Topology and Fractionalization in 2D Electron Systems

Transcription

1 Lectures on Mesoscopic Physics and Quantum Transport, June 1, 018 Topology and Fractionalization in D Electron Systems Xin Wan Zhejiang University xinwan@zju.edu.cn

2 Outline Two-dimensional Electron Systems Integer Quantum Hall Effect Topological Chern number Edge states Digression to topological insulators Fractional quantum Hall effect Laughlin wave function Quasiparticles with fractional charge Noise measurement

3 Electrons in a Magnetic Field e F L= v x B c F E =e E y vx Ey= B c Ey ne c jx = n e vx = Ey ρ xy B Resistivity/conductivity is a tensor!

4 Hall Effect (1879) ρxy =E y / j x Edwin Hall ( ) ρxy B ρxx = E x / j x Ey B = = jx nec Useful to measure either the carrier density (and type) or the magnetic field B

5 Quantum Well & Quasi-D Electrons z E1 E0 When temperature is considerably smaller than E1-E0, the motion of the electron along the zaxis is frozen in its ground state. The electron can only move in the x-y plane, hence its motion is quasi-twodimensional.

6 Two-Dimensional Electron Gases (DEGs) DEG GaAs/AlGaAs heterostructure

7 Quantum Motion of Electron in a B-Field D electrons in a perpendicular B: the quantization of the cyclotron motion a discrete spectrum with macroscopically large degeneracy Magnetic length: lb = ℏc B 1/ eb Cyclotron frequency: ωc = eb B mc Landau level degeneracy: Nφ = A = Φ B Φ0 π lb

8 DEGs: Algebraic Approach Coordination of electrons in a plane described by a complex z = x + iy Perpendicular magnetic field, choose symmetric gauge Hamiltonian (free spin-polarized electrons) H 0= 1 p e A m 1 H 0 =ℏ ω c a a+ ( Intra-LL a = b = lb= ( ( l B z + 1 z 4 lb l B z + 41l ℏc eb z B ) ) ) B Two sets of ladder operators Inter-LL + E cyclotron motion E ωc = eb mc ℏ ωc ℏ ωc guiding center motion 1 ( ) En = ℏ ω c n+ Density of States (DOS)

9 Integer Quantum Hall Effect (1980) Fermi energy Klaus von Klitzing disorder broadened Landau levels ρ xy = h ne ne σ xy = h Nobel Prize 1985: "for the discovery of the quantized Hall effect"

10 Landau Levels and IQHE Landau level filling factor n φ0 N ν= = B Nφ Fermi energy Landau level degeneracy disorder broadened Landau levels Incompressibility dμ dn (existence of a gap)

11 Difference in Quantum Numbers Quantum numbers related to symmetry Example: angular momentum rotational symmetry Degeneracy from the algebra of the group generators Structure destroyed by symmetry breaking Quantum numbers determined by topology Related to the winding number of a condensate wave function Survive relatively strong perturbation h/e = 5, (86) Ω

12 Berry Phase See, e.g., J. J. Sakurai, Modern QM, Supplement I H ( R(t )) Ψ ( R) =E ( R) Ψ ( R) Ψ (t ) =ei φ(t ) Ψ ( R(t )) Expect F i ℏ Ψ (t) = H (R(t )) Ψ (t ) t [ ℏ φ +i ℏ Ψ ( R(t)) =E ( R) Ψ ( R(t )) t ] E ( R) φ = ℏ +i Ψ ( R(t)) Ψ ( R(t)) t tf Ψ ( R) Ψ ( R) φ f φ i= t dt E ( R)/ ℏ+i d R R i dynamical geometrical + topological (path indep.)

13 Berry Connection and Berry Curvature tf Ψ ( R) Ψ ( R) φ f φ i = t dt E ( R)/ ℏ + i d R R i R) S d S Ω( R) d R A( Berry connection Berry curvature R) = i Ψ ( R) Ψ ( R) A( R = A( R) Ω( R) R Applications in solids: ψ n k ( r ) = e i k r un k ( r ) A n ( k ) = i u n k ( r ) k un k ( r ) Xiao, Chang, and Niu, RMP 8, 1959 (010)

14 Hall Conductance as Curvature Hall conductance: Kubo formula derived from the linear-response theory: i t L x, y = e x,y m v x n n v y m h.c. i e ℏ xy m ; x, y = L x L y n m E m E n Geometric interpretation of Hall conductance: local curvature in the boundary condition space or the flux space ψm ψm σ xy (m ; θ x, θ y ) ℑ θ x θ y

15 Hall Conductance as a Topological Invariant Hall conductance averaged over a torus of boundary conditions a quantized integral (Thouless et al.) m m e e xy m = xy m ; x, y = d x d y ℑ = C 1 m h x y h First Chern class of a principal U(1) fiber bundle over a torus Chern number (named after Chern Shiing-shen) 1 K da = (1 g ) S π Gauss-Bonnet-Chern formula Topological: small perturbation no change in Chern number Transitions between Chern numbers: level crossing (curvature diverges)

16 IQH Edge States: Halperin (198) Hard Wall V(r) B 1 0 Bπ r m mhc = m ϕ r0 = e r0 r Chiral, gapless edge excitation No backscattering!! Chiral Fermi liquid EF

17 Topological Insulators and Superconductors Integer quantum Hall system is an example of topological insulators, which have gapped bulk but gapless edge modes (skipping orbits). A new type of topological insulator has been introduced in so called quantum spin Hall effect. In fact, one can classify topological insulators and topological superconductors by time-reversal, particle-hole, and chiral symmetries. There are exactly five classes of topological insulators in each dimension. IQHE e σ xy = h Z top. insulator See Schnyder et al., Phys. Rev. B (008); also by A. Kitaev

18 The Ten-Fold Way Symmetry class A (GUE)

19 In a 1976 paper, Kawaji and Wakabayashi (1976) reported the observation of localized states in the energy gap between two Landau levels.. Phil Anderson, after hearing this from John Rowell, asked to see the data. But by the time I showed them to him in the Bell Labs tearoom, I had repeated the experiment and found them to be sample specific. I told this to Phil and he made a cryptic remark under his breath that there should be some commensuration energy anyway.. I assumed he meant: In the n < nf extreme quantum limit, at commensuration when n = n/nf = 1/i, some interaction energy might become dominant to drive the D system into some new ground state. I was not brave enough to ask him: What do you mean? But I felt affirmed that I should continue to concentrate on the extreme quantum limit.

20 Fractional Quantum Hall Effect (198) High quality sample Low temperature High magnetic field RH = VH h = I νe R= V xx I Fractional filling factor: interaction important! Daniel C. Tsui Horst L. Störmer Robert B. Laughlin Nobel Prize 1998: "for their discovery of a new form of quantum fluid with fractionally charged excitations."

21 Landau Level Wavefunctions Single-particle eigenstates generated by + n n, m = + m ( a ) (b ) n! m! 0,0 m z z /4 l ϕ ( z ) z 0, m = e m π l B m! lb 1 l B z + 1 z 4 lb b = lb z + 1 z 4 lb + The lowest Landau level (n = 0) wavefunctions in the symmetric gauge 0 m a = ( ) High Landau levels wavefunctions can be mapped from the LLL (or 0LL) ones by using the ladder operator a+ + ( ( B E E ωc = eb mc ℏ ωc ℏ ωc Assume full spin polarization Density of States (DOS) ) )

22 Laughlin State Disk Geometry In the LLL, electron-electron interaction is not a perturbation. l z /4 ϕ l (z ) z e z= x iy Basic requirement for an electron wave function in the LLL: antisymmetric function analytic function a universal Gaussian factor z 0 1 l z z z Laughlin state m i z i / 4 Ψ L = ( z i z j ) e i< j R l = l r l = l 1

23 Introducing Quasihole 9-electron Laughlin state H W = W c +0 c0

24 Constructing Bulk Quasiparticles Consider the following product (alternatively, a sum of edge states) N N m u(w)= ( z i w)= ( w) i m=0 N 1, w,, wn m sm = z i The wave function Ψw sm w i=1 =u(w1 )u(w ) u(wn ) Ψ Laughlin describes n quasiholes at {wi}. If we put q quasihole at the same location w, we seems to obtain a state of (N + 1) electrons but with one missing at w. q q Ψ 1qh = ( z w) ( z z ) w k i j k i< j This suggests that a quasihole carries a fractional charge of e/q.

25 Plasma Analogy How are particle distributed? βplasma H plasma Ψ Laughlin =e Boltzmann weight of a D plasma (classical charged particles) at temperature T = 1/(q) with uniform background charge. H plasma= ln z i z j + i< j 1 z i 4q i The D plasma is in a screened phase at high T (q < 70), so the Laughlin wave function describes a disk of fluid with uniform density. With a quasihole at w, u(w) Ψ Laughlin plasma + an impurity at w Plasma has to screen out the impurity potential. Exactly 1/q particles are missing from around the w, which is interpreted as that a quasihole has charge e /q fractional charge!

26 How to Measure the Charge in Theory? Adiabatically move a charged particle/quasiparticle around a loop. The wave function will pick up a geometric phase (Berry phase) that is proportional to the magnetic flux through the loop, with the prefactor being essentially the effective charge. Aharonov Bohm effect F Δ φ= π(φ/φ0) hc Φ0 = e

27 Berry Phase [Berry] When a system evolves in the parameter space, its ground state picks up dynamical and geometric (including topological) phases. tf Δ φ = dt ti E (R (t)) Ψ ( R) + i d R R Ψ ( R) ℏ Suppose we use real normalization constant and move quasihole at w. q i z i / 4 Ψ w= N (w, w 1, w,, wn ) ( z j w) ( z j wα ) ( z i z j ) e j j,α i< j w Ψ w= w ln N + w ln( z j w) Ψ w [ j ] F Use trick ρ( z )= i δ() (z z i) Ψ w w Ψw = w ln N Ψ w Ψw + d z w ln( z w) Ψw ρ Ψ w pi, if z inside w loop; 0, otherwise. i dw Ψ w w Ψ w = 0 + i d z dw w ln (z w) Ψ w ρ Ψ w

28 Berry Phase (continued) Therefore, we find Δ φ= π d z ρ( z) charge enclosed inside the loop Without quasiholes inside the loop Δ φ= π Φ hc/(e ν) A-B effect for e* = en around F With quasiholes inside, charge charge quasihole charge Δφ Δφ + topological π nquasihole q ( ) Ψ Ψe F i θ

29 How to Measure the Charge Experimentally? t t I = average # of events in t t In = rms fluctuation of # in t charge contribution per event charge contribution per event t * SI In e I

30 Fractional Charge in Shot Noise De-Picciotto et al., Nature 389, 16 (1997) Saminadayar et al., PRL 79, 56 (1997)