Funded by NSF, Harvard-MIT CUA, AFOSR, DARPA, MURI Takuya Kitagawa Harvard University Mark Rudner Harvard University Erez Berg Harvard University Yutaka 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 Fractional Quantum Hall effects Quantum Spin Hall effect Polyethethylene SSH 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

Outline 1. 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 DTQW Topology of DTQW 3. 2D Topological phase with quantum walk Quantum Hall system without Landau levels Quantum 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 emphasize it’s evolution operator

arXiv:

arXiv: v1

Quantum walk in 1D: Topological phase

Discrete quantum walk One step Evolution operator Spin rotation around y axis emphasize it’s evolution operator Translation

Effective Hamiltonian of Quantum Walk Interpret evolution operator of one step as resulting from Hamiltonian. Stroboscopic implementation of H eff 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 Diagram Split-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 brillouin zone chern number, for example counts the number of edge modes brillouin zone chern number, for example counts the number of edge modes

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

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 with is satisfiedby the choice of Introduce another index, A, B

Taketo be the DTQW for 2D triangular lattice Ifhas zero Chern number, the total system is in trivial phase of QSH phase 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 E q (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.