1. 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

2. 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)

3. Summary of the talk: Quantum Walks can be used to realize all Topological Insulators in 1D and 2D

4. 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

5. Discrete quantum walks

6. 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

8. arXiv:0911.1876

9. arXiv:0910.2197v1

10. Quantum walk in 1D: Topological phase

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

12. 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

13. 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

14. Symmetries of the effective Hamiltonian Chiral symmetry Particle-Hole symmetry For this DTQW, Time-reversal symmetry For this DTQW,

15. Classification of Topological insulators in 1D and 2D

16. Detection of Topological phases: localized states at domain boundaries

17. 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

18. Split-step DTQW

19. Phase Diagram Split-step DTQW

20. Apply site-dependent spin rotation for Split-step DTQW with site dependent rotations

21. Split-step DTQW with site dependent rotations: Boundary State

22. Quantum Hall like states: 2D topological phase with non-zero Chern number Quantum Hall system

23. 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

24. 2D triangular lattice, spin 1/2 “One step” consists of three unitary and translation operations in three directions big points

25. Phase Diagram

26. Chiral edge mode

27. Integer Quantum Hall like states with Quantum Walk

28. 2D Quantum Spin Hall-like system with time-reversal symmetry

29. Introducing time reversal symmetry Given, time reversal symmetry with is satisfiedby the choice of Introduce another index, A, B

30. 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

31. Quantum Spin Hall states with Quantum Walk

32. Classification of Topological insulators in 1D and 2D In fact...

33. 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

34. 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.