ(r) c Warning: Other “atomic units” with e = . √ 2 . . . . . Atomic Gaussian units 1 H= 2M ( )2 1 p − A(r) + QΦ(r) c Schrödinger Hamiltonian for the electron H= )2 1 ( e −i~∇ + A(r) − eΦ(r) 2me c me = 1, ~ = 1, e = 1, (c = 137) 1 a.u. of energy = 1 hartree = 2 rydberg = 27.21 eV 1 H= 2 ( )2 1 −i∇ + A(r) − Φ(r) c Warning: Other “atomic units” with e = . √ 2 . . . . . Figure from Kittel ISSP, Ch. 6 . . . . . . Hall effect (1879) 2. Symmetry considerations 2D system ric sketch " j x # " ! xx % j & $ % '! ( y ) ( xy (E.R. Hall) ffect (E.R. Hall) " Ex # udies of AHE % E ...) &$ Pugh, J. Smit, ( y ) ! xy # " E x # ! xx &) %( E y &) " * xx * xy # " j x # % ' * Edwin*R. &Hall % & xx ) ( j y ) ( xy E (Karplus and Luttinger): pin-orbit coupling From Kittel ISSP (carriers of mass m and charge −e) ' * xy * ! xx $ 2 xx 2 ) ( ! xy $ * 2 +)* 2 ( * xx + * xy xx xy " 1 1 with effect of scattering (Kohn, Luttinger) dv + v = −e E + v × ,B m H dt τ c mit) * xx $ ! xx 2 2 ! xx + ! xy * xy $ '! xy 2 2 + ! xy ! xx Steady-state: ! dv =0 dt . . . . . . Drude-Zener theory ( ) 2. Symmetry considerations eτ 1 v=− E+ v×B m c em ! xx ! xy # " E x # ! xy ! xx &) %( E y &) * xx ' * xy * xy # " j x # * xx &) %( j y &) * xx 2 2 x + * xy ! xx 2 2 xx + ! xy In 2d, set Ey = 0; ! xy $ * xy $ cyclotron frequency ' * xy 2 * xx2 + * xy '! xy 2 2 + ! xy ! xx ωc = eB mc " vx vy eτ = − Ex,H− ωc τ vy m = ωc τ vx . ! . . . . . Hall conductivity Current j = −ne v (n carrier density) jx jy ne2 τ Ex − ωc τ jy m = ωc τ jx = In zero B field jx = σ0 Ex , σ0 = ne2 τ m In a B field jx = jy = σ0 Ex = σxx Ex 1 + (ωc τ )2 ωc τ σ 0 Ex = σyx Ex 1 + (ωc τ )2 . . . . . . Conductivity vs. resistivity (classical & quantum) ( jx jy ) ( = σxx σyx ↔ −σyx σyy )( Ex Ey ) ↔ ρ = ( σ )−1 ρxx = At B = 0 2 σxx σxx , 2 + σyx ρxy = 2 σxx σyx 2 + σyx ρxx = 1/σxx In the nondissipative regime (j · E = 0) σxx = 0 and ρxx = 0 ρxy = 1/σyx . . . . . . Conductivity vs. resistivity (classical & quantum) ( jx jy ) ( = σxx σyx ↔ −σyx σyy )( Ex Ey ) ↔ ρ = ( σ )−1 ρxx = At B = 0 2 σxx σxx , 2 + σyx ρxy = 2 σxx σyx 2 + σyx ρxx = 1/σxx In the nondissipative regime (j · E = 0) σxx = 0 and ρxx = 0 ρxy = 1/σyx . . . . . . Conductivity vs. resistivity (classical & quantum) ( jx jy ) ( = σxx σyx ↔ −σyx σyy )( Ex Ey ) ↔ ρ = ( σ )−1 ρxx = At B = 0 2 σxx σxx , 2 + σyx ρxy = 2 σxx σyx 2 + σyx ρxx = 1/σxx In the nondissipative regime (j · E = 0) σxx = 0 and ρxx = 0 ρxy = 1/σyx . . . . . . Nondissipative limit (τ → ∞, classical Drude-Zener) σ0 = ne2 τ m At B = 0 σxx = σxx = σ0 At B ̸= 0 for ωc τ σ 0 1 + (ωc τ )2 σyx = diverges τ ≫ 1/ωc σxx = 0, ρxy = 1/σyx = σ0 1 + (ωc τ )2 1 B nec ρxx = 0 (longitudinal resistivity) mωc m eB = = ne2 ne2 mc (Hall resistivity) . . . . . . Nondissipative limit (τ → ∞, classical Drude-Zener) σ0 = ne2 τ m At B = 0 σxx = σxx = σ0 At B ̸= 0 for ωc τ σ 0 1 + (ωc τ )2 σyx = diverges τ ≫ 1/ωc σxx = 0, ρxy = 1/σyx = σ0 1 + (ωc τ )2 1 B nec ρxx = 0 (longitudinal resistivity) mωc m eB = = ne2 ne2 mc (Hall resistivity) . . . . . . Nondissipative limit (τ → ∞, classical Drude-Zener) σ0 = ne2 τ m At B = 0 σxx = σxx = σ0 At B ̸= 0 for ωc τ σ 0 1 + (ωc τ )2 σyx = diverges τ ≫ 1/ωc σxx = 0, ρxy = 1/σyx = σ0 1 + (ωc τ )2 1 B nec ρxx = 0 (longitudinal resistivity) mωc m eB = = ne2 ne2 mc (Hall resistivity) . . . . . . Multiplying and dividing by h In 2d resistance/resistivity and conductance/conductivity have the same dimensions: do they coincide? n = N/A (number of carrriers per unit area) 1 AB Φ 1 h B= = = nec Nec Nec ν e2 Φ magnetic flux through area A h/e2 ≃ 25813 Ω (natural resistance unit) ν dimensionless ρxy = ν= NΦ0 Φ filling factor, Φ0 = hc e flux quantum ν = (number of electrons)/(number of flux quanta) . . . . . . Multiplying and dividing by h In 2d resistance/resistivity and conductance/conductivity have the same dimensions: do they coincide? n = N/A (number of carrriers per unit area) 1 AB Φ 1 h B= = = nec Nec Nec ν e2 Φ magnetic flux through area A h/e2 ≃ 25813 Ω (natural resistance unit) ν dimensionless ρxy = ν= NΦ0 Φ filling factor, Φ0 = hc e flux quantum ν = (number of electrons)/(number of flux quanta) . . . . . . Multiplying and dividing by h In 2d resistance/resistivity and conductance/conductivity have the same dimensions: do they coincide? n = N/A (number of carrriers per unit area) 1 AB Φ 1 h B= = = nec Nec Nec ν e2 Φ magnetic flux through area A h/e2 ≃ 25813 Ω (natural resistance unit) ν dimensionless ρxy = ν= NΦ0 Φ filling factor, Φ0 = hc e flux quantum ν = (number of electrons)/(number of flux quanta) . . . . . . Multiplying and dividing by h In 2d resistance/resistivity and conductance/conductivity have the same dimensions: do they coincide? n = N/A (number of carrriers per unit area) 1 AB Φ 1 h B= = = nec Nec Nec ν e2 Φ magnetic flux through area A h/e2 ≃ 25813 Ω (natural resistance unit) ν dimensionless ρxy = ν= NΦ0 Φ filling factor, Φ0 = hc e flux quantum ν = (number of electrons)/(number of flux quanta) . . . . . . A typical experiment GaAs-GaAlAs heterojunction, at 30mK . . . . . . Outline 1 What topology is about 2 Topology shows up in electronic structure 3 Classical Hall effect 4 2d noninteracting electrons in a magnetic field 5 Quantum Hall Effect . . . . . . Hamiltonian in B field (flat substrate potential) N noninteracting (& spin-polarized) electrons in zero potential: N ]2 1 ∑[ e Ĥ = pi + A(ri ) 2me c i=1 Gaussian units me electron mass −e electron charge ( ) 1 e me pi + c A(ri ) velocity pi = −i~∇i canonical momentum B = ∇ × A(r) . . . . . . Landau gauge Everything in 2d; B uniform, along z. Ax = 0, Ay = Bx For each electron the Hamiltonian is [ ( )2 ] ∂ ~2 ∂2 e H(x, y ) = − 2 + −i + Bx 2me ∂y ~c ∂x Landau ansatz ψk (x, y ) = eiky φk (x) ~2 iky ′′ ~2 − e φk (x) + 2me 2me ( eB k+ x ~c )2 eiky φk (x) = εk eiky φk (x). Harmonic oscillator in 1d . . . . . . Landau gauge Everything in 2d; B uniform, along z. Ax = 0, Ay = Bx For each electron the Hamiltonian is [ ( )2 ] ∂ ~2 ∂2 e H(x, y ) = − 2 + −i + Bx 2me ∂y ~c ∂x Landau ansatz ψk (x, y ) = eiky φk (x) ~2 iky ′′ ~2 − e φk (x) + 2me 2me ( eB k+ x ~c )2 eiky φk (x) = εk eiky φk (x). Harmonic oscillator in 1d . . . . . . Landau gauge Everything in 2d; B uniform, along z. Ax = 0, Ay = Bx For each electron the Hamiltonian is [ ( )2 ] ∂ ~2 ∂2 e H(x, y ) = − 2 + −i + Bx 2me ∂y ~c ∂x Landau ansatz ψk (x, y ) = eiky φk (x) ~2 iky ′′ ~2 − e φk (x) + 2me 2me ( eB k+ x ~c )2 eiky φk (x) = εk eiky φk (x). Harmonic oscillator in 1d . . . . . . Landau oscillator ( ) ~2 ′′ ~2 eB 2 φk (x) + k+ x φk (x) = εk φk (x) 2me 2me ~c ( ) ( )2 2 ~2 ′′ 1 eB ~c − φ (x) + me x+ k φk (x) = εk φk (x) 2me k 2 me c eB − Harmonic oscillator ~c Center in xk = − eB k = −ℓ2 k ℓ = (~c/eB)1/2 “magnetic length” (diverges for B → 0) Frequency ωc = meBe c cyclotron frequency (classical, Gaussian units) . . . . . . Landau oscillator ( ) ~2 ′′ ~2 eB 2 φk (x) + k+ x φk (x) = εk φk (x) 2me 2me ~c ( ) ( )2 2 ~2 ′′ 1 eB ~c − φ (x) + me x+ k φk (x) = εk φk (x) 2me k 2 me c eB − Harmonic oscillator ~c Center in xk = − eB k = −ℓ2 k ℓ = (~c/eB)1/2 “magnetic length” (diverges for B → 0) Frequency ωc = meBe c cyclotron frequency (classical, Gaussian units) . . . . . . Landau oscillator ( ) ~2 ′′ ~2 eB 2 φk (x) + k+ x φk (x) = εk φk (x) 2me 2me ~c ( ) ( )2 2 ~2 ′′ 1 eB ~c − φ (x) + me x+ k φk (x) = εk φk (x) 2me k 2 me c eB − Harmonic oscillator ~c Center in xk = − eB k = −ℓ2 k ℓ = (~c/eB)1/2 “magnetic length” (diverges for B → 0) Frequency ωc = meBe c cyclotron frequency (classical, Gaussian units) . . . . . . Eigenvalues and eigenvectors Spectrum independent of k : εn = (n + 12 )ωc Ground-state orbitals (LLL): ψk (x, y ) = eiky φk (x) = eiky χ(x + ℓ2 k ) ( χ(x) = 1 πℓ2 )1/4 e−x 2 /(2ℓ2 ) Infinite degeneracy: one orbital for each k Electron confined in a vertical strip centered at ℓ2 k What about the current? Any unitary transformation of the LLL orbitals is an eigenfunction . . . . . . Eigenvalues and eigenvectors Spectrum independent of k : εn = (n + 12 )ωc Ground-state orbitals (LLL): ψk (x, y ) = eiky φk (x) = eiky χ(x + ℓ2 k ) ( χ(x) = 1 πℓ2 )1/4 e−x 2 /(2ℓ2 ) Infinite degeneracy: one orbital for each k Electron confined in a vertical strip centered at ℓ2 k What about the current? Any unitary transformation of the LLL orbitals is an eigenfunction . . . . . . Eigenvalues and eigenvectors Spectrum independent of k : εn = (n + 12 )ωc Ground-state orbitals (LLL): ψk (x, y ) = eiky φk (x) = eiky χ(x + ℓ2 k ) ( χ(x) = 1 πℓ2 )1/4 e−x 2 /(2ℓ2 ) Infinite degeneracy: one orbital for each k Electron confined in a vertical strip centered at ℓ2 k What about the current? Any unitary transformation of the LLL orbitals is an eigenfunction . . . . . . Eigenvalues and eigenvectors Spectrum independent of k : εn = (n + 12 )ωc Ground-state orbitals (LLL): ψk (x, y ) = eiky φk (x) = eiky χ(x + ℓ2 k ) ( χ(x) = 1 πℓ2 )1/4 e−x 2 /(2ℓ2 ) Infinite degeneracy: one orbital for each k Electron confined in a vertical strip centered at ℓ2 k What about the current? Any unitary transformation of the LLL orbitals is an eigenfunction . . . . . . Counting the states (discretize k) ( iky ψk (x, y ) = e χ(x − ℓ k ) 2 χ(x) = 1 πℓ2 )1/4 e−x ki+1 − ki = Periodic boundary conditions in y : 2 /(2ℓ2 ) 2π L 2πℓ2 L Horizontal distance between neighboring orbitals: Area covered by one state: 2πℓ2 Number of states in each LL: N = A 2πℓ2 Magnetic flux: hc Φ = AB = N 2πℓ2 B = N 2π~c e = N e = N Φ0 Flux quantum: Φ0 = hc e (Φ0 = h e in SI units) Φ0 a universal constant . . . . . . Counting the states (discretize k) ( iky ψk (x, y ) = e χ(x − ℓ k ) 2 χ(x) = 1 πℓ2 )1/4 e−x ki+1 − ki = Periodic boundary conditions in y : 2 /(2ℓ2 ) 2π L 2πℓ2 L Horizontal distance between neighboring orbitals: Area covered by one state: 2πℓ2 Number of states in each LL: N = A 2πℓ2 Magnetic flux: hc Φ = AB = N 2πℓ2 B = N 2π~c e = N e = N Φ0 Flux quantum: Φ0 = hc e (Φ0 = h e in SI units) Φ0 a universal constant . . . . . . Density of states At B = 0: At B ̸= 0: eA D(ε) = constant = 2πm h2 Φ/Φ0 states in each LL ( ) ∞ Φ ∑ 1 D(ε) = δ ε − (n + )~ωc Φ0 2 n=1 maximum filling for each LL is ν = 1. . . . . . . Density of states At B = 0: At B ̸= 0: eA D(ε) = constant = 2πm h2 Φ/Φ0 states in each LL ( ) ∞ Φ ∑ 1 D(ε) = δ ε − (n + )~ωc Φ0 2 n=1 maximum filling for each LL is ν = 1. . . . . . . Density of states How many states in the hatched region? ∫ ε ε+~ωc dε′ D(ε′ ) = ~ωc 2πme A Φ = 2 Φ0 h . . . . . . Outline 1 What topology is about 2 Topology shows up in electronic structure 3 Classical Hall effect 4 2d noninteracting electrons in a magnetic field 5 Quantum Hall Effect . . . . . . What the experiment shows In modern jargon: The plateaus are “topologically protected” . . . . . . Wavefunction “knotted” or “twisted” PERSPECTIVE INSIGHT NATURE|Vol 464|11 March 2010 a b Figure 1 | Metallic states are born when a surface unties ‘knotted’ electron wavefunctions. a, An illustration of topological change and the resultant surface state. The trefoil knot (left) and the simple loop (right) represent different insulating materials: the knot is a topological insulator, and the loop is an ordinary insulator. Because there is no continuous deformation by which one can be converted into the other, there must be a surface where the string is cut, shown as a string with open ends (centre), to pass between the two knots; more formally, the topological invariants cannot remain defined. If the topological invariants are always defined for an insulator, then the surface must be metallic. b, The simplest example of a knotted 3D electronic band structure (with two bands)35, known to mathematicians as the Hopf map. The full topological structure would also have linked fibres on each ring, in addition to the linking of rings shown here. The knotting in real topological insulators is more complex as these require a minimum of four electronic bands, but the surface structure that appears is relatively simple (Fig. 3). introduced in around 2003 it can lead to a quantum spin Hall effect, in which electrons with opposite spin angular momentum (commonly called spin up and spin down) move in opposite directions around the edge of the droplet in the absence of an external magnetic field2 (Fig. 2b). These simplified models were the first steps towards understanding topological insulators. But it was unclear how realistic the models were: in real materials, there is mixing of spin-up and spin-down electrons, of the quantum spin Hall effect (discussed earlier), which may allow 2D Knotted in reciprocal space in nontrivial ways The famous TKNN paper: D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. 49, 405 (1982). symmetry as an applied magnetic field would), in simplified models in the crystal) will always contain a quantum wire like that at the edge topological insulator physics to be observed in a 3D material . Integer numbers are very “robust” 9 There is also, however, a ‘strong’ topological insulator, which has a more subtle relationship to the 2D case; the relationship is that in two dimensions it is possible to connect ordinary insulators and topological insulators smoothly by breaking time-reversal symmetry7. Such a continuous interpolation can be. used to build a 3D. band structure that . . . . Is it possible to “cut the knots”? Wavefunction “knotted” or “twisted” PERSPECTIVE INSIGHT NATURE|Vol 464|11 March 2010 a b Figure 1 | Metallic states are born when a surface unties ‘knotted’ electron wavefunctions. a, An illustration of topological change and the resultant surface state. The trefoil knot (left) and the simple loop (right) represent different insulating materials: the knot is a topological insulator, and the loop is an ordinary insulator. Because there is no continuous deformation by which one can be converted into the other, there must be a surface where the string is cut, shown as a string with open ends (centre), to pass between the two knots; more formally, the topological invariants cannot remain defined. If the topological invariants are always defined for an insulator, then the surface must be metallic. b, The simplest example of a knotted 3D electronic band structure (with two bands)35, known to mathematicians as the Hopf map. The full topological structure would also have linked fibres on each ring, in addition to the linking of rings shown here. The knotting in real topological insulators is more complex as these require a minimum of four electronic bands, but the surface structure that appears is relatively simple (Fig. 3). introduced in around 2003 it can lead to a quantum spin Hall effect, in which electrons with opposite spin angular momentum (commonly called spin up and spin down) move in opposite directions around the edge of the droplet in the absence of an external magnetic field2 (Fig. 2b). These simplified models were the first steps towards understanding topological insulators. But it was unclear how realistic the models were: in real materials, there is mixing of spin-up and spin-down electrons, of the quantum spin Hall effect (discussed earlier), which may allow 2D Knotted in reciprocal space in nontrivial ways The famous TKNN paper: D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. 49, 405 (1982). symmetry as an applied magnetic field would), in simplified models in the crystal) will always contain a quantum wire like that at the edge topological insulator physics to be observed in a 3D material . Integer numbers are very “robust” 9 There is also, however, a ‘strong’ topological insulator, which has a more subtle relationship to the 2D case; the relationship is that in two dimensions it is possible to connect ordinary insulators and topological insulators smoothly by breaking time-reversal symmetry7. Such a continuous interpolation can be. used to build a 3D. band structure that . . . . Is it possible to “cut the knots”? Quantum Mechanics: Data for a torus Berry connection & Berry curvature Parametric Hamiltonian What data fully specify homogeneous on a closed surface (a torus) : magnetic field on torus? H(ϑ, φ) = H(ϑ + 2π, φ) = H(ϑ, φ + 2π) Ground nondegenerate eigenstate |ψ0 (ϑ, φ)⟩ Field, ,two fundamental loops and two fluxes Berry connection: Aϑ (ϑ, φ) = i ⟨ψ0 | ∂ ψ0 ⟩, ∂ϑ Aφ (ϑ, φ) = i ⟨ψ0 | ∂ Physical information (observable) ∂φ ψ0 ⟩ Berry curvature: ( Ω(ϑ, φ) = i ) ∂ ∂ ∂ ∂ ⟨ ψ0 | ψ0 ⟩ − ⟨ ψ0 | ψ0 ⟩ ∂ϑ ∂φ ∂φ ∂ϑ . . . . . . Data for a torus A flavor of the theory: Gauss-Bonnet-Chern theorem at data fully y homogeneous c field on torus? Berry curvature: two fundamental loops and two fluxes ( Ω(ϑ, φ) = i ) ∂ ∂ ∂ ∂ ⟨ ψ0 | ψ0 ⟩ − ⟨ ψ0 | ψ0 ⟩ ∂ϑ ∂φ ∂φ ∂ϑ Chern number (a.k.a. TKNN invariant): ∫ 2π Physical information ∫ (observable) 2π 1 2π dφ Ω(ϑ, φ) = C ∈ Z dϑ 0 0 Analogue of the genus in differential geometry . . . . . . Data for a torus A flavor of the theory: Gauss-Bonnet-Chern theorem at data fully y homogeneous c field on torus? Berry curvature: two fundamental loops and two fluxes ( Ω(ϑ, φ) = i ) ∂ ∂ ∂ ∂ ⟨ ψ0 | ψ0 ⟩ − ⟨ ψ0 | ψ0 ⟩ ∂ϑ ∂φ ∂φ ∂ϑ Chern number (a.k.a. TKNN invariant): ∫ 2π Physical information ∫ (observable) 2π 1 2π dφ Ω(ϑ, φ) = C ∈ Z dϑ 0 0 Analogue of the genus in differential geometry . . . . . . Role of disorder Current carried by delocalized states only . . . . . .