Wannier functions, Modern theory of polarization 1 / 51

Literature: 1 R. D. King-Smith and David Vanderbilt, Phys. Rev. B 47, 12847. 2 Nicola Marzari and David Vanderbilt, Phys. Rev. B 56, 12847. 3 Raffaele Resta, Phys. Rev. Lett. 80, 1800. 4 Raffaele Resta and David Vanderbilt, Theory of Polarization: A Modern Approach, in Physics of Ferroelectrics: a Modern Perspective, ed. by K.M. Rabe, C.H. Ahn, and J.-M. Triscone (Springer-Verlag, 2007, Berlin), pp. 31-68. (local preprint). 5 Jenő Sólyom. Fundamentals of the Physics of Solids: Volume II, Electronic properties (Springer Verlag, Berlin;Heiderberg, 2009). 6 Rui Yu, Xiao Liang Qi, Andrei Bernevig, Zhong Fang, and Xi Dai, Phys. Rev. B 84, 075119 (2011). 7 J. K. Asbóth, L. Oroszlány, and A. Pályi, arxiv:1509.02295 (A Short Course on Topological Insulators, Lecture Notes in Physics, Vol. 919, 2016, Springer Verlag). 2 / 51

Further reading: 1 E. Blount, Formalisms of band theory, Solid State Physics, 13, page 305-73. New York, Academic Press, (1962). 2 S. Kivelson, Phys. Rev. B 26, 4269 (1982). 3 Raffaele Resta, What makes an insulator different from a metal?, arxiv preprint cond-mat/0003014, 2000. 4 J. Zak, Phys. Rev. Lett. 62, 2747 (1989). 3 / 51

Overview Reminder: Bloch functions, periodic boundary condition Wannier functions: introduction Maximally localized Wannier functions Modern theory of polarization Polarization and Wannier centers Calculation of Berry phase in discrete k space Position operator and periodic boundary conditions Calculation of Wannier centers in 1D systems 4 / 51

Bloch functions, periodic boundary condition We want to describe the bulk properties of crystalline materials. In non-interacting approximation: N e ( ˆp 2 ) Ĥ = i +U(r i ) 2m e i=1 U(r) = U(r+R n ) is lattice-periodic potential, R n lattice vector. Hamiltonian is a sum of single-particle Hamiltonians, the solution of this N e electron problem can be obtained using the solutions of the Schrödinger equation ] [ 2 2 +U(r) Ψ i (r) = E i Ψ i (r) 2m e 5 / 51

Bloch functions, periodic boundary condition In order to capture the translational invariance of the bulk, we use use periodic boundary condition the solutions satisfy (Bloch theorem) Ψ k (r+r m ) = e ik Rm Ψ k (r) wavevector k can take on discrete values in the Brillouin zone Equivalent formulation: Ψ k (r) = e ik r u k (r) where u k (r+r m ) = u k (r) lattice periodic and [ ] (ˆp + k) 2 +U(r) u n,k (r) = E n,k u n,k (r) 2m e 6 / 51

Bloch functions, periodic boundary condition For equivalent wavenumber vectors k = k+g (G lattice vector of the reciprocal lattice) Ψ n,k = e ik r u nk (r) = e i(k G) r u nk (r) = e ik r u nk (r) where u nk (r) = u nk+g (r) = e ig r u nk (r). One can also show that E nk+g = E nk. 7 / 51

Wannier functions Since Ψ n,k+g (r) = e i(k+g) r u n,k+g = e ik r u n,k = Ψ n,k (r) for fixed r the Bloch function Ψ n,k (r) is periodic in the reciprocal space. 9 / 51

Wannier functions Since Ψ n,k+g (r) = e i(k+g) r u n,k+g = e ik r u n,k = Ψ n,k (r) for fixed r the Bloch function Ψ n,k (r) is periodic in the reciprocal space. it can be expanded into a Fourier series: Ψ n,k (r) = 1 N R j : Lattice vectors in real space R j Φ n (r,r j )e ik Rj 9 / 51

Wannier functions Φ n (r,r j ) is a function of r R j : Φ n (r+r n,r j +R n ) = = = 1 N k Bz 1 N k BZ 1 N k BZ = Φ n (r,r j ) e ik (R j+r n) Ψ nk (r+r n ) e ik (R j+r n) e ik Rn Ψ nk (r) e ik R j Ψ nk (r) 10 / 51

Wannier functions Φ n (r,r j ) is a function of r R j : Φ n (r+r n,r j +R n ) = = = 1 N k Bz 1 N k BZ 1 N k BZ = Φ n (r,r j ) e ik (R j+r n) Ψ nk (r+r n ) e ik (R j+r n) e ik Rn Ψ nk (r) e ik R j Ψ nk (r) For R n = R j : Φ n (r,r j ) = Φ n (r R j,0) it depends only on r R j 10 / 51

Wannier functions Orthonormality and completeness: Bloch functions: Ψ n,k (r)ψ n,k (r)dr = δ n,n δ k,k Ψ n,k (r)ψ n,k(r ) = δ(r r ) n,k Wannnier functions W n(r R j )W n (r R j )dr = δ Rj,R j δ n,n (See, e.g., Ref[5]). n,r j W n (r R j)w n (r R j ) = δ(r r ) 11 / 51

Wannier centers Define the Wannier center of W n (r R j ) as r (j) n = W n (r R j ) r W n (r R j ) In general, a non-trivial question if the the above expectation value is finite in extended systems localization properties of Wannier functions 12 / 51

Wannier centers in the Rice-Mele model Consider the Rice-Mele model (N site long, periodic boundary condition) Bloch functions: Ψ n (k) = k u n (k) 13 / 51

Wannier centers in the Rice-Mele model Consider the Rice-Mele model (N site long, periodic boundary condition) Bloch functions: Ψ n (k) = k u n (k) n = 1,2 (conduction and valence band), u n (k): eigenstates of the momentum-space Hamiltonian H(k) k = 1 N e ikxl, k x { 2π N N,22π N,...,N2π N } l=1 13 / 51

Wannier centers in the Rice-Mele model Consider the Rice-Mele model (N site long, periodic boundary condition) Bloch functions: Ψ n (k) = k u n (k) n = 1,2 (conduction and valence band), u n (k): eigenstates of the momentum-space Hamiltonian H(k) Position operator: k = 1 N e ikxl, k x { 2π N N,22π N,...,N2π N } ˆx = l=1 N m( m,a m,a + m,b m,b ) m=1 m: unit cell index; A, B, site index in a unit cell 13 / 51

Wannier centers in the Rice-Mele model ˆxW n (j) = ˆx 1 N k x e ijkx Ψ n (k x ) 1 = e ijkxˆx 1 N e ilkx l u n (k x ) N k x N l = 1 me imkx m u n (k x ) N k x e ijkx m 14 / 51

Wannier centers in the Rice-Mele model For N k x continous, use partial integration: ˆxW n (j) = 1 2π 2π 0 = 1 2π ( i) + 1 2π + i 2π 2π 0 2π 0 dk x e ijkx me imkx m u n (k x ) 2π 0 m dk x e i(m j)kx m u n (k x ) k x m dk x j e ijkx m u n (k x ) dk x m m m k x u n (k x ) 15 / 51

Wannier centers in the Rice-Mele model This can be simplified: ˆxW n (j) = ( i 2π + j 2π + i 2π e i(m j)kx m u n (k x ) ) m 2π 0 2π 0 dk x e ijkx m u n (k x ) m dk x m kx u n (k x ) m 2π 0 16 / 51

Wannier centers in the Rice-Mele model ˆxW n (j) = ( i 2π + j 2π + i 2π e i(m j)kx m u n (k x ) ) m 2π 0 2π 0 dk x e ijkx m u n (k x ) = j W n (j) m dk x m kx u n (k x ) m 2π 0 0 (periodic function) 17 / 51

Wannier centers in the Rice-Mele model ˆxW n (j) = One finds: ( i 2π + j 2π + i 2π e i(m j)kx m u n (k x ) ) m 2π 0 2π 0 dk x e ijkx m u n (k x ) = j W n (j) m dk x m kx u n (k x ) m 2π 0 0 (periodic function) W n (j) ˆx W n (j) = i 2π dk x u n (k x ) kx u n (k x ) +j 2π 0 17 / 51

Maximally localized Warnier functions Figure: Phys Rev B 56, 12847 (1997). 19 / 51

Maximally localized Warnier functions Gauge freedom in choosing the Bloch orbitals: Ψ nk (r) e iφn(k) Ψ nk (r) [u nk (r) e iφn(k) u nk (r)] describes the same electron density 20 / 51

Maximally localized Warnier functions Gauge freedom in choosing the Bloch orbitals: Ψ nk (r) e iφn(k) Ψ nk (r) [u nk (r) e iφn(k) u nk (r)] describes the same electron density Wannier functions are not unique! If there are band degeneracies for some k (symmetries, e.g., time reversal): not sufficient to consider isolated bands consider a set of J bands (composite bands) 20 / 51

Maximally localized Warnier functions (MLWF) Define for the Wannier functions at the origin R j = 0 the following: 1 Wannier center r n = W n (r) r W n (r) 2 second moment r 2 n = W n (r) r 2 W n (r) A measure of the spread (delocalization) of the Wannier functions: Ω = n [ r 2 n r 2 n] One can then try to minimize Ω with respect to the unitary transformations U (k) pn Maximally localized Wannier functions (MLWF) 21 / 51

Maximally localized Warnier functions (MLWF) One can decompose Ω = Ω 1 +Ω 2, where Ω 1 = r 2 n Rm r 0n 2 n R,m Ω 2 = n Rm 0n Rm r 0n 2 22 / 51

Maximally localized Warnier functions (MLWF) One can decompose Ω = Ω 1 +Ω 2, where Ω 1 = r 2 n Rm r 0n 2 n R,m One can show that Ω 2 = n Rm 0n Rm r 0n 2 Ω 1 is gauge invariant, i.e., does not depend on U (k) pn transformation Ω 2 is positive definite 22 / 51

Special case: MLWF in 1D Notation: W n (x R x ) = R x,n 23 / 51

Special case: MLWF in 1D Notation: W n (x R x ) = R x,n Consider a group of bands, e.g., the valence band(s): Projectors: P = R x,n R x,n = Ψ n,kx Ψ n,kx R x,n n,k x 23 / 51

Special case: MLWF in 1D Notation: W n (x R x ) = R x,n Consider a group of bands, e.g., the valence band(s): Projectors: P = R x,n R x,n = Ψ n,kx Ψ n,kx R x,n n,k x In 1D the eigenfunctions of the projected position operator P ˆx P are MLWF. Let 0m be an eigenfunction of P ˆx P with eigenvalue x 0m. R n ˆx 0m = R n P ˆx P 0m = x 0m δ R,0 δ m,n 23 / 51

Summary I bulk properties of materials periodic boundary conditions Bloch functions alternative description: Wannier functions Wannier-centers (expectation value of the position operator): in Rice-Mele model Berry-phase Maximally localized Wannier functions: in 1D eigenfunctions of the projected position operator P ˆxP 24 / 51

Modern theory of polarization Figure: Theory of Polarization: A Modern Approach, in Physics of Ferroelectrics: a Modern Perspective, ed. by K.M. Rabe, C.H. Ahn, and J.-M. Triscone (Springer-Verlag, 2007, Berlin), pp. 31-68. (local preprint) 26 / 51

Modern theory of polarization Macroscopic polarization P: fundamental concept in the phenomenological description of dielectrics 27 / 51

Modern theory of polarization Macroscopic polarization P: fundamental concept in the phenomenological description of dielectrics Some materials possess polarization without external field (ferroelectricity) or become polarized upon applying strain (pieozoelectricity) How to measure the polarization? Figure: Pieozoelectricity: surface or bulk effect? (Figure from Ref[4]). 27 / 51

Modern theory of polarization Fundamental relation: the change in polarization P is accompanied with a transient current j(t) flowing through the sample: dp dt = j(t) 28 / 51

Modern theory of polarization Fundamental relation: the change in polarization P is accompanied with a transient current j(t) flowing through the sample: Change in polarization: dp dt = j(t) P = P( t) P(0) = If the change is slow enough adiabatic limit t 0 j(t) 28 / 51

Modern theory of polarization Fundamental relation: the change in polarization P is accompanied with a transient current j(t) flowing through the sample: Change in polarization: dp dt = j(t) P = P( t) P(0) = t If the change is slow enough adiabatic limit adiabatic perturbation theory to calculate j(t) 0 j(t) 28 / 51

Formal description of the theory: polarization P Introduce the parameter λ [0, 1]: dimensionless adiabatic time 29 / 51

Formal description of the theory: polarization P Introduce the parameter λ [0, 1]: dimensionless adiabatic time P = λ = 0:initial state; λ = 1:final state 1 0 dλ dp dλ 29 / 51

Formal description of the theory: polarization P Introduce the parameter λ [0, 1]: dimensionless adiabatic time P = λ = 0:initial state; λ = 1:final state Assumptions: 1 0 dλ dp dλ system remains insulating for all values of λ system bulk retains crystalline periodicity for all λ 29 / 51

Formal description of the theory: current j (λ) 30 / 51

Formal description of the theory: current j (λ) Because of crystalline periodicity the solutions of the Schrödinger equation are Bloch-functions: Ψ (λ) k (r) = eik r u λ k (r) 30 / 51

Formal description of the theory: current j (λ) Because of crystalline periodicity the solutions of the Schrödinger equation are Bloch-functions: Ψ (λ) k (r) = eik r uk λ(r) Schrödinger equation for the lattice periodic part u (λ) k (r) [ ] (ˆp + k) 2 +U (λ) (r) u (λ) 2m n,k (r) = E(λ) n,k u(λ) n,k (r) e Adiabatic change: the current j (λ) can be calculated using adiabatic perturbation theory For a adiabatically changing time periodic 1D lattice [H(k,t) = H(k,t +T)] this is done in Chapter 5 of the Lecture Notes (Ref[7]) The same steps can be done here: t λ, λ does not need be periodic 30 / 51

Formal description of the theory: current j (λ) Current from a single filled band n: dp n (λ) dλ = j(λ) n = ie (2π) 3 dk[ k u (λ) n,k λu (λ) n,k λu (λ) n,k ku (λ) n,k ] BZ 31 / 51

Formal description of the theory: current j (λ) Current from a single filled band n: dp n (λ) dλ = j(λ) n = ie (2π) 3 dk[ k u (λ) n,k λu (λ) n,k λu (λ) n,k ku (λ) n,k ] BZ Total change in polarization ( number of pumped particles): P = ie (2π) 3 N n=1 BZ N: number of filled bands dk λ 0 dλ[ k u (λ) n,k λu (λ) n,k λu (λ) n,k ku (λ) n,k ] 31 / 51

Polarization as Berry phase For simplicity, consider a 1D system: P = ie (2π) N n=1 BZ dk λ 0 dλ[ k u (λ) n,k λu (λ) n,k λu (λ) n,k ku (λ) n,k ] 32 / 51

Polarization as Berry phase For simplicity, consider a 1D system: P = ie (2π) N n=1 BZ dk λ 0 dλ[ k u (λ) n,k λu (λ) n,k λu (λ) n,k ku (λ) n,k ] Make use of the following to re-write the integral for P partial integration with respect to λ k u (λ) n,k λu (λ) n,k = ku (λ) n,k λu (λ) n,k + u(λ) n,k k λ u (λ) n,k 32 / 51

Polarization as Berry phase Note, that 1 0 dλ dk BZ k u(λ) n,k λu (λ) n,k = 0 33 / 51

Polarization as Berry phase Note, that 1 dλ dk 0 BZ k u(λ) n,k λu (λ) n,k = 0 because the Bloch functions are periodic in reciprocal space: Ψ (λ) k,n (x) = Ψ(λ) k+g,n (x) 33 / 51

Polarization as Berry phase Note, that 1 dλ dk 0 BZ k u(λ) n,k λu (λ) n,k = 0 because the Bloch functions are periodic in reciprocal space: Ψ (λ) k,n (x) = Ψ(λ) k+g,n (x) u(λ) n,k (x) = eigx u (λ) n,k+g (x) u (λ) n,k λu (λ) n,k is periodic in k its integral over the BZ is zero. 33 / 51

Polarization as Berry phase P (λ) = ie (2π) N n=1 BZ dk u (λ) n,k k u (λ) n,k the right hand side is the integral of the Berry-connection over the BZ Berry phase (Zak s phase) 34 / 51

Polarization as Berry phase P (λ) = ie (2π) N n=1 BZ dk u (λ) n,k k u (λ) n,k the right hand side is the integral of the Berry-connection over the BZ Berry phase (Zak s phase) in case the path in λ space takes the crystal from its centrosymmetric reference state (P (λ=0) = 0) to its equilibrium polarized state spontaneous polarization Note: the total electrical polarization of any material has an ionic contribution as well (but this will not be important in the discussion of topological properties) 34 / 51

Polarization as Berry phase: gauge dependence There is a gauge freedom in defining u n,k : ũ n,k = e iβ(k) u n,k gives the same electron density. 35 / 51

Polarization as Berry phase: gauge dependence There is a gauge freedom in defining u n,k : ũ n,k = e iβ(k) u n,k gives the same electron density. β(k) is not arbitrary: Bloch functions obey Ψ n,k+g (x) = Ψ n,k (x) β(2π/a) β(0) = 2πj, j integer the polarization is well defined only modulo 2π: dk ũ (λ) n,k k ũ (λ) n,k = dk u (λ) n,k k u (λ) n,k +2iπj BZ BZ 35 / 51

Polarization and Wannier centers Remember the relation between the Bloch functions and Wannier functions M: number of unit cells Ψ n,k (x) = 1 M M m=1 W n (x R m )e ik Rm 37 / 51

Polarization and Wannier centers Remember the relation between the Bloch functions and Wannier functions M: number of unit cells Ψ n,k (x) = 1 M M m=1 u n,k (x) = 1 M M m=1 W n (x R m )e ik Rm W n (x R m )e ik (R m x) Using the Berry-phase formula, one can express the polarization in terms of the Wannier functions: P (λ) = e N W n (λ) (x) x W n (λ) (x) n=1 37 / 51

Tracking the Wannier centers for cyclic λ Until now, λ is not periodic 38 / 51

Tracking the Wannier centers for cyclic λ Until now, λ is not periodic The formalism remains valid for cyclic changes: H λ=1 = H λ=0 P cyc = 1 0 dλ dp dλ 38 / 51

Tracking the Wannier centers for cyclic λ Until now, λ is not periodic The formalism remains valid for cyclic changes: H λ=1 = H λ=0 P cyc = 1 0 dλ dp dλ The Wannier center must return to their initial location at the end of the cyclic evolution. But this is possible in two different ways: Figure: Fig.10 of Theory of Polarization: A Modern Approach 38 / 51

Tracking the Wannier centers in the Rice-Mele model In the case of Rice-Mele model, λ = t time, the adiabatic charge pumping can be visualized by tracking the Wannier centers: 39 / 51

Tracking the Wannier centers in the Rice-Mele model In the case of Rice-Mele model, λ = t time, the adiabatic charge pumping can be visualized by tracking the Wannier centers: W n (t) (j) ˆx W n (t) (j) = i 2π dk x u n (t) (k x ) kx u n (t) (k x ) +j 2π 0 Figure: Figure 4.5(a) of the Lecture Notes. Time evolution of the Wannnier centers of the bands. Solid line: valence band, dashed line: conduction band. The Chern number is 1. 39 / 51

Polarization as Berry phase: calculation in discrete k-space We only consider the 1D case P n (λ) = e 2π Φ(λ) n ; L length of sample Φ n (λ) = Im BZ dk u (λ) n,k k u (λ) n,k 41 / 51

Polarization as Berry phase: calculation in discrete k-space We only consider the 1D case P n (λ) = e 2π Φ(λ) n ; L length of sample Φ n (λ) = Im BZ dk u (λ) n,k k u (λ) n,k Typically, we need to calculate this on a discrete grid of k-points dk u n,k k u n,k dk k j dk u n,k k u n,k k=kj Note the following: u n,k+dk u n,k + k u n,k dk 41 / 51

Polarization as Berry phase: calculation in discrete k-space dk u n,k k u n,k dk k j ln[ u n,k u n,k+dk ] k=kj 42 / 51

Polarization as Berry phase: calculation in discrete k-space dk u n,k k u n,k dk k j ln[ u n,k u n,k+dk ] k=kj Take k j = 2πj J a, a lattice constant, J, j = 0,...J 1 integer, dk = 2π/(J a) 42 / 51

Polarization as Berry phase: calculation in discrete k-space dk u n,k k u n,k dk k j ln[ u n,k u n,k+dk ] k=kj Take k j = 2πj J a, a lattice constant, J, j = 0,...J 1 integer, dk = 2π/(J a) Φ (λ) n = Im j ln[ u (λ) n,k j u (λ) J 1 n,k j+1 ] = Imln u (λ) n,k j u (λ) n,k j+1 j=0 Note, the product u (λ) n,k 0 u (λ) n,k 1 u (λ) n,k 1 u (λ) n,k 2... does not depend on the phase of u (λ) n,k j s gauge independent 42 / 51

A subtle issue: position operator and periodic boundary conditions In 1D the eigenfunctions of the operator PˆxP, where P is a projector onto a set of bands, yields a localized set of Wannier functions. The eigenvalues of P ˆxP are the Wannier centers. The projectors are most easily calculated using Bloch functions: P = n,k x Ψ n,kx Ψ n,kx = n,k x u n,kx u n,kx The projectors u n,kx u n,kx are periodic in x because u n,kx (x) is periodic. However, the operator ˆx is not periodic. 44 / 51

A subtle issue: position operator and periodic boundary conditions Raffaele Resta [3]: if periodic boundary conditions are used, expectation values that involve the operator ˆx should be calculated using the unitary operator ˆX = e i 2π L ˆx L: imposed periodicity in the system to obtain well-localized Wannier functions while using periodic boundary conditions: find the eigenfunction(s) of ˆX P = P ˆXP P: projects onto the occupied bands (see Phys Rev B 84, 075119) 45 / 51

P ˆXP operator in 1D lattice system For a 1D system consisting of M unit cells: M ˆx = R m m,α m,α m=1 α α band index, R m labels the mth unit cell 47 / 51

P ˆXP operator in 1D lattice system For a 1D system consisting of M unit cells: M ˆx = R m m,α m,α m=1 α α band index, R m labels the mth unit cell M ˆX = e iδ kr m m,α m,α m=1 α δ k = 2π Ma a: lattice constant 47 / 51

P ˆXP operator in 1D lattice system For simplicity, consider a single occupied band and the corresponding Bloch functions Ψ k 48 / 51

P ˆXP operator in 1D lattice system For simplicity, consider a single occupied band and the corresponding Bloch functions Ψ k First, the matrix elements of ˆX: Ψ k ˆX Ψ k = M m =1 e i k R m = 1 M u(k ) u(k) M m u(k ) M m=1 = u(k ) u(k) δ k+δk k,0 M e iδ kr m e i k Rm m u(k) m=1 e i(k+δ k k )R m 48 / 51

P ˆXP operator in 1D lattice system For a system consisting of M unit cells, the k values are discrete and there are M x Bloch-wavefunctions Ψ km : k m = 2π Mam, m = 1...M 49 / 51

P ˆXP operator in 1D lattice system For a system consisting of M unit cells, the k values are discrete and there are M x Bloch-wavefunctions Ψ km : k m = 2π Mam, m = 1...M Taking matrix elements of ˆX P in the basis of Bloch wavefunctions: X P = 0 0 0... u(k 1 ) u(k M ) u(k 2 ) u(k 1 ) 0 0 0 0 u(k 3 ) u(k 2 ) 0 0 0 0 0 u(k 4 ) u(k 3 ) 0 0 0 0 0... 0 X P is a M M matrix One can easily check, that (X P ) M = w1, where w C and 1 is the unit matrix 49 / 51

P ˆXP operator in 1D lattice system (X P ) M = w1 where w = u(k 1 ) u(k M ) u(k 2 ) u(k 1 ) u(k 3 ) u(k 2 )... u(k M ) u(k M 1 ) w is a 1D Wilson loop, gauge independent 50 / 51

P ˆXP operator in 1D lattice system (X P ) M = w1 where w = u(k 1 ) u(k M ) u(k 2 ) u(k 1 ) u(k 3 ) u(k 2 )... u(k M ) u(k M 1 ) w is a 1D Wilson loop, gauge independent Similar to the discrete Berry-phase The eigenvalues λ m of X P are the M roots of w: w = w e iθ θ = Imln[w] λ m = M w e i(2πm+θ)/m, m = 1,...M Note: since ˆX P is not Hermitian, the eigenvectors are not orthogonal in general, only in the M limit (we will actually not need the eigenstates) 50 / 51

Wannier centers and the eigenvalues of P ˆXP Remember from earlier: 1) polarization of a single filled band in a 1D lattice x j Wannier centers, x j = W(j) x W(j), j is unit cell index 51 / 51

Wannier centers and the eigenvalues of P ˆXP Remember from earlier: 1) polarization of a single filled band in a 1D lattice x j Wannier centers, x j = W(j) x W(j), j is unit cell index 2) In discrete k space J 1 x Imln u n,kj u n,kj+1 j=0 j different discrete value of k in the 1D Brillouin zone If there are M unit cells, k is discretized as k m = 2π Ma m The Wannier centers x m can be obtained from λ m as x m = M 2π Imln[λ m] = θ Imln[w] +m = +m 2π 2π 51 / 51