Quantum walk approach
Topological Insulators and Anderson Localization
Norio Kawakami Department of Physics, Kyoto University
Hideaki Obuse(Karlsruhe)
Yuki Nishimura(M1, Kyoto)
Collaborators
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1. Introduction◆Definition of quantum walk (QW)◆Symmetry of QW and topological insulators
2. QW with spatial disorder◆ Topological phase in 1D chiral class
topologically protected edge states◆ Anderson transition
coexistence of edge, localized and critical states
3. QW with temporal disorder◆Time evolution◆ How robust the edge states ?
Contents
What is a quantum walk ?
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quantum mechanical time-evolution of particlesQuantum version of random walk
Random walkWalkers move to right (left ) with probability p (1-p)
Quantum walk
Walker’s position at t : Gaussian variance σ2 ∝ t(Random walks)
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◇Time evolution operator
A walker at n: internal degrees |L>, |R>
◇Coin operatorrotate spin, mix |L> and |R>
◇Shift operator
left right
spin-selective motion
Discrete-time QWQuantum walk
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Tim
e s
tep
Hadamard walk
Progress in experimentsso rapid !
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Experiments and proposals
◇ Optical lattices◇ Trapped ions
◇ Photons
◇ NMR
◇ Photosynthetic energy transfer(excitons)
etc
Quantum Walks
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Cold atoms
Science 2009
(Cs atoms)
1DOptical lattice
(Position space)
λ/2=433 nm
F=4, mF=4F=3, mF=3
|L>, |R>
symmetric antisymmetric
Final image
Initial image
10 steps
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PRL (2010)
Trapped Ions40Ca+
Position:Phase space23 steps
Jaynes-CummingsHamiltonian
S1/2, m=1/2D5/2, m=3/2
|L>, |R>
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PRL (2010)
Photons
Polarization
Position:spatial modes6 steps
(70 steps, 2011Erlangen)
Decoherence
Quantum to Classical
temporal-disorder
|L>, |R>
Quantum Classical
positionstep
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M. Karski et al., Science 325, 174 (2009)
H. Schmitz et al., Phys. Rev. Lett. 103, 090504 (2009)
F. Zahringer et al., Phys. Rev. Lett. 104, 100503 (2010)
A. Schreiber et al., Phys. Rev. Lett. 104, 050502 (2010)
M. Hilley , Science 329, 1477 (2010)
A. Peruzzo et al. , Science 329, 1500 (2010)
M. A. Broome et al, Phys. Rev. Lett. 104, 153602 (2010)
U. Schneider et al, arXiv:1005.3541 (2011)
Y. Zou et al, arXiv:1007.2245 (2011) …. etc
Quantum Walk Experimental realization
Cold atoms
Photons
Trapped ions
Trapped ions
Photons
Photons
Photons
Photons
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◇Developed in Quantum Computation
Quantum Walk
Mathematical
1.Toplogical insulators:
All the possible topological insulators (1D, 2D)
tuning the operator
Kitagawa et al 2010Dirac equationHamiltonian: Time-evolution
◇Condensed Matter Physics
New arena to studytopological states
coinshift
2.Applications to Mott breakdownZener Tunneling: modeled by QW
T.Oka et al 2005
Non-equilibrium dynamics of Mott phase
T. Fukui-NK 1998T.Oka et al. 2010
cf 1D Non-Hermitian Hubbard: Exact solutionCorrelated electron systems
e.g. Konno et al.
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Dynamics of 1D Quantum Walks
Purpose
Static and dynamical random defectsHow the dynamics and topological edge states are influenced.
Systematic Studies of Topological Insulators
・Various kinds of topological states・Edge states: robustness
Topological insulator: 1D chiral class
Quantum Walks
・Anderson localization etc
Complementary to solid state physics
coexistence of edge, localized and critical states
Symmetry of Quantum Walks
Topological insulators
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Symmetries : quantum walksRelevant to topological insulators
Hamiltonian
QW realized in many experiments: chiral symmetryEigen energies: ±E
chiral orthogonal
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1D Quantum Walk: topological phases and edge states
Topological insulators: d=1, 2, 3
Kitagawa, Rudner, Berg, Demler, PRA 2011
Schnyder, Ryu, Furusaki, Ludwig, PRB ’08, NJP ’10; Kitaev AIP conf. ’08.
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Shift operator in momentum space
Coin operator
Dispersion relation: Hamiltonian
Hamiltonian
QW in 1D has a massive dispersionω(k) is a quasi-energy
2πperiodicity
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Dispersion relation
Z=1 Topological Insulator1D chiral orthogonal class
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Observation of Edge StatesKitagawa, Broome, Fedrizzi, Rudner, et al., arXiv:1105.5334
=> 0, 1
(θ1- ,θ1+) => 1, 1
0
Topological #θ1-
Topological #θ1+
x
edge
May, 2011
Quantum Walk with RandomnessAnderson transition
vs. topological edge states
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Clean systemQuantum walks
◇Coin operator θn=π/4
Initial state:
Hadamard walk
quantum
classical
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reflecting boundary condition
Coin operator:
Edge state ?
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Clean system: boundary
t=80
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◇Disorder is introduced fluctuations of coin operators,
s(t) : spatial (temporal) disorder.
◇How dynamics and topological edge states are influenced. spatial or temporal random defects
θ=π/4We focus on QWs with
Randomness
◇Theoretical:Joye & M. Merkli(2010)Ahlbrecht et al. (2011)Chandrashekar (2011)
◇Experimental:Schreiber et al. (2011)
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Static disorder: with boundary
◇ Edge state: robust ?◇ Anderson localization occurs ?◇ Extended state exists ?
(spatial disorder)Topological Phase
Anomalous behavior !
quantum
classicalN=104
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Recurrence probability P(t): Variance v(t):
◇Constant P(t) for system with edge states w/wo static disorder.◇Clean system: variance v(t) grows quadratically.◇Disordered system: power-law behaviors of v(t) are observed.
⇒ existence of delocalized states?⇒ But, is it possible ?
Static disorder: boundary
Protected edge mode
t0.2
t2 Critical delocalized
mode
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Zero-energy Edge States
Gap is open at ω = 0. Gap is closed at ω = 0.
They cannot exist at ω = 0 simultaneously !
Coexistence ?Edge States & Anderson transition at ω =0
Divergence in DOS
Critical Delocalized State
Dyson 1953
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◆Edge state: robust ⇒ topological edge state.◆Divergence in DOS: ω = ±π/2
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Scaling: Density of States
Divergence in ρ(ω) at ω = ±π/2
Clear Scaling Behavior
Critical State ! ω = ±π/2
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Anderson transition with chiral symmetry occurs at ω = ±π/2.
Scaling: Localization length
Localization length near ω=π/2 : transfer matrix method System size L = 108 and various δθ.
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Why ω =±π/2 are so special ?
Mechanism of Anderson transitionin 1D chiral class at ω =±π/2
Anderson transition with critical mode at ±π/2 !!
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Spatially disordered QW:
Revisit DOS of QW
t2
t0.2
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Strong spatial disorder:
DOS at ω=0 always diverges
When edge statesdisappear ?
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Inverse localization length at ω = 0.(System size N = 108)
Bulk gap around ω = 0 is closed at δθ=2π.Anderson transitions occur at ω = 0.
Coexistence of edge states and delocalized states does not occur.
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Dynamics of 1D Quantum Walks
◆Robustness of edge states
◆Anderson transition in 1D chiral classes
with static disorderRich !
◆Realization of topological states
New research arenaTopological phasesAnderson localization, etc
Topologically protected
Localized, critical states
Kitagawa et al. 2011
All characteristics appear simultaneously !
Temporal DisorderQuantum vs. Classical
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QW with Temporal Disorder
◇ Coin operator Cn depends on time.
◇ QW with temporal disorder approaches random walk.Konno ’05
How robust are edge states of QW against temporal disorder?
temporal disorder.
Question
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Probability distribution
◆How robust are edge states against temporal disorder ?
◆Symmetric distribution due to no spatial disorder.Quantum to Classical
δ
t=80
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Averaged Probability Distribution at 104 time steps
◆Gaussian distributions => classical random walk.◆Small peaks => remnants of edge states.
δ (weak disorder)
(strong disorder)
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Survival Probabilityfor δθt = π/8 and π/4
Position Variancefor δθt = π/8 and π/4
Solid thin curves : QW without reflecting coin.
◆QW gradually approaches classical random walk.
◆At short time steps, edge states are still observable.
v(t)~t.
δ
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SummaryStatic/dynamical disorder on QW:
edge states due to topological phase.
◇Pure QW in the topological phase・ normal transport modes & edge modes at the reflecting boundary.
◇ QW with static disorder in the topological phase・localized modes of the Anderson localization・critical modes of the Anderson transition in the 1D chiral system・edge modes are robust for the disorder if δθ< 2π.
◇ Strong disorder・edge modes disappear due to gap closing.
◇ Temporal disorder・Edge states are not robust. ・Still, the edge states can survive for long-time steps.