Wannier functions, Modern theory of polarization 1 / 51

Transcription

1 Wannier functions, Modern theory of polarization 1 / 51

2 Literature: 1 R. D. King-Smith and David Vanderbilt, Phys. Rev. B 47, Nicola Marzari and David Vanderbilt, Phys. Rev. B 56, Raffaele Resta, Phys. Rev. Lett. 80, 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 (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, (2011). 7 J. K. Asbóth, L. Oroszlány, and A. Pályi, arxiv: (A Short Course on Topological Insulators, Lecture Notes in Physics, Vol. 919, 2016, Springer Verlag). 2 / 51

3 Further reading: 1 E. Blount, Formalisms of band theory, Solid State Physics, 13, page 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/ , J. Zak, Phys. Rev. Lett. 62, 2747 (1989). 3 / 51

4 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

5 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. 5 / 51

6 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

7 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 6 / 51

8 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

9 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

10 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 8 / 51

11 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

12 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

13 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 Reverse transformation: Φ n (r,r j ) = 1 N R j Φ n (r,r j )e ik Rj k BZ e ik R j Ψ n,k (r) 9 / 51

14 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

15 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

16 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 (See, e.g., Ref[5]). Φ n (r R j,0) = W n (r R j ) are Wannier functions 10 / 51

17 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

18 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 12 / 51

19 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

20 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 For explicit calculation of the Wannier functions in the SSH model, see Phys. Rev. B 26, 4269 (2016). 12 / 51

21 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

22 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) 13 / 51

23 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

24 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

25 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

26 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

27 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

28 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

29 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

30 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 can be expressed in terms of Berry-phase (Zak s phase) 17 / 51

31 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 can be expressed in terms of Berry-phase (Zak s phase) Wannier centers are equally spaced 17 / 51

32 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 18 / 51

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

34 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

35 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! 20 / 51

36 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

37 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) A more general U(J) gauge freedom: u nk (r) p U (k) pn u pk (r) 20 / 51

38 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) 21 / 51

39 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

40 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) This approach can be used to obtain MLWFs from DFT calculations in plane-wave basis. 21 / 51

41 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

42 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

43 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 minimalization of Ω means minimalization of Ω 2 22 / 51

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

45 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

46 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. 23 / 51

47 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

48 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 Ω 2 vanishes, Ω 1 gauge invariant 23 / 51

49 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 Ω 2 vanishes, Ω 1 gauge invariant 0m is a MLWF 23 / 51

50 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 Ω 2 vanishes, Ω 1 gauge invariant 0m is a MLWF Argument does not work in 3D, because P ˆx P, Pŷ P, PẑP do not commmute 23 / 51

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

52 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 25 / 51

53 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 (local preprint) 26 / 51

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

55 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) 27 / 51

56 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

57 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]). While the crystal is strained, a transient electrical current flows through the sample 27 / 51

58 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

59 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

60 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

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

62 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

63 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

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

65 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

66 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 30 / 51

67 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 30 / 51

68 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

69 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

70 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

71 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

72 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

73 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 P = ie (2π) N n=1 BZ dk [ u (λ) n,k ku (λ) n,k dλ ] k u(λ) n,k λu (λ) n,k 32 / 51

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

75 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

76 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) 33 / 51

77 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

78 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. One finds: where P (λ) = ie (2π) P = P (λ=1) P (λ=0) N n=1 BZ dk u (λ) n,k ku (λ) n,k 33 / 51

79 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

80 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 34 / 51

81 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

82 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

83 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 35 / 51

84 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

85 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 In 3D: P n = P n + er V cell, R a lattice vector. 35 / 51

86 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 36 / 51

87 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

88 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) 37 / 51

89 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

90 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 Polarization sum of the Wannier centers of the occupied bands (for one given R m ) 37 / 51

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

92 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

93 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

94 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

95 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 / 51

96 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 40 / 51

97 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

98 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 41 / 51

99 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

100 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 u n,k u n,k+dk 1+ u n,k k u n,k dk 41 / 51

101 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 u n,k u n,k+dk 1+ u n,k k u n,k dk ln[ u n,k u n,k+dk ] = ln[1+ u n,k k u n,k dk] u n,k k u n,k dk 41 / 51

102 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

103 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

104 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 42 / 51

105 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

106 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 Can be considered a 1D Wilson loop for a single non-degenerate band 42 / 51

107 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 43 / 51

108 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. 44 / 51

109 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 44 / 51

110 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

111 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. We want to work in the Hilbert space spanned by u n,kx (x). 44 / 51

112 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. We want to work in the Hilbert space spanned by u n,kx (x). We need to find a periodic form of PˆxP 44 / 51

113 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 45 / 51

114 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, ) 45 / 51

115 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, ) Note: in general ˆXP is not a Hermitian operator, only for L 45 / 51

116 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 46 / 51

117 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

118 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

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

120 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

121 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 ˆX P = k,k Ψ k Ψ k ˆX Ψ k Ψ k = k u(k +δ k ) u(k) Ψ k+δk Ψ k 48 / 51

122 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

123 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 = u(k 1 ) u(k M ) u(k 2 ) u(k 1 ) u(k 3 ) u(k 2 ) u(k 4 ) u(k 3 ) X P is a M M matrix 49 / 51

124 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 = u(k 1 ) u(k M ) u(k 2 ) u(k 1 ) u(k 3 ) u(k 2 ) u(k 4 ) u(k 3 ) 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

125 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 = u(k 1 ) u(k M ) u(k 2 ) u(k 1 ) u(k 3 ) u(k 2 ) u(k 4 ) u(k 3 ) 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 Note, we used that u(k 2 ) u(k 1 ), u(k 3 ) u(k 2 ) etc are complex numbers they commmute 49 / 51

126 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

127 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 50 / 51

128 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] 50 / 51

129 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 50 / 51

130 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

131 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

132 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 51 / 51

133 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