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)