Topological Physics in Band Insulators IV

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1 Topological Physics in Band Insulators IV Gene Mele University of Pennsylvania Wannier representation and band projectors

2 Modern view: Gapped electronic states are 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

3 Introduction: Exercise: consider three problems (of increasing complexity) and their solutions

4 P1. Polarization (dipole density) of a one dimensional classical lattice N P 1 1 d q x q x q L Na a a i i i i i1 i{1}

5 P1. Polarization (dipole density) of a one dimensional classical lattice N Ptot 1 1 d qi xi qi xi q 1 L Na i1 a i{1} a P is only defined modulo a unit of polarization (q in 1D)

6 P2. Polarization (dipole density) of a one dimensional quantum lattice P L 1 a i {1} ez x i i e n( x) dx a

7 P2. Polarization (dipole density) of a one dimensional quantum lattice P L 1 a i {1} ez x i i e n( x) dx a P depends on the choice of unit cell: with discrete jumps when classical cores cross boundary, and continuous variation due to quantum n(x) Is quantum P undefinable as a bulk quantity?

Quantum theory of polarization Consider adiabatic evolution from an unpolarized state P( ( T ) Z Z ) P( 0) J dt a ( T ) P e d d dk 2 0 0 n b T 0, k with Berry curvature, k i un( k) kun( k) kun( k)

8 Quantum theory of polarization Consider adiabatic evolution from an unpolarized state P( ( T ) Z Z ) P( 0) J dt a ( T ) P e d d dk n b T 0, k with Berry curvature, k i un( k) kun( k) kun( k) un( k) Stokes: loop integral of connection on (,k) circuit ie P( ( T ) Z Z ) P( 0) dk u ( k) u ( k) a b n k n 0 2 n 2 Bulk P is defined up to its quantum (e) even for a smoothly distributed n(x). This information is not in n(x) but in (i.e. in (x)) (King-Smith & Vanderbilt, Resta) e

9 Interpretation: Wannier-charge charge-centers centers are discrete Ground state average over n(x) contains the first moments of the charge densities in its Wannier functions. continuous P 1 L a i { 1} ez x i i ( ) 2 e x wn x dx n discrete (WC s locations are functions of Hamiltonian parameters)

P3. Determine the Z 2 index of a generic* time-reversal reversal-invariant band insulator Z 2 index counts the number (mod 2) of band inversions from the filled manifold of a T-invariant insulator.

10 P3. Determine the Z 2 index of a generic* time-reversal reversal-invariant band insulator Z 2 index counts the number (mod 2) of band inversions from the filled manifold of a T-invariant insulator. Calculate the overlap of the cell-periodic Bloch function u(-k) with its time reversed partner at -k w ( k ) u ( k ) u ( k ) mn m n Count the complex zeroes of Pf [w] within one half of Brillouin zone (difficulty: the overlap is k-nonlocal) *time reversal & lattice translations only

11 Some possible solutions: 1. Parity test: for crystal with inversion symmetry N ( 1) a a 2m (parity eigenvalues, 1) a1 m Requires an inversion symmetric space group (nongeneric) 2. Adiabatic continuity. Apply parity test to reference inversion-symmetric structure and check for no gap closure under a slow deformation to P-breaking structure of interest. 3. Gap closure: Compare the level ordering in band insulator computed without 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.

12 ik r H, T 0 n, k ( r ) e un( k; r ) R Wave packet is constructed from a BZ integral d n, k d (2 ) (2 ) ik r h( k ) e He V V ik r w ( r ) dk dk e u ( k; r ) n Wannier functions: definitions And its discrete lattice translates: V ik r R wn ( r R) dk e un ( k; r ) d (2 ) m, R n, R ' m n R R,, ' ik R e w ( ) n, k n r R R n ik r This is a reconstruction of from a lattice sum

13 Wannier functions: nonuniqueness 1. U(1) gauge freedom V ik r wn ( r ) dk e un( k; r ) d (2 ) u k r e u k r i ( k ) n( ; ) n( ; ) superposition from eigenfunctions of disconnected sectors, h(k) exponentially localized wf's require a smooth ( k ) ( k G) 2. U(N) freedom index switching at a band crossing generates power law tails in its single band Wannier functions. removable by specification of a k-differentiable (i.e. composite manifold: u ( k; r ) U ( k ) u ( k; r ) m mn n smooth)

14 Maximally localized wf s Degree of localization is measured by the spread functional 2 r r r n n n nocc Task: Minimize with respect to U(k) to obtain the smoothest possible k-dependence of its quasi-bloch states and the optimum localization of its Wannier representation. Physics contained in a projected subspace Pˆ n, R n, R nk nk k, nocc R, nocc Qˆ Iˆ Pˆ These band projectors are invariant under choice of U(k)

Topological obstruction A smooth periodic gauge is impossible for a band insulator with a nonzero Chern number (i.e. one with a nonzero Hall conductance).

15 Topological obstruction A 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* i 0 when n k n k G d uk kuk 2,, k C 0 in Chern insulator smooth uk kuk u u k k k jumps here vortex *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)

16 Alternative route to WF s: : Band Projection Pˆ n, R n, R nk nk k, nocc R, nocc Choose M (localized) trial functions with N lattice translates Ground state projected image: With symmetric orthogonalization: 1/ 2 m m, n n m, n m n Projected M-band problem P ˆ w S ; S (MM for each k)

17 Comments 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 ) Projection: choose M localized trial functions whose valence band projections are minimally inflated by the projector v nk Note: while projectors of the form are invariant under U(N). the charge "centers" of individual wf's are not (clearly gauge dependent) but the "sum of charge centers" Pˆ n, R n, R M x, n n nk nk k, nocc R, nocc x R, n Xˆ R, n, n are gauge invariant modulo R

18 Special 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 S z -conserving models (e.g. Haldane 2 ) the ground state has disconnected sectors each of which contains a topological obstruction 3. k-space criterion (Pfaffian) for TI requires that we work in a globally smooth gauge. natural vs. smooth gauge

19 Time Reversal Polarization simple model: parametric pump for one dimensional lattice H[ t T ] H[ t] H t H t i K 1 [ ] [ ] ; y T-invariant H T-invariant k ' s

20 Time Reversal Polarization simplest model: parametric pump for one dimensional lattice 1 1 P ( t) P (0) dt dk Ft, k dk dk 2 A A 2 A t 0 final initial

21 Time Reversal Polarization For termination at T/2 the loop integral can be reduced to a half zone in a (locally smooth) gauge that explicitly respects T-symmetry at t=0,t/2 i II k, m k, m I k, m u e u II i k, m u e u I k, m k, m

22 Time Reversal Polarization I II A( k) i u u A ( k) A ( k) nk k nk P P P P P P I I II II P is the ordinary charge polarization (and vanishes if C 0). P for a half period measures the bulk flow of time reversed partners to the boundaries (integer mod 2)

23 TI Surface States Parametric pump ( k, t) Band topology ( k, k ) 1 2 P integrated over a half zone distinguishes the binding/liberation of its Kramers partners vs ordinary topological

24 Wannier states for Z 2 insulators Impossible for TI using the TRG. If the gauge is smooth over the full zone then Z 2 =0 (ordinary insulator) so that Z 2 =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.

25 WF s by band projection Pˆ n, R n, R nk nk k, nocc R, nocc The projection recipe is algorithmic Choose M (localized) trial functions with N lattice translates Ground state projected image: With symmetric orthogonalization: 1/ 2 m m, n n m, n m n P ˆ w S ; S (MM for each k) The obstruction appears here

Chern insulator: breaks T Fourier spectrum of its overlap matrix

26 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 Ref: Thonhauser and Vanderbilt (2006) Chern insulator

27 Example 2: Spin-full graphene Ordinary insulator: sublattice potential breaks P Z 2 odd insulator: spin-orbit term gaps the K point Trial states: Kramers pair A, z, A, z

28 Example 2: Spin-full graphene Ordinary insulator: sublattice potential breaks P Z 2 odd insulator: spin-orbit term gaps the K point Trial states: Kramers pair A, z, A, z Trial states: TR-polarized pair A, x, B, x Soluyanov and Vanderbilt (2011) Breaking Kramers symmetry in real space avoids the the Z 2 obstruction.

29 Schematically Ordinary insulators and topological insulators are both nearsighted gapped phases with Wannier representations as Kramers partners (ordinary) or as time-reversal polarized partners (topological)

30 Local Diagnostics for Topological Order In an M-band Chern insulator M 1 C Im d k u k n k u k n k n1 BZ x y after some algebra 4 C ImTr Px Py ; A P Ground state projector C=0 for finite system with open boundary conditions (trivial topology) P is a short ranged operator (insulator is nearsighted) X (r,r') d r" P(r,r") x" P(r",r') (Note: projected translations: X, Y 0) c "Chern number density": replace global trace by a local trace 4 C ImTr 2 r ' (r,r ') (r ',r) (r,r ') (r ',r) c Px Py i d X Y Y X A Bianco and Resta (2011)

31 Local Diagnostics for Topological Order d r C(r) 0 with C(r) 0 Chern density is a local marker for topological order Bianco and Resta (2011)

32 Local Diagnostics for Topological Order d r C(r) 0 with C(r) 0 Chern density is a local marker for topological order C ( x) C( x) n ( x) n( x) Chern ribbon Normal/Chern interface Bianco and Resta (2011)

33 Some References: Wannier Functions (Review): N. Marzari, A. Mostofi, J.R.Yates, I. Souza and D. Vanderbilt arxiv: (to appear in Rev. Mod. Phys.) Wannier Representations of Z2 insulators: A.A. Soluyanov and D.H. Vanderbilt, Phys. Rev. B 83, (2011) Smooth Gauge for Z2 insulators. A.A. Soluyanov and D.H. Vanderbilt, Phys. Rev B 85, (2012) Real space marker for Chern insulators: R. Bianco and R. Resta, Phys. Rev. B 84, (2011)

First-Principles Calculation of Topological Invariants (Wannier Functions Approach) Alexey A. Soluyanov

First-Principles Calculation of Topological Invariants (Wannier Functions Approach) Alexey A. Soluyanov ES'12, WFU, June 8, 212 The present work was done in collaboration with David Vanderbilt Outline: