Maximally localized Wannier functions for periodic Schrödinger operators: the MV functional and the geometry of the Bloch bundle

Transcription

1 Maximally localized Wannier functions for periodic Schrödinger operators: the MV functional and the geometry of the Bloch bundle Gianluca Panati Università di Roma La Sapienza 4ème Rencontre du GDR Dynamique Quantique IMT, Toulouse, France, du 8 au 10 Février 2012

2 Bloch orbitals The electronic ground state of a periodic system is usually described in terms of Bloch orbitals, i. e. simultaneous generalized eigenfunctions of the periodic 1-particle Hamiltonian H per = + V Γ where V Γ (x + γ) = V Γ (x), for all γ Γ = Span Z (e 1,..., e d ) = Z d, and of the lattice translations {T γ } γ Γ (T γ ψ) (x) = ψ(x γ). While convenient for many purposes, these orbitals (= generalized eigenfunctions) have the disadvantage that are not localized in position space.

3 ... and Wannier orbitals An alternative representation in terms of localized orbitals has been introduced by Gregory Wannier in The main advantage of this approach is that Wannier functions are localized in position space. Thus (i) they provide an intuitive (visual) insight into the structure of chemical bonds in crystals

4 Amplitude isosurface contours for maximally-localized Wannier functions in Si (left panel) and GaAs (right panel). Red and blue contours are for isosurfaces of identical absolute value but opposite signs; Si and As atoms are in green, Ga in cyan. Notice that breaking of inversion symmetry in GaAs polarizes the WFs towards the more electronegative As anion. (Courtesy of N. Marzari, I. Souza and D. Vanderbilt)

5 ... and Wannier orbitals An alternative representation in terms of localized orbitals has been introduced by Gregory Wannier in The main advantage of this approach is that Wannier functions are localized in position space. Thus (i) they provide an intuitive (visual) insight into the structure of chemical bonds in crystals

6 ... and Wannier orbitals An alternative representation in terms of localized orbitals has been introduced by Gregory Wannier in The main advantage of this approach is that Wannier functions are localized in position space. Thus (i) they provide an intuitive (visual) insight into the structure of chemical bonds in crystals (ii) in computational physics, localized Wannier functions are crucial to develop numerical methods whose cost scales linearly with the size of the conning box [Gödecker] (iii) they are crucial in the theory of polarization of crystalline solids [King- Smith & Vanderbilt, Resta] and of orbital magnetization [Ceresoli & Resta, Thonhauser & Vanderbilt]. All these advantages rely on the assumption that Wannier functions are indeed exponentially localized. Is this always the case? Is there an alghoritm to obtain them?

7 For constant coecients dierential operators one introduces the Fourier transform, which yields the duality Position space Dirac's deltas {x δ a (x)} a R d Momentum space Plane waves {k e ik a } a R d As far as dierential operators with periodic coecients are concerned Position space Wannier functions {x w n,γ (x)} n N,γ Γ Momentum space Bloch waves {k ψ n (k, )} n N

8 I Periodic systems and Bloch-Floquet transform

9 The Bloch-Floquet transform Conguration space Momentum space Lattice Γ Γ Fundamental domain Y B T d Y = Rd /Γ T d = ˆR d /Γ Hilbert space H f = L 2 (Y )

10 The Bloch-Floquet transform Assumption on V Γ : we make the following Kato-type assumption on the Γ-periodic potential: V Γ L 2 loc(r d ) for d 3, V Γ L p loc (Rd ) with p > d/2 for d 4, to assure that H = + V Γ is self-adjoint in L 2 (R d ) on the domain W 2,2 (R d ).

11 The Bloch-Floquet transform The modied Bloch-Floquet transform is dened as 1 (Ũψ)(k, y) = B 1/2 Ũ : L 2 (R d ) L 2 (B) L 2 (T d Y ) }{{} H f e i(y+γ) k ψ(y + γ), k, y R d. γ Γ

12 The Bloch-Floquet transform Notice that (Ũψ)(k + λ, y) = e iy λ (Ũψ)(k, y) λ Γ so that the transformed function is determined by the values assumed on k B.

13 In modied BF representation H = + V Γ becomes a bered operator Ũ H Ũ 1 = B H per (k) dk in L 2 (B, H f ) = L 2 (B) H f, H per (k) = 1 2 ( i y+k) 2 +V Γ (y) acting on D = W 2,2 (T d Y ) L 2 (T d Y ) = H f. The band structure: σ( ) Solution of the eigenvalue problem: H per (k) H per (k)ϕ n (k, y) = E n (k)ϕ n (k, y) E 5 E 4 E 3 Eigenvalue: Eigenvector: Eigenprojector: E n (k) ϕ n (k, ) H f = L 2 (T d Y, dy) P n (k) = ϕ n (k) ϕ n (k) B E 2 E 1 k Total projector: P n = {P n (k)} k B Bloch function: ϕ n L 2 (B, H f )

14 Denition. A (normalized) Bloch function corresponding to the nth Bloch band is any ϕ n L 2 (B, H f ) such that H per (k)ϕ n (k, y) = E n (k)ϕ n (k, y) ϕ n (k, ) Hf = 1 for a.e. k B ϕ n (k + λ, y) = e iy λ ϕ n (k, y) λ Γ, k B. The last condition is called equivariance or pseudoperiodicity. It will become crucial when requiring some regularity on the map k ϕ n (k, ).

15 II Wannier functions: the isolated band case

16 Wannier functions and gauge freedom Let ϕ n be a Bloch function corresponding to an isolated Bloch band E n. Notice that the choice of ϕ n is not unique, since the function ϕ n (k, y) = e iϑ(k) ϕ n (k, y) is also an eigenfunction of H per (k) corresponding to E n (k) (Bloch gauge). Denition. The Wannier function w n L 2 (R d ) corresponding to a Bloch function ϕ n is the preimage of ϕ n by the Bloch-Floquet transform, i. e. w n (x) := (Ũ 1 ϕ n ) (x) = 1 B 1/2 B e ik x ϕ n (k, x)dk, x R d. Note: it is misleading to talk about the Wannier function for the nth band.

17 Some elementary properties of the Wannier functions: (i) the translated Wannier functions are written as w n,γ (x) w n (x γ) = 1 B 1/2 B e ik γ ϕ n (k, x) dk, γ Γ. (ii) if the norm ϕ n (k, ) L 2 (Y ) is k-independent, then the functions {w n,γ } γ Γ are mutually orthogonal in L 2 (R d ). (iii) under this condition the family {w n,γ } γ Γ is a complete orthonormal basis of Ran P n. (iv) if I n := Ran E n R is isolated from the rest of the spectrum of H, then Ran P n is the spectral projector of H corresponding to the interval I n R. Globally, the family of all Wannier functions {w n,γ } n N,γ Γ is a complete orthonormal basis of L 2 (R d ).

18 Question (A): to which extent are the Wannier functions localized?

19 Question (A): to which extent are the Wannier functions localized? Localization in position space Smoothness in momentum space

20 Question (A): to which extent are the Wannier functions localized? Localization in position space Decay faster than any polynomial Decay faster than an exponential Smoothness in momentum space Innite dierentiability Analyticity in a complex strip

21 Question (A ): how smooth is the Bloch function? The question is ill-posed, it crucially depends on the choice of the phase. In numerical simulations the phase is random. Therefore one has to readjust phases a posteriori in order to obtain reasonably localized Wannier functions. Question (A ): is it possible to choose the phase (Bloch gauge) of the eigenfunctions ϕ n (k, ) so that the corresponding Wannier function is exponentially localized? It seems very easy... Since the band E n is assumed to be an isolated band one has that k P n (k) is smooth (even analytic in a complex strip Ω α C d ) k ϕ n (k, ) can be chosen locally smooth

22 ... but a topological obstruction might appear!!

23 Answer(A): the answer is positive for an isolated Bloch band. d = 1 W. Kohn (1959) d > 1 d > 1 G. Nenciu (1983), B. Heler & J. Sjöstrand (1989). de Cloiseaux (1964) requiring space-reection symmetry The result depends crucially on the fact that the Hamiltonian H = + V Γ is real, i. e. the system is time-reversal symmetric (TR). For a non TR-symmetric operator, e. g. H B = 1 2 ( i x + A B (x)) 2 + V Γ, counterexamples appear already for d = 2. [Dubrovin, Novikov, Lyskova]. On the other hand, in some cases one may recover the result by exploiting magnetic TR symmetry [De Nittis & Lein].

24 III Wannier functions: the multi-band case

25 Eigenvalue crossings In dimension d = 3 there are generically no isolated Bloch bands. However, in insulators there is a spectral gap. Then it is interesting to consider the family of bands which are below the gap. Let σ (k) R be an interval including at every k the relevant family of bands.

26 The Bloch functions do not have a smooth continuation across the crossing points (except in dimension d = 1). Thus no hope to obtain directly exponentially localized Wannier functions. Vision (de Cloiseaux): let us consider the relevant family of bands as a unity. Let P (k) be the orthogonal projector on the relevant family of bands, i. e. P (k) = ϕ n (k) ϕ n (k). {n:e n (k) σ (k)} Denition. A function k χ(k, ) H f is called a quasi-bloch function [de Cloiseaux 64] if χ L 2 (B, H f ) P (k)χ(k, ) = χ(k, ), χ(k, ) 0 for a.e. k B and it is Γ -equivariant,i. e. χ(k + λ, y) = e iy λ χ(k, y) λ Γ, k B.

27 Let m := dim Ran P (k) < +. Denition. A Bloch frame is a family {χ a } a=1,...,m of quasi-bloch functions such that {χ 1 (k, ),..., χ m (k, )} is an orthonormal basis of Ran P (k) k B. Denition. The composite Wannier functions corresponding to a Bloch frame {χ a } a=1,...,m are dened as w a (x) := ) (Ũ 1 χ a (x), a {1,..., m}. Notice that the family {w a,γ } a=1,...,m;γ Γ is an orthonormal basis of the spectral subspace corresponding to the energy window Ran σ.

28 Composite Wannier functions and Bloch gauge A Bloch frame is xed only up to a k-dependent unitary matrix U U(C m ), i. e. m χ a (k) = U a,b (k)χ b (k) b=1 is still a Bloch frame if {χ a } a=1,...,m is a Bloch frame. Question (B): is there a choice of Bloch gauge which makes the composite Wannier functions exponentially localized? Question (B ): does exist a family of quasi-bloch functions k χ a (k) such that (B 1 ) each map χ a : R d H f has a analytic extension to a strip; (B 2 ) the set {χ 1 (k),..., χ m (k)} is an (orthonormal) basis spanning RanP (k) for every k B. First answer to (B): yes for dimension d = 1 [G. Nenciu, 1983].

29 The geometric viewpoint Vision: the problem is equivalent to a geometric one. We are interested to prove the triviality of a Hermitian vector bundle over the d-dimensional torus T d.

30 The geometric viewpoint Vision: the problem is equivalent to a geometric one. Moreover the obstruction might appear only at the topological level Continuous solution Steenrod Smooth solution Stein Analytic solution

31 Time-reversal symmetry H per ( k) = CH per (k)c First Chern class of the vector bundle is zero d 3 (technical) Existence of continuous and periodic global frame [Stein] Exponentially localized Wannier functions Existence of analytic and periodic global frame

32 Second answer to (B): yes in dimension d 3. G. Panati. Triviality of Bloch and Bloch-Dirac bundles, Annales Henri Poincaré 8, (2007). Ch. Brouder, G. Panati, M. Calandra, Ch. Mourougane and N. Marzari. Exponential localization of Wannier functions in insulators, Phys. Rev. Lett. 98, (2007).

33 IV Maximally localized Wannier functions: the Marzari-Vanderbilt functional

34 The Marzari-Vanderbilt localization functional In the 90's, the long-lasting uncertainty about the existence of exponentially localized composite WFs, and the need of an approach suitable for numerical simulations, forced the solid state physics community to change the perspective: one writes a convenient localization functional and look for its minimizers [Marzari & Vanderbilt 97].

35 The Marzari-Vanderbilt localization functional In the 90's, the long-lasting uncertainty about the existence of exponentially localized composite WFs, and the need of an approach suitable for numerical simulations, forced the solid state physics community to change the perspective: one writes a convenient localization functional and look for its minimizers [Marzari & Vanderbilt 97]. Denition For a single-band normalized Wannier function w L 2 (R d ), F MV (w) := d Var ( X j ; w(x) 2 dx ) = x 2 w(x) 2 dx R d j=1 which is well-dened at least whenever R d x 2 w(x) 2 dx < +. d ( ) 2 x j w(x) 2 dx, R d More generally, for a system of L 2 -normalized composite Wannier functions w = {w 1,..., w m } L 2 (R d ) the Marzari-Vanderbilt localization functional is F MV (w) := m a=1 R d x 2 w a (x) 2 dx m a=1 j=1 j=1 d ( ) 2 x j w a (x) 2 dx. R d

36 Denition A system of maximally localized composite Wannier functions (MLWF) is: a minimizer {w 1,..., w m } of the Marzari-Vanderbilt localization functional in the set W m := (D(H) D(X)) m under the constraint that {ϕ 1,..., ϕ m }, for ϕ a = Ũw a, is a Bloch frame.

37 Denition A system of maximally localized composite Wannier functions (MLWF) is: a minimizer {w 1,..., w m } of the Marzari-Vanderbilt localization functional in the set W m := (D(H) D(X)) m under the constraint that {ϕ 1,..., ϕ m }, for ϕ a = Ũw a, is a Bloch frame. Since [Marzari-Vanderbilt 97] this approach has been extremely successful in computational physics, see the recent review Marzari et al., Maximally localized Wannier functions: theory and applications, submitted to Rev. Mod. Physics, arxiv: There are has been convincing numerical evidence that MLWF are exponentially localized, i. e. there exist β > 0 such that R d e 2β x w a (x) 2 dx < + Beyond numerical results, no mathematical proof... a {1,..., m}.

38 Natural mathematical problems (MV 1 ) (Existence) prove that there exists a system of maximally localized composite Wannier functions; (MV 2 ) (Localization) prove that any maximally localized composite Wannier function is exponentially localized

39 The MV functional in momentum space Since the modied Bloch-Floquet transform is an isometry and it satises (ŨX j g)(k, y) = i k j (Ũ g)(k, y) the MV functional can be rewritten in terms of the Bloch frame ϕ = {ϕ 1,..., ϕ m } as F MV (ϕ) = m a=1 { d j=1 B dk T Y ϕ a (k, y) k j 2 dy ( B dk ϕ a (k, y) i ϕ ) } 2 a (k, y) dy T Y k j. The minimization space W = D(H) D(X) is mapped by the Bloch-Floquet transform into H τ L 2 loc(r d, W 2,2 (T Y )) W 1,2 loc (Rd, L 2 (T Y )) =: W.

40 The MV functional for the unitary gauge It is convenient to use the existence of a real-analytic Bloch frame χ = {χ 1,..., χ m } and rewrite the functional in terms of the unknown change of gauge, i. e. ϕ a (k, ) = b χ b (k, ) U b,a (k) with U b,a (k) = χ b (k) ϕ a (k) Hf Here χ is xed, and the minimization variable is the map U W 1,2 (T d, U(C m )) where T d R d /Γ

41 The MV functional for the unitary gauge For the given reference frame χ, the localization functional in terms of the gauge U reads F MV (U; χ) = + + d j=1 d j=1 m a=1 T d T d [ tr tr ( d j=1 ( U (k) U ) (k) k j k j + m m χ a (k, ) k j ) a=1 [(U(k) U k j (k) U k j (k)u (k) T d 2 H f ] ] A j (k) dk + [ ( )] 2 U U (k) (k) + A j (k)u(k) dk). k j aa Here the matrix coecients A j L 2 (T d ; u(m)) are given by the formula [ Aj (k) ] χ cb = c (k, ) χ b(k, ) χc (k, ) χ b (k, ) k j k j H f H f. dk +

42 The MV functional for the unitary gauge For the given reference frame χ, the localization functional in terms of the gauge U reads F MV (U; χ) = + + d j=1 d j=1 m a=1 T d T d [ tr tr ( d j=1 ( U (k) U ) (k) k j k j + m m χ a (k, ) k j ) a=1 [(U(k) U k j (k) U k j (k)u (k) T d 2 H f ] ] A j (k) dk + [ ( )] 2 U U (k) (k) + A j (k)u(k) dk). k j aa dk + Beautiful!!! The former functional is a perturbation of the Dirichlet energy for maps U : T d U(Cm )! In other words, stationary points are harmonic maps.

43 In summary inf { F MV (w) : {w 1,..., w m } W Ũ w is a Bloch frame } = inf { FMV (U; χ) : U W 1,2 (T d; U(C m )) }. Therefore, problem (MV 1 ) is equivalent to show that the r.h.s. is attained. Analogously, problem (MV 2 ) corresponds to show that any minimizer of F MV ( ; χ) is real-analytic, provided that χ is also real-analytic.

44 Theorem. [Panati & Pisante 11] Let σ be a family of m Bloch bands for the operator + V Γ satisfying the gap condition, and let {P (k)} k R d be the corresponding family of spectral projectors. Assume d 2 and m 1, or d 1 and m = 1, or d = 3 and 1 m 15. Then: (MV 1 ) there exist composite Wannier functions {w 1,..., w m } W which minimize the localization functional under the constraint that the corresponding quasi-bloch functions are an orthonormal basis for Ran P (k) for each k B. (MV 2 ) for any system of maximally localized composite Wannier function w = {w 1,..., w m } there exists β > 0 such that e β x w a is in L 2 (R d ) for every a {1,..., m}, i. e. the composite Wannier function w a is exponentially localized. Conjecturally, we expect that the parameter β appearing in the latter claim does not depend on the minimizer w, and that the claim holds true for any β < α, where α is the width of the analyticity strip for k P (k). We also expect that the result holds true for any m N even for d = 3.

45 Snapshots from the proof The existence of a minimizer is standard, it follows essentially from the direct method of calculus of variations We prove the analyticity of the minimizers of F MV by adapting ideas and methods from the regularity theory for harmonic maps [Chan Wang & Yang][Lin & Wang] The crucial step is to prove that any minimizer of F MV is continuous. In the 2-dimensional case, this fact is a consequence of the hidden structure of the nonlinear terms in the Euler Lagrange equation for the F MV functional. In the 3-dimensional case, the continuity follows instead from the deeper fact that minimizers at smaller and smaller scales look like minimizing harmonic maps from T d to U(Cm ). We are able to prove that, for m 15, the latter are actually real-analytic, by showing constancy of the tangent maps as in [Schoen & Uhlenbeck 84]. As a consequence, we obtain the continuity of the minimizers of F MV.

46 The Euler-Lagrange equations Let ϕ C (T d ; M m(c)) and for ε 0 xed let U(k) + εϕ(k) be a free variation of U in the direction ϕ. In a suciently small tubular neighborhood O of U(m) in M m (C) there is a well dened nearest point projection map Π : O variations are U ε (k) := Π(U(k)+εϕ(k)) = U(k) ( I + ε 1 2 U(m), so the projected [ U 1 (k)ϕ(k) (U 1 (k)ϕ(k)) ]) +o(ε). Lemma. A map U W 1,2 (T d ; U(m)) satises d dε F MV (U ε ; χ) ε=0 = 0 if and only if U is a weak solution of the Euler-Lagrange equation d U U + U 1 U + k j=1 j k j d [ ( ) U + + A j U k j j=1 d [ U U 1 A j U A ] j U U A j k j k j k j ( )] U G j + UG j U 1 + A j U = 0. k j j=1 +

47 Here the constant (purely imaginary) diagonal matrices {G j } M m (C) are dened as [ ] U G j = diag U (k) (k) + A j (k)u(k) dk. k j T d

48 Continuity in the 2-dimensional case Crucial lemma. [P & Pisante] borrowing ideas from [Lin & Wang] Let d 2, m 2 and let U W 1,2 (T d ; U(m)). Assume B j L 2 (T d ; u(m)) for j {1,..., d} and div B = 0 in D (T d ). If U is a weak solution to U = d j=1 U k j Bj + f for some p > 1, then U W 2,1 (T d ; U(m)). and U 1 f L p (T d ; u(m)) The Euler-Lagrange equations can be recast in the form above. For d = 2 one has continuity by Sobolev embedding.

49 Continuity in the 3-dimensional case Let Ω R d an open set, d 3, and let U W 1,2 loc (Ω ; U(m)), m 2. Dene the Dirichlet energy E(U; Ω) = Ω 1 2 d tr j=1 ( ) U U dk, Ω Ω. k j k j The condition of stationarity for E easily implies that U is a weakly harmonic map, i. e. U is a weak solution of U + d j=1 U k j U 1 U k j = 0. We aim to prove that, when d = 3 and m 2 any local minimizer U W 1,2 loc (R3 ; U(m)) which is degree-zero homogeneous, i. e. is constant. U(k) = ω ( k k ) for some ω C (S 2 ; U(m))

50 Proposition [P & Pisante] ( Let m 2 and ω C (S 2 ; U(m)) a harmonic map. If U(k) = ω a local minimizer of the Dirichlet energy then the homogeneous energy satises E(ω) π 2 m. k k ) is Here the homogeneous energy is the Dirichlet energy for maps S 2 U(m) E(ω) = S ω 1 dω 2 dv ol. Proposition [Valli 88]. The energy E(ω) of any harmonic map ω : S 2 is an integer multiple of 8π. U(m) Obvious corollary. If m 15 then E(ω) = 0, so ω is constant.

51 Details, proofs & applications in the preprint: Bloch bundles, Marzari-Vanderbilt functional and maximally localized Wannier functions, arxiv: Thank you for you attention!!