Topological Insulators and Beyond Kai Sun University of Maryland, College Park.
Topological Insulators and Beyond
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Transcript of Topological Insulators and Beyond
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Topological Insulators and Beyond
Kai SunUniversity of Maryland, College Park
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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
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Definition
• Many• A state of matter whose ground state wave-
function has certain nontrivial topological structure– the property of a state – Hamiltonian and excitations are of little
importance
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Family tree
Resonating Valence Bond State•Frustrated spin system•Orbital motion of ultracold dipole molecule on a special lattice
Quantum Hall StateFraction Quantum Hall
Anomalous Quantum Hall
Quantum Spin Hall
Anomalous Quantum Spin Hall
Topological superconductors
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Family tree
Resonating Valence Bond State•Frustrated spin system•Orbital motion of ultracold dipole molecule on a special lattice
Quantum Hall StateFraction Quantum Hall
Anomalous Quantum Hall
Quantum Spin Hall
Anomalous Quantum Spin Hall
Topological superconductors
Topological insulators
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Magnetic Monopole
Gauge Transformation
Vector potential cannot be defined globally
Matter field
wave-function on each semi-sphere is single valued
Magnetic flux for a compact surface:
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2D noninteracting fermions
• Hamiltonian:
• A gauge-like symmetry:
• “Gauge” field: (Berry connection)
• “Magnetic” field: (Berry phase)
• Compact manifold: (to define flux) Brillouin zone: T2
• Only for insulators: no Fermi surfaces• Quantized flux (Chern number)
Haldane, PRL 93, 206602 (2004).
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Two-band model (one “gauge” field)Hamiltonian:
Kernel:
withDispersion relation:
with
With i=x, y or z
For insulators:
Topological index for 2D insulators :
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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)
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Interactions
• Ward identity:
• Hall Conductivity:
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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
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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
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Naïve picture
• Described by an integer topological index• Hall conductivity being zero• No chiral charge edge current• Have a chiral spin edge currentHowever, life is not always so simple• Spin is not a conserved quantity
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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)
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1D Edge of a 2D insulator (Z2 Topological classification)
Topological protected edge states
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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.
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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.
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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:
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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)
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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 well-
defined-local gauge field• What is left? Gauge field
• Insulators can be described by the gauge field only
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Gauge field
• Original gauge theory:• 2+1D (anomalous) Quantum Hall state
• 3D time-reversal symmetry preserved
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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)