Les états de bord d un. isolant de Hall atomique

Transcription

1 Les états de bord d un isolant de Hall atomique séminaire Atomes Froids 2/9/22 Nathan Goldman (ULB), Jérôme Beugnon and Fabrice Gerbier

2 Outline Quantum Hall effect : bulk Landau levels and edge states Quantum Hall effect on a lattice : Hofstadter model Hall conductivity as topological invariant Bulk-edge correspondence Realization with cold atoms and detection of edge states angular momentum spectroscopy of edge states experimental scheme to realize the Hofstadter model shelving technique to detect edge states

3 Quantum Hall effect 2D electron gases (very pure semiconductors) in Hall geometry: large perpendicular magnetic field Hall current perpendicular to applied voltage Transverse resistance Longitudinal resistance Around certain magic values of magnetic field (QH plateaux) : Longitudinal resistance vanishes Hall conductance assumes quantized values H = n e2 h

4 Hall conductivity Hall geometry : current flows along x Hall voltage develops along y Vy Ix Resistivity tensor : E = j Conductivity tensor : = Ideal system without impurities B en = el B en el = en el B B en el ~ E field current density ~ voltage current xx = xy = e B n el

5 Landau levels and integer quantum Hall effect Hamiltonian for a 2D electron in a uniform magnetic field : H = (p + ea)2 2m e Energy eigenstates of the form : n(r) =e ik xx Landau gauge : n (y) A = By n= n=2 Plane wave along x, wavevector x Harmonic oscillator wavefunction along y, centered at the guiding center position y = ~k x eb En,m cyclotron frequency c = eb m magnetic length l B = eb Sequence of Landau levels with high orbital degeneracy n=2 n= n= n=4 n=8 ħωc guiding center position x

6 Bulk system : Hall conductivity Consider one Landau level (LL), with periodic boundary conditions along x : y y = ~k x is between -Ly/2 and Ly/2 eb eb 2 N states = ~ L y / = eb L h L xl y Ly x Electron density for one filled LL : n el = eb h xy = e B n el This alone is not sufficient to explain the existence of conductance plateaus : Lx One must invoke disorder : localized states between each LL The localized states tend to be denser between two LL They do not contribute to transport

7 Strip geometry : edge states y Assume hard wall potential along y : what happens near the edges? Recall y = ~k x eb Ly x.8.6 Lx Energy (cyclotron units) LL n= edge states y /l B LL n=

8 Properties of edge states y They carry current : group velocity vg non zero states on opposite edges at the same energy carry opposite currents in equilibrium same chirality v g/k Ly x explains robustness of edge currents : no state available for backscattering in presence, e.g., of impurities Lx red: current flow

9 Properties of edge states Add weak electric field along y, producing an electrostatic potential eey v b,kx = n,kx n (k x x L x h (E n(k x +2 /L) E n (k x )) Energy (cyclotron units).8.6 I (n) x = e L x X vn,kx e h (E n(k max ) E n (k min )) no current EF They carry current : goup.6 velocity non zero y /l B y /l B This reproduces the bulk result, with the added information that the Hall current is flowing at the edge of the sample only (bulk is an insulator) e Energy (cyclotron units) V y.8 current EF

10 Bulk-edge correspondence Hall current carried by chiral edge states. Edge states and bulk Hall conductivity are connected. This connexion between bulk properties and current-carrying edge states is a general (one could even say defining) feature of topological phases. Topological phase : state of matter defined by an integer-valued topological invariant, usually physically related to transport properties, characterizing the topology of the Hilbert space. To make this connection more precise, we consider now the same situation on a lattice

11 Outline Quantum Hall effect : bulk Landau levels and edge states Quantum Hall effect on a lattice : Hofstadter model Hall conductivity as topological invariant Bulk-edge correspondence Realization with cold atoms and detection of edge states angular momentum spectroscopy of edge states experimental scheme to realize the Hofstadter model shelving technique to detect edge states

12 Harper hamiltonian : lattice and magnetic field Harper hamiltonian on a tight-binding lattice : H = J X hr,r i e i r,r ĉ r ĉ r +h.c. Harper, 956; Azbel 964; Hofstadter, 976;Thouless et al., 983; Kohmoto; Osadchy-Avron 2... Aharonov-Bohm phase: r,r = e r r A dl Landau gauge: By A = x x+d A x dx = Bd x y r,r+dex =2 ebd xy h =2 y d y Finite flux: A dl = Bd x d y r,r =2

13 Hofstadter s butterfly: interplay between lattice and vector potentials H = J X n,m e i2 m ĉ n+,mĉn,m +h.c. J X n,m ĉ n,m+ĉn,m +h.c. If α=, we have a single Bloch band of width 8J. Bloch theorem does not apply in general for arbitrary α. When α=p/q rational: Translation by q sites reproduces the same model : we recover lattice translational invariance, but with an enlarged unit cell of size q dx*dy Crystal momentum in the «magnetic Brillouin zone» ]-π/q,π/q] * ]-π,π] Eigenstates along y («magnetic Bloch functions») have q components the Bloch band splits in q subbands contained within the original one for α=

14 Hofstadter s butterfly: interplay between lattice and vector potentials H = J X n,m e i2 m ĉ n+,mĉn,m +h.c. J X n,m ĉ n,m+ĉn,m +h.c. When α=p/q rational: q sub-bands, width <<8J Recursive structure, discovered by Azbel and Hofstadter Very different from Landau levels (which appear for α<<) /2 /3 /4 Flux α -4J 4J Energy E

15 top x Closer look at α=/3 Energy spectrum on a cylinder (kx=) (a) bottom (b) 3 3y 2 2 bottom top - - top bottom kx (c) Similar structure as in the «bulk» : (d) chiral edge states are present in the bandgaps, with opposite currents flowing at the edges 4 4 5

16 Hall conductivity as a topological invariant : TKNN formula Key result from Thouless, Kohmoto, Nightingale, den Niijs (TKNN) : H = n = i 2 xy = e2 h Z BZ X n n h@ kx u n ky u n (k)i provided the Fermi energy is in a gap (k x $ k y ) d 2 k Berry curvature associated with the band eigenstates Chern number = topological index (integer-valued) characterizes the topology of the Hilbert space (more precisely, of the subspace associated with each energy band) : non zero only if the Bloch eigenstates have a vortex structure in k space always defined, but related to Hall conductivity only if the Fermi energy lies in the gap between two bands

17 Bulk-edge correspondence There is a deep relation between the Chern number characterizing the bulk material and the edge states appearing at its edge : (a) 3 2 number of edge states branches (per physical edge) : H - chirality of edge states : sign( -2 H) kx (c) Although the Chern number does not always exist in the present form (e.g., spin Hall topological insulators or interacting systems), other topological invariants can be defined in these situations. 4 When non-zero, chiral edge states always appear at the boundaries. 4