ON EFFECTIVE HAMILTONIANS FOR ADIABATIC PERTURBATIONS. Mouez Dimassi Jean-Claude Guillot James Ralston

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1 ON EFFECTIVE HAMILTONIANS FOR ADIABATIC PERTURBATIONS Mouez Dimassi Jean-Claude Guillot James Ralston Abstract We construct almost invariant subspaces and the corresponding effective Hamiltonian for magnetic Bloch bands. We assume that the magnetic and electric potentials are slowly varying perturbations of the potential of a constant magnetic field and a periodic lattice potential, respectively. 1. Introduction In [5] we constructed wave packets for adiabatic perturbations of Schrödinger operators in periodic media. The recent work of Panati-Spohn-Teufel, [14], led us to consider the relation of those constructions to effective Hamiltonians. In 3 we sketch a simple derivation of effective Hamiltonians for these problems, and in 4 we show how one could predict the form of the effective Hamiltonians from the wave packets in [5]. The main simplification in our method is the omission of the Floquet-Bloch transformation. This transformation has many nice properties. In particular, it is unitary, and this makes it useful in studying spectral properties of operators. In the work of Helffer-Sjöstrand [10] and Gérard-Martinez-Sjöstrand [8] this transformation was used quite effectively in the computation of spectra, both of perturbed and effective Hamiltonians. However, if one is simply interested in effective Hamiltonians, the Floquet-Bloch transformation requires that one transform the Hamiltonian by a Fourier integral unitary operator only to transform it back at the end of the calculation.

2 3 M. DIMASSI J.-C. GUILLOT J. RALSTON The method that we use here applies when the eigenspaces of the unperturbed Hamiltonian depend smoothly on quasi-momentum and form trivial bundles over a fundamental domain for the dual lattice.. Preliminaries The Hamiltonian for an electron in a crystal lattice Γ in R 3 in the presence of a constant magnetic field ω = (ω 1, ω, ω 3 ) is given by H 0 = 1 ( ih m x + eω x ) + V (x), (1) where V is a smooth, real-valued potential, periodic with respect to Γ. Here m and e are the mass and charge of the electron. To simplify notation we will use units in which h = m = e = 1. We will assume that Γ is generated by the basis {e 1, e, e 3 } for R 3, Γ = e 1 Z + e Z + e 3 Z, () and let E be the fundamental domain { 3 j=1 t j e j, t j [0, 1)}. We will use the dual lattice Γ = e 1Z + e Z + e 3Z, where e j e k = πδ jk, with the fundamental domain E = { 3 j=1 t j e j, t j [0, 1)}. To realize H 0 as a self-adjoint operator in L (R 3 ) we define it first on the Schwartz functions S(R 3 ), and then take the Friedrichs extension. The resulting operator commutes with the magnetic translations introduced by Zak [19], for γ Γ. We assume that T γ f(x) = e i ω x,γ / f(x γ) (3) ω, Γ Γ 4πZ. With this assumption G = {T γ, γ Γ} is an abelian group, and we can reduce H 0 by the eigenspaces of G, i.e. setting D k = {u H loc (R3 ), T γ u = e ik γ u, γ Γ}, (4)

3 ON EFFECTIVE HAMILTONIANS FOR ADIABATIC considered as a subspace of L (E), H 0 restricted to D k is self-adjoint with compact resolvant. We denote its spectrum by E 1 (k) E (k)... Then by standard results the spectrum of H 0 as an operator in L (R 3 ) is equal to k E m=1 E m (k). Note that, since D k+γ = D k for γ Γ, E m (k + γ ) = E m (k). Standard perturbation theory shows that the function E m (k) is continuous for k R 3 and real analytic in a neighborhood of any k such that E m 1 (k) < E m (k) < E m+1 (k) (5) The closed interval Λ m = k E E m (k) is known as the m-th magnetic Bloch band in the spectrum of H 0. In what follows it will be convenient to replace H 0 acting on D k by H 0 (k) = e ik x H 0 e ik x = ( i x + ω x + k) + V (x) with the domain D = {u H loc (R3 ), T γ u = u, γ Γ}. for all k. As with D k, we consider D as a subspace of L (E). Assumption A. For a given m we will assume that E m satisfies (5) for all k. Under this assumption we can choose the eigenfunction Ψ(x, k) associated to E m (k) to be a real-analytic function of k with values in D, such that E Ψ(x, k) dx = 1, and Assumption B. We assume that H 0 (k)ψ(k) = E m (k)ψ(k) for all k. Ψ(x, k + γ ) = e iγ x Ψ(x, k), γ Γ.

4 34 M. DIMASSI J.-C. GUILLOT J. RALSTON This assumption makes the complex line bundle of the eigenspaces a trivial bundle over the torus the torus, R 3 /Γ. In general one has Ψ(x, k + γ ) = e i(γ x+θ(k,γ )) Ψ(x, k), γ Γ, where θ(k, γ ) is real-valued, and determines the structure of the family of eigenspaces as a complex line bundle. Since θ(k, m 1 e 1 + m e + m 3e 3 ) = m 1θ(k, e 1 ) + m θ(k, e ) + m 3θ(k, e 3 ), when θ(k, γ ) is nonzero, the derivatives of Ψ with k may be unbounded and Ψ may not belong to the class of symbolds B which we introduce below. Thus we need Assumption B. Remark 1. The general method of constructing effective Hamiltonians which we give here will apply under the weaker hypothesis: for a given m there exist p and q such that E m p 1 (k) < E m p (k) and E m+q (k) < E m+q+1 (k) for all k, and the corresponding eigenspaces form a trivial bundle over R 3 /Γ. However, in this case the effective Hamiltonian will be a matrix operator acting on functions with values in C p+q+1, as in [7],[8] and [14]. 3. Main Result The adiabatically perturbed Hamiltonian is ( H ɛ = i x + ω x + A(ɛx)) + V (x) + W (ɛx), where W and A = (A 1, A, A 3 ) are smooth, and bounded together with all of their derivatives. As before, we define H ɛ first on S(R 3 ), and then take the Friedrichs extension to get a self-adjoint operator in L (R 3 ). The essential step in applying multi-scale techniques is simply to consider y = ɛx as a new independent variable in H ɛ. Let ( H ɛ = i x iɛ + ω x + A(y)) + V (x) + W (y).

5 ON EFFECTIVE HAMILTONIANS FOR ADIABATIC Then, for u(x, y) we can define w(x) = u(x, ɛx) and conclude that [ H ɛ u](x, ɛx) = [H ɛ w](x). (6) The identity (6) enables us to solve the Schrödinger equation for H ɛ uniformly in ɛ by solving the Schrödinger equation for H ɛ uniformly in (y, ɛ). The latter might sound more difficult, but it turns out not to be. Let B denote the subspace of C (R 3 R 3 R 3 ) consisting of functions of the form P (x, y, k, ɛ) = P 0 (x, y, k) + ɛp 1 (x, y, k) + + ɛ N P N (x, y, k) such that P (x + γ, y, k, ɛ) = e i ω x,γ / P (x, y, k, ɛ) and sup α β k P j(, y, k) L (E) <, for all α, β N 3. y,k To P B we associate the ɛ-pseudo-differential operator P (x, y, ɛd y, ɛ)f(x, y, ɛ) = (πɛ) 3 e ik (y z)/ɛ P (x, y, k, ɛ)f(z)dzdk, f S(R 3 ). Note that here we are using the standard quantization as opposed to the Weyl quantization. Our main result is the following: Theorem. For every N N there exist P N = F 0 + ɛf ɛ N F N B and H N eff = h 0 + ɛh ɛ N h N B, with the h j s independent of x, such that H ɛ (P N (x, y, ɛd y, ɛ)u) P N (x, y, ɛd y, ɛ)(h N eff (y, ɛd y)u) = O(ɛ N+1 ) (7) for u S(R 3 ). Moreover, considered as an operator from L (R 3 ) into L (E R 3 ), P N is approximately isometric, i.e. PN P N = I + O(ɛ N+1 ). We interpret Heff N as the effective Hamiltonian up to order ɛ N. The leading term in its symbol is h 0 (y, k) = E m (k + A(y)) + W (y). This is the well-known Peierls substitution, [15]. The symbol of h 1 is given by h 1 (y, k) = 1 i E m(k + A(y)) L B i Ψ(, k + A(y), k Ψ(, k + A(y)). (8)

6 36 M. DIMASSI J.-C. GUILLOT J. RALSTON Here B(y) = A(y) and ( L = Im M(y, k) Ψ, Ψ, M(y, k) Ψ, Ψ, M(y, k) Ψ, Ψ ). k k 3 k 3 k 1 k 1 k with M(y, k) = H 0 (k) h 0 (y, k). The function Ψ(x, k + A(y)) is given by Ψ(x, k + A(y)) = Ψ(x, k + A(y)) ẏ + Ψ(x, k + A(y)) k k and ẏ and k are defined by the Hamiltonian system ẏ = (E m(k + A(y)) + W (y)) k k = (E m(k + A(y)) + W (y)). Thus one recognizes i Ψ(, k + A(y)), Ψ(, k + A(y)) as the term generating the Berry phase precession, cf. [13], [17]. The vector L is an angular momentum and L B contributes the Rammal-Wilkinson term to the energy, cf. [1]. Comparing (8) with [14, ()] (in the case l = 1), one sees that they agree completely when one takes into account the difference in the choice of sign in the magnetic potential, A(y), and the use of Weyl quantization in [14]. The sign of the Berry phase term in (8) may appear inconsistent with [5, (9)], but it is not. In [5] (ẏ, k) was the vector field from the Hamiltonian E m (k +A(y)). The proof of the theorem is given in [6], and it is short. One treats x as a parameter, and uses the ɛ-pseudo-differential calculus in y to construct the symbols of the pairs (h 0, F 0 ), (h 1, F 1 ),..., successively so that (7) holds to order O(ɛ N+1 ). Relation to Wave Packets. In [5] and [9], instead of introducing effective Hamiltonians, we constructed wave packets. These packets are nonetheless related to effective Hamiltonians in that one can compute what the effective Hamiltonian must be assuming that there is one from the packets. To see this one can proceed as follows. The packets have the form (here s = ɛt and W (y) = 0) u(x, y, s, ɛ) = e iφ(y,s)/ɛ [f(y, s)ψ(x, φ + A(y)) + O(ɛ)],

7 ON EFFECTIVE HAMILTONIANS FOR ADIABATIC where φ and f are solutions of φ s = E m( φ + A(y)) and f s = E m k ( φ + A(y)) f + (D il B + Ψ, Ψ )f. (9) Here all functions of (k, y) are evaluated at k = k(y, s) = y φ, and D = (1/) y ( k E m ( y φ+a(y))). Assuming that the evolution of f is governed by an effective Hamiltonian H eff = h 0 (y, ɛd y )+ɛh 1 (y, ɛd y )+O(ɛ ), we must have (on bounded intervals in s) [e ish eff /ɛ (e iφ(,0)/ɛ f(, 0)](y, s) = e iφ(y,s)/ɛ f(y, s) + O(ɛ ). (10) Differentiating (10) with respect to s, one concludes i ɛ H eff(e iφ(y,s)/ɛ f(y, s)) = ( i φ ɛ s f + f s )eiφ(y,s)/ɛ + O(ɛ). (11) Using the symbol expansion from the pseudo-differential calculus ɛ[ 1 i e iφ/ɛ H eff (e iφ/ɛ f) = h 0 (y, k)+ h 0 f (y, k) k (y)+ 1 h 0 (y, i j,l k j k k) φ (y)+h 1 (y, l j k)]f(y)+o(ɛ ). (1) l Substituting (1) into (11) and comparing the result with (9), one recovers the formulas given earlier for h 0 (y, k) and h 1 (y, k). Acknowledgements. We wish to thank Professors Panati, Spohn and Teufel for sending us the preprint version of [14] and for several helpful discussions of this work. We especially thank Stefan Teufel for pointing out the necessity of Assumption B here. References [1] Bellissard, J.; Rammal, R., An algebric semi-classical approach to Bloch electrons in a magnetic field, J. Pysique France 51(1990), 1803.

8 38 M. DIMASSI J.-C. GUILLOT J. RALSTON [] Buslaev, V. S., Semi-classical approximation for equations with periodic coefficients, Russian. Math. Surveys, 4 (1987), [3] Chang, M. C.; Niu, Q., Berry phase, hyperorbits, and the Hofstadter spectrum, Phys. Rev. lett. 75(1996), [4] Chang, M. C.; Niu, Q., Berry phase, hyperorbits, and the Hofstadter spectrum: semiclassical in magnetic Bloch bands, Phys. Rev. B 53(1996) [5] Dimassi, M.; Guillot, J.-C.; Ralston, J., Semi-Classical Asymptotics in Magnetic Bloch Bands, J. Phys. A: Math. G., 35 (00), [6] Dimassi, M.; Guillot, J.-C.; Ralston, J., On effective Hamiltonians for adiabatic perturbations of magnetic Schrödinger operators, Asymptotic Analysis 40 (004), [7] Dimassi, M.; Sjöstrand, J., Spectral asymptotics in the semi-classical limit, London Math. Soc. Lecture Note Series, 68. Cambridge University Press, Cambridge, [8] Gérard, C.; Martinez, A.; Sjöstrand, J., A Mathematical Approach to the effective Hamiltonian in perturbed periodic Problems, Commun. Math. Phys., 14 (1991), [9] Guillot, J. C.; Ralston, J.; Trubowitz, E., Semi-classical methods in solid state physics, Commun. Math. Phys., 116 (1988), [10] Helffer, B.; Sjöstrand, J., On diamagnetism and the de Haas - van Alphen effect, Annales I.H.P. (Physique théorique) 5(1990), [11] Hôvermann, F.; Spohn, H.; Teufel, S., Semi-classical limit for the Schrödinger equation with a short scale periodic potential, Comm. Math. Phys. 15 (001), no. 3,

9 ON EFFECTIVE HAMILTONIANS FOR ADIABATIC [1] Horn, W., Semi-classical construction in solid state physics, Commun. P.D.E. 16(1993) [13] Kohomoto, M., Berry s phase of Bloch electrons in electromagnetic fields, J. Phys. Soc. Japan 6(1993), [14] Panati, G.; Spohn, H.; Teufel, S., Effective dynamics for Bloch electrons: Peierls substitution and beyond, Commun. Math. Phys. 4 (003), [15] Peierls, R., Zur Theorie des diamagnetimus von leitungselektronen, Z. Phys. 80(1933), [16] Ralston, J., Magnetic breakdown, Astérisque 10(199), [17] Simon, B., Holonomy, the quantum adiabatic theorem, and Berry s phase, Phys. Rev. Lett. 51(1983), [18] Sundaram, G.; Niu, Q., Wave packet dynamics in slowly perturbed crystals: Gradient corrections and Berry phase effects, Phys. Rev. B 59(1999), [19] Zak, J., Dynamics of electrons in solids in external fields, Phy. Rev. 168(1968), M. Dimassi and J.-C. Guillot J. Ralston Département de Mathématiques University of California Université Paris 13 Los Angeles, CA Villetaneuse, France USA dimassimath.univ-paris13.fr ralstonmath.ucla.edu guillotmath.univ-paris13.fr