AFOSR DARPA MURI Takuya Kitagawa Harvard University Mark Rudner Harvard University Erez Berg Harvard University Yutaka Shikano ID: 511312
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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 LukinSlide2
Topological states of matter
Integer and Fractional
Quantum Hall effectsQuantum 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)Slide3
Summary of the talk: Quantum Walks can be used to realize all Topological Insulators in 1D and 2DSlide4
Outline
1. Introduction to quantum walk
What is (discrete time) quantum walk (DTQW)?Experimental realization of quantum walk2. 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 systemSlide5
Discrete quantum walksSlide6
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 operatorSlide7Slide8
arXiv:0911.1876Slide9
arXiv:0910.2197v1Slide10
Quantum walk in 1D: Topological phase
Slide11
Discrete quantum walk
One stepEvolution operatorSpin rotation around y axis
emphasize it’s evolution operator
TranslationSlide12
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 HamiltonianSlide13
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” fieldSlide14
Symmetries of the effective Hamiltonian
Chiral symmetry
Particle-Hole symmetry
For this DTQW,
Time-reversal symmetry
For this DTQW, Slide15
Classification of Topological insulators in 1D and 2DSlide16
Detection of Topological phases:
localized states at domain boundariesSlide17
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 expectedSlide18
Split-step DTQWSlide19
Phase Diagram
Split-step DTQWSlide20
Apply site-dependent spin rotation for
Split-step DTQW with site dependent rotationsSlide21
Split-step DTQW with site dependent
rotations: Boundary StateSlide22
Quantum Hall like states:
2D topological phase with non-zero Chern number
Quantum Hall systemSlide23
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 modesSlide24
2D triangular lattice, spin 1/2
“One step” consists of three unitary and translation operations in three directions
big pointsSlide25
Phase DiagramSlide26
Chiral edge modeSlide27
Integer Quantum Hall like states with Quantum WalkSlide28
2D Quantum Spin Hall-like system
with time-reversal symmetry
Slide29
Introducing time reversal symmetry
Given , time reversal symmetry with
is satisfied
by the choice of
Introduce another index, A, BSlide30
Take
to be the DTQW for 2D triangular lattice
If
has 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 phaseSlide31
Quantum Spin Hall states with Quantum WalkSlide32
Classification of Topological insulators in 1D and 2D
In fact...Slide33
Extension to many-body systems
Can one prepare adiabatically topologically nontrivial
states starting with trivial states? YesCan one do adiabatic switching of the Hamiltonians implemented stroboscopically? Yes
k
E
q
(k)
Topologically trivial
Topologically nontrivial
Gap has to closeSlide34
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.