The semiclassical. model for adiabatic slow-fast systems and the Hofstadter butterfly

Transcription

1 The semiclassical. model for adiabatic slow-fast systems and the Hofstadter butterfly Stefan Teufel Mathematisches Institut, Universität Tübingen Archimedes Center Crete,

2 1. Reminder: Semiclassics for scalar symbols Consider an ε-pseudodifferential Operator ĥ ε = h(x, iε x ) that is the ε-weyl-quantization of a symbol h : R 2n R, i.e. it acts on functions ψ S(R n ) L 2 (R n ) as (ĥε ψ)(x) := 1 (2πε) n R 2n eip (x y)/ε h ( 1 2 (x + y), p ) ψ(y) dp dy.

3 1. Reminder: Semiclassics for scalar symbols Consider an ε-pseudodifferential Operator ĥ ε = h(x, iε x ) that is the ε-weyl-quantization of a symbol h : R 2n R, i.e. it acts on functions ψ S(R n ) L 2 (R n ) as (ĥε ψ)(x) := 1 (2πε) n R 2n eip (x y)/ε h ( 1 2 (x + y), p ) ψ(y) dp dy. Example: The symbol of the Schrödinger operator ĥε = ε2 2 x + V (x) is h(q, p) = 1 2 p2 + V (q).

4 1. Reminder: Semiclassics for scalar symbols There are two very useful ways of approximating functions of ĥ ε solely in terms of the classical system given by the symbol h.

5 1. Reminder: Semiclassics for scalar symbols There are two very useful ways of approximating functions of ĥ ε solely in terms of the classical system given by the symbol h. Semiclassical approximation of stationary expectations Let f Cb (R) and a S(R2n ), then ( ) tr(f(ĥε ) â ε ) = 1 (2πε) n dqdp f(h(q, p)) a(q, p) + R 2n O(ε2 ).

6 1. Reminder: Semiclassics for scalar symbols There are two very useful ways of approximating functions of ĥ ε solely in terms of the classical system given by the symbol h. Semiclassical approximation of stationary expectations Let f Cb (R) and a S(R2n ), then ( ) tr(f(ĥε ) â ε ) = 1 (2πε) n dqdp f(h(q, p)) a(q, p) + R 2n O(ε2 ). Semiclassical approximation of Heisenberg evolution (Egorov Thm) Let a C b (R2n ) and a(t) := e iĥε t ε â ε e iĥε t ε the corresponding Heisenberg observable. Then uniformly on bounded time intervals. a(t) Op W (a Φ t ) = O(ε 2 ) Here Φ t : R 2n R 2n is the classical flow of h.

7 2. Adiabatic slow-fast systems Consider a system with Hilbert space L 2 (R n ) H f = L 2 (R n, H f ), where H f is the Hilbert space of some quantum system.

8 2. Adiabatic slow-fast systems Consider a system with Hilbert space L 2 (R n ) H f = L 2 (R n, H f ), where H f is the Hilbert space of some quantum system. For an operator-valued symbol H : R 2n L(H f ) we define Ĥε acting on functions ψ L 2 (R n, H f ) again as (Ĥε ψ)(x) := 1 (2πε) n R 2n eip (x y)/ε H ( 1 2 (x + y), p ) ψ(y) dp dy.

9 2. Adiabatic slow-fast systems Consider a system with Hilbert space L 2 (R n ) H f = L 2 (R n, H f ), where H f is the Hilbert space of some quantum system. For an operator-valued symbol H : R 2n L(H f ) we define Ĥε acting on functions ψ L 2 (R n, H f ) again as (Ĥε ψ)(x) := 1 (2πε) n Example: The molecular Hamiltonian R 2n eip (x y)/ε H ( 1 2 (x + y), p ) ψ(y) dp dy. ε2 2 x 1 2 y + V (x, y) = Ĥε on L 2 (R n x R m y ) = L 2 (R n x, L 2 (R m y )) is the Weyl-quantization of the operator-valued symbol H(q, p) = 1 2 p2 1 2 y + V (q, y).

10 2. Adiabatic slow-fast systems Since H(q, p) is operator-valued, it cannot be interpreted as a classical Hamiltonian function.

11 2. Adiabatic slow-fast systems Since H(q, p) is operator-valued, it cannot be interpreted as a classical Hamiltonian function. But H(q, p) is self-adjoint for each (q, p) R 2n. Its eigenvalues e(q, p) are real-valued and thus define Hamiltonian functions on phase space.

12 2. Adiabatic slow-fast systems Since H(q, p) is operator-valued, it cannot be interpreted as a classical Hamiltonian function. But H(q, p) is self-adjoint for each (q, p) R 2n. Its eigenvalues e(q, p) are real-valued and thus define Hamiltonian functions on phase space. It is well known by now that on certain subspaces the two results presented for the scalar case before hold for the Hamiltonian function e(q, p), but only with O(ε) instead of O(ε 2 ).

13 2. Adiabatic slow-fast systems Since H(q, p) is operator-valued, it cannot be interpreted as a classical Hamiltonian function. But H(q, p) is self-adjoint for each (q, p) R 2n. Its eigenvalues e(q, p) are real-valued and thus define Hamiltonian functions on phase space. It is well known by now that on certain subspaces the two results presented for the scalar case before hold for the Hamiltonian function e(q, p), but only with O(ε) instead of O(ε 2 ). The following results show, that an approximation up to errors of order O(ε 2 ) can be obtained by suitably modifying the classical system.

14 2. Adiabatic slow-fast systems Adiabatic projections: (Littlejohn-Flynn, Emmrich-Weinstein, Brummelhuis-Nourrigat, Martinez-Nenciu-Sordoni, Panati-Spohn-T.) If H(q, p) has an eigenvalue e(q, p) with spectral projection π 0 (q, p) that is separated by a gap from the remainder of the spectrum of H(q, p), then there exists a unique symbol π ε (q, p) with π ε (q, p) = π 0 (q, p) + O(ε) such that π ε is an orthogonal projection, i.e. ( π ε ) 2 = π ε and ( π ε ) = π ε, that commutes with the Hamiltonian Ĥε up to small errors, [Ĥε, π ε ] = O(ε ). Hence Ran π ε is almost invariant under the group e iĥε t/ε, [e iĥε t/ε, π ε ] = O(ε t ).

15 3. Results: The modified classical system Definition 1: The modified classical system With an isolated simple eigenvalue e(q, p) of H(q, p) with spectral projection π 0 (q, p) we associate a classical Hamiltonian system (h ε, ω ε ) with Hamilton function and symplectic form h ε (q, p) := e(q, p) + εm(q, p) ω ε (q, p) = ω 0 + εω(q, p).

16 3. Results: The modified classical system Definition 1: The modified classical system With an isolated simple eigenvalue e(q, p) of H(q, p) with spectral projection π 0 (q, p) we associate a classical Hamiltonian system (h ε, ω ε ) with Hamilton function and symplectic form h ε (q, p) := e(q, p) + εm(q, p) ω ε (q, p) = ω 0 + εω(q, p). Here and M(z) = i 2 tr H f ({π 0 (z) H(z) π 0 (z)}) Ω(z) αβ = i tr Hf ( π0 (z)[ zα π 0 (z), zβ π 0 (z)] ) with z = (q, p) and α, β = 1,..., 2n.

17 3. Results: Semiclassical approximations (jointly with H. Stiepan) Theorem 1 (Equilibrium expectations) Let f C 0 (R) and a S(R2n ), then Tr( π ε f(h ε ) â 1) = 1 (2πε) n ( R 2n dλε f(h ε (q, p)) a(q, p) + O(ε 2 ) where λ ε is the Liouville measure of ω ε. )

18 3. Results: Semiclassical approximations (jointly with H. Stiepan) Theorem 1 (Equilibrium expectations) Let f C 0 (R) and a S(R2n ), then Tr( π ε f(h ε ) â 1) = 1 (2πε) n ( R 2n dλε f(h ε (q, p)) a(q, p) + O(ε 2 ) where λ ε is the Liouville measure of ω ε. ) Theorem 2 (Egorov) Let a C b (R2n ) and a(t) := e ihε t ε (â 1) e ihε t ε the corresponding Heisenberg observable. Then π ε ( a(t) Op W (a Φ t ε )) π ε = O(ε 2 ) uniformly on bounded time intervals. Here Φ t ε : R 2n R 2n is the classical flow of (h ε, ω ε ).

19 4. Example: The Hofstadter Hamiltonian Consider a single particle on the lattice Z 2 subject to a constant magnetic field B = ( ) 0 B B 0. Its Hamiltonian is H = Tn B n =1 on l 2 (Z 2 ) with magnetic translations (T B n ψ) j = e i 2 n Bj ψ j n.

20 4. Example: The Hofstadter Hamiltonian Consider a single particle on the lattice Z 2 subject to a constant magnetic field B = ( ) 0 B B 0. Its Hamiltonian is H = Tn B n =1 on l 2 (Z 2 ) with magnetic translations (T B n ψ) j = e i 2 n Bj ψ j n. For a rational magnetic field B = B 0 = 2π p q a Bloch-Floquet transformation turns H into a matrix-valued multiplication operator H 0 (k) = 2 cos(k 2 ) e ik 1 0 e ik 1 e ik 1 2 cos(k 2 + B 0 ) e ik e ik 1 2 cos(k 2 + 2B 0 ) e ik 1 e ik 1 0 e ik 1 2 cos(k 2 + (q 1)B 0 ) on L 2 ([0, 2π/q) [0, 2π); C q ).

21 4. Example: The Hofstadter Hamiltonian

22 4. Example: The Hofstadter Hamiltonian

23 4. Example: The Hofstadter Hamiltonian

24 4. Example: The Hofstadter Hamiltonian

25 4. Example: The Hofstadter Hamiltonian For a magnetic field B = B 0 +εb and in the presence of an electric potential V ε (n) = V (εn) the same transformation yields the Hamiltonian H ε = H 0 (k b iε k 2, k b iε k 1 ) + V (iε k ) =: H 0 (k b iε k) + V (iε k ) on the same space L 2 ([0, 2π/q) [0, 2π); C q ).

26 4. Example: The Hofstadter Hamiltonian For a magnetic field B = B 0 +εb and in the presence of an electric potential V ε (n) = V (εn) the same transformation yields the Hamiltonian H ε = H 0 (k b iε k 2, k b iε k 1 ) + V (iε k ) =: H 0 (k b iε k) + V (iε k ) on the same space L 2 ([0, 2π/q) [0, 2π); C q ). Define the matrix-valued function H(k, r) := H 0 (k br 2, k br 1) + V (r) on phase space T 2 k R2 r, then H ε is the ε-weyl quantization of H, i.e. H ε = Op W (H) = H(k, iε k ).

27 4. Example: The Hofstadter Hamiltonian For a magnetic field B = B 0 +εb and in the presence of an electric potential V ε (n) = V (εn) the same transformation yields the Hamiltonian H ε = H 0 (k b iε k 2, k b iε k 1 ) + V (iε k ) =: H 0 (k b iε k) + V (iε k ) on the same space L 2 ([0, 2π/q) [0, 2π); C q ). Define the matrix-valued function H(k, r) := H 0 (k br 2, k br 1) + V (r) on phase space T 2 k R2 r, then H ε is the ε-weyl quantization of H, i.e. H ε = Op W (H) = H(k, iε k ). Now ε 0 corresponds to a semiclassical limit, however, for a Hamiltonian with a matrix-valued symbol and the previous theory applies.

28 4. The modified classical system in solids For the Hofstadter Hamiltonian at fixed B 0 one finds h ε (k, r) := e(k br) + V (r) + ε b M(k br) and Ω(k, r) = Ω(k br), where e, M, Ω are smooth and 2π/q-periodic functions.

29 4. The modified classical system in solids For the Hofstadter Hamiltonian at fixed B 0 one finds h ε (k, r) := e(k br) + V (r) + ε b M(k br) and Ω(k, r) = Ω(k br), where e, M, Ω are smooth and 2π/q-periodic functions. The corresponding Hamiltonian equations of motion are ṙ = (1 εbω(κ)) κ h ε (κ, r) + εω(κ) V (r) κ = V (r) + b ṙ where κ = k br is the kinetic momentum.

30 4. The modified classical system in solids For the Hofstadter Hamiltonian at fixed B 0 one finds h ε (k, r) := e(k br) + V (r) + ε b M(k br) and Ω(k, r) = Ω(k br), where e, M, Ω are smooth and 2π/q-periodic functions. The corresponding Hamiltonian equations of motion are ṙ = (1 εbω(κ)) κ h ε (κ, r) + εω(κ) V (r) κ = V (r) + b ṙ where κ = k br is the kinetic momentum. The Liouville measure is dλ ε = (1 + εbω(κ))dqdκ

31 4. The modified classical system in solids

32 4. The modified classical system in solids

33 4. The modified classical system in solids

34 4. The modified classical system in solids

35 4. The modified classical system in solids

36 5. Application: thermodynamic potential The thermodynamic pressure in the Hofstadter model is ( ( p(β, µ) := β 1 lim ε 2 Λ R 2 Λ Tr χ Λ (εx) ln 1 + e β(ĥε µ) )).

37 5. Application: thermodynamic potential The thermodynamic pressure in the Hofstadter model is ( ( p(β, µ) := β 1 lim ε 2 Λ R 2 Λ Tr χ Λ (εx) ln 1 + e β(ĥε µ) For B 0 = 2π p q β p(β, µ, b) = = lim Λ R 2 1 Λ (2π) 2 = 1 (2π) 2 q j=1 and ε = b we have the semiclassical approximation q j=1 T 2 d 2 k ( dλ b j χ Λ(r) ln 1 + e β(hb j (k,r) µ)) + O(b 2 ) T 2 R n ( 1 + bωj (k) ) ln ( 1 + e β(e j(k)+bm j (k) µ) ) + O(b 2 ) )).

38 5. Application: thermodynamic potential The thermodynamic pressure in the Hofstadter model is ( ( p(β, µ) := β 1 lim ε 2 Λ R 2 Λ Tr χ Λ (εx) ln 1 + e β(ĥε µ) For B 0 = 2π p q β p(β, µ, b) = = lim Λ R 2 1 Λ (2π) 2 = 1 (2π) 2 q j=1 and ε = b we have the semiclassical approximation q j=1 T 2 d 2 k ( dλ b j χ Λ(r) ln 1 + e β(hb j (k,r) µ)) + O(b 2 ) T 2 R n ( 1 + bωj (k) ) ln ( 1 + e β(e j(k)+bm j (k) µ) ) + O(b 2 ) )). From this one can compute e.g. the density ρ(β, µ, b) = p(β, µ, b) µ = 1 (2π) 2 q j=1 T 2 d 2 k ( 1 + bωj (k) ) f β,µ (h b j (k)) + O(b2 ).

39 5. Application: thermodynamic potential The resulting formula for the magnetization is p(β, µ, b) m(β, µ, b) = = b q 1 (2π) 2 d 2 k ((1 + bω j ) M j f β,µ (h b j ) + β (1 1 ln + e β(hb j µ)) ) Ω j j=1 T 2 which is correct for b = 0, m(β, µ, 0) = 1 (2π) 2 q j=1 d 2 ( k Mj f β,µ (e j ) + β 1 ln ( 1 + e β(e j µ) ) ) Ω j. T 2,

40 5. Application: thermodynamic potential The resulting formula for the magnetization is p(β, µ, b) m(β, µ, b) = = b q 1 (2π) 2 d 2 k ((1 + bω j ) M j f β,µ (h b j ) + β (1 1 ln + e β(hb j µ)) ) Ω j j=1 T 2 which is correct for b = 0, m(β, µ, 0) = 1 (2π) 2 q j=1 d 2 ( k Mj f β,µ (e j ) + β 1 ln ( 1 + e β(e j µ) ) ) Ω j. T 2, This formula was obtained rigorously also as a special case of a general formula for disordered systems (Schulz-Baldes, T., 2012).

41 5. Application: thermodynamic potential The resulting formula for the magnetization is p(β, µ, b) m(β, µ, b) = = b q 1 (2π) 2 d 2 k ((1 + bω j ) M j f β,µ (h b j ) + β (1 1 ln + e β(hb j µ)) ) Ω j j=1 T 2 which is correct for b = 0, m(β, µ, 0) = 1 (2π) 2 q j=1 d 2 ( k Mj f β,µ (e j ) + β 1 ln ( 1 + e β(e j µ) ) ) Ω j. T 2, This formula was obtained rigorously also as a special case of a general formula for disordered systems (Schulz-Baldes, T., 2012). For β = this formula was first found by Gat and Avron in 2003 and the general formula by Niu et al. in 2005.

42 5. Application: Quantum Hall current In order to compute the current from a fully occupied band observe that Ĵ(t) = d dt x(t) = 1 d ε dtˆr(t) and that according to the Egorov theorem the symbol of ˆr(t) is approximately r(t) = Φ t r + O(ε 2 ).

43 5. Application: Quantum Hall current In order to compute the current from a fully occupied band observe that Ĵ(t) = d dt x(t) = 1 d ε dtˆr(t) and that according to the Egorov theorem the symbol of ˆr(t) is approximately r(t) = Φ t r + O(ε 2 ). If one applies a constant electric field εe, the Hamiltonian becomes h ε (k) = e(k br) + r E + ε b M(k br) and its Hamiltonian vector field is ṙ = (1 εbω(k)) k h ε (k) + εω(k)e.

44 5. Application: Quantum Hall current If one applies a constant electric field εe, the Hamiltonian becomes h ε (k) = e(k br) + r E + ε b M(k br) and its Hamiltonian vector field is ṙ = (1 εbω(k)) k h ε (k) + εω(k)e. Thus the current density from a filled band is J(b) = lim ε 2 Λ R 2 Λ Tr (ˆπ χ Λ (εx) Ĵ ) = 1 (2π) 2 ε T 2 d 2 k (1 + εbω(k)) ṙ(k) + O(ε) = E 1 (2π) 2 d 2 k Ω(k) + O(ε) T 2 and thus independent of b at leading order.

45 6. Sketch of the proof Proposition 1 It holds that and thus π ε #h ε #π ε π ε #H#π ε = O(ε 2 ) ˆπ ε ĥ ε ˆπ ε ˆπ ε Ĥ ε ˆπ ε = O(ε 2 ).

46 6. Sketch of the proof Proposition 1 It holds that and thus π ε #h ε #π ε π ε #H#π ε = O(ε 2 ) ˆπ ε ĥ ε ˆπ ε ˆπ ε Ĥ ε ˆπ ε = O(ε 2 ). The proof is essentially a computation using the expansion formula A#B A 0 B 0 + ε(a 1 B 0 + A 0 B 1 i 2 {A 0, B 0 }) + O(ε 2 ) of the Moyal product.

47 6. Sketch of the proof Theorem 1 (Equilibrium expectations) Let f C 0 (R) and a S(R2n ), then Tr( π ε f(h ε ) â 1) = 1 (2πε) n R 2n dλε f(h ε (q, p)) a(q, p) + O(ε 2 n ). Proof: Assume dimh f <. Then Tr (ˆπf(Ĥ) â ) = Tr (ˆπf(Ĥ) ˆπ â ) + O(ε ) = Tr (ˆπf(ˆπĤˆπ) ˆπ â ) + O(ε ) = Tr (ˆπf(ˆπĥˆπ) ˆπ â ) + O(ε 2 n ) = Tr (ˆπf(ĥ) ˆπ â ) + O(ε 2 n ).

48 6. Sketch of the proof For Weyl operators it holds without error that Tr(Â ˆB) = 1 (2πε) n dqdp tr(a(q, p) B(q, p)) and for scalar symbols the functional calculus for pseudo-differential operators implies that f(ĥ) f(h) = O(ε 2 ).

49 6. Sketch of the proof For Weyl operators it holds without error that Tr(Â ˆB) = 1 (2πε) n dqdp tr(a(q, p) B(q, p)) and for scalar symbols the functional calculus for pseudo-differential operators implies that f(ĥ) f(h) = O(ε 2 ). Hence Tr (ˆπf(ĥ) ˆπ â ) = = 1 (2πε) n 1 (2πε) n dqdp tr (f(h(q, p)) (π#a#π)(q, p)) + O(ε 2 n ) dqdp f(h(q, p)) tr ((π#a#π)(q, p)) + O(ε 2 n ).

50 6. Sketch of the proof For Weyl operators it holds without error that Tr(Â ˆB) = 1 (2πε) n dqdp tr(a(q, p) B(q, p)) and for scalar symbols the functional calculus for pseudo-differential operators implies that f(ĥ) f(h) = O(ε 2 ). Hence Tr (ˆπf(ĥ) ˆπ â ) = = Using 1 (2πε) n 1 (2πε) n dqdp tr (f(h(q, p)) (π#a#π)(q, p)) + O(ε 2 n ) dqdp f(h(q, p)) tr ((π#a#π)(q, p)) + O(ε 2 n ). π 0 π 1 π 0 = i 2 π 0{π 0, π 0 }π 0 one computes tr (π#a#π) = a (1 + iεtr (π 0 {π 0, π 0 })) + O(ε 2 ).

51 6. Sketch of the proof Theorem 2 (Egorov) Let a C b (R2n ) and A(t) := e ihε t ε (â 1) e ihε t ε the corresponding Heisenberg observable. Then π ε ( A(t) Op W (a Φ t ε) ) π ε = O(ε 2 ) uniformly on bounded time intervals. Proof: Let a(t) := a Φ t ε then one finds that [ˆπĤˆπ, ˆπ a(t)ˆπ ] = ε i ˆπ [ ĥ, a(t) ] ˆπ + ε i ˆπ [ [ĥ, ˆπ], [ a(t), ˆπ] ] ˆπ + O(ε 2 ) i ε = ˆπ Op W ( {h, a(t)} iεtr (π 0 [{h, π 0 }, {a(t), π 0 }]) I) ˆπ.

52 6. Sketch of the proof In order to compute t a(t) recall that the Hamiltonian vector-field X h with respect to ω ε is X α h = (ω ε) αβ β h and thus by definition of a(t) and Ω we have a(t) t = ( q a p a ) T ( ε Ω pp E n + εω qp E n + εω pq ε Ω qq ) ( q h p h = {h, a(t)} iε tr Hf (π 0 [{h, π 0 }, {a(t), π 0 }]). )

53 7. Literature Heuristic derivation of the semiclassical model in solids: - Sundaram, Niu, Phys. Rev. B (1999) Magnetization of the Hofstadter model: - Gat, Avron, Phys. Rev. Lett. (2003) Mathematical derivation of Egorov for B 0 = 0: - Panati, Spohn, Teufel, Comm. Math. Phys. (2003) Colored Butterfly: - Osadchy, Avron, J. Math. Phys. (2001) Numerous other applications of the semiclassical model in solids: - Xiao, Chang, Niu, Rev. Mod. Phys. (2010).

54 7. References Mathematical details of the general construction: - Stiepan, Teufel, arxiv: v2 (2012). Mathematical details for magnetic Bloch bands: - Stiepan, Teufel, in preparation. Synopsis including also time dependent Hamiltonians: - Teufel, arxiv: v1, to appear in EPL (2012).

55 7. References Mathematical details of the general construction: - Stiepan, Teufel, arxiv: v2 (2012). Mathematical details for magnetic Bloch bands: - Stiepan, Teufel, in preparation. Synopsis including also time dependent Hamiltonians: - Teufel, arxiv: v1, to appear in EPL (2012). Thanks!