Funded by NSF, Harvard-MIT CUA, AFOSR, DARPA, MURI

Takuya Kitagawa Harvard UniversityMark Rudner Harvard UniversityErez Berg Harvard UniversityYutaka Shikano Tokyo Institute of Technology/MIT Eugene Demler Harvard University

Exploration of Topological Phases with Quantum Walks

Thanks to Mikhail Lukin

Topological states of matter

Integer and FractionalQuantum Hall effects

Quantum Spin Hall effectPolyethethyleneSSH model

Geometrical character of ground states:Example: TKKN quantization of Hall conductivity for IQHE

Exotic properties:quantized conductance (Quantum Hall systems, Quantum Spin Hall Sysytems)fractional charges (Fractional Quantum Hall systems, Polyethethylene)

PRL (1982)

Summary of the talk: Quantum Walks can be used to realize all Topological

Insulators in 1D and 2D

Outline1. Introduction to quantum walk

What is (discrete time) quantum walk (DTQW)?Experimental realization of quantum walk

2. 1D Topological phase with quantum walk

Hamiltonian formulation of DTQWTopology of DTQW3. 2D Topological phase with

quantum walk Quantum Hall system without Landau levelsQuantum spin Hall system

Discrete quantum walks

Definition of 1D discrete Quantum Walk

1D lattice, particle starts at the origin

Analogue of classical random walk.Introduced in

quantum information:

Q Search, Q computations

Spin rotation

Spin-dependent Translation

arXiv:0911.1876

arXiv:0910.2197v1

Quantum walk in 1D: Topological

phase

Discrete quantum walk

One stepEvolution operator

Spin rotation around y axis

Translation

Effective Hamiltonian of Quantum WalkInterpret evolution operator of one step as resulting from Hamiltonian.

Stroboscopic implementation of Heff

Spin-orbit coupling in effective Hamiltonian

From Quantum Walk to Spin-orbit Hamiltonian in 1d

Winding Number Z on the plane defines the topology!

Winding number takes integer values, and can not be changed unless the system goes through gapless phase

k-dependent“Zeeman” field

Symmetries of the effective Hamiltonian

Chiral symmetry

Particle-Hole symmetry

For this DTQW, Time-reversal symmetry

For this DTQW,

Classification of Topological insulators in 1D and 2D

Detection of Topological phases:localized states at domain boundaries

Phase boundary of distinct topological phases has bound

states!Bulks are insulators

Topologically distinct, so the “gap” has to close

near the boundary

a localized state is expected

Split-step DTQW

Phase DiagramSplit-step DTQW

Apply site-dependent spin rotation for

Split-step DTQW with site dependent rotations

Split-step DTQW with site dependent rotations: Boundary State

Quantum Hall like states:2D topological phase

with non-zero Chern number

Quantum Hall system

Chern Number This is the number that characterizes the

topology of the Integer Quantum Hall type states

Chern number is quantized to integers

2D triangular lattice, spin 1/2“One step” consists of three unitary and translation operations in three directions

Phase Diagram

Chiral edge mode

Integer Quantum Hall like states with Quantum Walk

2D Quantum Spin Hall-like system

with time-reversal symmetry

Introducing time reversal symmetry

Given , time reversal symmetry withis satisfiedby the choice

of

Introduce another index, A, B

Take to be the DTQW for 2D triangular lattice

If has non-zero Chern number, the total system is in non-trivial phase of QSH phase

Quantum Spin Hall states with Quantum Walk

Classification of Topological insulators in 1D and 2D

In fact...

Extension to many-body systems

Can one prepare adiabatically topologically nontrivial states starting with trivial states? Yes

Can one do adiabatic switching of the Hamiltonians implemented stroboscopically? Yes

k

Eq(k)

Topologically trivial Topologically nontrivial

Gap has to close

Conclusions

• Quantum walk can be used to realize all of the classified topological insulators in

1D and 2D.

• Topology of the phase is observable through the localized states at phase boundaries.