Magnetic symmetries:

Transcription

1 Magnetic symmetries: applications and prospects Giuseppe De Nittis (AGM, Université de Cergy-Pontoise) Friedrich Schiller Universität, Mathematisches Institut, Jena. April 18, 2012 Joint work with: M. Lein (Universität Tübingen) Reference article: J. Math. Phys. 52, (2011) Grant N.: ANR-08-BLAN

2 From symmetries to Bloch-bundle H = separable Hilbert space, U : Z N U (H ) = unit. rep. (wandering syst. + algebraically. comp.), H L (H ), H = H (not necessarily bounded), DEFINITION H has a Z N -symmetry iff [f (H);U(n)] = 0, n Z N, f L (R).

3 From symmetries to Bloch-bundle H = separable Hilbert space, U : Z N U (H ) = unit. rep. (wandering syst. + algebraically. comp.), H L (H ), H = H (not necessarily bounded), DEFINITION H has a Z N -symmetry iff [f (H);U(n)] = 0, n Z N, f L (R). THEOREM (G. D. & G. Panati, 2012) Assume that H has a Z N -symmetry, then: (i) a Bloch-Floquet (type) unitary decomposition exists: H T N dt H (t), f (H) T N dt f (H) t ; (ii) if p C ( σ(h) ) such that p(h) = p(h) 2 (gap condition) then: π : E (p) T N, E (p) = [p(h) t H (t)] t T N is a Hermitian vector (Hilbert) bundle which is uniquely determined.

4 Topology of the Bloch-bundle Topology of the total space: U (t,ψ) (O,ε) := { (t,ψ ) : t O, ψ H p (t ), ψ ψ < ε } with O t any neighborhood and H p (t ) := p(h) t H (t ).

5 Topology of the Bloch-bundle Bloch functions and continuous sections: ψ : t H p (t) is norm-continuous and T N -periodic iff ψ Γ [ E (p) ] (continuous sections).

6 Topology of the Bloch-bundle Continuous Bloch-frame: {ψ 1,...,ψ m } Γ [ E (p) ] (orth.) basis of H p(t), t T N, m := rk [ E (p) ] There exists iff E (p) is trivial.

7 Topology of the Bloch-bundle Oka s principle: continuous Bloch-frame analytic Bloch-frame The base space T N is a Stein manifold.

8 Periodic Schrödinger operators H = L 2 (R d ), U(n)ψ(x) = ψ(x n), n Z d, ψ L 2 (R d ), H = H(x, i x ) translation invariant (Z d -symmetry). The rank of the Bloch-bundle π : E (p) T d agrees with the number of the energy bands selected by the gap.

9 Periodic Schrödinger operators H = L 2 (R d ), U(n)ψ(x) = ψ(x n), n Z d, ψ L 2 (R d ), H = H(x, i x ) translation invariant (Z d -symmetry). The rank of the Bloch-bundle π : E (p) T d agrees with the number of the energy bands selected by the gap.

10 Periodic Schrödinger operators H = L 2 (R d ), U(n)ψ(x) = ψ(x n), n Z d, ψ L 2 (R d ), H = H(x, i x ) translation invariant (Z d -symmetry). The rank of the Bloch-bundle π : E (p) T d agrees with the number of the energy bands selected by the gap.

11 Triviality of the Bloch-bundle DEFINITION Let H = H(x, i x ) be a Z d -translationary invariant Schrödinger operator acting on L 2 (R d ) and p(h) be a spectral projector in a gap which selects m (crossing) bands. The associated Bloch-bundle π : E (p) T d is trivial iff E (p) T d C m.

12 Triviality of the Bloch-bundle DEFINITION Let H = H(x, i x ) be a Z d -translationary invariant Schrödinger operator acting on L 2 (R d ) and p(h) be a spectral projector in a gap which selects m (crossing) bands. The associated Bloch-bundle π : E (p) T d is trivial iff E (p) T d C m. THEOREM (F. B. Peterson, 1959) Let π : E T d be a rank m Hermitian vector bundle which satisfies d 2m, (stable rank condition). Then E is trivial iff all the Chern classes c j [E ] H 2j (T d,z), with j = 1,..., d/2, are trivial. Moreover d = 1 triviality.

13 Triviality of the Bloch-bundle DEFINITION Let H = H(x, i x ) be a Z d -translationary invariant Schrödinger operator acting on L 2 (R d ) and p(h) be a spectral projector in a gap which selects m (crossing) bands. The associated Bloch-bundle π : E (p) T d is trivial iff E (p) T d C m. Consequences of the triviality: Existence of an exponentially localized system of Wannier functions for the subspace H p = p(h) L 2 (R d ); Let H ε be an adiabatic perturbation such that H = H ε=0, then one can construct an effective operator h ε eff (H ε,p) = p(h) U ε H ε U 1 ε p(h) + O(ε ).

14 Exponentially localized Wannier system H periodic, Σ σ(h) gapped energy band, p Σ spec. projection.

15 Exponentially localized Wannier system H periodic, Σ σ(h) gapped energy band, p Σ spec. projection. Bloch-Floquet (or Wannier) decomposition: L 2 (R d ) W dk L 2 W...W 1 (V ), p Σ T d T d dk p Σ (k).

16 Exponentially localized Wannier system H periodic, Σ σ(h) gapped energy band, p Σ spec. projection. Bloch-Floquet (or Wannier) decomposition: L 2 (R d ) W Generalized Bloch frame: dk L 2 W...W 1 (V ), p Σ T d ψ j : T d L 2 (V ), s.t. p Σ (k) = with m = Tr L 2 (V )[ pσ (k) ]. m j=1 T d dk p Σ (k). ψ j (k) ψ j (k)

17 Exponentially localized Wannier system H periodic, Σ σ(h) gapped energy band, p Σ spec. projection. Bloch-Floquet (or Wannier) decomposition: L 2 (R d ) W Generalized Bloch frame: dk L 2 W...W 1 (V ), p Σ T d ψ j : T d L 2 (V ), s.t. p Σ (k) = with m = Tr L 2 (V )[ pσ (k) ]. Wannier system for Σ: Let w (j) r m j=1 T d dk p Σ (k). ψ j (k) ψ j (k) {w (1),...,w (m) } L 2 (R d ) w (j) := W 1 [ψ j ]. := U(r)w (j), then w (i) s ;w (j) r L 2 (R d ) = δ ij δ sr

18 Exponentially localized Wannier system Completeness: p Σ H p Σ = = m i,j=1 m i,j=1 p Σ = m h (i,j) s r w (i) s s,r Z d ( s Z d h (i,j) w (j) n j=1 n Z d w (j) r 0 w (i) s w (j) w (j) n s + s r =1 where h (i,j) s r := U(s r) w (i) ;H w (j) L 2 (R d ). h (i,j) 1 w (i) s w (j) r +... )

19 Exponentially localized Wannier system Completeness: p Σ H p Σ = = m i,j=1 m i,j=1 p Σ = m h (i,j) s r w (i) s s,r Z d ( s Z d h (i,j) w (j) n j=1 n Z d w (j) r 0 w (i) s w (j) w (j) n s + s r =1 where h (i,j) s r := U(s r) w (i) ;H w (j) L 2 (R d ). Exponentially localized Wannier functions: sup r Z d χ K r w (j) L 2 (R d ) eb r < + for some b > 0 and for some compact K R d. h (i,j) 1 w (i) s w (j) r +... )

20 Exponentially localized Wannier system Localization vs. triviality: Exp. loc. Wannier system {w (1),...,w (m) } Analytic Bloch frame {ψ 1,...,ψ m } Triviality of Bloch bundle E T d. (Paley-Wiener theorem) (Oka s principle)

21 Exponentially localized Wannier system Localization vs. triviality: Exp. loc. Wannier system {w (1),...,w (m) } Analytic Bloch frame {ψ 1,...,ψ m } Triviality of Bloch bundle E T d. (Paley-Wiener theorem) (Oka s principle) Effective tight-binding model: If the Wannier system is Exp. loc., then h (i,j) l 1 if l 1 p Σ H p Σ m i,j=1 h (i,j) 0 ( s Z d h (i,j) 0 w (i) s w (j) s ) + small errors

22 The rôle of the time-reversal symmetry Let H per be the Z d -periodic Schrödinger operator on L 2 (R d ) H per = 1 2 x + V, [V ;U(n)] = 0, n Z d with V real and sufficiently regular. Let (Cψ)(x) = ψ(x) the time-reversal (anti-unitary) operator.

23 The rôle of the time-reversal symmetry Let H per be the Z d -periodic Schrödinger operator on L 2 (R d ) H per = 1 2 x + V, [V ;U(n)] = 0, n Z d with V real and sufficiently regular. Let (Cψ)(x) = ψ(x) the time-reversal (anti-unitary) operator. Then [C;H per ] = 0, (time-reversal symmetry). THEOREM (B. Simon, 1983; G. Panati, 2007) Let p(h per ) be a spectral projector in a gap which selects m bands and π : E (p) T d the related Bloch-bundle. Then c 1 [ E (p)] = 0.

24 The rôle of the time-reversal symmetry Let H per be the Z d -periodic Schrödinger operator on L 2 (R d ) H per = 1 2 x + V, [V ;U(n)] = 0, n Z d with V real and sufficiently regular. Let (Cψ)(x) = ψ(x) the time-reversal (anti-unitary) operator. Consequences of the Peterson s Thorem: d=1: E (p) is trivial for any rank m (W. Kohn, 1959). Very special case: i) C plays no rôle; ii) no magnetic field; m=1: E (p) is trivial for any spatial dimension d (G. Nenciu, 1983; B. Helffer & J. Sjöstrand, 1989): Lin(T d c ) 1 H 1 (T d,z); d=2,3: E (p) is trivial for any rank m > 1 (G. Panati, 2007). Stable rank condition verified and d/2 = 1; d=4: stable rank condition verified if m > 1 but c 2 [ E (p)] =?; d>5: if d > 2m the analysis of Chern classes does not suffice.

25 Magnetic symmetries Let Hper A be the Z d -periodic magnetic Schrödinger operator Hper A = 1 ( i x + A ) 2 + V, [Aj ;U(n)] = 0, n Z d 2 with V = A 0,A = (A 1,...,A d ) real and sufficiently regular.

26 Magnetic symmetries Let Hper A be the Z d -periodic magnetic Schrödinger operator Hper A = 1 ( i x + A ) 2 + V, [Aj ;U(n)] = 0, n Z d 2 with V = A 0,A = (A 1,...,A d ) real and sufficiently regular. Since the presence of A [C;Hper] A 0, (breaking of the time-reversal symmetry). Main question: Is it possible to recover a new (anti-unitary) symmetry C A such that [C A ;H A per] = 0? If yes, the full analysis valid for A = 0 should be still valid!

27 Magnetic symmetries Let H A per be the Z d -periodic magnetic Schrödinger operator Hper A = 1 ( i x + A ) 2 + V, [Aj ;U(n)] = 0, n Z d 2 with V = A 0,A = (A 1,...,A d ) real and sufficiently regular. THEOREM (G. D. & M. Lein, 2011) Let R O(R d ) and U R unitary (resp. anti-unitary) on L 2 (R d ) such that: a) if V is an operator of multiplication then the same holds for U R V U 1 R ; b) U R x U 1 R = R x. Consider the unitary (resp. anti-unitary) operator UR A = Φ A U R where ( ΦA ψ ) (x) = e i [0,x] f R;A(y) dy ψ(x) with f R;A = R 1 (U R A U 1 ) R A. Then (i) UR A ( i x + A ) UR A 1 ( = R i x + A ) ; (ii) [U R ;V ] = 0 [UR A;HA per] = 0.

28 Magnetic parity and magnetic time-reversal symmetry Magnetic parity: Let (Πψ)(x) = ψ( x) be the parity operator: (Π A ψ)(x) = e +i [ ] [0,x] A(y)+A( y) dy ψ( x). Magnetic time-reversal symmetry: Let (Cψ)(x) = ψ(x): (C A ψ)(x) = e i2 [0,x] A(y) dy ψ(x).

29 Magnetic parity and magnetic time-reversal symmetry Magnetic parity: Let (Πψ)(x) = ψ( x) be the parity operator: (Π A ψ)(x) = e +i [ ] [0,x] A(y)+A( y) dy ψ( x). Magnetic time-reversal symmetry: Let (Cψ)(x) = ψ(x): (C A ψ)(x) = e i2 [0,x] A(y) dy ψ(x). Since V is real [C A ;H A per] = 0; If V (x) = V ( x) (center of inversion) [Π A ;H A per] = 0; If A is Z d -periodic [C A ;U(n)] = 0 = [Π A ;U(n)] for all n Z d (hence they factorize through the Bloch-Floquet transform).

30 Magnetic parity and magnetic time-reversal symmetry Magnetic parity: Let (Πψ)(x) = ψ( x) be the parity operator: (Π A ψ)(x) = e +i [ ] [0,x] A(y)+A( y) dy ψ( x). Magnetic time-reversal symmetry: Let (Cψ)(x) = ψ(x): (C A ψ)(x) = e i2 [0,x] A(y) dy ψ(x). THEOREM (G. D. & M. Lein, 2011) Let A,V be Z d -periodic, p(h A per) a projector for m bands and E (p) the associated rank m Bloch-bundle. Then: (i) E (p) f E (p), where f (z) = z for all z T d ; (ii) E (p) E (p) if in addition V (x) = V ( x). Moreover, both (i) or (ii) suffice to prove that: c 2j 1 [ E (p) ] = 0, j = 1,..., d/2.

31 Open problems and prospects d = 4 is the last case saturating the stable rank condition. The triviality of the Bloch-bundle for m (crossing) bands (with projection p) is implied by c 2 = 0 or equivalently 1 lim L 4 Tr( ) χ L W p χ L = 0 L where χ L is the indicator function of [ L/2,L/2] 4, and W p = δ 12 (p)δ 34 (p) δ 13 (p)δ 24 (p) + δ 14 (p)δ 23 (p) with δ ij (p) = (2π) 2[ [x i ;p];[x j ;p] ].

32 Open problems and prospects d = 4 is the last case saturating the stable rank condition. The triviality of the Bloch-bundle for m (crossing) bands (with projection p) is implied by c 2 = 0 or equivalently 1 lim L 4 Tr( ) χ L W p χ L = 0 L where χ L is the indicator function of [ L/2,L/2] 4, and W p = δ 12 (p)δ 34 (p) δ 13 (p)δ 24 (p) + δ 14 (p)δ 23 (p) with δ ij (p) = (2π) 2[ [x i ;p];[x j ;p] ]. The discrete case seems to escape the previous analysis. Haldane showed a discrete model with zero-flux magnetic field and non-zero Chern charges (F. D. M. Haldane, 1988).

33 Open problems and prospects d = 4 is the last case saturating the stable rank condition. The triviality of the Bloch-bundle for m (crossing) bands (with projection p) is implied by c 2 = 0 or equivalently 1 lim L 4 Tr( ) χ L W p χ L = 0 L where χ L is the indicator function of [ L/2,L/2] 4, and W p = δ 12 (p)δ 34 (p) δ 13 (p)δ 24 (p) + δ 14 (p)δ 23 (p) with δ ij (p) = (2π) 2[ [x i ;p];[x j ;p] ]. The discrete case seems to escape the previous analysis. Haldane showed a discrete model with zero-flux magnetic field and non-zero Chern charges (F. D. M. Haldane, 1988). It is expected that the magnetic rotations could play for the QHE on S 2 (Dirac monopol) the same rôle that the magnetic translations play for the QHE on R 2. Link with the non-commutative spheres.

34 ! Thank you for your attention!