Integer quantum Hall transition on a tight-binding lattice Thomas Vojta, Martin Puschmann, Philipp Cain, and Michael Schreiber Department of Physics, Missouri University of Science and Technology, USA Institute of Physics, Chemnitz University of Technology, Germany Prague, July 19, 2019
Outline • Quantum Hall effect • Exponent puzzle of the quantum Hall transition • Tight-binding model in magnetic field • Quantum Hall transition: critical behavior • Conclusions
Hall effect • voltage (the Hall voltage ) appears transverse to electric current in conductor with magnetic field applied perpendicular to current • discovered 1879 by Edwin Hall Images: Wikipedia
(Integer) quantum Hall effect • Hall conductance quantized , σ H = I/V n = n ( e 2 /h ) ( n integer) • discovered 1980 by Klaus von Klitzing Image: oer.physics.manchester.ac.uk/AQM2
Landau levels and Landau bands Electrons in magnetic field: • discrete energy levels (Landau levels) E n = ( n + 1 / 2) eB/m • highly degenerate Effects of quenched disorder: • degeneracy lifted, Landau levels broadened into bands • all states localized except at single critical energy in each band center Integer quantum Hall transition: • continuous transition between two localized phases • localization length diverges at critical energies, ξ ∼ | E − E c | ν
Semiclassical picture: Chalker-Coddington network • smooth random potential, high magnetic field ⇒ electrons move along equipotential lines • close to saddle points of potential, electron can tunnel between classical trajectories Chalker-Coddington network • square lattice of saddle points, each with some tunneling amplitude • saddle points connected by links, each associated with a random phase change Images: Huckestein, RMP (1995)
Exponent puzzle Experiment: • high-mobility Al x Ga 1 − x As heterostructures • correlation length exponent ν = 2 . 38(5)
Exponent puzzle Experiment: • high-mobility Al x Ga 1 − x As heterostructures • correlation length exponent ν = 2 . 38(5) Theory: • best numerical results based on Chalker-Coddington network • several studies give ν ≈ 2 . 60(2)
Exponent puzzle Experiment: • high-mobility Al x Ga 1 − x As heterostructures • correlation length exponent ν = 2 . 38(5) Theory: • best numerical results based on Chalker-Coddington network • several studies give ν ≈ 2 . 60(2) What is the reason for the disagreement between the best experimental and theoretical values for the correlation length exponent? • Electron-electron interactions? • Too regular structure of CC network? [Gruzberg et al., PRB 95 , 125414 (2017)]
Quantum Hall effect Exponent puzzle of the quantum Hall transition Tight-binding model in magnetic field Quantum Hall transition: critical behavior Conclusions
Anderson Hamiltonian Disorder-free case, W = 0 � � H = u j | j �� j | + exp( iϕ jk ) | j �� k | Hofstadter butterfly j � j,k � • u j ∈ [ − W/ 2 , W/ 2] : random potentials W : disorder strength • Peierls phase shift in hopping term � k � ϕ jk = e 0 x direction A · d r = � ± 2 π Φ x j y direction j Φ : magnetic flux through unit cell • vector potential in Landau gauge A = Bx e y • self-similar spectrum • magnetic field B = B e z = curl A • nonzero Landau level width • free-electron gas approximation for small flux
Recursive Green function approach Quantities: • Green function η → 0 (( E + iη ) I − H ) − 1 G ( E ) = lim calculated recursively layer by layer • localized wave function � ψ (0) ψ ( L x ) � ∼ exp( − γL x ) • Lyapunov exponent ln tr | G 1 L x | 2 γ = − lim 2 L x L x →∞ • dimensionless Lyapunov exponent Γ = γL Simulation parameters : L x = 10 6 , L up to 768, data averaged over up to 200 strips
Quantum Hall effect Exponent puzzle of the quantum Hall transition Tight-binding model in magnetic field Quantum Hall transition: critical behavior Conclusions
Overview: Density of states + Lyapunov exponent 25 L [bottom to top] 8 16 20 32 64 (a) 15 Γ 10 5 0 0 1 2 3 4 E Magnetic flux Φ = 0 . 1 , disorder strength W = 0 . 5
Dimensionless Lyapunov exponent close to transition 1.2 (b) 1.1 4 1.0 0.9 0.8 3.415 3.420 3.425 3.430 3 L [bottom to top (at edges)] Γ 8 12 16 24 32 48 2 64 96 128 192 256 384 512 768 1 3.410 3.415 3.420 3.425 3.430 3.435 E Magnetic flux Φ = 0 . 1 , disorder strength W = 0 . 5
System size dependence of Γ at E min Φ [bottom to top] 1 / 3 1 / 4 1.0 1 / 5 1 / 10 1 / 20 1 / 100 1 / 1000 Γ( E min ) 0.8 0.2 0.1 ∆Γ [%] ∆Γ [%] Φ = 1 / 10 0.1 0.0 -0.1 0.0 Φ = 1 / 5 -0.2 -0.1 0.6 0.1 0.2 0.3 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 ( L/L B ) − 0 . 38
System size dependence of curvature Γ ′′ at E min 10 3 Φ [bottom to top] 1 / 3 1 / 4 1 / 5 1 / 10 10 2 1 / 20 1 / 100 1 / 1000 Γ ′′ ( E min ) 10 3.0 2 . 4 2 1 Γ ′′ ( L/L B ) - 2.9 10 − 1 2.8 2.7 2 1 2 2 2 3 2 4 2 5 2 6 2 7 2 8 2 9 10 − 2 2 − 1 2 0 2 1 2 2 2 3 2 4 2 5 2 6 2 7 2 8 2 9 L/L B
Finite-size scaling analysis Details [Slevin et al., PRL 82, 382 (1999)] : 1.2 (b) 1.1 4 • energy and system-size dependence of 1.0 0.9 dimensionless Lyapunov exponent 0.8 3.415 3.420 3.425 3.430 3 L [bottom to top (at edges)] Γ( x r L ν , x i L − y ) Γ = Γ 8 12 16 24 n i n r a ij x i i x 2 j 32 48 2 i !(2 j )! L 2 j/ν − iy 64 96 � � r = 128 192 256 384 i =0 j =0 512 768 1 • relevant and irrelevant scaling variables 3.410 3.415 3.420 3.425 3.430 3.435 E x r , x i [ e = ( E/E c − 1) ] Final exponent estimates: m i m r b k c l k ! e k , l ! e l � � x r = e + x i = 1+ • correlation length exponent ν = 2 . 58(3) k =2 l =1 • irrelevant exponent y = 0 . 35(4) • combined fit gives E c , ν , y and expansion coefficients Logarithmic corrections to scaling [ x i / ( b + x i log( L )) instead of x i L − y ] also fit the data well, yielding ν ≈ 2 . 60 .
Quantum Hall effect Exponent puzzle of the quantum Hall transition Tight-binding model in magnetic field Quantum Hall transition: critical behavior Quantum Hall transition on random lattices Conclusions
Random Voronoi-Delaunay lattices • lattice constructed from set of points at independent random positions in the plane Voronoi cell of site: • contains all points in the plane closer to given site than to any other • sites whose Voronoi cells share an edge considered neighbors Delaunay triangulation : • graph consisting of all bonds connecting pairs of neighbors • dual lattice to Voronoi lattice
Properties of random Voronoi lattices • lattice sites at independent random positions • local coordination number q i fluctuates 2d: � q � = 6 , σ q ≈ 1 . 33 3d: � q � = 2 + (48 / 35) π 2 ≈ 15 . 54 , σ q ≈ 3 . 36 • random connectivity (topology) generates disorder in physical system 0 10 -1 10 -2 10 -3 10 P -4 10 -5 10 -6 10 -7 10 4 6 8 10 12 14 16 18 q
Quantum Hall transition on random Voronoi-Delaunay lattices • tight-binding model on VD lattice • lattice not bipartite ⇒ asymmetric DOS Exponent estimates: • correlation length exp. ν = 2 . 62(2) • irrelevant exponent y = 0 . 39(8) ⇒ agree with square lattice results
Conclusions • tight-binding models on square lattice and random VD lattice feature integer quantum Hall transitions • correlation length critical exponent ν ≈ 2 . 58(3) , agrees with that of the semiclassical Chalker-Coddington network ⇒ Chalker-Coddington network correctly captures physics of disordered noninteracting electrons • disagreement between best experimental and theoretical estimates for ν persists • points to electron-electron interaction as culprit M. Puschmann et al., Phys. Rev. B 99 , 121301(R) (2019)
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