topological physics in band insulators ivmele/qcmt/ppts/windsorlec4v4.pdfp3. determine the z 2 index...
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Gene MeleUniversity of Pennsylvania
Wannier representation and band projectors
Topological Physics inTopological Physics inBand Insulators IVBand Insulators IV
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Modern view: Gapped electronic states Modern view: Gapped electronic states are equivalentare equivalent
Kohn (1964): insulator is exponentially insensitivity to boundary conditions
weak coupling strong coupling “nearsighted”, local
Postmodern: Gapped electronic states are distinguished by topological invariants
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Introduction:Introduction:
Exercise: consider three problems (of increasing complexity) and their solutions
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P1. Polarization (dipole density) of a one P1. Polarization (dipole density) of a one dimensional classical latticedimensional classical lattice
1 {1}
1 1N
i i i ii i
P dq x q x qL Na a a
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P1. Polarization (dipole density) of a one P1. Polarization (dipole density) of a one dimensional classical latticedimensional classical lattice
1 {1}
1 1 1N
toti i i i
i i
P dq x q x qL Na a a
P is only defined modulo a unit of polarization (q in 1D)
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P2. Polarization (dipole density) of a one P2. Polarization (dipole density) of a one dimensional quantum latticedimensional quantum lattice
{1}
1 ( )ii a
iP eZ xL
e n x dxa
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P2. Polarization (dipole density) of a one P2. Polarization (dipole density) of a one dimensional quantum latticedimensional quantum lattice
{1}
1 ( )ii a
iP eZ xL
e n x dxa
P depends on the choice of unit cell:with discrete jumps when classical cores cross boundary, and continuousvariation due to quantum n(x)
Is quantum P undefinableas a bulk quantity?
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Quantum theory of polarizationQuantum theory of polarization
0
( )
,0 0
( ( ) ) ( 0)
2
Consider adiabatic evolution from an unpolarized stateT
a b
T
kn
P T Z Z P J dt
P ed d dk
, ( ) | ( ) ( ) | ( )
with Berry curvature
k n k n k n ni u k u k u k u k
0( ( ) ) ( 0) ( ) | ( )
2 2
Stokes: loop integral of connection on ( ,k) circuit
a b n k nn
ie eP T Z Z P dk u k u k
Bulk P is defined up to its quantum (e) even for a smoothlydistributed n(x). This information is not in n(x) but in (i.e. in (x))(King-Smith & Vanderbilt, Resta)
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Interpretation: Interpretation: WannierWannier--chargecharge--centers are discretecenters are discrete
Ground state average over n(x) contains the first moments ofthe charge densities in its Wannier functions.
{ }
2
1
1 ( )nn
i ii
e x w x dxP eZ xL a
continuous
discrete
(WC’s locations are functions ofHamiltonian parameters)
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P3. Determine the ZP3. Determine the Z22 index of a generic* index of a generic* timetime--reversalreversal--invariant band insulator invariant band insulator
*time reversal & lattice translations only
Z2 index counts the number (mod 2) of band inversions from the filledmanifold of a T-invariant insulator.
Calculate the overlap of the cell-periodic Bloch function u(-k) with its time reversedpartner at -k
( ) ( ) | | ( )mn m nw k u k u k
Count the complex zeroes of Pf [w] within one half of Brillouin zone(difficulty: the overlap is k-nonlocal)
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Some possible solutions: Some possible solutions:
1
( 1)N
aa
2a mm
(parity eigenvalues, 1)
1. Parity test: for crystal with inversion symmetry
Requires an inversion symmetric space group (nongeneric)
2. Adiabatic continuity. Apply parity test to reference inversion-symmetricstructure and check for no gap closure under a slow deformation to P-breakingstructure of interest.
3. Gap closure: Compare the level ordering in band insulator computedwithout and with spin orbit coupling. Band inversion requires a gap closure and (may) denote transition to topological phase, verified by surface spectrum.
4. Transformation to Wannier representation.
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,, 0 ( ) ( ; )ik rn k nRH T r e u k r
( ) ik r ik rh k e He
Wave packet is constructed from a BZ integral
,( ) k k ( ; )(2 ) (2 )
ik rn nd dn k
V Vw r d d e u k r
And its discrete lattice translates:
( ) k ( ; )(2 )
ik r Rn nd
Vw r R d e u k r
, , ', | , ' m n R Rm R n R
, ( )ik Rnn k
Re w r R
WannierWannier functions: definitionsfunctions: definitions
This is a reconstruction of from a lattice sum
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( )( ; ) ( ; )i kn nu k r e u k r
WannierWannier functions: functions: nonuniquenessnonuniqueness
1. U(1) gauge freedom
superposition from eigenfunctionsof disconnected sectors, h(k)
( ) k ( ; )(2 )
ik rn nd
Vw r d e u k r
( ) ( )exponentially localized wf's require a smooth k k G
2. U(N) freedom
index switching at a band crossing generatespower law tails in its single band Wannier functions.
( ; ) ( ) ( ; )m mn nu k r U k u k r
removable by specification of a k-differentiable (i.e. composite manifol
smooth): d
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Maximally localized Maximally localized wfwf’’ss
Task: Minimize with respect to U(k) to obtain thesmoothest possible k-dependence of its quasi-Bloch states and the optimum localization of its Wannier representation.
2
occn nn
nr r r
Physics contained in a projected subspace
occ occnk nk
n nP n R n R
Q I P
k, R,
ˆ , ,
ˆ ˆ ˆ
These band projectors are invariant under choice of U(k)
Degree of localization is measured by the spread functional
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Topological obstructionTopological obstructionA smooth periodic gauge is impossible for a band insulator with a nonzero Chern number (i.e. one with a nonzero Hall conductance).
Chern insulators are not Wannier-representable*
02 0
when in Chern insulator
n k n k Gk k k
i d u uC
, ,k |
jumps herek k ku u|
vortex
smooth k k ku u|
*It is perhaps not surprising that the existence of a representation of the magnetic subband In terms of propertly localized states precludes the existence of a Hall current. - Thouless (1984)
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Alternative route to Alternative route to WFWF’’ss: Band Projection: Band Projection
occ occnk nk
n nP n R n R
k, R,
ˆ , ,
Projected “M-band” problem
1 2
Choose M (localized) trial functions with N lattice translates
Ground state projected image:
With symmetric orthogonalization: (M M for each k)m m n n m n m n
P
w S S
/, ,
ˆ
; |
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CommentsComments
The k-space construction and the projection method are complementary
K-space: find smoothest possible quasi Bloch states whose superposition produces most localized wavepackets (minimize )
nkv
Projection: choose M localized trial functions whose valenceband projections are minimally inflated by the projector
occ occNote: while projectors of the form
are invariant under U(N).nk nk
n nP n R n R
k, R,
ˆ , ,
the charge "centers" of wf's are not (clearly gauge dependent
ind)
ividual nx R n X R n ,ˆ, | | ,
M
nn
x R ,
but the "sum of charge centers" are gauge invariant modulo
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Special considerations for topological insulatorsSpecial considerations for topological insulators
1. TI’s are time-reversal invariant (Chern number = 0)thus the topological obstruction to the construction of exponentially localized wf’s is formally absent.
2. In Sz-conserving models (e.g. Haldane2) the ground state has disconnected sectors each of whichcontains a topological obstruction
3. k-space criterion (Pfaffian) for TI requires that we work in a globally smooth gauge.
natural vs. smooth gauge
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Time Reversal PolarizationTime Reversal Polarization
simple model: parametric pump for one dimensional lattice
HT-invariant 'T-invariant k s
1
[ ] [ ][ ] [ ] ; y
H t T H tH t H t i K
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Time Reversal PolarizationTime Reversal Polarization
simplest model: parametric pump for one dimensional lattice
, 0
1 1( ) (0)2 2t kA t
P t P dt dk dk dk F A A
initial
final
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Time Reversal PolarizationTime Reversal Polarization
For termination at T/2 the loop integral can be reduced to a half zonein a (locally smooth) gauge that explicitly respects T-symmetry at t=0,T/2
,
,
, ,
, ,
k m
k m
iI IIk m k m
iII Ik m k m
u e u
u e u
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Time Reversal PolarizationTime Reversal Polarization
( ) | ( ) ( )I IInk k nk
I II
I II
k i u u k kP P PP P P
A A A
0).P
C
is the ordinary charge
polarization (and vanishes if
P for a measures the bulk flow of time reversed partners to the boundaries (intege
ha
r
lf peri
mod
od
2)
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TI Surface StatesTI Surface States
1 2( , ) ( , )k t k kParametric pump Band topology
P integrated over a distinguishesthe binding/liberation of its Kramers
hp
alf zart
oneners
vs
ordinary topological
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WannierWannier states for Zstates for Z22 insulatorsinsulators
Impossible for TI using the TRG. If the gaugeis smooth over the full zone then Z2=0 (ordinary insulator) so that Z2=1 (topological insulator) can occur only if the time reversal gauge contains an unremovable singularity.
Nonetheless TI’s are gapped states with Chern number =0, i.e. in an unobstructed gapped state, thus Wannier representable.
TOPOLOGICAL INSULATOR IS WANNIER-REPRESENTABLE THOUGH NOT IN THE TIME REVERSAL SYMMETRIC GAUGE.
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WFWF’’ss by band projectionby band projection
occ occnk nk
n nP n R n R
k, R,
ˆ , ,
1 2
Choose M (localized) trial functions with N lattice translates
Ground state projected image:
With symmetric orthogonalization: (M M for each k)m m n n m n m n
P
w S S
/, ,
ˆ
; |
The projection recipe is algorithmic
The obstruction appears here
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Example 1: Haldane Model Example 1: Haldane Model Spinless tight binding model on honeycomb lattice
with two distinct gapped phases
Ordinary insulator: breaks P Chern insulator: breaks T
Fourier spectrum of its overlap matrix eigenvalues
Ordinary insulator
Chern insulator
Ref: Thonhauser and Vanderbilt (2006)
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Example 2: SpinExample 2: Spin--full full graphenegraphene
Ordinary insulator: sublattice potential breaks PZ2 odd insulator: spin-orbit term gaps the K point
, , ,Trial states: Kramers pair z zA A
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Example 2: SpinExample 2: Spin--full full graphenegraphene
Ordinary insulator: sublattice potential breaks PZ2 odd insulator: spin-orbit term gaps the K point
, , ,Trial states: Kramers pair z zA A
, , ,Trial states: TR-polarized pair x xA B
Soluyanov and Vanderbilt (2011)
Breaking Kramers symmetry in real space avoids thethe Z2 obstruction.
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SchematicallySchematicallyOrdinary insulators and topological insulators are both nearsightedgapped phases with Wannier representations as Kramerspartners (ordinary) or as time-reversal polarized partners (topological)
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Local Diagnostics for Topological OrderLocal Diagnostics for Topological Order
k k
1
1 Im kM
n n
n x yBZ
u uC dk k
In an M-band Chern insulator
4 ImTr ;C Px PyA
P
after some algebra
Ground state projector
C=0 for finite system with open boundary conditions (trivial topology)
(r, r ') r" (r, r") " (r", r') , 0)
P
X d P x P X Y is a short ranged operator (insulator is nearsighted)
(Note: projected translations:
4 ImTr 2 r ' (r, r ') (r ', r) (r, r ') (r ', r)cc
Px Py i d X Y Y XA
"Chern number density": replace global trace by a local trace
C
Bianco and Resta (2011)
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Local Diagnostics for Topological OrderLocal Diagnostics for Topological Order
Bianco and Resta (2011)
r (r) 0 (r) 0d with C C Chern density is a local markerfor topological order
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Local Diagnostics for Topological OrderLocal Diagnostics for Topological Order
Bianco and Resta (2011)
r (r) 0 (r) 0d with C C Chern density is a local markerfor topological order
( )x C ( )x C
( )x n ( )x n
Chern ribbon Normal/Chern interface
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Some References:
Wannier Functions (Review): N. Marzari, A. Mostofi, J.R.Yates, I. Souza and D. Vanderbilt arXiv:1112.5411 (to appear in Rev. Mod. Phys.)
Wannier Representations of Z2 insulators: A.A. Soluyanov and D.H. Vanderbilt, Phys. Rev. B 83, 035108 (2011)
Smooth Gauge for Z2 insulators. A.A. Soluyanov and D.H. Vanderbilt, Phys. Rev B 85, 115415 (2012)
Real space marker for Chern insulators: R. Bianco and R. Resta, Phys. Rev. B 84, 241106 (2011)