funded by nsf, harvard-mit cua, afosr, darpa, muri
Post on 24-Feb-2016
72 Views
Preview:
DESCRIPTION
TRANSCRIPT
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.
top related