NETWORK MODELS
FOR THE QUANTUM HALL EFFECT
AND ITS GENERALISATIONS
John Chalker
Physics Department, Oxford University
Outline
• Network models
Quantum lattice models for single-particle systems with disorder
• Symmetry classes
Discrete symmetries and additions to Wigner-Dyson classification
• Random-bond Ising model and network model
Classical spin order and quantum delocalisation
• Classical percolation and network model
Classical and quantum delocalistion
Network Models
Ingredients
Motivation
2D lattice
Links:
zi
V(x,y)
zf
zf = eiφ zi
Nodes:
z3
B
Evolution
z2
operator
charged particle
z1
in magnetic field
0
@
z3
z4
1
0
A=@
W = W1 W2
z4
cos(α)
− sin(α)
sin(α)
cos(α)
10
A@
z1
z2
1
A
W1 : links
W2 : nodes
Two-dimensional U(1) model
Random link phases + uniform scattering angle α at nodes
Delocalisation transition as α varied
Model
α=0
Localisation length
ξ
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Limiting cases
π/4
α=π/2
ξ ∼ |α − π/4|−ν
ν ≃ 2.3
Chalker and Coddington, 1988
α
U(1) model in other geometries
One dimension
Cayley tree
ξ = 1/ ln(csc α)
Three dimensions
insulator
α=0
metal
Hall
insulator
α=π/2
Symmetry Classes
Dyson random matrix ensembles
Additional symmetry classes
Altland and Zirnbauer 1997
Orthogonal
with time-reversal symmetry
Hamiltonian H with discrete symmetry
Energy levels in pairs ±E
X −1 H ∗ X = −H
Symplectic
with time-reversal symmetry
and Kramers degeneracy
Unitary
without time-reversal symmetry
Given Hψ
= Eψ , define ψ̃ = Xψ ∗ .
Then H ψ̃
‘Class D’
‘Class C’
= −E ψ̃ .
X=1
X = iσy
Generalisations of network models
Amplitudes
Link phases
zi → n-component vector
eiφ → n × n unitary matrices U
Without further restrictions: U(n) model
not time-reversal invariant, so member of unitary symmetry class
With discrete symmetries:
H ∗ = −H so link phases U ∼ eiH are real
Class D:
O(n) model
Class C:
related to random bond Ising model
σy H ∗ σy = −H so link phases ∈ Sp(n), with Sp(2) ∼ SU(2)
SU(2) model
related to classical percolation
Ising model and network model
Relation between models
H
fi
gi
V
Phase diagram
h
T
v
Nishimori Line
f i+1
g i+1
PARA
T
0
HIsing −
X
Jij σi σj
hiji
FERRO
T
N
C

 +J probability p
Jij =
 −J probability (1 − p)
sin αij = tanh 2βJij
0
p
C
p
0.5
Cho and Fisher, 1998; Gruzberg, Read and Ludwig, 2001; Merz and Chalker 2002.
SU(2) network model and classical random walks
Feynman path expansion for Green function
[G(ζ)]r1,r2 =
X
G(ζ) = (1 − ζW )−1
ζ n Apath
n−step paths
with weight
SU(2) Averages
Apath
n

 cos(α) 
Q
∼ links Ulink
 ± sin(α) 
hU n i =



 1 n=0
−1/2 n = ±2


 0 otherwise
– keep only paths that cross each link 0 or 2 times.
Gruzberg, Ludwig and Read, 1999; Beamond, Cardy and Chalker, 2002
Quantum to classical mapping
Disorder-average for quantum system → average over classical paths
Classical
Quantum
sin α
cos α
or
p
1−p
Classical probabilities
Quantum amplitudes
+ random SU(2) phases
p = sin2 (α)
1 − p = cos2 (α)
Application
Eigenphase density of evolution operator, W :
ρ(θ) =
1
2π
[1 −
P
n
pn cos(2nθ)]
where W has eigenvalues eiθ and pn is classical return probability after n steps
SU(2) network model and percolation
Quantum
Classical
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sin α
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cos α
Quantum amplitudes
+ random SU(2) phases
Consequences:
and
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Classical probabilities
p = sin2 (α), 1 − p = cos2 (α)
ξQuantum ∼ |α − π/4|−4/3
ρ(θ) ∼ |θ|1/7 at α = π/4
Summary
Network models
Single quantum particle moving on lattice with randomness
Distinct phases for α
→ 0 and α → π/2
– distinguished by nature of edge states, separated by critical point
Symmetry classes via link phases: U(n), O(n) and Sp(n)
Discrete symmetries define classes additional to Wigner-Dyson ones
Mappings to problems from classical statistical physics
Random bond Ising model and O(1) model
Classical percolation and SU(2) model
Selected references
U(1) model
J. T. Chalker and P. D. Coddington, J. Phys C 21, 2665 (1988)
Review
B. Kramer, T. Ohtsuki and S. Kettemann, Phys. Rep. 417, 211 (2005)
O(1) model and Ising model
I. A. Gruzberg, N. Read, and A. W. W. Ludwig, Phys. Rev. 63, 104422 (2001)
F. Merz and J. T. Chalker, Phys. Rev. B65, 54424 (2002)
Classical percolation and SU(2) model
I. A. Gruzberg, A. W. W. Ludwig, and N. Read, Phys. Rev. Lett. 82, 4254 (1999)
E. Beamond, J. L. Cardy, and J. T. Chalker, Phys. Rev. B65, 214301 (2002)