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Gene MeleUniversity of Pennsylvania
Topological Insulators in Two and Three Dimensions
Topological Physics inTopological Physics inBand Insulators IIBand Insulators II
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The canonical list of electric forms of matter is actually incomplete
Conductor
Insulator
Superconductor
18th century
20th century
Topological Insulator
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Electronic States of MatterElectronic States of Matter
Topological Defects in (CH)x
Self conjugate state fromDirac mass inversion
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Summary of First Lecture: The unsual spin charge relation appearsin the strong coupling limit, where it is a property of atoms and decoupled dimers.
This is adiabatically connected to a continuum limit where it arises as a transition in the ground state topology.
Summary of Second Lecture: This transition occurs at the boundary between a topological insulator and an ordinary insulator.
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Electronic States of MatterElectronic States of Matter
Topological Insulators
This novel electronic state of matter is gapped in the bulk and supports the transport of spin and charge in gapless edge states that propagate at the sample boundaries. The edge states are …insensitive to disorder because their directionality is correlated with spin.
2005 Charlie Kane and GMUniversity of Pennsylvania
Electron spin admits a topologicallydistinct insulating state
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Electronic States of MatterElectronic States of Matter
This state is realized in three dimensional materials where spin orbit coupling produces a bandgap “inversion.”
It has boundary modes (surface states) with a 2D Dirac singularity protected by time reversal symmetry.
Bi2Se3 is a prototype.
.Hasan/Cava (2009)
Topological Insulators
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GrapheneGraphene: the Parent Phase: the Parent Phase
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The dispersion of a free particle in 2D..
…is replacedby an unconventional E(k) relation on thegraphene lattice
……. it has a critical electronic state. it has a critical electronic state
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The low energy theory is described byThe low energy theory is described byan effective mass theory for an effective mass theory for masslessmassless electronselectrons
( ) • ( )Bloch Wavefunction Wavefunction s at K r
eff FH r iv r ( ) ( )
NOTE: Here the “spin” degree of freedom describes the sublatticepolarization of the state, called pseudospin. In addition electrons carrya physical spin ½ and an isospin ½ describing the valley degeneracy.
It is a massless Dirac Theory in 2+1 Dimensions
D.P. DiVincenzo and GM (1984)
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A continuum of structures all with √3 x √3 period hybridizes the two valleys
Gapping the Dirac PointGapping the Dirac PointValley mixing from brokentranslational symmetry
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Gapping the Dirac PointGapping the Dirac PointValley mixing from brokentranslational symmetry
Kekule0
'0
ix
ix
eH
e
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BN
0'
0z
z
H
Gapping the Dirac PointGapping the Dirac PointCharge transfer from broken
inversion symmetry
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Gapping the Dirac PointGapping the Dirac PointOrbital currents from modulated flux
(Broken T-symmetry)
Gauged second neighbor hopping breaks T. “Chern insulator” with Hall conductance e2/h
FDM Haldane “Quantum Hall Effect without Landau Levels” (1988)
FDMH
0'
0z
z
H
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Topological ClassificationTopological Classification
1 2
21 2
1 ( , ) 04 k k
S
n d k d k k d d
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FDMH
0'
0z
z
H
1 2
21 2
1 ( , ) 14 k k
S
n d k d k k d d
Topological ClassificationTopological Classification
2
xyeh
“Chern Insulator” with (has equal contributionsfrom two valleys)
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2 ' 0 ''
( ) cos sin
2 cos cos sin sin
n x n y zn
n n zn
H k t k a k a M
t k b k b
Orthodoxy: Spectrum Gapped only for Broken Symmetry States
Crucially, this ignores the electron spin
:na triad of nearest neighbor bond vectors
' :nb triad of directed “left turn”
second neighbor bond vectors
Breaks P
Breaks TBreaks e-h symmetry
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Coupling orbital motion to the electron spin
SOH s V p
( ) ( )V r V r T Microscopic
Lattice model † † ( )SO m n n m m nH i r r
Spin orbit field Bond vector
Intersite hopping with spin precession
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Coupling orbital motion to the electron spin
0, 0xy z ˆ ˆRH s n p n s p
†1 ˆn mn mit n s d
Breaking mirror symmetry with a perpendicular spin orbit field
Modifies first neighbor coupling by spin dependent potential
R R x z y y xs s
Renormalizes Fermi velocity and can fission the Dirac point
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0, 0xy z ˆ ˆ( )SO z z effH s n p p s n p a
0effa d
/ 2
0effa d
† †2
† †
2 † †
cos
sin
i in m m n
n m m n
n m m n
t e c c e c c
c c c ct
i c c c c
Preserve mirror symmetry with a parallel spin orbit field
Generates a spin-dependent Haldane-type mass (two copies)
SO SO z z zs
Coupling orbital motion to the electron spin
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Mass Terms (amended)Mass Terms (amended)
z
z z
z z zs x z y y xs s
,x x x y Kekule: valley mixing
Heteropolar (breaks P)
Modulated flux (breaks T)
Spin orbit (Rashba, broken z→-z)
Spin orbit (parallel)**This term respects all symmetries and is therefore present, though possibly weak
spinless
For carbon definitely weak, but still important
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Topologically different statesTopologically different states
1 2
21 2
1 ( , )4 k k
S
n d k d k k d d
Charge transfer insulator Spin orbit coupled insulator
0n 1 ( 1) 0n
Topology of Chern insulatorin a T-invariant state
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Boundary ModesBoundary Modes
Ballistic propagation through one-way edge state
Counter propagating spinpolarized edge statesIntrinsic SO-Graphene
model on a ribbon
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Quantum Spin Hall EffectQuantum Spin Hall EffectIts boundary modes are spin filtered
propagating surface states (edge states)
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CommentsThe H2 model conserves Sz and is oversimplified. Spin, unlike charge, is not conserved.
But the edge state picture is robust!
Boundary modes: Kramers pair
(a) Band crossing protectedby T-reversal symmetry
(b) Elastic backscattering eliminated by T-symmetry
QSHE: quantum but not quantized
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More commentsCounter-propagating surface modes reflect thebulk topological order. They can only be eliminatedby a phase transition to a non-topological phase.
weak sublatticesymmetry breaking
strong sublatticesymmetry breaking
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Symmetry ClassificationSymmetry Classification
Conductors: unbroken state1
Insulators: broken translational symmetry:bandgap from Bragg reflection2
Superconductor: broken gauge symmetry
Topological Insulator ?
1possibly with mass anisotropy
2band insulators
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Symmetry ClassificationSymmetry ClassificationOrdinary insulators and topological insulators are distinguishedby a two-valued (even-odd) surface index.
Kramers Theorem: T-symmetry requires E(k,) =E(-k,)
But at special points k and -k are identified (TRIM)
even: ordinary (trivial) odd: topological
Kane and GM (2005)
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Bulk SignatureThe surface modes reflect bulk topological order
distinguished by a bulk symmetry
1 2
21 2
1 ( , )4 k k
S
n d k d k k d d
e.g. TKKN invariant = Chern number = Hall conductance
T-reversal symmetry requires n=0“Spin Chern number” in Sz conserving model
is nontopologicalTI index is defined mod 2
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BulkBulk timetime--reversal invariant reversal invariant momentamomenta
Symmetry-protected twofold degeneracy at opposingpoints (d and –d) on Bloch sphere
2 2G GH H
Comparison of T reversal pairs allowstopological classification of ground state
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Diagnostic for Topological Order:Diagnostic for Topological Order:
Periodic part of Bloch state: ( ) ( ; )ik rn nu k e k r
Q. How different are and ? ( )n Nu k
( )n Nu k
A. For a trivial atomic insulator they are the same
A. For N bands quantify by ( ) ( ) | | ( )mn m nw k u k u k
Antisymmetric: periodic complex-valued ( ) Pf (w)P k
( ) 0P k
points (vortices) at k
but never at TRIM (k=-k)
Kane and GM (2005)
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PfaffianPfaffian TestTestCount the zeroes of in one half of Brillouin zone
Zero: Trivial, like an atomic insulator
Even: Adiabatically connected to atomic insulatorby pairwise annihilation of its zeroes
Odd: Can’t be adiabatically connected to atomicinsulator since is forbidden at TRIM.( ) 0P k
Direct integration requires a smooth gauge and is awkward
( )P k
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PointwisePointwise Integration RulesIntegration Rules( 1) a
a
Pf( ( )) 1det ( )
aa
a
ww
0 0)a Atomic insulator: all (or
0 : exchange Kramers partners a b
1 2 3 40Gauge Invariant Products: a a a a
Track sign changes of ’s between TRIM
0 :
"conventional"
aa 0 :a
a
"topological"
Fu, Kane and GM (2007)
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With inversion symmetryWith inversion symmetryOrdinary insulators and topological insulators are distinguishedby a two-valued ( = 0,1) bulk index.
1
( 1)N
aa
a mm
(parity eigenvalues, 1)
Fu, Kane and GM (2007)
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Example: one orbital diamond lattice
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Example: BixSb1-x
Fu Kane (2007)
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Some References:
Review Article: M.Z. Hasan and C.L. Kane Rev. Mod. Phys. 82, 3045 (2010)
QSH in Graphene: C.L. Kane and E.J. MelePhys. Rev. Lett. 95, 226801 (2005)
Z2 insulators: C.L. Kane and E.J. MelePhys. Rev. Lett. 95, 146802 (2005)
Three Dimensional TI’s. L. Fu, C.L. Kane and E.J. MelePhys. Rev. Lett. 98, 106803 (2007)
Inversion symmetric TI’s. L. Fu and C.L. KanePhys. Rev. B 76, 045302 (2007)