Nobel Lecture, Aula Magna, Stockholm University, December 8, 2016 Topological Quantum Matter F. Duncan M. Haldane Princeton University • The TKNN formula (on behalf of David Thouless) • The Chern Insulator and the birth of “topological insulators” • Quantum Spin Chains and the “lost preprint” • In high school chemistry, we learn that electrons bound to the nucleus of an atom move in closed orbits around the nucleus , and quantum mechanics then fixes their energies to only be one of a discrete set of energy levels. E 4p 4s 3d 3s 3p The rotational symmetry of the 2p spherical atom means that there 2s are some energy levels at which there are more than one state 1s • This picture (which follows from the Heisenberg uncertainty principle) is completed by the Pauli exclusion principle, which says that no two electrons can be in the same state or “orbital” E 4p 4s 3d !"3s 3p !"2!p"!" An additional ingredient is that electrons !2"s have an extra parameter called “spin” !"1s which takes values “up” ( !) and “down” (") Occupied orbitals of the This allows two electrons (one !, one") Calcium atom to occupy each orbital (12 electrons) • If electrons which are not bound to atoms are free to move on a two-dimensional surface, with a magnetic field normal to the surface, they also move in circular orbits because there a magnetic force at right angles to the direction in which they move • In high magnetic fields, all electrons have spin ! pointing in the direction of the magnetic field magnetic field B force F = evB velocity v e F surface on which the x center of electrons circular orbit move As in atoms, the (kinetic) energy of the electron can only take one of a finite set of values, and now determines the radius of the orbit (larger radius = larger kinetic energy) • As with atoms, we can draw an energy-level diagram: (spin direction is fixed in each level) E ! 2 unlike atoms, the number of orbitals in each " E1 “Landau level” is huge! E ! 0 number of orbitals is proportional to area of surface! Total magnetic flux through surface B area degeneracy of = = ⇥ Landau level (London) quantum of magnetic flux h/e • For a fixed density of electrons let’s choose the magnetic field B just right, so the lowest level is filled: E2 empty E 1 ∆ energy gap E # # # # # # # # # # # # # 0 filled independent of electron density if if n Landau eB details levels are exactly filled 2⇡~ • This appears to describe the integer quantum Hall states discovered by Klaus von Klitzing (Nobel Laureate 1985) • BUT: seems to need the magnetic field to be “fine-tuned”. • In fact, this is a “topological state” with extra physics at edges of the system that fix this problem • counter-propagating “one- way” edge states (Halperin) • confined system with edge must have edge states! Fermi level ∆ # bulk gap # pinned at edge E0 # # # # # # # # # # # # # # don’t need to fine-tune magnetic field K. von Klitzing • The integer quantum Hall effect (1980) was the first “topological quantum state” to be experimentally discovered (Nobel Laureate1985, Klaus von Klitzing) MAGNETI C FIELD ( T ) current • Hall conductance jx = σ E H y E~ e2 σH = ⌫ 2⇡~ dissipationless current ⌫ = integer flows at right angles to electric field (number of filled Landau levels) • Von Klitzing’s system is much dirtier that the theoretical toy model and work in the early1980‘s focussed on difficult problems of disorder, random potentials and localized states • David Thouless had the idea to study the effect of a periodic potential in perturbing the flat Landau levels of the integer quantum Hall states: Bob Laughlin (Laureate 1998 for fractional QHE ) gave a clear argument V for quantization of the Hall conductance in this case edge edge flat potential in bulk x V random potential in bulk (realistic but difficult) x V Toy model! periodic potential in bulk smeared Landau level x The TKNN or TKN2 paper Quantized Conductance in a Two-Dimensional Periodic Potential D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs Physical Review Letters 49, 405 (1982) • In 1982 David Thouless with three postdoc collaborators (TKNN) asked how the presence of a periodic potential would affect the integer quantum Hall effect of an electron moving in a uniform magnetic field • They found a remarkable formula ..... • David became particularly interested in an interesting “toy model”, of a crystal in a magnetic field, a family of models including the “Hofstadter Butterfly” 1 2 U0 ~!c H = p eA(r) + U(r) ⌧ 2m| − | U 0 U(r)=U0(cos(2⇡x/a) + cos(2⇡y/a)) 0 EF • Harper’s equation (square -1 1 symmetry) or -2 2 • “Hofstadter’s Butterfly” splits the lowest Landau level into 0 bands separated by gaps. The band are very narrow, and 2 • -2 the gaps wide, for low magnetic 1 -1 flux per cell (like Landau levels) 0 U color-coded Hall conductance − 0 ! 1 magnetic flux per unit cell simple Landau level limit colored “butterfly” courtesy of D. Osadchy and J. Avron • TKNN pointed out that Laughlin’s argument just EF 0 required a bulk gap at the Fermi energy for the Hall -1 1 conductance to be -2 2 quantized as integers So it should work in gaps • between bands of the 2 -2 “butterfly” 1 -1 • so what was the integer? 0 magnetic flux per unit cell 0 ! 1 simple Landau level limit colored “butterfly” courtesy of D. Osadchy and J. Avron • Bloch’s theorem for a particle in a periodic potential ik r kn(r)=un(k, r)e · periodic factor that varies over the unit cell of the potential • Starting from the fundamental Kubo formula for electrical conductivity, TKNN obtained a remarkable formula that does not depend in any way on the energy bands, but just on the Bloch wavefunctions: 2 ie 2 2 @un⇤ @un @un⇤ @un σH = d k d r 2⇡h Brillouin @k @k − @k @k n unit cell 1 2 2 1 X Z zone Z ✓ ◆ TKNN first form Sum over fully-occupied bands below the Fermi energy • Shortly after the TKNN paper was published, Michael Berry (1983) (Lorentz Medal, 2014) discovered his famous geometric phase in adiabatic quantum mechanics. • (The Berry phase is geometric, not topological, but many consider this extremely influential work a contender for a Nobel prize). • Berry’s example: a spin S aligned along an axis S ! direction of spin moves on closed path on unit sphere ⌦ˆ Γ eiΦΓ = eiS! Berry phase solid angle enclosed is ambiguous modulo 4! so 2S must be an integer • the mathematical physicist Barry Simon (1983) then recognized the TKNN expression as an integral over the (Berry) curvature associated with the Berry’s phase, on a compact manifold: the Brillouin zone. • This is mathematical extension of Carl Friedrich Gauss’s* 1828 Theorema Egregium “remarkable theorem” *foreign member of Royal Swedish Academy of Sciences • geometric properties (such as curvature) are local properties • but integrals over local geometric properties may characterize global topology! Gauss-Bonnet (for a closed surface) d2r(Gaussian curvature) = 4⇡(1 genus) − Z product of 1 principal radii of =2⇡(Euler characteristic) cuvature R1R2 Risultato della ricerca immagini di Google per http://www.psdg... http://www.google.it/imgres?imgurl=http://www.psdgraphics.c... 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