Topological Physics in Band Insulators IV...Vanderbilt, Phys. Rev B 85, 115415 (2012) Real space...
Transcript of Topological Physics in Band Insulators IV...Vanderbilt, Phys. Rev B 85, 115415 (2012) Real space...
Gene MeleUniversity of Pennsylvania
Wannier representation and band projectors
Topological Physics inTopological Physics inBand Insulators IVBand Insulators IV
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
Introduction:Introduction:
Exercise: consider three problems (of increasing complexity) and their solutions
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
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)
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
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?
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)
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)
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)
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.
,, 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
( )( ; ) ( ; )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
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
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)
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
; |
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
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
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
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
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
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)
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
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.
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
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)
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
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.
SchematicallySchematicallyOrdinary insulators and topological insulators are both nearsightedgapped phases with Wannier representations as Kramerspartners (ordinary) or as time-reversal polarized partners (topological)
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)
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
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
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)