Quantum walk approach

Topological Insulators and Anderson Localization

Norio Kawakami Department of Physics, Kyoto University

Hideaki Obuse(Karlsruhe)

Yuki Nishimura(M1, Kyoto)

Collaborators

京都

Kyoto University

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 ?

京都

Kyoto University

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)

京都

Kyoto University

◇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

京都

Kyoto University

Tim

e s

tep

Hadamard walk

Progress in experimentsso rapid !

京都

Kyoto University

Experiments and proposals

◇ Optical lattices◇ Trapped ions

◇ Photons

◇ NMR

◇ Photosynthetic energy transfer(excitons)

etc

Quantum Walks

京都

Kyoto University

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

京都

Kyoto University

PRL (2010)

Trapped Ions40Ca+

Position:Phase space23 steps

Jaynes-CummingsHamiltonian

S1/2, m=1/2D5/2, m=3/2

|L>, |R>

京都

Kyoto University

PRL (2010)

Photons

Polarization

Position:spatial modes6 steps

(70 steps, 2011Eｒlangen)

Decoherence

Quantum to Classical

temporal-disorder

|L>, |R>

Quantum Classical

positionstep

京都

Kyoto University

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

京都

Kyoto University

◇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.

京都

Kyoto University

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

京都

Kyoto University

Symmetries : quantum walksRelevant to topological insulators

Hamiltonian

QW realized in many experiments: chiral symmetryEigen energies: ±E

chiral orthogonal

京都

Kyoto University

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.

京都

Kyoto University

Shift operator in momentum space

Coin operator

Dispersion relation: Hamiltonian

Hamiltonian

QW in 1D has a massive dispersionω(k) is a quasi-energy

2πperiodicity

京都

Kyoto University

Dispersion relation

Z=1 Topological Insulator1D chiral orthogonal class

京都

Kyoto University

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

京都

Kyoto University

Clean systemQuantum walks

◇Coin operator θn=π/4

Initial state:

Hadamard walk

quantum

classical

京都

Kyoto University

reflecting boundary condition

Coin operator:

Edge state ?

京都

Kyoto University

Clean system: boundary

t=80

京都

Kyoto University

◇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)

京都

Kyoto University

Static disorder: with boundary

◇ Edge state: robust ?◇ Anderson localization occurs ?◇ Extended state exists ?

(spatial disorder)Topological Phase

Anomalous behavior !

quantum

classicalN=104

京都

Kyoto University

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

京都

Kyoto University

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

京都

Kyoto University

◆Edge state: robust ⇒ topological edge state.◆Divergence in DOS： ω = ±π/2

京都

Kyoto University

Scaling: Density of States

Divergence in ρ(ω) at ω = ±π/2

Clear Scaling Behavior

Critical State ! ω = ±π/2

京都

Kyoto University

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.

京都

Kyoto University

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 !!

京都

Kyoto University

Spatially disordered QW:

Revisit DOS of QW

t2

t0.2

京都

Kyoto University

Strong spatial disorder:

DOS at ω=0 always diverges

When edge statesdisappear ?

京都

Kyoto University

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.

京都

Kyoto University

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

京都

Kyoto University

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

京都

Kyoto University

Probability distribution

◆How robust are edge states against temporal disorder ?

◆Symmetric distribution due to no spatial disorder.Quantum to Classical

δ

t=80

京都

Kyoto University

Averaged Probability Distribution at 104 time steps

◆Gaussian distributions => classical random walk.◆Small peaks => remnants of edge states.

δ (weak disorder)

(strong disorder)

京都

Kyoto University

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

京都

Kyoto University

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.