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 2D lattice Links: Motivation z z i f V(x,y) z f = e iφ z i Nodes: Evolution z B z 3 2 operator charged particle W = W 1 W 2 z z in magnetic field 1 4 W 1 : links 0 1 0 1 0 1 cos( α ) sin( α ) @ z 3 @ z 1 W 2 : nodes A = @ A A − sin( α ) cos( α ) z 4 z 2
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 ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� α ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� π/4 ������� ������� ������� ������� ������� ������� ������� ������� Limiting cases ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� α=π/2 ������� ������� ξ ∼ | α − π/ 4 | − ν ν ≃ 2 . 3 Chalker and Coddington, 1988
U(1) model in other geometries One dimension Cayley tree ξ = 1 / ln(csc α ) Three dimensions Hall insulator insulator metal α=0 α=π/2
Symmetry Classes Dyson random matrix ensembles Additional symmetry classes Altland and Zirnbauer 1997 Hamiltonian H with discrete symmetry Orthogonal Energy levels in pairs ± E with time-reversal symmetry X − 1 H ∗ X = − H Symplectic Given Hψ = Eψ , define ˜ with time-reversal symmetry ψ = Xψ ∗ . Then H ˜ ψ = − E ˜ and Kramers degeneracy ψ . Unitary X = 1 ‘Class D’ without time-reversal symmetry X = iσ y ‘Class C’
Generalisations of network models z i → n-component vector Amplitudes e iφ → n × n unitary matrices U Link phases 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 ∼ e iH are real Class D: O(n) model related to random bond Ising model σ y H ∗ σ y = − H so link phases ∈ Sp(n), with Sp(2) ∼ SU(2) Class C: SU(2) model related to classical percolation
Ising model and network model Relation between models Phase diagram H V f i h T g i v Nishimori Line f i+1 g i+1 PARA T 0 � H Ising − J ij σ i σ j FERRO � ij � T C N + J probability p J ij = − J probability (1 − p ) p sin α ij = tanh 2 βJ ij p 0 0.5 C Cho and Fisher, 1998; Gruzberg, Read and Ludwig, 2001; Merz and Chalker 2002.
SU(2) network model and classical random walks G ( ζ ) = (1 − ζW ) − 1 Feynman path expansion for Green function � ζ n A path [ G ( ζ )] r 1 ,r 2 = n − step paths n cos( α ) A path ∼ � links U link with weight ± sin( α ) SU(2) Averages 1 n = 0 � U n � = − 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 α sin or p 1−p α cos Classical probabilities Quantum amplitudes p = sin 2 ( α ) 1 − p = cos 2 ( α ) + random SU(2) phases Application 1 Eigenphase density of evolution operator, W : ρ ( θ ) = 2 π [1 − � n p n cos(2 nθ )] where W has eigenvalues e iθ and p n is classical return probability after n steps
SU(2) network model and percolation Classical Quantum ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� α sin ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� α cos ��� ��� ��� ��� ��� ��� ��� ��� Quantum amplitudes ��� ��� ��� ��� Classical probabilities + random SU(2) phases p = sin 2 ( α ) , 1 − p = cos 2 ( α ) ξ Quantum ∼ | α − π/ 4 | − 4 / 3 Consequences: ρ ( θ ) ∼ | θ | 1 / 7 at α = π/ 4 and
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)
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