Exploration of Topological Phases
with Quantum Walks
Takuya Kitagawa
Mark Rudner
Erez Berg
Yutaka Shikano
Eugene Demler
Harvard University
Harvard University
Harvard University
Tokyo Institute of Technology/MIT
Harvard University
Thanks to Mikhail Lukin
Funded by NSF, Harvard-MIT CUA,
AFOSR, DARPA, MURI
Topological states of matter
Polyethethylene
SSH model
Integer and Fractional
Quantum Hall effects
Quantum Spin Hall effect
Exotic properties:
quantized conductance (Quantum Hall systems, Quantum Spin Hall Sysytems)
fractional charges (Fractional Quantum Hall systems, Polyethethylene)
Geometrical character of ground states:
Example: TKKN quantization of
Hall conductivity for IQHE
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
Spin rotation
Spindependent
Translation
Analogue of classical
random walk.
Introduced in quantum
information:
Q Search, Q computations
arXiv:0911.1876
arXiv:0910.2197v1
Quantum walk in 1D:
Topological phase
Discrete quantum walk
Spin rotation around y axis
Translation
One step
Evolution operator
Effective Hamiltonian of Quantum Walk
Interpret 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
k-dependent
“Zeeman” field
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 phas
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
Topologically distinct,
insulators
so the “gap” has to close
near the boundary
a localized state is expected
Split-step DTQW
Split-step DTQW
Phase Diagram
Split-step DTQW with site dependent rotations
Apply site-dependent spin
rotation for
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
Introduce another index, A, B
Given
, time reversal symmetry with
is satisfiedby the choice of
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
In fact...
Classification of Topological insulators in 1D and
2D
Extension to many-body systems
Can one do adiabatic switching of the Hamiltonians
implemented stroboscopically? Yes
Can one prepare adiabatically topologically nontrivial
states starting with trivial states? Yes
Topologically trivial
Topologically nontrivial
Eq(k)
Gap has to close
k
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.