Topological Insulators and Beyond
Kai Sun
University of Maryland, College Park
Outline
• Topological state of matter
• Topological nontrivial structure and
topological index
• Anomalous quantum Hall state and the Chern
number
• Z2 topological insulator with time-reversal
symmetry
• Summary
Definition
• Many
• A state of matter whose ground state wavefunction has certain nontrivial topological
structure
– the property of a state
– Hamiltonian and excitations are of little
importance
Family tree
Resonating Valence
Bond State
•Frustrated spin system
•Orbital motion of ultracold dipole
molecule on a special lattice
Quantum Hall State
Fraction Quantum Hall
Anomalous Quantum Hall
Quantum Spin Hall
Anomalous Quantum Spin Hall
Topological superconductors
Family tree
Resonating Valence
Bond State
•Frustrated spin system
•Orbital motion of ultracold dipole
molecule on a special lattice
Quantum Hall State
Fraction Quantum Hall
Anomalous Quantum Hall
Quantum Spin Hall
Anomalous Quantum Spin Hall
Topological insulators
Topological superconductors
Magnetic Monopole
Vector potential cannot be defined globally
Gauge Transformation
Matter field
wave-function on each semi-sphere is single valued
Magnetic flux for a compact surface:
2D noninteracting fermions
• Hamiltonian:
• A gauge-like symmetry:
• “Gauge” field: (Berry connection)
• “Magnetic” field: (Berry phase)
Haldane, PRL 93, 206602 (2004).
• Compact manifold: (to define flux) Brillouin zone: T2
• Only for insulators: no Fermi surfaces
• Quantized flux (Chern number)
Two-band model (one “gauge” field)
Hamiltonian:
Kernel:
Dispersion relation:
For insulators:
Topological index for 2D insulators :
with
With i=x, y or z
with
Implications
• Theoretical:
– wavefunction and the “gauge field” cannot be
defined globally
– Chern number change sign under time-reversal
– Time-reversal symmetry is broken
• Experimentally
– Integer Hall conductivity (without a magnetic field)
– (chiral) Edge states
• Stable against impurites (no localization)
Interactions
• Ward identity:
• Hall Conductivity:
3D Anomalous Hall states?
• No corresponding topological index available
in 3D (4D has)
• No Quantum Hall insulators in 3D (4D has)
• But, it is possible to have stacked 2D layers of
QHI
Time-reversal symmetry preserved
insulator with topological ordering?
• Idea:
Spin up and spin down electrons are both in a
(anomalous) quantum Hall state and have
opposite Hall conductivity (opposite Chern
number)
• Result:
– Hall conductivity cancels
– Under time-reversal transformation
• Spin up and down exchange
• Chern number change sign
• Whole system remains invariant
Naïve picture
• Described by an integer topological index
• Hall conductivity being zero
• No chiral charge edge current
• Have a chiral spin edge current
However, life is not always so simple
• Spin is not a conserved quantity
Time-reversal symmetry for fermions and
Kramers pair
• For spin-1/2 particles, T2=-1
– T flip spin:
– T2 flip spin twice
– Fermions: change sign if the spin is rotated one
circle.
• Every state has a degenerate partner (Kramers
pair)
1D Edge of a 2D insulator
(Z2 Topological classification)
Topological protected edge states
Z2 topological index
• Bands appears in pairs (Kramers pair)
– Total Chern number for each pair is zero
• For the occupied bands: select one band from
each pair and calculate the sum of all Chern
numbers.
• This number is an integer.
• But due to the ambiguous of selecting the
bands, the integer is well defined up to mod 2.
Another approach
• T symmetry need only half the BZ
• However, half the BZ is not a compact
manifold.
• Need to be extended (add two lids for the
cylinder)
• The arbitrary of how to extending cylinder into
a closed manifold has ambiguity of mod 2.
4-band model with
inversion symmetry
• 4=2 (bands)x 2 (spin)
• Assumptions:
• High symmetry points in the BZ:
invariant under k to –k
• Two possible situations
– P is identity: trivial insulator
– P is not identity:
• Parity at high symmetry points:
• Topological index:
3D system
• 8 high symmetry points
– 1 center+1 corner+3 face center+3 bond center
• Strong topological index
• Three weak-topological indices (stacks of 2D
topologycal insulators)
Interaction and topological gauge field
theory
• Starting by Fermions + Gauge field
• Integrate out Fermions
– For insulators, fermions are gapped
– Integrate out a gapped mode the provide a welldefined-local gauge field
• What is left? Gauge field
• Insulators can be described by the gauge field
only
Gauge field
• Original gauge theory:
• 2+1D (anomalous) Quantum Hall state
• 3D time-reversal symmetry preserved
Summary
• 3D Magnetic Monopole:
– integer topological index: monopole charge
• 2D Quantum Hall insulator
– integer topological: integer: Berry phase
– Quantized Hall conductivity and a chiral edge state
• 2D/3D Quantum Spin Hall insulator (with T symmetry)
– Z2 topological index (+/-1 or say 0 and 1)
– Chiral spin edge/surface state
• Superconductor can be classified in a similar way (not
same due to an extra particle-hole symmetry)