Integer quantum Hall transition on a tight-binding lattice Thomas - - PowerPoint PPT Presentation

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, σH = I/Vn = n(e2/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)

En = (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 − Ec|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 AlxGa1−xAs

heterostructures

• correlation length exponent

ν = 2.38(5)

Exponent puzzle

Experiment:

• high-mobility AlxGa1−xAs

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 AlxGa1−xAs

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

H =

• j

uj |jj| +

• j,k

exp(iϕjk) |jk|

• uj ∈ [−W/2, W/2]: random potentials

W: disorder strength

• Peierls phase shift in hopping term

ϕjk = e

• k

j

A · dr =

• x direction

±2πΦxj y direction Φ: magnetic flux through unit cell

• vector potential in Landau gauge A = Bxey
• magnetic field B = Bez = curlA

Disorder-free case, W = 0 Hofstadter butterfly

• self-similar spectrum
• nonzero Landau level width
• free-electron gas

approximation for small flux

Recursive Green function approach

Quantities:

• Green function

G(E) = lim

η→0((E + iη)I − H)−1

calculated recursively layer by layer

• localized wave function

ψ(0)ψ(Lx) ∼ exp(−γLx)

• Lyapunov exponent

γ = − lim

Lx→∞

ln tr|G1Lx|2 2Lx

• dimensionless Lyapunov exponent

Γ = γL Simulation parameters: Lx = 106, 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

(a)

1 2 3 4 5 10 15 20 25

E Γ

8 16 32 64

L [bottom to top]

Magnetic flux Φ = 0.1, disorder strength W = 0.5

Dimensionless Lyapunov exponent close to transition

(b)

3.410 3.415 3.420 3.425 3.430 3.435 1 2 3 4

E Γ

8 12 16 24 32 48 64 96 128 192 256 384 512 768

L [bottom to top (at edges)]

3.415 3.420 3.425 3.430 0.8 0.9 1.0 1.1 1.2 Magnetic flux Φ = 0.1, disorder strength W = 0.5

System size dependence of Γ at Emin

0.0 0.1 0.2 0.3 0.4 0.6 0.8 1.0

(L/LB)−0.38 Γ(Emin)

1/3 1/4 1/5 1/10 1/20 1/100 1/1000

Φ [bottom to top] Φ = 1/10

0.1 0.2 0.3

• 0.1

0.0 0.1 0.2

∆Γ [%] Φ = 1/5

0.1 0.2 0.3 0.4

• 0.2
• 0.1

0.0 0.1

∆Γ [%]

System size dependence of curvature Γ′′ at Emin

2−1 20 21 22 23 24 25 26 27 28 29 10−2 10−1 1 10 102 103

L/LB Γ′′(Emin)

1/3 1/4 1/5 1/10 1/20 1/100 1/1000

Φ [bottom to top]

21 22 23 24 25 26 27 28 29 2.7 2.8 2.9 3.0

Γ′′(L/LB)-

2 2.4

Finite-size scaling analysis

(b)

3.410 3.415 3.420 3.425 3.430 3.435 1 2 3 4

E Γ

8 12 16 24 32 48 64 96 128 192 256 384 512 768

L [bottom to top (at edges)]

3.415 3.420 3.425 3.430 0.8 0.9 1.0 1.1 1.2

Final exponent estimates:

• correlation length exponent ν = 2.58(3)
• irrelevant exponent y = 0.35(4)

Details [Slevin et al., PRL 82, 382 (1999)]:

• energy and system-size dependence of

dimensionless Lyapunov exponent Γ = Γ(xrLν, xiL−y) =

ni

• i=0

nr

• j=0

aijxi

ix2j r

i!(2j)! L2j/ν−iy

• relevant and irrelevant scaling variables

xr, xi [e = (E/Ec − 1)] xr = e+

mr

• k=2

bk k!ek, xi = 1+

mi

• l=1

cl l!el

• combined fit gives Ec, ν, y and

expansion coefficients

Logarithmic corrections to scaling [xi/(b + xi log(L)) instead of xiL−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 qi fluctuates

2d: q = 6, σq ≈ 1.33 3d: q = 2 + (48/35)π2 ≈ 15.54, σq ≈ 3.36

• random connectivity (topology) generates disorder in physical system

4 6 8 10 12 14 16 18

q

10

• 7

10

• 6

10

• 5

10

• 4

10

• 3

10

• 2

10

• 1

10

P

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)