The Localization Dichotomy for Periodic Schro dinger Operators

Transcription

1 The Localization Dichotomy for Periodic Schro dinger Operators Gianluca Panati Dipartimento di Matematica G. Castelnuovo based on joint work with D. Monaco, A. Pisante and S. Teufel Tosio Kato Centennial Conference University of Tokio, September 8, 2017

2 Chern insulators and the Quantum Anomalous Hall effect Left panel: transverse and direct resistivity in a Quantum Hall experiment Right panel: transverse and direct resistivity in a Chern insulator (histeresis cycle) Picture: c A. J. Bestwick (2015) G. Panati Tosio Kato Centennial Conference The Localization Dichotomy 2 / 31

3 G. Panati Tosio Kato Centennial Conference The Localization Dichotomy 3 / 31

4 Chern insulators: Energy gap at the Fermi energy band insulator There are no dissipative currents at zero temperature. nevertheless, dissipationless currents may exist at T = 0. In particular, a non-vanishing Hall current appears with vanishing external magnetic field (histeresis cycle) G. Panati Tosio Kato Centennial Conference The Localization Dichotomy 4 / 31

5 Chern insulators: Energy gap at the Fermi energy band insulator There are no dissipative currents at zero temperature. nevertheless, dissipationless currents may exist at T = 0. In particular, a non-vanishing Hall current appears with vanishing external magnetic field (histeresis cycle) Theoretically predicted by Haldane in 1988:. Panati Tosio Kato Centennial Conference The Localization Dichotomy 4 / 31

6 Chern insulators: Energy gap at the Fermi energy band insulator There are no dissipative currents at zero temperature. nevertheless, dissipationless currents may exist at T = 0. In particular, a non-vanishing Hall current appears with vanishing external magnetic field (histeresis cycle) Theoretically predicted by Haldane in 1988: Only TR-breaking is necessary to have a non-zero Hall conductivity. Panati Tosio Kato Centennial Conference The Localization Dichotomy 4 / 31

7 Mathematical questions about Chern insulators General goal: mathematical understanding of the QAHE and the QHE (including disorder, electrons interaction,... ) G. Panati Tosio Kato Centennial Conference The Localization Dichotomy 5 / 31

8 Mathematical questions about Chern insulators General goal: mathematical understanding of the QAHE and the QHE (including disorder, electrons interaction,... ) Our question: Relation between dissipationless transport and localization of electronic states. G. Panati Tosio Kato Centennial Conference The Localization Dichotomy 5 / 31

9 Mathematical questions about Chern insulators General goal: mathematical understanding of the QAHE and the QHE (including disorder, electrons interaction,... ) Our question: Relation between dissipationless transport and localization of electronic states. What is the relevant notion of localization for periodic quantum systems? G. Panati Tosio Kato Centennial Conference The Localization Dichotomy 5 / 31

10 Mathematical questions about Chern insulators General goal: mathematical understanding of the QAHE and the QHE (including disorder, electrons interaction,... ) Our question: Relation between dissipationless transport and localization of electronic states. Spectral type?? For ergodic random Schrödinger operators localization is measured by the spectral type (σ pp, σ ac, σ sc ) [AW]. However, for periodic systems the spectrum is generically purely absolutely continuous. G. Panati Tosio Kato Centennial Conference The Localization Dichotomy 5 / 31

11 Mathematical questions about Chern insulators General goal: mathematical understanding of the QAHE and the QHE (including disorder, electrons interaction,... ) Our question: Relation between dissipationless transport and localization of electronic states. Spectral type?? For ergodic random Schrödinger operators localization is measured by the spectral type (σ pp, σ ac, σ sc ) [AW]. However, for periodic systems the spectrum is generically purely absolutely continuous. [AW] M. Aizenman, S. Warzel: Random operators, AMS (2015). [AS 2 ] J. Avron, R. Seiler, B. Simon: Commun. Math. Phys. 159 (1994). [NN] A. Nenciu, G. Nenciu: Phys. Rev B 47 (1993). G. Panati Tosio Kato Centennial Conference The Localization Dichotomy 5 / 31

12 Mathematical questions about Chern insulators General goal: mathematical understanding of the QAHE and the QHE (including disorder, electrons interaction,... ) Our question: Relation between dissipationless transport and localization of electronic states. Spectral type?? For ergodic random Schrödinger operators localization is measured by the spectral type (σ pp, σ ac, σ sc ) [AW]. However, for periodic systems the spectrum is generically purely absolutely continuous. Kernel of the Fermi projector?? For gapped periodic 1-body Hamiltonians one has P µ (x, y) e λgap x y as a consequence of Combes-Thomas theory [AS 2, NN]. [AW] M. Aizenman, S. Warzel: Random operators, AMS (2015). [AS 2 ] J. Avron, R. Seiler, B. Simon: Commun. Math. Phys. 159 (1994). [NN] A. Nenciu, G. Nenciu: Phys. Rev B 47 (1993).. Panati Tosio Kato Centennial Conference The Localization Dichotomy 5 / 31

13 The localization of electrons is conveniently expressed by Composite Wannier functions (CWFs). Panati Tosio Kato Centennial Conference The Localization Dichotomy 6 / 31

14 The localization of electrons is conveniently expressed by Composite Wannier functions (CWFs) The Fermi projector P µ reads, for {w a,γ } γ Z d,1 a m a system of CWFs, m P µ = w a,γ w a,γ. a=1 γ Γ. Panati Tosio Kato Centennial Conference The Localization Dichotomy 6 / 31

15 The localization of electrons is conveniently expressed by Composite Wannier functions (CWFs) The Fermi projector P µ reads, for {w a,γ } γ Z d,1 a m a system of CWFs, P µ = Notice that crucially m w a,γ w a,γ. a=1 γ Γ P µ (x, y) e λgap x y { exist CWFs such that w a,γ (x) e c x γ. Panati Tosio Kato Centennial Conference The Localization Dichotomy 6 / 31

16 The localization of electrons is conveniently expressed by Composite Wannier functions (CWFs) The Fermi projector P µ reads, for {w a,γ } γ Z d,1 a m a system of CWFs, P µ = Notice that crucially m w a,γ w a,γ. a=1 γ Γ P µ (x, y) e λgap x y { exist CWFs such that w a,γ (x) e c x γ (GAP CONDITION) (TOPOLOGICAL TRIVIALITY). Panati Tosio Kato Centennial Conference The Localization Dichotomy 6 / 31

17 Theorem [MPPT]: The Localization Dichotomy for periodic systems Under assumptions specified later, the following holds: either there exists α > 0 and a choice of CWFs w = ( w 1,..., w m ) satisfying m e 2β x w a (x) 2 dx < + for every β [0, α); R d a=1 G. Panati Tosio Kato Centennial Conference The Localization Dichotomy 7 / 31

18 Theorem [MPPT]: The Localization Dichotomy for periodic systems Under assumptions specified later, the following holds: either there exists α > 0 and a choice of CWFs w = ( w 1,..., w m ) satisfying m e 2β x w a (x) 2 dx < + for every β [0, α); R d a=1 or for every possible choice of CWFs w = (w 1,..., w m ) one has X 2 w = m a=1 R d x 2 w a (x) 2 dx = +. G. Panati Tosio Kato Centennial Conference The Localization Dichotomy 7 / 31

19 Theorem [MPPT]: The Localization Dichotomy for periodic systems Under assumptions specified later, the following holds: either there exists α > 0 and a choice of CWFs w = ( w 1,..., w m ) satisfying m e 2β x w a (x) 2 dx < + for every β [0, α); R d a=1 or for every possible choice of CWFs w = (w 1,..., w m ) one has X 2 w = m a=1 R d x 2 w a (x) 2 dx = +. G. Panati Tosio Kato Centennial Conference The Localization Dichotomy 7 / 31

20 Theorem [MPPT]: The Localization Dichotomy for periodic systems Under assumptions specified later, the following holds: either there exists α > 0 and a choice of CWFs w = ( w 1,..., w m ) satisfying m e 2β x w a (x) 2 dx < + for every β [0, α); R d a=1 or for every possible choice of CWFs w = (w 1,..., w m ) one has X 2 w = m a=1 R d x 2 w a (x) 2 dx = +. Intermediate regimes are forbidden!!. Panati Tosio Kato Centennial Conference The Localization Dichotomy 7 / 31

21 Plan of the talk 1 Introduction 2 Mathematical models for Chern and Quantum Hall insulators Haldanium Magnetic periodic Schrödinger operators The Bloch bundle and its topology 3 Wannier functions and their localization 4 The Localization-Topology Correspondence 5 Consequences for transport properties Application to interacting electron gases Application to flat band superconductivity. Panati Tosio Kato Centennial Conference The Localization Dichotomy 8 / 31

22 Haldanium: the prototypical Chern insulator. Panati Tosio Kato Centennial Conference The Localization Dichotomy 9 / 31

23 Haldanium: the prototypical Chern insulator Let C R 2 be the hexagonal structure. For ψ l 2 (C) and a R 2, set { ψ x a if x a C (T aψ) x = 0 otherwise.. Panati Tosio Kato Centennial Conference The Localization Dichotomy 9 / 31

24 Haldanium: the prototypical Chern insulator Let C R 2 be the hexagonal structure. For ψ l 2 (C) and a R 2, set { ψ x a if x a C (T aψ) x = 0 otherwise. A discrete Laplacean on C (tight-binding model of graphene) reads H 0,0 = 3 j=1 t 1 ( T dj + T d j ) + 3 j=1 t 2 ( T aj + T a j ). Panati Tosio Kato Centennial Conference The Localization Dichotomy 9 / 31

25 Haldanium: the prototypical Chern insulator Relevant symmetries: Let C R 2 be the hexagonal structure. Z 2 -symmetry: periodicity 2π/3-rotation For ψ symmetry l 2 (C) and a R 2, set { Z 2 -symmetry: A ψ B x a if x a C (T aψ) x = TR-symmetry (antiunitary) 0 otherwise. A discrete Laplacean on C (tight-binding model of graphene) reads H 0,0 = 3 j=1 t 1 ( T dj + T d j ) + 3 j=1 t 2 ( T aj + T a j ). Panati Tosio Kato Centennial Conference The Localization Dichotomy 9 / 31

26 Haldanium: the prototypical Chern insulator Relevant symmetries: Let C R 2 be the hexagonal structure. Z 2 -symmetry: periodicity 2π/3-rotation For ψ symmetry l 2 (C) and a R 2, set { Z 2 -symmetry: A ψ B x a if x a C (T aψ) x = TR-symmetry (antiunitary) 0 otherwise. A discrete Laplacean on C (tight-binding model of graphene) reads H 0, M = 3 j=1 t 1 ( T dj + T d j ) + 3 j=1 t 2 ( T aj + T a j ) +MV (x) where V (x) = { +1 if x A 1 if x B. Panati Tosio Kato Centennial Conference The Localization Dichotomy 9 / 31

27 Haldane Hamiltonian TR-symmetry is broken while preserving Z 2 -symmetry, i. e. periodicity with respect to Γ = Span Z {a 1, a 2 }: the magnetic flux per unit cell is zero. Topology appears after Fourier transform F : l 2 (C) l 2 (Z 2 ) C 2 ( FHφ,M F 1) (k) = where {σ 0,..., σ 3 } are the Pauli matrices. 3 R a (k) σ a a=0 F L 2 (T 2 ; C 2 ). R 0 (k) = 2t 2 (cos φ) j cos(k a j ) R 1 (k) = t 1 (1 + cos(k a 1 ) + cos(k a 2 )) R 2 (k) = t 1 (sin(k a 1 ) sin(k a 2 )) R 3 (k) = M 2t 2 (sin φ) sin(k a j ). j. Panati Tosio Kato Centennial Conference The Localization Dichotomy 10 / 31

28 Haldane Hamiltonian TR-symmetry is broken while preserving Z 2 -symmetry, i. e. periodicity with respect to Γ = Span Z {a 1, a 2 }: the magnetic flux per unit cell is zero. Topology appears after Fourier transform F : l 2 (C) l 2 (Z 2 ) C 2 ( FHφ,M F 1) (k) = where {σ 0,..., σ 3 } are the Pauli matrices. 3 R a (k) σ a a=0 R 0 (k) = Explicit 2t 2 (cos φ) expressions cos(k a j ) may dependj on the R 1 (k) = choice t 1 (1 of + cos(k the unit a 1 ) + cell cos(k a 2 )) R 2 (k) = t 1 (sin(k a 1 ) sin(k a 2 )) R 3 (k) = M 2t 2 (sin φ) j sin(k a j ). F L 2 (T 2 ; C 2 ).. Panati Tosio Kato Centennial Conference The Localization Dichotomy 10 / 31

29 The phase diagram of the Haldane model The system is gapped for all the values of the parameters (φ, M t 2 ) R 2, except those in a -shaped curve, where a conical intersection of eigenvalues appear: Phase diagram for the family of Haldane operators H φ,m. G. Panati Tosio Kato Centennial Conference The Localization Dichotomy 11 / 31

30 The phase diagram of the Haldane model The system is gapped for all the values of the parameters (φ, M t 2 ) R 2, except those in a -shaped curve, where a conical intersection of eigenvalues appear: Phase diagram for the family of Haldane operators H φ,m. In the gapped phase, the eigenprojector on the lower eigenvalue P (k) defines a complex line bundle over T 2, which is characterized by its Chern number c 1 (P ), where ( ) c 1 (P) := 1 2πi tr T 2 C 2 P(k) [ k1 P(k), k2 P(k)] dk 1 dk 2 [AS 2 ] J. Avron, R. Seiler, B. Simon: Commun. Math. Phys. 159 (1994). G. Panati Tosio Kato Centennial Conference The Localization Dichotomy 11 / 31

31 Magnetic periodic Schrödinger operators Chern insulators: For d {2, 3}, consider the magnetic Schrödinger operator H Γ = 1 2 ( i x A Γ (x)) 2 + V Γ (x) acting in L 2 (R d ) where A Γ and V Γ are periodic with respect to Γ = Span Z {a 1,..., a d }.. Panati Tosio Kato Centennial Conference The Localization Dichotomy 12 / 31

32 Magnetic periodic Schrödinger operators Chern insulators: For d {2, 3}, consider the magnetic Schrödinger operator H Γ = 1 2 ( i x A Γ (x)) 2 + V Γ (x) acting in L 2 (R d ) where A Γ and V Γ are periodic with respect to Γ = Span Z {a 1,..., a d }. Hence the modified Bloch-Floquet transform (U ψ)(k, y) := γ Γ e ik (y γ) (T γ ψ)(y), y R d, k R d provides a simultaneous decomposition of H Γ and T γ : UH Γ U 1 = B dk H(k) with H(k) = 1 2( i y A Γ (y) + k ) 2 + VΓ (y).. Panati Tosio Kato Centennial Conference The Localization Dichotomy 12 / 31

33 Magnetic periodic Schrödinger operators Chern insulators: For d {2, 3}, consider the magnetic Schrödinger operator H Γ = 1 2 ( i x A Γ (x)) 2 + V Γ (x) acting in L 2 (R d ) where A Γ and V Γ are periodic with respect to Γ = Span Z {a 1,..., a d }. Hence the modified Bloch-Floquet transform (U ψ)(k, y) := γ Γ e ik (y γ) (T γ ψ)(y), y R d, k R d provides a simultaneous decomposition of H Γ and T γ : UH Γ U 1 = B dk H(k) with H(k) = 1 2( i y A Γ (y) + k ) 2 + VΓ (y). The operator H(k) acts in L 2 per(r d ) := { ψ L 2 loc (Rd ) : T γ ψ = ψ for all γ Γ }.. Panati Tosio Kato Centennial Conference The Localization Dichotomy 12 / 31

34 The Bloch bundle and its topology For each fixed k R d, the operator H(k) has compact resolvent, so pure point spectrum accumulating at +. Question: how can I read from the picture whether TR-symmetry is broken? Spectrum of H(k) as a function of k j (left). Spectrum of H Γ = H(k)dk (right). G. Panati Tosio Kato Centennial Conference The Localization Dichotomy 13 / 31

35 The Bloch bundle and its topology For each fixed k R d, the operator H(k) has compact resolvent, so pure point spectrum accumulating at +. An isolated family of J Bloch bands. Notice that the spectral bands may overlap. G. Panati Tosio Kato Centennial Conference The Localization Dichotomy 14 / 31

36 The Bloch bundle and its topology For each fixed k R d, the operator H(k) has compact resolvent, so pure point spectrum accumulating at +. Topology is encoded in the orthogonal projections on an isolated family of bands P (k) = i 2π C (H(k) z1) 1 (k) dz = n I u n (k, ) u n (k, ). An isolated family of J Bloch bands. Notice that the spectral bands may overlap. where C (k) intersects the real line in E (k) and E + (k) (GAP CONDITION). G. Panati Tosio Kato Centennial Conference The Localization Dichotomy 14 / 31

37 A model for Quantum Hall insulators Quantum Hall insulators: For d {2, 3}, consider A b (x) = b 2c x ê with ê a unit vector H Γ,b = 1 ( i 2 x A b (x)) 2 + V Γ (x) acting in L 2 (R d ) where V Γ is periodic with respect to Γ = Span Z {a 1,..., a d }. Assume moreover the commensurability condition for the magnetic field B = bê: 1 c B (γ γ ) 2πQ for all γ, γ Γ Ordinary translations are replaced by the magnetic translations [Zak64] (T b γ ψ)(x) := e iγ A b(x) ψ(x γ) γ Γ. These commute with H Γ,b, but satisfy the pseudo-weyl relations T b γ T b γ = e i c B (γ γ ) T b γ T b γ γ, γ Γ. The Z d -symmetry is recovered at the price of considering a smaller lattice Γ b Γ. G. Panati Tosio Kato Centennial Conference The Localization Dichotomy 15 / 31

38 Assumptions Consider A = A Γ + A b (periodic + linear) with A b satisfying the commensurability condition, and set H(κ) = 1 2( i y A(y) + κ ) 2 + VΓ (y) acting in H b f, κ C d, where H b f := { ψ L 2 loc (Rd ) : T b γ ψ = ψ, for all γ Γ b }. Assumption 1: The magnetic potential A = A Γ + A b and the scalar potential V Γ are such that the family of operators H(κ) is an entire analytic family in the sense of Kato with compact resolvent.. Panati Tosio Kato Centennial Conference The Localization Dichotomy 16 / 31

39 Assumptions Consider A = A Γ + A b (periodic + linear) with A b satisfying the commensurability condition, and set H(κ) = 1 2( i y A(y) + κ ) 2 + VΓ (y) acting in H b f, κ C d, where H b f := { ψ L 2 loc (Rd ) : T b γ ψ = ψ, for all γ Γ b }. Assumption 1: The magnetic potential A = A Γ + A b and the scalar potential V Γ are such that the family of operators H(κ) is an entire analytic family in the sense of Kato with compact resolvent. If A = A Γ is Γ-periodic, with fundamental unit cell Y, then it is sufficient to assume either: A L (Y ; R 2 ) when d = 2 or A L 4 (Y ; R 3 ) when d = 3, and div A, V Γ L 2 loc (Rd ) when 2 d 3; A L r (Y ; R 2 ) with r > 2 and V Γ L p (Y ) with p > 1 when d = 2, or A L 3 (Y ; R 3 ) and V Γ L 3/2 (Y ) when d = 3.. Panati Tosio Kato Centennial Conference The Localization Dichotomy 16 / 31

40 Lemma: Let P (k) be the spectral projector of H(k) corresponding to a set σ (k) R such that the gap condition is satisfied. Then the family {P (k)} k R d has the following properties: (P 1 ) the map k P (k) is analytic from R d to B(Hf b); (P 2 ) the map k P (k) is τ-covariant, i. e. P (k + λ) = τ(λ) P (k) τ(λ) 1 k R d, λ Γ b. where τ : Γ b Zd U(Hf b ) is a unitary representation.. Panati Tosio Kato Centennial Conference The Localization Dichotomy 17 / 31

41 Lemma: Let P (k) be the spectral projector of H(k) corresponding to a set σ (k) R such that the gap condition is satisfied. Then the family {P (k)} k R d has the following properties: (P 1 ) the map k P (k) is analytic from R d to B(Hf b); (P 2 ) the map k P (k) is τ-covariant, i. e. P (k + λ) = τ(λ) P (k) τ(λ) 1 k R d, λ Γ b. where τ : Γ b Zd U(Hf b ) is a unitary representation. For the sake of simpler slides, here we consider the case τ 1.. Panati Tosio Kato Centennial Conference The Localization Dichotomy 17 / 31

42 Lemma: Let P (k) be the spectral projector of H(k) corresponding to a set σ (k) R such that the gap condition is satisfied. Then the family {P (k)} k R d has the following properties: (P 1 ) the map k P (k) is analytic from R d to B(Hf b); (P 2 ) the map k P (k) is τ-covariant, i. e. P (k + λ) = τ(λ) P (k) τ(λ) 1 k R d, λ Γ b. where τ : Γ b Zd U(Hf b ) is a unitary representation. In view of (P 1 ) and (P 2 ) the ranges of P (k) define a (smooth) Hermitian vector bundle over T d := R d /Γ b. For d = 2, it is characterized by the Chern number ) c 1 (P) := 1 2πi Tr T 2 H b f (P(k) [ k1 P(k), k2 P(k)] dk 1 dk 2 For the sake of simpler slides, here we consider the case τ 1.. Panati Tosio Kato Centennial Conference The Localization Dichotomy 17 / 31

43 Wannier functions and their localization IDEA: Wannier functions provide a reasonable compromise between localization in energy and localization in position space, as far as compatible with the uncertainty principle.. Panati Tosio Kato Centennial Conference The Localization Dichotomy 18 / 31

44 Wannier functions and their localization IDEA: Wannier functions provide a reasonable compromise between localization in energy and localization in position space, as far as compatible with the uncertainty principle. They are associated at the energy window corresponding to P ( ) via Definition: A Bloch frame is a collection {φ 1,..., φ m } L 2 (T d, Hf b ) such that: (φ 1 (k),..., φ m (k)) is an orthonormal basis of Ran P (k) for a.e. k R d. Panati Tosio Kato Centennial Conference The Localization Dichotomy 18 / 31

45 Wannier functions and their localization IDEA: Wannier functions provide a reasonable compromise between localization in energy and localization in position space, as far as compatible with the uncertainty principle. They are associated at the energy window corresponding to P ( ) via Definition: A Bloch frame is a collection {φ 1,..., φ m } L 2 (T d, Hf b ) such that: (φ 1 (k),..., φ m (k)) is an orthonormal basis of Ran P (k) for a.e. k R d In general, a Bloch frame mixes different Bloch bands φ a (k) = n I u n (k) U na (k) }{{} Bloch funct. for some unitary matrix U(k).. Panati Tosio Kato Centennial Conference The Localization Dichotomy 18 / 31

46 Wannier functions and their localization IDEA: Wannier functions provide a reasonable compromise between localization in energy and localization in position space, as far as compatible with the uncertainty principle. They are associated at the energy window corresponding to P ( ) via Definition: A Bloch frame is a collection {φ 1,..., φ m } L 2 (T d, Hf b ) such that: (φ 1 (k),..., φ m (k)) is an orthonormal basis of Ran P (k) for a.e. k R d In general, a Bloch frame mixes different Bloch bands φ a (k) = The ambiguity in the choice is dubbed u n (k) U na (k) for some unitary matrix U(k). }{{} Bloch gauge n I Bloch funct. ambiguity.. Panati Tosio Kato Centennial Conference The Localization Dichotomy 18 / 31

47 The Bloch bundle RanP (k) Competition between regularity and periodicity is encoded by the Bloch bundle E = (E T d ). φ 1 (k) φ2 (k) Existence of continuous Bloch frames is topologically obstructed if c 1 (P ) 0. k G. Panati Tosio Kato Centennial Conference The Localization Dichotomy 19 / 31

48 Definition (CWFs): The composite Wannier functions {w 1,..., w m } L 2 (R d ) associated to a Bloch frame {φ 1,..., φ m } are defined as w a (x) := ( U 1 ) 1 φ a (x) = dk e ik x φ a (k, x). B b B b G. Panati Tosio Kato Centennial Conference The Localization Dichotomy 20 / 31

49 Definition (CWFs): The composite Wannier functions {w 1,..., w m } L 2 (R d ) associated to a Bloch frame {φ 1,..., φ m } are defined as w a (x) := ( U 1 ) 1 φ a (x) = dk e ik x φ a (k, x). B b B b Localization of CWFs in position space BF transform Smoothness of Bloch frame in momentum space G. Panati Tosio Kato Centennial Conference The Localization Dichotomy 20 / 31

50 Definition (CWFs): The composite Wannier functions {w 1,..., w m } L 2 (R d ) associated to a Bloch frame {φ 1,..., φ m } are defined as w a (x) := ( U 1 ) 1 φ a (x) = dk e ik x φ a (k, x). B b B b Localization of CWFs in position space BF transform Smoothness of Bloch frame in momentum space R d x 2s w a,γ (x) 2 dx φ a 2 H s (T d,hb f ) G. Panati Tosio Kato Centennial Conference The Localization Dichotomy 20 / 31

51 Definition (CWFs): The composite Wannier functions {w 1,..., w m } L 2 (R d ) associated to a Bloch frame {φ 1,..., φ m } are defined as w a (x) := ( U 1 ) 1 φ a (x) = dk e ik x φ a (k, x). B b B b Localization of CWFs in position space BF transform Smoothness of Bloch frame in momentum space R d x 2s w a,γ (x) 2 dx φ a 2 H s (T d,hb f ) Hence, by Sobolev theorem, for s > d/2 the existence of s-localized CWFs is topologically obstructed. But, for 2 d 3, still there might exist 1-localized CWFs, i. e. X 2 w < +. G. Panati Tosio Kato Centennial Conference The Localization Dichotomy 20 / 31

52 Definition (CWFs): The composite Wannier functions {w 1,..., w m } L 2 (R d ) associated to a Bloch frame {φ 1,..., φ m } are defined as w a (x) := ( U 1 ) 1 φ a (x) = dk e ik x φ a (k, x). B b B b Localization of CWFs in position space BF transform Smoothness of Bloch frame in momentum space R d x 2s w a,γ (x) 2 dx φ a 2 H s (T d,hb f ) Hence, by Sobolev theorem, NO! forins > thed/2 topologically the existence of s-localized CWFs is non-trivial topologically case, obstructed. does NOT But, for 2 d 3, still there exist mightany exist system 1-localized of CWFs, i. e. X 2 w < +. 1-localized composite Wannier functions!. Panati Tosio Kato Centennial Conference The Localization Dichotomy 20 / 31

53 Grassmanian reinterpretation G m (H) := { P B(H) : P 2 = P = P, Tr P = m } (Grassmann manifold) W m (H) := {J : C m H linear isometry} (Stiefel manifold), Notice that J = m a=1 ψa ea, where {ψa} H is an m-frame. There is a natural map π : W m(h) G m(h) sending each m-frame Ψ = {ψ 1,..., ψ m} into the orthogonal projection on its linear span, namely π : J JJ = m a=1 ψa ψa.. Panati Tosio Kato Centennial Conference The Localization Dichotomy 21 / 31

54 Grassmanian reinterpretation G m (H) := { P B(H) : P 2 = P = P, Tr P = m } (Grassmann manifold) W m (H) := {J : C m H linear isometry} (Stiefel manifold), Notice that J = m a=1 ψa ea, where {ψa} H is an m-frame. There is a natural map π : W m(h) G m(h) sending each m-frame Ψ = {ψ 1,..., ψ m} into the orthogonal projection on its linear span, namely π : J JJ = m a=1 ψa ψa. At least formally, W m (H) is a principal bundle over G m (H) with projection π and fiber U(C m ). The data P and Φ correspond to a commutative diagram: where P C (T d ; B(H)) and Φ H s (T d ; H m ).. Panati Tosio Kato Centennial Conference The Localization Dichotomy 21 / 31

55 Grassmanian reinterpretation G m (H) := { P B(H) : P 2 = P = P, Tr P = m } (Grassmann manifold) W m (H) := {J : C m H linear isometry} (Stiefel manifold), Notice that J = m a=1 ψa ea, where {ψa} H is an m-frame. There is a natural map π : W m(h) G m(h) sending each m-frame Ψ = {ψ 1,..., ψ m} into the orthogonal projection on its linear span, namely π : J JJ = m a=1 ψa ψa. At least formally, W m (H) is a principal bundle over G m (H) with projection π and fiber U(C m ). The data P and Φ correspond to a commutative diagram: where P C (T d ; B(H)) and Φ H s (T d ; H m ).. Panati Tosio Kato Centennial Conference The Localization Dichotomy 21 / 31

56 Grassmanian reinterpretation G m (H) := { P B(H) : P 2 = P = P, Tr P = m } (Grassmann manifold) W m (H) := {J : C m H linear isometry} (Stiefel manifold), Notice that J = m a=1 ψa ea, where {ψa} H is an m-frame. There is a natural map π : W m(h) G m(h) sending each m-frame Ψ = {ψ 1,..., ψ m} into the orthogonal projection on its linear span, namely π : J JJ = m a=1 ψa ψa. At least formally, W m (H) is a principal bundle over G m (H) with projection π and fiber U(C m ). The data P and Φ correspond to a commutative diagram: c 1 (P) = 1 ) Tr H (P(k) [ k1 P(k), k2 P(k)] dk 1 dk 2 2πi T 2 = 1 m 2 Im k1 φ a (k), k1 φ a (k) 2πi H dk 1 dk 2. where P C (T d T ; B(H)) 2 a=1 and Φ H s (T d ; H m ).. Panati Tosio Kato Centennial Conference The Localization Dichotomy 21 / 31

57 Grassmanian reinterpretation G m (H) := { P B(H) : P 2 = P = P, Tr P = m } (Grassmann manifold) W m (H) := {J : C m H linear isometry} (Stiefel manifold), Notice that J = m a=1 ψa ea, where {ψa} H is an m-frame. There is a natural map π : W m(h) G m(h) sending each m-frame Ψ = {ψ 1,..., ψ m} into the orthogonal projection on its linear span, namely π : J JJ = m a=1 ψa ψa. At least formally, W m (H) is a principal bundle over G m (H) with projection π and fiber U(C m ). The data P and Φ correspond to a commutative diagram: where P C (T d ; B(H)) and Φ H s (T d ; H m ). Since the set of m-frames is not a linear space, the approximation of a given Sobolev map by smooth maps might be topologically obstructed.. Panati Tosio Kato Centennial Conference The Localization Dichotomy 21 / 31

58 Approximation of a given Sobolev map Theorem [Hang & Lin] Let 2 d 3. Consider a compact, boundaryless, smooth submanifold M R ν. If d = 3, assume moreover that the homotopy group π 2 (M) is trivial. Then, every Sobolev map Ψ H 1 (T d, M) can be approximated by a sequence {Ψ (l) } l N C (T d, M) such that Ψ (l) H 1 Ψ as l. [HL] Hang, F.; Lin, F.H. : Topology of Sobolev mappings. II. Acta Math [CHN] Cornean, H.; Herbst, I., Nenciu, G. : Ann. H. Poincaré G. Panati Tosio Kato Centennial Conference The Localization Dichotomy 22 / 31

59 Approximation of a given Sobolev map Theorem [Hang & Lin] Let 2 d 3. Consider a compact, boundaryless, smooth submanifold M R ν. If d = 3, assume moreover that the homotopy group π 2 (M) is trivial. Then, every Sobolev map Ψ H 1 (T d, M) can be approximated by a sequence {Ψ (l) } l N C (T d, M) such that Ψ (l) H 1 Ψ as l. The result can be applied to the (finite-dimensional) Stiefel manifold W m (C n ), using that π 2 (W m (C n )) = 0 whenever n m + 2 (here C n is regarded as a Galerkin truncation of H). [HL] Hang, F.; Lin, F.H. : Topology of Sobolev mappings. II. Acta Math [CHN] Cornean, H.; Herbst, I., Nenciu, G. : Ann. H. Poincaré G. Panati Tosio Kato Centennial Conference The Localization Dichotomy 22 / 31

60 Approximation of a given Sobolev map Theorem [Hang & Lin] Let 2 d 3. Consider a compact, boundaryless, smooth submanifold M R ν. If d = 3, assume moreover that the homotopy group π 2 (M) is trivial. Then, every Sobolev map Ψ H 1 (T d, M) can be approximated by a sequence {Ψ (l) } l N C (T d, M) such that Ψ (l) H 1 Ψ as l. The result can be applied to the (finite-dimensional) Stiefel manifold W m (C n ), using that π 2 (W m (C n )) = 0 whenever n m + 2 (here C n is regarded as a Galerkin truncation of H). One has to reduce to a finite-dimensional space V n C n [HL] Hang, F.; Lin, F.H. : Topology of Sobolev mappings. II. Acta Math [CHN] Cornean, H.; Herbst, I., Nenciu, G. : Ann. H. Poincaré G. Panati Tosio Kato Centennial Conference The Localization Dichotomy 22 / 31

61 Approximation of a given Sobolev map Theorem [Hang & Lin] Let 2 d 3. Consider a compact, boundaryless, smooth submanifold M R ν. If d = 3, assume moreover that the homotopy group π 2 (M) is trivial. Then, every Sobolev map Ψ H 1 (T d, M) can be approximated by a sequence {Ψ (l) } l N C (T d, M) such that Ψ (l) H 1 Ψ as l. The result can be applied to the (finite-dimensional) Stiefel manifold W m (C n ), using that π 2 (W m (C n )) = 0 whenever n m + 2 (here C n is regarded as a Galerkin truncation of H). One has to reduce to a finite-dimensional space V n C n One has to reduce covariance to periodicity, while staying in a k-independent fiber Hilber space [CHN] [HL] Hang, F.; Lin, F.H. : Topology of Sobolev mappings. II. Acta Math [CHN] Cornean, H.; Herbst, I., Nenciu, G. : Ann. H. Poincaré G. Panati Tosio Kato Centennial Conference The Localization Dichotomy 22 / 31

62 Approximation of a given Sobolev map Theorem [Hang & Lin] Let 2 d 3. Consider a compact, boundaryless, smooth submanifold M R ν. If d = 3, assume moreover that the homotopy group π 2 (M) is trivial. Then, every Sobolev map Ψ H 1 (T d, M) can be approximated by a sequence {Ψ (l) } l N C (T d, M) such that Ψ (l) H 1 Ψ as l. The result can be applied to the (finite-dimensional) Stiefel manifold W m (C n ), using that π 2 (W m (C n )) = 0 whenever n m + 2 (here C n is regarded as a Galerkin truncation of H). One has to reduce to a finite-dimensional space V n C n One has to reduce covariance to periodicity, while staying in a k-independent fiber Hilber space [CHN] In our specific case, there is no obstruction to construct a Bloch frame with regularity H s (T d ; H m ) for every s < 1, essentially via parallel transport [construction in the paper]. [HL] Hang, F.; Lin, F.H. : Topology of Sobolev mappings. II. Acta Math [CHN] Cornean, H.; Herbst, I., Nenciu, G. : Ann. H. Poincaré G. Panati Tosio Kato Centennial Conference The Localization Dichotomy 22 / 31

63 The constructive theorem Theorem 1 [Monaco, GP, Pisante, Teufel 17] Assume d 3. Consider a magnetic periodic Schrödinger operator in L 2 (R d ) satisfying the previous assumptions (Kato smallness + commensurability + gap). Then we construct a Bloch frame in H s (T d ; H m ) for every s < 1 and, correspondingly, a system of CWFs {w a,γ } such that m a=1 R d x 2s w a,γ (x) 2 dx < + γ Γ, s < 1. G. Panati Tosio Kato Centennial Conference The Localization Dichotomy 23 / 31

64 The Localization-Topology Correspondence Theorem 2 [Monaco, GP, Pisante, Teufel 17] Assume d 3. Consider a magnetic periodic Schrödinger operator in L 2 (R d ) satisfying the previous assumptions (Kato smallness + commensurability + gap). The following statements are equivalent: Finite second moment: there exist CWFs {w a,γ } such that m a=1 R d x 2 w a,γ (x) 2 dx < + γ Γ; Exponential localization: there exist CWFs {w a,γ } and α > 0 such that m a=1 R d e 2β x w a,γ (x) 2 dx < + γ Γ, β [0, α); Trivial topology: the family {P (k)} k corresponds to a trivial vector bundle. G. Panati Tosio Kato Centennial Conference The Localization Dichotomy 24 / 31

65 Synopsis: symmetry, localization, transport, topology G. Panati Tosio Kato Centennial Conference The Localization Dichotomy 25 / 31

66 References First column: existence of exponentially localized CWFs [Ko] Kohn, W., Phys. Rev. 115(1959). (m = 1, d = 1, even) [dc] des Cloizeaux, J., Phys. Rev. 135 (1964). (m = 1, any d, even) [NeNe 1 ] Nenciu, A.; Nenciu, G., J. Phys. A 15 (1982). (any m, d = 1) [Ne 1 ] Nenciu, G., Commun. Math. Phys. 91 (1983). (m = 1, any d) [HeSj] Helffer, B.; Sjöstrand, J., LNP (1989). (m = 1, any d) [Pa] Panati, G., Ann. Henri Poincaré 8 (2007). (any m, d 3, abstract) [BPCM] Brouder Ch. et al., Phys. Rev. Lett. 98 (2007). (any m, d 3, abstract) [FMP] Fiorenza, D. et al., Ann. Henri Poincaré (2016). (any m, d 3, construct) [CHN] Cornean, H.D. et al., Ann. Henri Poincaré (2016).(any m, d 2, construct). Panati Tosio Kato Centennial Conference The Localization Dichotomy 26 / 31

67 References First column: existence of exponentially localized CWFs [Ko] Kohn, W., Phys. Rev. 115(1959). (m = 1, d = 1, even) [dc] des Cloizeaux, J., Phys. Rev. 135 (1964). (m = 1, any d, even) [NeNe 1 ] Nenciu, A.; Nenciu, G., J. Phys. A 15 (1982). (any m, d = 1) [Ne 1 ] Nenciu, G., Commun. Math. Phys. 91 (1983). (m = 1, any d) [HeSj] Helffer, B.; Sjöstrand, J., LNP (1989). (m = 1, any d) [Pa] Panati, G., Ann. Henri Poincaré 8 (2007). (any m, d 3, abstract) [BPCM] Brouder Ch. et al., Phys. Rev. Lett. 98 (2007). (any m, d 3, abstract) [FMP] Fiorenza, D. et al., Ann. Henri Poincaré (2016). (any m, d 3, construct) [CHN] Cornean, H.D. et al., Ann. Henri Poincaré (2016).(any m, d 2, construct) Second column: power-law decay of CWFs in the topological phase [Th] Thouless, J. Phys. C 17 (1984) (conjectured w(x) x 2 for d = 2) [TV] Thonhauser, Vanderbilt, Phys. Rev. B (2006). (numerics on Haldane model). Panati Tosio Kato Centennial Conference The Localization Dichotomy 26 / 31

68 References First column: existence of exponentially localized CWFs [Ko] Kohn, W., Phys. Rev. 115(1959). (m = 1, d = 1, even) [dc] des Cloizeaux, J., Phys. Rev. 135 (1964). (m = 1, any d, even) [NeNe 1 ] Nenciu, A.; Nenciu, G., J. Phys. A 15 (1982). (any m, d = 1) [Ne 1 ] Nenciu, G., Commun. Math. Phys. 91 (1983). (m = 1, any d) [HeSj] Helffer, B.; Sjöstrand, J., LNP (1989). (m = 1, any d) [Pa] Panati, G., Ann. Henri Poincaré 8 (2007). (any m, d 3, abstract) [BPCM] Brouder Ch. et al., Phys. Rev. Lett. 98 (2007). (any m, d 3, abstract) [FMP] Fiorenza, D. et al., Ann. Henri Poincaré (2016). (any m, d 3, construct) [CHN] Cornean, H.D. et al., Ann. Henri Poincaré (2016).(any m, d 2, construct) Second column: power-law decay of CWFs in the topological phase [Th] Thouless, J. Phys. C 17 (1984) (conjectured w(x) x 2 for d = 2) [TV] Thonhauser, Vanderbilt, Phys. Rev. B (2006). (numerics on Haldane model) Our result applies to all d 3 and is largely model-independent, since it covers both continuous and tight-binding models.. Panati Tosio Kato Centennial Conference The Localization Dichotomy 26 / 31

69 Trasport of electrons in a periodic background For non-interacting electrons, the Hall conductivity in linear response approximation is given by the Kubo formula [TKNN, Hal, BGKS] σ xy (U = 0) = e2 h c 1(P µ ) where P µ is the Fermi projector, and U 0 the strength of the interaction. [TKNN] D.J. Thouless, M. Kohmoto, M. Nightingale, de Nijs Phys. Rev. Lett. (1982) [Hal] F.D.M. Haldane: Phys. Rev. Lett. (1988) [BGKS] J.M. Bouclet, F.Germinet, A. Klein, J.H. Schenker: J. Funct. Analysis (2005) [GMP] A. Giuliani, V. Mastropietro, M. Porta: Commun. Math. Phys. (2016). Panati Tosio Kato Centennial Conference The Localization Dichotomy 27 / 31

70 Trasport of electrons in a periodic background For non-interacting electrons, the Hall conductivity in linear response approximation is given by the Kubo formula [TKNN, Hal, BGKS] σ xy (U = 0) = e2 h c 1(P µ ) where P µ is the Fermi projector, and U 0 the strength of the interaction. In the Haldane-Hubbard model, for moderate strength of the interaction U [0, U 0 ) one has [GMP] σ xy (U) = σ xy (U = 0) [TKNN] D.J. Thouless, M. Kohmoto, M. Nightingale, de Nijs Phys. Rev. Lett. (1982) [Hal] F.D.M. Haldane: Phys. Rev. Lett. (1988) [BGKS] J.M. Bouclet, F.Germinet, A. Klein, J.H. Schenker: J. Funct. Analysis (2005) [GMP] A. Giuliani, V. Mastropietro, M. Porta: Commun. Math. Phys. (2016). Panati Tosio Kato Centennial Conference The Localization Dichotomy 27 / 31

71 Localization Dichotomy for interacting electrons? Consider a gas of interacting electrons in 2d in a periodic background.. Panati Tosio Kato Centennial Conference The Localization Dichotomy 28 / 31

72 Localization Dichotomy for interacting electrons? Consider a gas of interacting electrons in 2d in a periodic background. Suppose that the 1-particle density matrix γ (1) N an orthogonal projector γ (1) ; converges, in the thermodynamic limit, to the Γ-periodicity is not broken in the thermodynamic limit.. Panati Tosio Kato Centennial Conference The Localization Dichotomy 28 / 31

73 Localization Dichotomy for interacting electrons? Consider a gas of interacting electrons in 2d in a periodic background. Suppose that the 1-particle density matrix γ (1) N an orthogonal projector γ (1) ; converges, in the thermodynamic limit, to the Γ-periodicity is not broken in the thermodynamic limit. Then γ (1) = B P (k)dk with orth. projectors P (k) B(H b f ) It is reasonable to define interacting Wannier functions as the preimage via U of an orthonormal frame spanning Ran P (k).. Panati Tosio Kato Centennial Conference The Localization Dichotomy 28 / 31

74 Localization Dichotomy for interacting electrons? Consider a gas of interacting electrons in 2d in a periodic background. Suppose that the 1-particle density matrix γ (1) N an orthogonal projector γ (1) ; converges, in the thermodynamic limit, to the Γ-periodicity is not broken in the thermodynamic limit. Then γ (1) = B P (k)dk with orth. projectors P (k) B(H b f ) It is reasonable to define interacting Wannier functions as the preimage via U of an orthonormal frame spanning Ran P (k). The Localization Dichotomy applies to this case as well!. Panati Tosio Kato Centennial Conference The Localization Dichotomy 28 / 31

75 Interacting electrons in the rhf-approximation Mathematically, the previous hypotheses have been proved, as far as I know, only in the reduced Hartree-Fock approximation assuming TR-symmetry [CLL, CDL] and in specific discrete models. It would be interesting to prove analogous results in a background which breaks TR-symmetry (magnetic dipole moments of ionic cores) [CLL] I. Catto, C. Le Bris C., P.-L. Lions : Ann. I. H. Poincaré 18 (2001). [CDL] Cancès, A. Deleurence, M. Lewin : Commun. Math. Phys. 281 (2008).. Panati Tosio Kato Centennial Conference The Localization Dichotomy 29 / 31

76 Application to flat band superconductivity. Panati Tosio Kato Centennial Conference The Localization Dichotomy 30 / 31

77 Application to flat band superconductivity The superfluid weight D s has two contributions: ( ( D s,conv := 2 E (k) k i k j vanishes for flat bands; )k=k 0 1 m eff )i,j D s,geom proportional to the quantum metric tensor.. Panati Tosio Kato Centennial Conference The Localization Dichotomy 30 / 31

78 Application to flat band superconductivity The superfluid weight D s has two contributions: ( ( D s,conv := 2 E (k) k i k j vanishes for flat bands; )k=k 0 1 m eff )i,j D s,geom proportional to thetransport-topology quantum metric tensor. inequality D s c 1 (P ). Panati Tosio Kato Centennial Conference The Localization Dichotomy 30 / 31

79 Details, proofs & applications in the papers: Optimal decay of Wannier functions in Chern and Quantum Hall insulators, arxiv: , submitted to Commun. Math. Phys. The Localization Dichotomy for gapped periodic quantum systems arxiv: , mainly devoted to physical applications. Thank you for your attention!!. Panati Tosio Kato Centennial Conference The Localization Dichotomy 31 / 31