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 ID: 280441
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Slide1
Topological Insulators and Beyond
Kai SunUniversity of Maryland, College ParkSlide2
Outline
Topological state of matterTopological nontrivial structure and topological indexAnomalous quantum Hall state and the Chern numberZ2 topological insulator with time-reversal symmetrySummarySlide3
DefinitionMany
A state of matter whose ground state wave-function has certain nontrivial topological structurethe property of a state Hamiltonian and excitations are of little importanceSlide4
Family tree
Resonating Valence Bond StateFrustrated spin systemOrbital motion of ultracold dipole molecule on a special latticeQuantum Hall StateFraction Quantum HallAnomalous Quantum Hall
Quantum Spin Hall
Anomalous Quantum Spin Hall
Topological superconductorsSlide5
Family tree
Resonating Valence Bond StateFrustrated spin systemOrbital motion of ultracold dipole molecule on a special latticeQuantum Hall State
Fraction Quantum Hall
Anomalous Quantum Hall
Quantum Spin Hall
Anomalous
Quantum Spin Hall
Topological superconductors
Topological insulatorsSlide6
Magnetic Monopole
Gauge TransformationVector potential cannot be defined globallyMatter field
wave-function on each semi-sphere is single valued
Magnetic flux for a compact surface: Slide7
2D
noninteracting fermionsHamiltonian:A gauge-like symmetry:“Gauge” field: (Berry connection)“Magnetic” field: (Berry phase)Compact manifold: (to define flux) Brillouin zone: T2Only for insulators: no
Fermi surfacesQuantized flux (
Chern number)
Haldane, PRL 93, 206602 (2004).Slide8
Two-band model (one “gauge” field)
Hamiltonian:Kernel:
withDispersion relation:
with
With
i
=x, y or z
For insulators:
Topological index for
2D insulators
:Slide9
Implications
Theoretical: wavefunction and the “gauge field” cannot be defined globallyChern number change sign under time-reversalTime-reversal symmetry is brokenExperimentallyInteger Hall conductivity (without a magnetic field)(chiral) Edge statesStable against impurites (no localization)Slide10
InteractionsWard identity:
Hall Conductivity:Slide11
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 QHISlide12
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 transformationSpin up and down exchangeChern number change signWhole system remains invariantSlide13
Naïve picture
Described by an integer topological indexHall conductivity being zeroNo chiral charge edge currentHave a chiral spin edge currentHowever, life is not always so simpleSpin is not a conserved quantitySlide14
Time-reversal symmetry for fermions and Kramers
pairFor spin-1/2 particles, T2=-1T flip spin:T2 flip spin twiceFermions: change sign if the spin is rotated one circle.Every state has a degenerate partner (Kramers pair)Slide15
1D Edge of a 2D insulator (Z2 Topological classification)
Topological protected edge statesSlide16
Z2 topological index
Bands appears in pairs (Kramers pair)Total Chern number for each pair is zeroFor 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.Slide17
Another approach
T symmetry need only half the BZHowever, 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.Slide18
4-band model with inversion symmetry
4=2 (bands)x 2 (spin)Assumptions:High symmetry points in the BZ: invariant under k to –kTwo possible situationsP is identity: trivial insulatorP is not identity: Parity at high symmetry points:Topological index:Slide19
3D system8 high symmetry points
1 center+1 corner+3 face center+3 bond centerStrong topological indexThree weak-topological indices (stacks of 2D topologycal insulators)Slide20
Interaction and topological gauge field theory
Starting by Fermions + Gauge fieldIntegrate out FermionsFor insulators, fermions are gappedIntegrate out a gapped mode the provide a well-defined-local gauge fieldWhat is left? Gauge fieldInsulators can be described by the gauge field onlySlide21
Gauge fieldOriginal gauge theory:
2+1D (anomalous) Quantum Hall state3D time-reversal symmetry preservedSlide22
Summary
3D Magnetic Monopole: integer topological index: monopole charge2D Quantum Hall insulatorinteger topological: integer: Berry phaseQuantized Hall conductivity and a chiral edge state2D/3D Quantum Spin Hall insulator (with T symmetry)Z2 topological index (+/-1 or say 0 and 1)Chiral spin edge/surface stateSuperconductor can be classified in a similar way (not same due to an extra particle-hole symmetry)