Wavepackets in inhomogeneous periodic media: Effective particle-field dynamics and Berry curvature

Watson, Alexander B.; Lu, Jianfeng; Weinstein, Michael I.

doi:10.1063/1.4976200

We consider a model of an electron in a crystal moving under the influence of an external electric field: Schrödinger’s equation with a potential which is the sum of a periodic function and a general smooth function. We identify two dimensionless parameters: (re-scaled) Planck’s constant and the ratio of the lattice spacing to the scale of variation of the external potential. We consider the special case where both parameters are equal and denote this parameter ϵ. In the limit $ϵ ↓ 0$ ⁠, we prove the existence of solutions known as semiclassical wavepackets which are asymptotic up to “Ehrenfest time” $t \sim \ln 1 / ϵ$ ⁠. To leading order, the center of mass and average quasi-momentum of these solutions evolve along trajectories generated by the classical Hamiltonian given by the sum of the Bloch band energy and the external potential. We then derive all corrections to the evolution of these observables proportional to ϵ. The corrections depend on the gauge-invariant Berry curvature of the Bloch band and a coupling to the evolution of the wave-packet envelope, which satisfies Schrödinger’s equation with a time-dependent harmonic oscillator Hamiltonian. This infinite dimensional coupled “particle-field” system may be derived from an “extended” ϵ-dependent Hamiltonian. It is known that such coupling of observables (discrete particle-like degrees of freedom) to the wave-envelope (continuum field-like degrees of freedom) can have a significant impact on the overall dynamics.

I. INTRODUCTION

In this work we study the non-dimensionalized time-dependent Schrödinger equation for $ψ^{ϵ} (x, t) : ℝ^{d} \times [0, \infty) \to ℂ$ ⁠,

\begin{matrix} i ϵ \partial_{t} ψ^{ϵ} & = & - \frac{1}{2} ϵ^{2} Δ_{x} ψ^{ϵ} + V (\frac{x}{ϵ}) ψ^{ϵ} + W (x) ψ^{ϵ}, \\ ψ^{ϵ} (x, 0) & = & ψ_{0}^{ϵ} (x), \end{matrix}

(1.1)

where $ϵ$ is a positive real parameter which we assume to be small: $ϵ ≪ 1$ ⁠. We assume throughout that the function V is smooth and periodic with respect to a d-dimensional lattice Λ so that

V (z + v) = V (z) for all v \in Λ, z \in ℝ^{d},

(1.2)

and that W is sufficiently smooth with uniformly bounded derivatives. Equation (1.1) is a well-studied model in condensed matter physics of the dynamics of an electron in a crystal under the independent-particle approximation,¹ whose periodic effective potential due to the atomic nuclei is specified by V, under the influence of the external electric field generated by a “slowly varying” potential W.

In this work, we rigorously derive a family of explicit asymptotic solutions of (1.1) known as semiclassical wavepackets. We then derive the equations of motion of the center of mass and average quasi-momentum of these solutions, including corrections proportional to ϵ.

At order $ϵ^{0}$ ⁠, the mean position and momentum of the semi-classical wavepacket evolve along the classical trajectories associated with the “Bloch band” Hamiltonian $H_{n} := E_{n} (p) + W (q)$ ⁠, where $p \mapsto E_{n} (p)$ is the dispersion relation associated with the nth spectral (Bloch) band of the periodic Schrödinger operator $- \frac{1}{2} Δ_{z} + V (z)$ ⁠. The order $ϵ$ corrections to the leading order equations of motion depend on the gauge-invariant Berry curvature of the Bloch band and the wavepacket envelope. Through order ϵ, the system governing appropriately defined mean position $Q^{ϵ} (t)$ ⁠, mean momentum $P^{ϵ} (t)$ ⁠, and wave-amplitude profile $a^{ϵ} (y, t)$ is a closed system of Hamiltonian type (Theorem 1.2). When d = 3, this system takes the form

\begin{matrix} \begin{matrix} {\dot{Q}}^{ϵ} (t) \\ {\dot{P}}^{ϵ} (t) \end{matrix} \begin{matrix} = \\ = \end{matrix} \underset{\begin{matrix} Dynamics generated by \\ “Bloch band” Hamiltonian \\ H_{n} := E_{n} (P^{ϵ}) + W (Q^{ϵ}) \end{matrix}}{\underset{⏟}{\begin{matrix} {\nabla_{P}}^{ϵ} H_{n} (Q^{ϵ} (t), P^{ϵ} (t)) \\ - {\nabla_{Q}}^{ϵ} H_{n} (Q^{ϵ} (t), P^{ϵ} (t)) \end{matrix}}} \begin{matrix} + \\ + \end{matrix} \begin{matrix} ϵ { \\ ϵ { \end{matrix} \underset{\begin{matrix} Anomalous velocity due to \\ Berry curvature \end{matrix}}{\underset{⏟}{\begin{matrix} - {\dot{P}}^{ϵ} (t) \times \nabla_{P^{ϵ}} \times A_{n} (P^{ϵ} (t)) \end{matrix}}} \underset{\begin{matrix} “Particle - field” \\ coupling to \\ envelope a^{ϵ} \end{matrix}}{\underset{⏟}{\begin{matrix} + C_{1} [a^{ϵ}] (t) \\ + C_{2} [a^{ϵ}] (t) \end{matrix}}} \begin{matrix} } \\ } \end{matrix}, \\ i \partial_{t} a^{ϵ} = H_{n}^{ϵ} (t) a^{ϵ}; H_{n}^{ϵ} (t) := \underset{\begin{matrix} Quantum harmonic oscillator Hamiltonian \\ with parametric forcing through Q^{ϵ} (t), P^{ϵ} (t) \end{matrix}}{\underset{⏟}{- \frac{1}{2} \nabla_{y} \cdot D_{P^{ϵ}}^{2} E_{n} (P^{ϵ} (t)) \nabla_{y} + \frac{1}{2} y \cdot D_{Q^{ϵ}}^{2} W (Q^{ϵ} (t)) y}}, \end{matrix}

(1.3)

where $D_{P^{ϵ}}^{2} E_{n}, D_{Q^{ϵ}}^{2} W$ denote the Hessian matrices and $A_{n}$ is the Berry connection (1.26). For the explicit forms of C₁[a], C₂[a] and the generalization of (1.3) to arbitrary dimensions $d \geq 1$ ⁠, see (1.45). The derivation of the form of the anomalous velocity displayed in (1.3) is given in Remark 1.11.

The “particle-field” dynamical system (1.3) appears to be new and contains terms which are not accounted for in the works of Niu et al.² The system reduces, in the case of Gaussian initial data and zero periodic background V = 0, to that presented in Proposition 4.4 of Ref. 3 (see also Ref. 4).

The asymptotic solutions and effective Hamiltonian system (1.3) provide an approximate description of the dynamics of the full PDE (1.1) up to “Ehrenfest time” $t \sim \ln 1 / ϵ$ ⁠, known to be the general limit of applicability of wavepacket, or coherent state, approximations.⁵ The validity of the approximation relies on an extension of the result of Carles and Sparber⁶ (Theorem 1.1)

Our methods are applicable when the wavepacket is spectrally localized in a Bloch band which has crossings (degeneracies), as long as the distance in phase space between the average quasi-momentum of the wavepacket and any crossing is uniformly bounded below independent of $ϵ$ (see Assumption 1.1). We do not attempt a description of wavepacket dynamics when this distance $↓ 0$ (propagation through a band crossing), or at an avoided crossing where the separation between bands is proportional to ϵ. We believe that both of these cases may be studied by adapting the work of Hagedorn and Joye^7,8 on wavepacket dynamics in the Born-Oppenheimer approximation of molecular dynamics to the model (1.1).

Our methods are also applicable, with some modifications (see Section I E), to potentials with the general two-scale form $U (\frac{x}{ϵ}, x)$ ⁠, where U is periodic in its first argument,

U (z + v, x) = U (z, x) for all z, x \in ℝ^{d}, v \in Λ

(1.4)

and U(z, x) is “nonseparable,” i.e., cannot be written as the sum of a periodic potential V(z) and an “external” potential W(x). For details, see Ref. 9. For ease of presentation we consider in this work only the “separable” case (1.1).

The semiclassical wavepacket ansatz was introduced by Heller¹⁰ and Hagedorn¹¹ to study the uniform background case (V = 0) of (1.1). See also related work on Gaussian beams.¹² Hagedorn then extended this theory to the case where the potential W(x) is replaced by an x-dependent operator in his study of the Born-Oppenheimer approximation of molecular dynamics.⁷ Semiclassical wavepacket solutions of (1.1) in the periodic background case (⁠ $V \neq 0$ ⁠) were then constructed by Carles and Sparber.⁶

The anomalous velocity term in (1.3) was first derived by Karplus and Luttinger.¹³ For a derivation in terms of Berry curvature of the Bloch band, see the work of Chang and Niu¹⁴ (see also Ref. 2). It was then derived rigorously by Panati, Spohn, and Teufel¹⁵ (see also Ref. 16). This term is responsible for the “intrinsic contribution” to the anomalous Hall effect, which occurs in solids with broken time-reversal symmetry (see the work of Nagaosa et al.¹⁷ and references therein). The anomalous velocity due to Berry curvature is better known in optics as the spin Hall effect of light and was experimentally observed by Bliokh et al.¹⁸

A. Dimensional analysis, derivation of equation (1.1)

In this section we derive the non-dimensionalized equation (1.1) starting with the Schrödinger equation in physical units,

i ℏ \partial_{t} ψ = - \frac{ℏ^{2}}{2 m} Δ_{x} ψ + V (x) ψ + W (x) ψ,

(1.5)

where $ℏ$ is the reduced Planck constant and m is the mass of an electron. This analysis is based on those given in Refs. 16 and 19. Define l as the lattice constant, and let $τ$ denote the quantum time scale,

τ = \frac{m l^{2}}{ℏ} .

(1.6)

Let L, T denote macroscopic length and time scales. We assume that the periodic potential V acts on the “fast quantum scale” and the W acts on the “slow macroscopic scale,”

V (x) = \frac{m l^{2}}{τ^{2}} \tilde{V} (\frac{x}{l}), W (x) = \frac{m L^{2}}{T^{2}} \tilde{W} (\frac{x}{L}) .

(1.7)

After re-scaling x, t by the macroscopic length and time scales,

\tilde{x} := \frac{x}{L}, \tilde{t} := \frac{t}{T}, \tilde{ψ} (\tilde{x}, \tilde{t}) := ψ (x, t),

(1.8)

(1.5) becomes

\frac{i ℏ}{T} \partial_{\tilde{t}} \tilde{ψ} = - \frac{ℏ^{2}}{2 m L^{2}} Δ_{\tilde{x}} \tilde{ψ} + \frac{m l^{2}}{τ^{2}} \tilde{V} (\frac{L \tilde{x}}{l}) ψ + \frac{m L^{2}}{T^{2}} \tilde{W} (\tilde{x}) ψ .

(1.9)

We now identify two dimensionless parameters. Let h denote a scaled Planck’s constant, and $ϵ$ the ratio of the lattice constant to the macroscopic scale,

h := \frac{ℏ T}{m L^{2}}, ϵ := \frac{l}{L} .

(1.10)

Writing (1.9) in terms of $h, ϵ$ and dropping the tildes we arrive at

i h \partial_{t} ψ^{h, ϵ} = - \frac{h^{2}}{2} Δ_{x} ψ^{h, ϵ} + \frac{h^{2}}{ϵ^{2}} V (\frac{x}{ϵ}) ψ^{h, ϵ} + W (x) ψ^{h, ϵ},

(1.11)

where we have written $ψ^{h, ϵ} (x, t)$ to emphasize the dependence of the solution on both parameters. We obtain the problem depending only on $ϵ$ (1.1) by setting $h = ϵ$ ⁠. Therefore, the limit $ϵ ↓ 0$ in (1.1) corresponds to sending to zero the ratio of the lattice spacing l to the scale of inhomogeneity L and Planck’s constant (appropriately re-scaled) to zero at the same rate.

Remark 1.1. Other scalings of the Schrödinger (1.11) have been considered. For example, the scaling corresponding to h fixed and $ϵ ↓ 0$ is considered in Refs. 20–22, and for the nonlinear Schrödinger/Gross-Pitaevskii (NLS/GP) equation in Refs. 23 and 24. Here, the dynamics are governed by a homogenized effective mass Schrödinger equation (linear, respectively, nonlinear). The articles, Refs. 22 and 24, concern the bifurcations of bound states of (1.11) or NLS/GP from spectral band edges into spectral gaps of the periodic potential, V. Another scaling where such band-edge bifurcations arise due to an oscillatory, localized, and mean-zero potential, W, is considered in Refs. 25–28. In this case, a subtle higher order effective potential correction to the classical homogenized Schrödinger operator is required to capture the bifurcation.

B. Statement of results

In order to state our results we require some background on the spectral theory of the Schrödinger operator,

H := - \frac{1}{2} Δ_{z} + V (z),

(1.12)

where V is periodic with respect to a d-dimensional lattice Λ.^29,30 Let $Λ^{*}$ denote the dual lattice to $Λ$ and define the first Brillouin zone $B$ to be a fundamental period cell. Consider the family of self-adjoint eigenvalue problems parameterized by $p \in B$ ⁠,

H (p) χ (z; p) = E (p) χ (z; p),

χ (z + v; p) = χ (z; p) for all z \in ℝ^{d}, v \in Λ,

(1.13)

H (p) := \frac{1}{2} {(p - i \nabla_{z})}^{2} + V (z) .

For fixed p, known as the quasi-momentum, the spectrum of the operator (1.13) is real and discrete and the eigenvalues can be ordered with multiplicity,

E_{1} (p) \leq E_{2} (p) \leq \dots \leq E_{n} (p) \leq \dots .

(1.14)

For fixed p, the associated normalized eigenfunctions $χ_{n} (z; p)$ are a basis of the space,

L_{p e r}^{2} := {f \in L_{l o c}^{2} : f (z + v) = f (z) for all v \in Λ, z \in ℝ^{d}} .

(1.15)

Varying p over the Brillouin zone, the maps $p \mapsto E_{n} (p)$ are known as the spectral band functions. Their graphs are called the dispersion surfaces of H. The set of all dispersion surfaces as p varies over $B$ is called the band structure of H (1.12). Any function in $L^{2} (ℝ^{d})$ can be written as a superposition of Bloch waves,

{Φ_{n} (z; p) = e^{i p z} χ_{n} (z; p) : n \in ℕ, p \in B},

(1.16)

see (2.8). Moreover, the L²-spectrum of the operator (1.12) is the union of the real intervals swept out by the spectral band functions E_n(p),

σ {(H)}_{L^{2} (ℝ^{d})} = \cup_{n \in ℝ} {E_{n} (p) : p \in B} .

(1.17)

The map $p \mapsto E_{n} (p)$ extends to a map on $ℝ^{d}$ which is periodic with respect to the reciprocal lattice $Λ^{*}$ ⁠,

for any b \in Λ^{*}, E_{n} (p + b) = E_{n} (p) .

(1.18)

If the eigenvalue E_n(p) is simple, then (up to a constant phase shift) $χ_{n} (z; p + b) = e^{- i b \cdot z} χ_{n} (z; p)$ ⁠. A more detailed account of the Floquet-Bloch theory which we require, in particular results on the regularity of the maps $p \to E_{n} (p), χ_{n} (z; p)$ ⁠, can be found in Section II.

We will make the following assumptions throughout.

Assumption 1.1. (Uniformly isolated band assumption). Let En(p) be an eigenvalue band function of the periodic Schrödinger operator (1.12). Assume that $(q_{0}, p_{0}) \in ℝ^{d} \times ℝ^{d}$ are such that the flow generated by the classical Hamiltonian $H_{n} (q, p) := E_{n} (p) + W (q)$ ,

\dot{q} (t) = \nabla_{p} E_{n} (p (t)),

\dot{p} (t) = - \nabla_{q} W (q (t)),

(1.19)

q (0), p (0) = q_{0}, p_{0}

has a unique smooth solution $(q (t), p (t)) \in ℝ^{d} \times ℝ^{d}, \forall t \geq 0$ , and that there exists a constant $M > 0$ such that

\underset{m \neq n}{i n f} | E_{m} (p (t)) - E_{n} (p (t)) | \geq M for all t \geq 0.

(1.20)

That is, the nth spectral band is uniformly isolated along the classical trajectory (q(t), p(t)).

Assumption 1.2. $\sum_{| α | = 1,2,3,4} | \partial_{x}^{α} W (x) | \in L^{\infty} (ℝ^{d})$ ⁠.

Remark 1.2. An example of W satisfying Assumption 1.2 is the “Stark” potential $W (x) = E \cdot x$ for any constant vector $E \in ℝ^{d}$ . Assumption 1.2 may be significantly weakened. For example, a refinement of our methods would allow us to deal with any function W with finite order polynomial growth at infinity. This larger class of admissible potentials would include the quantum harmonic oscillator potential $W (x) = \frac{1}{2} x \cdot M x$ , where M is any real positive definite $d \times d$ matrix.

Remark 1.3. Our methods may be adapted to work with time-dependent external potentials W(x, t) which are smooth in x and continuous in t as long as Assumption 1.1 holds with W(q) replaced everywhere by W(q,t) and there exists a constant $C > 0$ such that for all $t \geq 0 : \sum_{| α | = 1, 2, 3, 4} | \partial_{x}^{α} W (x, t) | \leq C$ .

C. Dynamics of semiclassical wavepackets in a periodic background

Our first result is an extension of Theorem 1.7 of Carles and Sparber.⁶

Theorem 1.1. Let Assumptions 1.1 and 1.2 hold. Let $a_{0} (y), b_{0} (y) \in S (ℝ^{d})$ . Let S(t) denote the classical action along the path (q(t),p(t)),

S (t) = \int_{0}^{t} p (t^{'}) \cdot \nabla_{p} E_{n} (p (t^{'})) - E_{n} (p (t^{'})) - W (q (t^{'})) d t^{'} .

(1.21)

Let a(y, t) satisfy

\begin{matrix} i \partial_{t} a (y, t) = H (t) a (y, t), \\ a (y, 0) = a_{0} (y), \end{matrix}

(1.22)

where

H (t) := - \frac{1}{2} \nabla_{y} \cdot D_{p}^{2} E_{n} (p (t)) \nabla_{y} + \frac{1}{2} y \cdot D_{q}^{2} W (q (t)) y .

(1.23)

And let b(y, t) satisfy

\begin{matrix} i \partial_{t} b (y, t) = H (t) b (y, t) + I (t) a (y, t), \\ b (y, 0) = b_{0} (y), \end{matrix}

(1.24)

where $H (t)$ is as in (1.23) and

I (t) := - \frac{1}{6} \nabla_{p} [\nabla_{y} \cdot D_{p}^{2} E_{n} (p (t)) \nabla_{y}] \cdot (- i \nabla_{y}) + \frac{1}{6} \nabla_{q} [y \cdot D_{q}^{2} W (q (t)) y] \cdot y + \nabla_{p} [\nabla_{q} W (q (t)) \cdot A_{n} (p (t))] \cdot (- i \nabla_{y}) + \nabla_{q} [\nabla_{q} W (q (t)) \cdot A_{n} (p (t))] \cdot y,

(1.25)

where $A_{n} (p)$ denotes the nth band Berry connection,

A_{n} (p) := i ⟨ χ_{n} (\cdot; p)) | \nabla_{p} χ_{n} (\cdot; p) ⟩ {L^{2}}_{(Ω)},

(1.26)

where $Ω$ denotes a fundamental period cell of the lattice $Λ$ ⁠. Let $ϕ_{B} (t)$ be the Berry phase associated with transport of $χ_{n}$ along the path $p (t) \in B$ given by

ϕ_{B} (t) = \int_{0}^{t} \dot{p} (t^{'}) \cdot A_{n} (p (t^{'})) d t^{'} = \int_{p_{0}}^{p (t)} A_{n} (p) \cdot d p .

(1.27)

Then, there exists a constant $ϵ_{0} > 0$ such that for all $0 < ϵ \leq ϵ_{0}$ the following holds. Let $ψ^{ϵ} (x, t)$ be the unique solution of the initial value problem (1.1) with “Bloch wavepacket” initial data,

\begin{matrix} i ϵ \partial_{t} ψ^{ϵ} = - \frac{1}{2} ϵ^{2} Δ_{x} ψ^{ϵ} + V (\frac{x}{ϵ}) ψ^{ϵ} + W (x) ψ^{ϵ} \\ ψ^{ϵ} (x, 0) = ϵ^{- d / 4} e^{i p_{0} \cdot (x - q_{0}) / ϵ} {a_{0} (\frac{x - q_{0}}{ϵ^{1 / 2}}) χ_{n} (\frac{x}{ϵ}; p_{0}) + ϵ^{1 / 2} [(- i \nabla_{y}) a_{0} (\frac{x - q_{0}}{ϵ^{1 / 2}}) \cdot \nabla_{p} χ_{n} (\frac{x}{ϵ}; p_{0}) + b_{0} (\frac{x - q_{0}}{ϵ^{1 / 2}}) χ_{n} (\frac{x}{ϵ}; p_{0})]} . \end{matrix}

(1.28)

Then, for all $t \geq 0$ the solution evolves as a modulated “Bloch wavepacket” plus a corrector $η^{ϵ} (x, t)$ ,

ψ^{ϵ} (x, t) = ϵ^{- d / 4} e^{i S (t) / ϵ} e^{i p (t) \cdot (x - q (t)) / ϵ} e^{i ϕ_{B} (t)} {a (\frac{x - q (t)}{ϵ^{1 / 2}}, t) χ_{n} (\frac{x}{ϵ}; p (t)) + ϵ^{1 / 2} [(- i \nabla_{y}) a (\frac{x - q (t)}{ϵ^{1 / 2}}, t) \cdot \nabla_{p} χ_{n} (\frac{x}{ϵ}; p (t)) + b (\frac{x - q (t)}{ϵ^{1 / 2}}, t) χ_{n} (\frac{x}{ϵ}; p (t))]} + η^{ϵ} (x, t),

(1.29)

where the corrector $η^{ϵ}$ satisfies the following estimate:

{∥ η^{ϵ} (\cdot, t) ∥}_{L^{2} (ℝ^{d})} \leq C ϵ e^{c t},

(1.30)

where $c > 0, C > 0$ are constants independent of $ϵ, t$ . It follows that

\sup_{t \in [0, \tilde{C} \ln 1 / ϵ]} {∥ η^{ϵ} (\cdot, t) ∥}_{L^{2} (ℝ^{d})} = o (ϵ^{1 / 2}),

(1.31)

where $\tilde{C}$ is any constant satisfying $\tilde{C} < \frac{1}{2 c}$ .

Remark 1.4. We include the pre-factor $ϵ^{- d / 4}$ throughout so that the $L_{x}^{2} (ℝ^{d})$ norm of $ψ^{ϵ} (x, t)$ is of order 1 as $ϵ ↓ 0$ .

Remark 1.5. We have improved the error bound of Carles and Sparber⁶⁶ from $C ϵ^{1 / 2} e^{c t}$ to $C ϵ e^{c t}$ by including correction terms in the asymptotic expansion proportional to $ϵ^{1 / 2}$ . Note that we must also assume that the initial data are well-prepared up to terms proportional to $ϵ^{1 / 2}$ (1.28). By keeping more terms in the expansion we may produce approximations where the corrector can be bounded by $C ϵ^{N / 2} e^{c t}$ for any positive integer N. The only changes in the proof are that we include corrections to the initial data proportional to $ϵ^{N / 2}$ ⁠, and that Assumption 1.2 is replaced by $\sum_{| α | = 1, 2, \dots, N + 2} | \partial_{x}^{α} W (x) | \in L^{\infty} (ℝ^{d})$ .

Remark 1.6. Keeping terms proportional to $ϵ^{1 / 2}$ in the expansion will allow us to calculate corrections to the dynamics of physical observables proportional to $ϵ$ ; see Theorem 1.2 and Section IV.

Remark 1.7. The time scale of validity of the approximation (1.29), $t \sim \ln 1 / ϵ$ ⁠, is known as “Ehrenfest time.” Without additional assumptions, this is known to be the general limit of validity of wavepacket, or coherent state, approximations. Note that including higher order terms (proportional to powers of $ϵ^{1 / 2}$ ) in the approximation does not extend the time scale of validity. Under further assumptions on the classical dynamics, coherent state approximations have been shown to be valid over the longer time scale $t = o (1 / \sqrt{ϵ})$ ; see Refs. 5 and 31 for further discussion.

There exists a family of time-dependent Gaussian explicit solutions of the envelope equation (1.22). Consider (1.22) with initial data,

a_{0} (y) = \frac{N}{{[\det A_{0}]}^{1 / 2}} \exp (i \frac{1}{2} y \cdot B_{0} A_{0}^{- 1} y),

(1.32)

where $N \in ℂ$ is an arbitrary non-zero constant, and A₀, B₀ are $d \times d$ complex matrices satisfying

\begin{matrix} A_{0}^{T} B_{0} - B_{0}^{T} A_{0} = 0, \\ \bar{A_{0}^{T}} B_{0} - \bar{B_{0}^{T}} A_{0} = 2 i I . \end{matrix}

(1.33)

Remark 1.8. The conditions (1.33) imply that

the matrices B0, A₀ are invertible,
the matrix $B_{0} A_{0}^{- 1}$ is complex symmetric: ${(B_{0} A_{0}^{- 1})}^{T} = B_{0} A_{0}^{- 1}$ ⁠,
the imaginary part of the matrix $B_{0} A_{0}^{- 1}$ is symmetric, positive definite, and satisfies $Im B_{0} A_{0}^{- 1} = {(A_{0} \bar{A_{0}^{T}})}^{- 1}$ ⁠,

and are equivalent to the condition that the matrix

Y := (\begin{matrix} Re A_{0} & Im A_{0} \\ Re B_{0} & Im B_{0} \end{matrix}) is symplectic: Y^{T} J Y = J, where J := (\begin{matrix} 0 & - I \\ I & 0 \end{matrix}) .

(1.34)

The proofs of (1)-(3) are given in Refs. 11, 32, and 33 .

Note that it follows from assertion (3) of Remark 1.8 that a₀(y) (1.32) satisfies $| a_{0} (y) | \leq C e^{- c {| y |}^{2}}$ for constants $C > 0, c > 0$ ⁠. We have then the following.

Proposition 1.1. (Gaussian wavepackets). The initial value problem (1.22) with initial data a0(y) given by (1.32) has the unique solution for all $t \geq 0$ ,

a (y, t) = \frac{N}{{[\det A (t)]}^{1 / 2}} \exp (i \frac{1}{2} y \cdot B (t) A^{- 1} (t) y) .

(1.35)

Here, the complex matrices A(t), B(t) satisfy

\begin{matrix} \dot{A} (t) = D_{p}^{2} E_{n} (p (t)) B (t), \dot{B} (t) = - D_{q}^{2} W (q (t)) A (t), \\ A (0) = A_{0}, B (0) = B_{0} . \end{matrix}

(1.36)

Moreover, for all $t \geq 0$ ⁠, the matrices A(t), B(t) satisfy (1.33) with A0 replaced by A(t) and B0 replaced by B(t). Thus (see Remark 1.8), $| a (y, t) | \leq C (t) e^{- c (t) {| y |}^{2}}$ where $C (t) > 0, c (t) > 0$ for all $t \geq 0$ . More generally, we may construct a basis of $L^{2} (ℝ^{d})$ of solutions of the envelope equation (1.22), consisting of products of Gaussians with polynomials, known as the “semiclassical wavepacket” basis.^11,32,33

Remark 1.9. Our convention for the complex matrices A, B follows that introduced in Ref. 33, with A, B standing for Q, P in Ref. 33 , respectively. Note that our convention is not to be confused with that introduced in Ref. 11; our choice of B corresponds to iB in Ref. 11.

D. Dynamics of observables associated with the asymptotic solution

We now deduce consequences for the physical observables associated with the solution $ψ^{ϵ} (x, t)$ of (1.28), using the asymptotic form (1.29). Denote the solution of (1.28) through order $ϵ^{1 / 2}$ by

\begin{matrix} {\tilde{ψ}}^{ϵ} (y, z, t) := ϵ^{- d / 4} e^{i S (t) / ϵ} e^{i p (t) \cdot y / ϵ^{1 / 2}} e^{i ϕ_{B} (t)} {a (y, t) χ_{n} (z; p (t)) \\ + ϵ^{1 / 2} [(- i \nabla_{y}) a (y, t) \cdot \nabla_{p} χ_{n} (z; p (t)) + b (y, t) χ_{n} (z; p (t))]} . \end{matrix}

(1.37)

Thus,

ψ^{ϵ} (x, t) = {{\tilde{ψ}}^{ϵ} (y, z, t) |}_{z = \frac{x}{ϵ}, y = \frac{x - q (t)}{ϵ^{1 / 2}}} + η^{ϵ} (x, t),

(1.38)

where $η^{ϵ}$ is the corrector which satisfies the bound (1.30). Define the physical observables

\begin{matrix} Q^{ϵ} (t) := \frac{1}{N^{ϵ} (t)} \int_{ℝ^{d}} x {| {\tilde{ψ}}^{ϵ} (y, z, t) |}_{z = \frac{x}{ϵ}, y = \frac{x - q (t)}{ϵ^{1 / 2}}}^{2} d x, \\ P^{ϵ} (t) := \frac{1}{N^{ϵ} (t)} \int_{ℝ^{d}} \bar{{\tilde{ψ}}^{ϵ} (y, z, t)} (- i ϵ {}^{1 / 2}\nabla_{y}) {\tilde{ψ}}^{ϵ} (y, z, t) |_{z = \frac{x}{ϵ}, y = \frac{x - q (t)}{ϵ^{1 / 2}}} d x, \end{matrix}

(1.39)

where $N^{ϵ} (t)$ is the normalization factor,

N^{ϵ} (t) = \int_{ℝ^{d}} {| {\tilde{ψ}}^{ϵ} (y, z, t) |}_{z = \frac{x}{ϵ}, y = \frac{x - q (t)}{ϵ^{1 / 2}}}^{2} d x .

(1.40)

We will refer to $Q^{ϵ} (t), P^{ϵ} (t)$ as the center of mass and average quasi-momentum of the wavepacket. We will see (Theorem 1.2) that $Q^{ϵ} (t) = q (t) + o (1), P^{ϵ} (t) = p (t) + o (1)$ up to “Ehrenfest time” $t \sim \ln 1 / ϵ$ ⁠.

Remark 1.10. In the uniform background case V = 0, solutions of (1.13) are independent of z: $χ_{n} (z; p) = 1$ for all $p \in B$ . The asymptotic solution (1.37) obtained in this case is therefore independent of z, and our definition of $P^{ϵ} (t)$ reduces to the usually defined momentum observable,

\int_{ℝ^{d}} \bar{ψ^{ϵ} (x, t)} (- i ϵ \nabla_{x}) ψ^{ϵ} (x, t) d x .

(1.41)

In the periodic background case $V \neq 0$ , $P^{ϵ}$ (1.39) corresponds to the quasi-momentum and may be measured in experiments.³⁴³⁴Let $a^{ϵ} (y, t)$ satisfy the equation of a quantum harmonic oscillator, with parametric forcing defined by $(Q^{ϵ} (t), P^{ϵ} (t))$ ⁠,

\begin{matrix} i \partial_{t} a^{ϵ} = H^{ϵ} (t) a^{ϵ}, H^{ϵ} (t) := - \frac{1}{2} \nabla_{y} \cdot D_{P^{ϵ}}^{2} E_{n} (P^{ϵ} (t)) \nabla_{y} + \frac{1}{2} y \cdot D_{Q^{ϵ}}^{2} W (Q^{ϵ} (t)) y, \\ a^{ϵ} (y, 0) = a_{0} (y) . \end{matrix}

(1.42)

Note that we have replaced dependence on (q(t),p(t)) in equation (1.22) with dependence on $Q^{ϵ} (t), P^{ϵ} (t)$ ⁠.

For simplicity of presentation of the following theorem we assume that

\begin{matrix} {⟨ a_{0} (y) | y a_{0} (y) ⟩}_{L_{y}^{2} (ℝ^{d})} = {⟨ a_{0} (y) | (- i \nabla_{y}) a_{0} (y) ⟩}_{L_{y}^{2} (ℝ^{d})} = 0, \\ {∥ a_{0} (y) ∥}_{L_{y}^{2} (ℝ^{d})} = 1. \end{matrix}

(1.43)

The result holds for general $a_{0} (y) \in S (ℝ^{d})$ ⁠; see Section IV.

Theorem 1.2. Let ${\tilde{ψ}}^{ϵ} (y, z, t)$ denote the asymptotic solution (1.37) including corrections proportional to $ϵ^{1 / 2}$ . Let $Q^{ϵ} (t), P^{ϵ} (t)$ denote the observables (1.39). Then, there exists an $ϵ_{0} > 0$ such that for all $0 < ϵ \leq ϵ_{0}$ ⁠, and for all $t \in [0, {\tilde{C}}^{'} \ln 1 / ϵ]$ where ${\tilde{C}}^{'} > 0$ is a constant independent of $t, ϵ$ ⁠.

$Q^{ϵ} (t), P^{ϵ} (t)$ satisfy
$\begin{matrix} Q^{ϵ} (t) = q (t) + ϵ [{⟨ b (y, t) | y a (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})} + {⟨ a (y, t) | y b (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})}] + ϵ A_{n} (p (t)) + o (ϵ), \\ P^{ϵ} (t) = p (t) + ϵ [{⟨ b (y, t) | (- i \nabla_{y}) a (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})} + {⟨ a (y, t) | (- i \nabla_{y}) b (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})}] + o (ϵ), \end{matrix}$
(1.44)
where $A_{n} (p)$ is the nth band Berry connection (1.26), and a(y, t),b(y, t) satisfy (1.22) and (1.24), respectively.
Let $a^{ϵ} (y, t)$ satisfy (1.42). Then,
$\begin{matrix} {\dot{Q}}_{α}^{ϵ} (t) = \partial_{P_{α}^{ϵ}} E_{n} (P^{ϵ} (t)) - ϵ {\dot{P}}_{β}^{ϵ} (t) F_{n, α β} (P^{ϵ} (t)) \\ + ϵ \frac{1}{2} \partial_{P_{α}^{ϵ}} {⟨ \nabla_{y} a^{ϵ} (y, t) | \cdot D_{P^{ϵ}}^{2} E_{n} (P^{ϵ} (t)) \nabla_{y} a^{ϵ} (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})} + o (ϵ) \\ {\dot{P}}_{α}^{ϵ} (t) = - \partial_{Q_{α}^{ϵ}} W (Q^{ϵ} (t)) \\ - ϵ \frac{1}{2} \partial_{Q_{α}^{ϵ}} {⟨ y a^{ϵ} (y, t) | \cdot D_{Q^{ϵ}}^{2} W (Q^{ϵ} (t)) y a^{ϵ} (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})} + o (ϵ), \end{matrix}$
(1.45)
where,
$F_{n, α β} (P^{ϵ} (t))$
denotes the Berry curvature of the n-th band,:
$F_{n, α β} (P^{ϵ}) := \partial {}_{P_{α}^{ϵ}}A_{n, β} (P^{ϵ}) - \partial {}_{P_{β}^{ϵ}}A_{n, α} (P^{ϵ}),$
(1.46)
where $A_{n} (P^{ϵ})$ is the nth band Berry connection (1.26). When d = 3 the anomalous velocity $- {\dot{P}}_{β}^{ϵ} (t) F_{n, α β} (P^{ϵ} (t))$ may be re-written using the cross product as in (1.3); see Remark 1.11.
After dropping the terms of $o (ϵ)$ in (1.45), Equations (1.45) and (1.42) form a closed, coupled “particle-field” system for $Q^{ϵ} (t), P^{ϵ} (t), a^{ϵ} (y, t)$ .
Let
$\begin{matrix} Q^{ϵ} (t) := Q^{ϵ} (t) - ϵ A_{n} (P^{ϵ} (t)), \\ P^{ϵ} (t) := P^{ϵ} (t) . \end{matrix}$
(1.47)

Let $𝔞^{ϵ} (y, t)$ denote the solution of (1.42) with co-efficients evaluated at $Q^{ϵ} (t), P^{ϵ} (t)$ rather than $Q^{ϵ} (t), P^{ϵ} (t)$ ⁠.

Then, after dropping terms of $o (ϵ)$ , $Q^{ϵ} (t), P^{ϵ} (t), 𝔞^{ϵ} (y, t)$ satisfy a closed, coupled “particle-field” system which is expressible as an $ϵ$ -dependent Hamiltonian system,

\begin{matrix} {\dot{Q}}^{ϵ} = \nabla_{P^{ϵ}} H^{ϵ}, {\dot{P}}^{ϵ} = - \nabla_{Q^{ϵ}} H^{ϵ}, \\ i \partial_{t} 𝔞^{ϵ} = \frac{δ H^{ϵ}}{δ \bar{𝔞^{ϵ}}}, \end{matrix}

(1.48)

with Hamiltonian

\begin{matrix} H^{ϵ} (P^{ϵ}, Q^{ϵ}, \bar{𝔞^{ϵ}}, 𝔞^{ϵ}) := E_{n} (P^{ϵ}) + W (Q^{ϵ}) + ϵ \nabla_{Q^{ϵ}} W (Q^{ϵ}) \cdot A_{n} (P^{ϵ}) \\ + ϵ \frac{1}{2} {⟨ \nabla_{y} 𝔞^{ϵ} | \cdot D_{P^{ϵ}}^{2} E_{n} (P^{ϵ}) \nabla_{y} 𝔞^{ϵ} ⟩}_{L_{y}^{2} (ℝ^{d})} + ϵ \frac{1}{2} {⟨ y 𝔞^{ϵ} | \cdot D_{Q^{ϵ}}^{2} W (Q^{ϵ}) y 𝔞^{ϵ} ⟩}_{L_{y}^{2} (ℝ^{d})} . \end{matrix}

(1.49)

Remark 1.11. In three spatial dimensions (d = 3), the anomalous velocity may be re-written using the cross product

\begin{matrix} - {\dot{P}}_{β}^{ϵ} (t) F_{n, α β} (P^{ϵ} (t)) & = - {\dot{P}}_{β}^{ϵ} (t) (\partial_{P_{α}^{ϵ}} A_{n, β} (P^{ϵ} (t)) - \partial_{P_{β}^{ϵ}} A_{n, α} (P^{ϵ} (t))) \\ = - (δ_{α γ} δ_{β ϕ} - δ_{α ϕ} δ_{β γ}) {\dot{P}}_{β}^{ϵ} (t) \partial_{P_{γ}^{ϵ}} A_{n, ϕ} (P^{ϵ} (t)) \\ = - ϵ_{η α β} ϵ_{η γ ϕ} {\dot{P}}_{β}^{ϵ} (t) \partial_{P_{γ}^{ϵ}} A_{n, ϕ} (P^{ϵ} (t)) \\ = - {({\dot{P}}^{ϵ} (t) \times \nabla_{P^{ϵ}} \times A_{n} (P^{ϵ} (t)))}_{α}, \end{matrix}

(1.50)

where $ϵ$ and $δ$ are the Levi-Civita and Kronecker delta symbols, respectively, and each equality follows from well-known properties of these symbols; see Section I F (1.70)–(1.73). In this case the curl of the Berry connection $\nabla_{P^{ϵ}} \times A_{n} (P^{ϵ})$ is often referred to as the Berry curvature, see, for example, Ref. 16.

Remark 1.12. Equations (1.45) agree with those derived elsewhere (for example, (3.5)-(3.6) of Ref. 2) up to the terms which depend on the wavepacket envelope $a^{ϵ}$ ⁠. The change of variables (1.47) was introduced in Ref. 16 to transform between the Hamiltonian system for the characteristics of a “corrected” eikonal ((4.9)-(4.10) in that work) and a gauge-invariant system ((4.11)-(4.12) in that work).

Corollary 1.1.

Choose initial data a0(y) of the form (1.32)–(1.33) with $N = π^{- d / 4}$ so that ${∥ a_{0} (y) ∥}_{L_{y}^{2}} = 1$ ⁠. Then, $a^{ϵ} (y, t)$ , the solution of the initial value problem (1.42), is given by
$a^{ϵ} (y, t) = \frac{N}{{[\det A^{ϵ} (t)]}^{1 / 2}} \exp (i \frac{1}{2} y \cdot B^{ϵ} (t) A^{ϵ}^{- 1} (t) y),$
(1.51)
where $A^{ϵ} (t), B^{ϵ} (t)$ satisfy
$\begin{matrix} {\dot{A}}^{ϵ} (t) = D_{P^{ϵ}}^{2} E_{n} (P^{ϵ} (t)) B^{ϵ} (t), {\dot{B}}^{ϵ} (t) = - D_{Q^{ϵ}}^{2} W (Q^{ϵ} (t)) A^{ϵ} (t), \\ A^{ϵ} (0) = A_{0}, B^{ϵ} (0) = B_{0} . \end{matrix}$
(1.52)
After the change of variables (1.47), the full coupled system governing $(Q^{ϵ}, P^{ϵ}, A^{ϵ} (t), B^{ϵ} (t))$ governed by (1.45) (with $o (ϵ)$ terms dropped) and (1.52), is expressible as a Hamiltonian system
$\begin{matrix} {\dot{Q}}^{ϵ} = \nabla {}_{P^{ϵ}}H^{ϵ},^{ϵ} = - \nabla {}_{\mathcaly Q^{ϵ}}H^{ϵ}, \\ {\dot{A}}^{ϵ} (t) = 4 \frac{\partial H^{ϵ}}{\partial \bar{B^{ϵ}}}, {\dot{B}}^{ϵ} (t) = - 4 \frac{\partial H^{ϵ}}{\partial \bar{A^{ϵ}}}, \end{matrix}$
(1.53)
with Hamiltonian
$H^{ϵ} (P^{ϵ}, Q^{ϵ}, \bar{A^{ϵ}}, A^{ϵ}, \bar{B^{ϵ}}, B^{ϵ}) := E_{n} (P^{ϵ}) + W (Q^{ϵ}) + ϵ \nabla_{Q^{ϵ}} W (Q^{ϵ}) \cdot A_{n} (P^{ϵ}) + ϵ \frac{1}{4} Tr [{(\bar{B^{ϵ}})}^{T} D_{P^{ϵ}}^{2} E_{n} (P^{ϵ}) B^{ϵ}] + ϵ \frac{1}{4} Tr [{(\bar{A^{ϵ}})}^{T} D_{Q^{ϵ}}^{2} W (Q^{ϵ}) A^{ϵ}] .$
(1.54)

Remark 1.13. In the special case where the periodic background potential V = 0 the Bloch band dispersion function $E_{n} (P^{ϵ})$ reduces to the “free” dispersion relation $\frac{1}{2} {(P^{ϵ})}^{2}$ and the Hamiltonian (1.54) takes on the simple form

H^{ϵ} (P^{ϵ}, Q^{ϵ}, \bar{A^{ϵ}}, A^{ϵ}, \bar{B^{ϵ}}, B^{ϵ}) := \frac{1}{2} {(P^{ϵ})}^{2} + W (Q^{ϵ}) + ϵ \frac{1}{4} Tr [{(\bar{A^{ϵ}})}^{T} D_{Q^{ϵ}}^{2} W (Q^{ϵ}) A^{ϵ}] + ϵ \frac{1}{4} Tr [{(\bar{B^{ϵ}})}^{T} B^{ϵ}] .

(1.55)

The system (1.53), with Hamiltonian $H^{ϵ}$ given by (1.55), has been derived by other methods: see Proposition 4.4 and Equations (32b)-(32c) of Ref. 3. It was shown furthermore in Ref. 4 that corrections to the dynamics of $Q^{ϵ}, P^{ϵ}$ proportional to $ϵ$ due to “field-particle” coupling to the $A^{ϵ}, B^{ϵ}$ system can lead to qualitatively different dynamical behavior. In particular, the coupling may destabilize periodic orbits of the unperturbed (⁠ $ϵ = 0$ ⁠) system; see Section 9 of Ref. 4.

Remark 1.14. In Remark 1.7 we commented that the general time scale of validity of our results is up to “Ehrenfest time” $t \sim \ln 1 / ϵ$ , and that under further assumptions on the classical dynamics we expect that this time scale may be extended up to $t = o (1 / \sqrt{ϵ})$ . Note that the Berry curvature terms and the new “field-particle” coupling terms occur at the same order in $ϵ$ ⁠. It is an interesting question to determine their impact on the dynamics for t greater than the “Ehrenfest time.”

E. Discussion of results, relation to previous work

The $ϵ ↓ 0$ limit of (1.1) has been studied by other methods: for example, by space-adiabatic perturbation theory^15,35–37 and by studying the propagation of Wigner functions associated to the solution of (1.1).^19,38,39 The Wigner function approach is notable in that it has been used to study the propagation of wavepacket solutions of (1.1) through band crossings.^40,41 It was shown in Ref. 16 that the anomalous velocity due to Berry curvature can be derived by a multiscale WKB-like ansatz by studying the characteristic equations of a corrected eikonal equation. The Hamiltonian structure of equations (1.45) without field-particle coupling terms was studied in Ref. 42

The effective system (1.45), in particular the “particle-field” coupling that we derive, is original to this work. Such coupled “particle-field” models arise naturally in many settings where a coherent structure interacts with a linear or nonlinear wave-field; see, for example, Ref. 43 and references therein.

The results detailed in Section I B generalize to the case where the potential has the more general form $U (\frac{x}{ϵ}, x)$ ⁠, where U is periodic in its first argument,

U (z + v, x) = U (z, x) for all z, x \in ℝ^{d}, v \in Λ .

(1.56)

If U(z, x) is not expressible as the sum of a periodic potential V(z) and a smooth potential W(x) we will say that U is “non-separable.” In this case we must work with an x-dependent Bloch eigenvalue problem,

\begin{matrix} H (p, x) χ_{n} (z; p, x) = E_{n} (p, x) χ_{n} (z; p, x), \\ χ_{n} (z + v; p, x) = χ_{n} (z; p, x) for all z, x \in ℝ^{d}, v \in Λ, \\ H (p, x) := \frac{1}{2} {(p - i \nabla_{z})}^{2} + U (z, x) . \end{matrix}

(1.57)

For details, see Ref. 9. Related problems were considered in Refs. 44 and 45. An interesting example of a potential of this type is that of a domain wall modulated honeycomb lattice potential, which was shown to support “topologically protected” edge states in Ref. 46.

F. Notation and conventions

Where necessary to avoid ambiguity, we will use index notation, making the standard convention that repeated indices are summed over from 1 to d where d is the spatial dimension. Thus, in the expression
$\partial_{p_{α}} \partial_{p_{β}} f (p) (- i \partial_{y_{α}}) (- i \partial_{y_{β}}) g (y),$
(1.58)
it is understood that we are summing over $α \in {1, \dots, d}, β \in {1, \dots, d}$ ⁠.
Where there is no danger of confusion, we will use the standard conventions
$v_{β} w_{β} = v \cdot w, v_{β} v_{β} = v \cdot v .$
(1.59)
$Δ_{x} = \nabla_{x}^{2}$ denotes the d-dimensional Laplacian.
$D_{x}^{2}$ denotes the d-dimensional Hessian matrix with respect to x.
We will adopt multi-index notation where appropriate so that
$\sum_{| α | = l} | \partial_{x}^{α} f (x) | \in L^{\infty} (ℝ^{d})$
(1.60)
means all derivatives of order l of f(x) are uniformly bounded.
It will be useful to introduce the energy spaces for every $l \in ℕ$ ⁠,
$Σ^{l} (ℝ^{d}) := {f \in L^{2} (ℝ^{d}) : ∥ f ∥_{Σ^{l}} := \sum_{| α | + | β | \leq l} ∥ y^{α} {(- i \partial_{y})}^{β} f (y) ∥_{L_{y}^{2}} < \infty,} .$
(1.61)
The space of Schwartz functions $S (ℝ^{d})$ is the space of functions defined as
$S (ℝ^{d}) := \cap_{l \in ℕ} Σ^{l} (ℝ^{d}) .$
(1.62)
We will refer throughout to the space of L²-integrable functions which are periodic on the lattice $Λ$ ⁠,
$L_{p e r}^{2} := {f \in L_{l o c}^{2} (ℝ^{d}) : for all z \in ℝ^{d}, v \in Λ, f (z + v) = f (z)} .$
(1.63)
We will write a fundamental period cell in $ℝ^{d}$ of the lattice $Λ$ as $Ω$ ⁠.
We will make use of the Sobolev norms on a fundamental period cell $Ω$ for integers $s \geq 0$ ⁠,
${∥ f (z) ∥}_{H_{z}^{s}} := \sum_{| j | \leq s} {∥ {(\partial_{z})}^{j} f (z) ∥}_{L_{z}^{2}} .$
(1.64)
It will also be useful to introduce the “shifted” Sobolev norms, for arbitrary $p \in ℝ^{d}$ ⁠,
${∥ f (z) ∥}_{H_{z, p}^{s}} := \sum_{| j | \leq s} {∥ {(p - i \partial_{z})}^{j} f (z) ∥}_{L_{z}^{2}} .$
(1.65)
Define the dual lattice to $Λ$ ⁠,
$Λ^{*} := {b \in ℝ^{d} : \exists v \in Λ : b \cdot v = 2 π n, n \in ℤ} .$
(1.66)
We will refer to a fundamental cell in $ℝ_{p}^{d}$ of the dual lattice $Λ^{*}$ as the Brillouin zone, or $B$
We make the standard convention for the L²-inner product,
${⟨ f | g ⟩}_{L^{2} (D)} := \int_{D} \bar{f (x)} g (x) d x .$
(1.67)
We will make the conventions,
$\begin{matrix} f^{ϵ} (x, t) = O (ϵ^{K} e^{c t}) ⟺ \exists c > 0, C > 0, independent of t, ϵ such that {∥ f^{ϵ} (x, t) ∥}_{L_{x}^{2}} \leq C ϵ^{K} e^{c t}, \\ g^{ϵ} (t) = O (ϵ^{K} e^{c t}) ⟺ \exists c > 0, C > 0, independent of t, ϵ such that | g^{ϵ} (t) | \leq C ϵ^{K} e^{c t} . \end{matrix}$
(1.68)
Let A be a complex matrix. Then we will write A^T for its transpose, $\bar{A}$ for its complex conjugate, and $Tr A$ for its trace. Using index notation,
${(A_{α β})}^{T} := A_{β α}, {(\bar{A})}_{α β} := \bar{A_{α β}}, Tr A := A_{α α} .$
(1.69)
The Kronecker delta $δ_{α β}$ is defined,
$δ_{α β} = {\begin{matrix} + 1 & when α = β \\ 0 & when α \neq β \end{matrix} .$
(1.70)
In dimension d = 3, the Levi-Civita symbol $ϵ_{α β γ}$ is defined,
$ϵ_{α β γ} := {\begin{matrix} + 1 & when (α, β, γ) \in {(1,2,3), (2,3,1), (3,1,2)} \\ - 1 & when (α, β, γ) \in {(1,3,2), (2,3,1), (3,2,1)} \\ 0 & when α = β, β = γ, or γ = α \end{matrix}$
(1.71)
and satisfies the identities
$\begin{matrix} ε_{α β γ} = ε_{β γ α} = ε_{γ α β}, \\ ε_{ϕ α β} ε_{ϕ γ ϕ} = δ_{α γ} δ_{β ϕ} - δ_{β γ} δ_{α ϕ} . \end{matrix}$
(1.72)
The cross product of 3-vectors $v, w$ may then be written as
${(v \times w)}_{α} = ϵ_{α β γ} v_{β} w_{γ} .$
(1.73)

II. SUMMARY OF RELEVANT FLOQUET-BLOCH THEORY

In this section we recall the spectral theory of the operator,

H := - \frac{1}{2} Δ_{z} + V (z),

(2.1)

where V is periodic with respect to the lattice $Λ$ ⁠.^29,30 For $p \in ℝ^{d}$ ⁠, define the spaces of p-pseudo-periodic L² functions as follows:

L_{p}^{2} := {f \in L_{l o c}^{2} : f (z + v) = e^{i p \cdot v} f (z) for all z \in ℝ^{d}, v \in Λ} .

(2.2)

Let $Λ^{*}$ denote the lattice dual to $Λ$ ⁠,

Λ^{*} := {b \in ℝ^{d} : \exists v \in Λ : v \cdot b = 2 π n, n \in ℤ},

(2.3)

and since the p–pseudo-periodic boundary condition is invariant under $p \to p + b$ where $b \in Λ^{*}$ ⁠, the dual lattice to $Λ$ ⁠, it is natural to restrict to a fundamental cell, $B$ ⁠.

We now consider the family of eigenvalue problems depending on the parameter $p \in B$ ⁠,

\begin{matrix} H Φ (z; p) = E (p) Φ (z; p), \\ Φ (z + v; p) = e^{i p \cdot v} Φ (z; p) for all z \in ℝ^{d}, v \in Λ . \end{matrix}

(2.4)

We can also define the space $L_{l o c}^{2}$ functions which are periodic with respect to the lattice,

L_{p e r}^{2} := {f (z) \in L_{l o c}^{2} (ℝ^{d}) : \forall v \in Λ, f (z + v) = f (z)} .

(2.5)

Then solving the eigenvalue problem (2.4) is equivalent via $Φ (z; p) = e^{i p \cdot z} χ (z; p)$ to solving the family of eigenvalue problems,

\begin{matrix} H (p) χ (z; p) = E (p) χ (z; p), \\ χ (z + v; p) = χ (z; p) for all z \in ℝ^{d}, v \in Λ, \\ H (p) = \frac{1}{2} {(p - i \nabla_{z})}^{2} + V (z) \end{matrix}

(2.6)

For fixed p, the operator H(p) with periodic boundary conditions is self-adjoint and has compact resolvent. So, for each $n \in ℕ$ ⁠, there exists an eigenpair $E_{n} (p), χ_{n} (z; p)$ ⁠. The eigenvalues are real and can be ordered with multiplicity,

E_{1} (p) \leq E_{2} (p) \leq \dots \leq E_{n - 1} (p) \leq E_{n} (p) \leq E_{n + 1} (p) \leq \dots,

(2.7)

and the set of normalized eigenfunctions ${χ_{n} (z; p) : n \in ℕ}$ is complete in $L_{p e r}^{2}$ ⁠. The set of Floquet-Bloch waves ${Φ_{n} (z; p) = e^{i p z} χ_{n} (z; p) : n \in ℕ, p \in B}$ are complete in $L^{2} (ℝ^{d})$ ⁠,

g \in L^{2} (ℝ^{d}) ⟹ g (x) = \sum_{n \geq 1} \int_{𝔹} {\tilde{g}}_{n} (p) Φ_{n} (x; p) d p, where {\tilde{g}}_{n} (p) := {⟨ Φ_{n} (\cdot; p) | g (\cdot) ⟩}_{L^{2} (ℝ^{d})},

(2.8)

where the sum converges in L². The $L^{2} (ℝ^{d})$ spectrum of the operator (2.1) is obtained by taking the union of the closed real intervals swept out as p varies over the Brillouin zone $B$ ⁠,

σ {(H)}_{L^{2} (ℝ^{d})} = \cup_{n \in ℝ} \bar{{E_{n} (p) : p \in B}} .

(2.9)

Our results require sufficient regularity of the maps,

E_{n} : B \to ℝ, p \mapsto E_{n} (p),

(2.10)

χ_{n} : B \to L_{p e r}^{2}, p \mapsto χ_{n} (z; p) .

(2.11)

Definition 2.1. We will call an eigenvalue band En(p) of the problem (2.6) isolated at a point $p \in B$ if

\underset{m \neq n}{i n f} | E_{m} (p) - E_{n} (p) | > 0.

(2.12)

We have in this case the following.

Theorem 2.1. (Smoothness of isolated bands). Let $E_{n} (p), χ_{n} (z; p)$ satisfy the eigenvalue problem (2.6). Let the band En(p) be isolated at a point p0 in the sense of Definition 2.1. Then the maps (2.10) are smooth in a neighborhood of the point p0.

When bands are not isolated we have the following situation.

Definition 2.2. Let $E_{m} (p), E_{n} (p) : m, n \in ℕ, m \neq n$ be eigenvalue bands of the eigenvalue problem (2.6). If $p^{*} \in B$ is such that

E_{m} (p^{*}) = E_{n} (p^{*}),

(2.13)

we will say that the bands En(p) and Em(p) have a band crossing at $p^{*}$ .

In a neighborhood of a crossing, the band functions E_n(p), E_m(p) are only Lipschitz continuous, and the eigenfunction maps $p \mapsto χ_{n} (z; p), χ_{m} (z; p)$ may be discontinuous.²⁹ This loss of regularity occurs at conical degeneracies, which appear, for example, in the band structure of honeycomb lattice potentials,^47,48 and in the dispersion surfaces of plane waves for homogeneous anisotropic media.⁴⁹ An in-depth study of conical crossings which appear in the study of the Born-Oppenheimer approximation of molecular dynamics was given in Ref. 7.

It will be convenient to extend the maps $p \mapsto E_{n} (p), χ_{n} (z; p)$ to maps on all of $ℝ^{d}$ ⁠. Let $p \in B$ ⁠, and let $b \in Λ^{*}$ denote a reciprocal lattice vector. Then we have that

\begin{matrix} H (p + b) (e^{- i b \cdot z} χ_{n} (z; p)) = e^{- i b \cdot z} H (p) χ_{n} (z; p) \\ = e^{- i b \cdot z} E_{n} (p) χ_{n} (z; p) = E_{n} (p) (e^{- i b \cdot z} χ_{n} (z; p)), \\ for all v \in Λ, e^{- i b \cdot (z + v)} χ_{n} (z + v; p) \\ = e^{- i b \cdot v} e^{- i b \cdot z} χ_{n} (z; p) = e^{- i b \cdot z} χ_{n} (z; p), \end{matrix}

(2.14)

so that if $χ_{n} (z; p)$ satisfies (2.6) with eigenvalue E_n(p), then $e^{- i b \cdot z} χ_{n} (z; p)$ satisfies (2.6) with p replaced by p + b, with the same eigenvalue. It then follows that the map $p \mapsto E_{n} (p)$ extends to a periodic function with respect to the reciprocal lattice $Λ^{*}$ ⁠. If the eigenvalue E_n(p) is simple, then (up to a constant phase shift) $χ_{n} (z; p + b) = e^{- i b \cdot z} χ_{n} (z; p)$ ⁠.

III. PROOF OF THEOREM 1.1 BY MULTISCALE ANALYSIS

A. Derivation of asymptotic solution (1.29) via multiscale expansion

Following the work of Hagedorn,^7,11 and Carles and Sparber,⁶ we seek a solution of (1.1) of the form

ψ^{ϵ} (x, t) = {ϵ^{- d / 4} e^{i S (t) / ϵ} e^{i p (t) \cdot y / ϵ^{1 / 2}} f^{ϵ} (y, z, t) |}_{z = \frac{x}{ϵ}, y = \frac{x - q (t)}{ϵ^{1 / 2}}} + η^{ϵ} (x, t) .

(3.1)

Substituting (3.1) into (1.1) gives an inhomogeneous time-dependent Schrödinger equation for $η^{ϵ} (x, t)$ ⁠, with a source term $r^{ϵ} (x, t)$ which depends on S(t), q(t), p(t), and $f^{ϵ} (y, z, t)$ ⁠,

\begin{matrix} i ϵ \partial_{t} η^{ϵ} (x, t) = [- \frac{ϵ^{2}}{2} Δ_{x} + V (\frac{x}{ϵ}) + W (x)] η^{ϵ} (x, t) + r^{ϵ} [S, q, p, f^{ϵ}] (x, t), \\ η^{ϵ} (x, 0) = η_{0}^{ϵ} [S, q, p, f^{ϵ}] (x) = ψ^{ϵ} (x, 0) - {ϵ^{- d / 4} e^{i S (0) / ϵ} e^{i p (0) \cdot y / ϵ^{1 / 2}} f^{ϵ} (z, y, 0) |}_{z = \frac{x}{ϵ}, y = \frac{x - q (0)}{ϵ^{1 / 2}}} . \end{matrix}

(3.2)

The idea behind the proof of Theorem 1.1 is to choose the functions S(t), q(t), p(t), and $f^{ϵ} (y, z, t)$ so that

\begin{matrix} r^{ϵ} [S, q, p, f^{ϵ}] (x, t) = O (ϵ^{2}), \\ η_{0}^{ϵ} [S, q, p, f^{ϵ}] (x) = O (ϵ) . \end{matrix}

(3.3)

We will derive $S (t), q (t), p (t), f^{ϵ} (y, z, t)$ by a systematic formal analysis. This is the content of Sections III A 1–III A 3. Proving rigorous bounds on the residual will be the content of Section III B. The bound (1.30) on $η^{ϵ} (x, t)$ will then follow from applying the standard a priori L² bound for solutions of the time-dependent inhomogeneous Schrödinger equation.

Before starting on the formal asymptotic analysis, we note some exact manipulations which will ease calculations below. The residual $r^{ϵ} (x, t)$ has the explicit form

r^{ϵ} (x, t) = {ϵ^{- d / 4} e^{i S (t) / ϵ} e^{i p (t) \cdot y / ϵ^{1 / 2}} {ϵ [\frac{1}{2} {(- i \nabla_{y})}^{2} - i \partial_{t}] + ϵ^{1 / 2} [(p (t) - i \nabla_{z}) \cdot (- i \nabla_{y}) - \dot{q} (t) \cdot (- i \nabla_{y}) + \dot{p} (t) \cdot y] + [\dot{S} (t) - \dot{q} (t) p (t) + \frac{1}{2} {(p (t) - i \nabla_{z})}^{2} + V (z) + W (q (t) + ϵ^{1 / 2} y)]} f^{ϵ} (y, z, t) |}_{z = \frac{x}{ϵ}, y = \frac{x - q (t)}{ϵ^{1 / 2}}} .

(3.4)

Since W is assumed smooth, we can replace $W (q (t) + ϵ^{1 / 2} y)$ by its Taylor series expansion in $ϵ^{1 / 2} y$ ⁠,

W (q (t) + ϵ^{1 / 2} y) = W (q (t)) + ϵ {}^{1 / 2}\nabla_{q} W (q (t)) \cdot y + ϵ \frac{1}{2} \partial_{q_{α}} \partial_{q_{β}} W (q (t)) y_{α} y_{β} + ϵ^{3 / 2} \frac{1}{6} \partial_{q_{α}} \partial_{q_{β}} \partial_{q_{γ}} W (q (t)) y_{α} y_{β} y_{γ} + ϵ^{2} \int_{0}^{1} \frac{{(τ - 1)}^{4}}{4!} \partial_{q_{α}} \partial_{q_{β}} \partial_{q_{γ}} \partial_{q_{δ}} W (q (t) + τ ϵ^{1 / 2} y) d τ y_{α} y_{β} y_{γ} y_{δ} .

(3.5)

We expand $f^{ϵ} (y, z, t)$ as a formal power series,

f^{ϵ} (y, z, t) = f^{0} (y, z, t) + ϵ^{1 / 2} f^{1} (y, z, t) + \dots

(3.6)

and assume that for all $j \in {0,1,2, \dots}$ the f^j(y, z, t) are periodic with respect to the lattice in z and have sufficient smoothness and decay in y,

\begin{matrix} for all v \in Λ, f^{j} (y, z + v, t) = f^{j} (y, z, t), \\ f^{j} (y, z, t) \in Σ_{y}^{R - j} (ℝ^{d}) . \end{matrix}

(3.7)

The $Σ^{l}$ -spaces are defined in (1.61). $R > 0$ is a fixed positive integer which we will take as large as required. Recall the notation

H (p) := \frac{1}{2} {(p - i \nabla_{z})}^{2} + V (z) .

(3.8)

Substituting (3.5) and (3.6) then gives

r^{ϵ} (x, t) = {ϵ^{- d / 4} e^{i S (t) / ϵ} e^{i p (t) \cdot y / ϵ^{1 / 2}} {ϵ^{2} [\int_{0}^{1} \frac{{(τ - 1)}^{4}}{4!} \partial_{q_{α}} \partial_{q_{β}} \partial_{q_{γ}} \partial_{q_{δ}} W (q (t) + τ ϵ^{1 / 2} y) d τ y_{α} y_{β} y_{γ} y_{δ}] + ϵ^{3 / 2} [\frac{1}{6} \partial_{q_{α}} \partial_{q_{β}} \partial_{q_{γ}} W (q (t)) y_{α} y_{β} y_{γ}] + ϵ [\frac{1}{2} {(- i \nabla_{y})}^{2} + \frac{1}{2} \partial_{q_{α}} \partial_{q_{β}} W (q (t)) y_{α} y_{β} - i \partial_{t}] + ϵ^{1 / 2} [(p (t) - i \nabla_{z}) \cdot (- i \nabla_{y}) + \nabla_{q} W (q (t)) \cdot y - \dot{q} (t) \cdot (- i \nabla_{y}) + \dot{p} (t) \cdot y] + [\dot{S} (t) - \dot{q} (t) \cdot p (t) + H (p (t))]} {f^{0} (y, z, t) + ϵ^{1 / 2} f^{1} (y, z, t) + \dots} |}_{z = \frac{x}{ϵ}, y = \frac{x - q (t)}{ϵ^{1 / 2}}} .

(3.9)

In order to prove Theorem 1.1 it will be sufficient to choose the $f^{j} (y, z, t), j \in {0, \dots, 3}$ so that terms of orders $ϵ^{j / 2}, j \in {0, \dots, 3}$ vanish. With this choice of $f^{j}, j \in {0, \dots, 3}$ we will then prove rigorously in Section III B that $r^{ϵ} (x, t)$ can be bounded by $C ϵ^{2} e^{c t}$ for constants $c > 0, C > 0$ independent of $ϵ, t$ ⁠. There will then be no loss of accuracy in the approximation by taking $f^{j} (y, z, t) = 0, j \geq 4$ ⁠.

1. Analysis of leading order terms

Recall that we assume each $f^{j}, j \in {0,1,2, \dots}$ to be periodic with respect to the lattice $Λ$ in z (3.7). Collecting terms of order 1 in (3.9) and setting equal to zero therefore gives the following self-adjoint elliptic eigenvalue problem in z:

\begin{matrix} H (p (t)) f^{0} (y, z, t) = [- \dot{S} (t) + \dot{q} (t) \cdot p (t)] f^{0} (y, z, t), \\ for all v \in Λ, f^{0} (y, z + v, t) = f^{0} (y, z, t), \\ f^{0} (y, z, t) \in Σ_{y}^{R} (ℝ^{d}) . \end{matrix}

(3.10)

Under Assumption 1.1, E_n(p(t)) is a simple eigenvalue with eigenfunction $χ_{n} (z; p (t))$ for all $t \geq 0$ ⁠. Projecting equation (3.10) onto the subspace of

L_{p e r}^{2} := {f \in L_{l o c}^{2} (ℝ^{d}) : \forall v \in Λ, f (z + v) = f (z)},

(3.11)

spanned by $χ_{n} (z; p (t))$ ⁠, implies

\dot{S} (t) = \dot{q} (t) \cdot p (t) - E_{n} (p (t))

(3.12)

which, after matching with the initial data (1.28), implies (1.21). Equation (3.10) then becomes

\begin{matrix} [H (p (t)) - E_{n} (p (t))] f^{0} (y, z, t) = 0, \\ for all v \in Λ, f^{0} (y, z + v, t) = f^{0} (y, z, t), \\ f^{0} (y, z, t) \in Σ_{y}^{R} (ℝ^{d}), \end{matrix}

(3.13)

which has the general solution

f^{0} (y, z, t) = a^{0} (y, t) χ_{n} (z; p (t)),

(3.14)

where a⁰(y, t) is an arbitrary function in $Σ_{y}^{R} (ℝ^{d})$ ⁠, to be fixed at higher order in the expansion.

2. Analysis of order $ϵ^{1 / 2}$ terms

Collecting terms of order $ϵ^{1 / 2}$ in (3.9), substituting the form of $\dot{S} (t)$ (3.12), and setting equal to zero gives the following inhomogeneous self-adjoint elliptic equation in z for f¹(y, z, t),

\begin{matrix} [H (p (t)) - E_{n} (p (t))] f^{1} (y, z, t) = ξ^{1} (y, z, t), \\ for all z \in Λ, f^{1} (y, z + v, t) = f^{1} (y, z, t), f^{1} (y, z, t) \in Σ_{y}^{R - 1} (ℝ^{d}), \\ ξ^{1} := - [(p (t) - i \nabla_{z}) \cdot (- i \nabla_{y}) + \nabla_{q} W (q (t)) \cdot y - \dot{q} (t) \cdot (- i \nabla_{y}) + \dot{p} (t) \cdot y] f^{0} (y, z, t) . \end{matrix}

(3.15)

Before solving (3.15) we remark on our general strategy for solving equations of this type.

Remark 3.1. Collecting terms of orders $ϵ^{j / 2}$ for each $j \in {1,2, \dots}$ and setting equal to zero, we obtain inhomogeneous self-adjoint elliptic equations of the form

\begin{matrix} [H (p (t)) - E_{n} (p (t))] f^{j} (y, z, t) = ξ^{j} [f^{0}, f^{1}, \dots, f^{j - 1}] (y, z, t), \\ f o r a l l z \in Λ, f^{j} (y, z + v, t) = f^{j} (y, z, t), f^{j} (y, z, t) \in Σ_{y}^{R - j} (ℝ^{d}) . \end{matrix}

(3.16)

Our strategy for solving (3.16) will be the same for each j. Under Assumption 1.1, the eigenvalue E_n(p(t)) is simple with eigenfunction $χ_{n} (z; p (t))$ for all $t \geq 0$ ⁠. By the Fredholm alternative, equation (3.16) is solvable if and only if

f o r a l l t \geq 0, {⟨ χ_{n} (z; p (t)) | ξ^{j} (y, z, t) ⟩}_{L_{z}^{2} (Ω)} = 0.

(3.17)

We will first use identities derived in Appendix Afrom the eigenvalue equation $[H (p) - E_{n} (p)] χ_{n} (z; p) = 0$ to write $ξ^{j} (y, z, t)$ as a sum,

ξ^{j} (y, z, t) = {\tilde{ξ}}^{j} (y, z, t) + [H (p (t)) - E_{n} (p (t))] u^{j} (y, z, t) .

(3.18)

Note that by self-adjointness of H(p(t)) − E_n(p(t)), condition (3.17) is equivalent to the same condition with $ξ^{j} (y, z, t)$ replaced by ${\tilde{ξ}}^{j} (y, z, t)$ ⁠,

f o r a l l t \geq 0, {⟨ χ_{n} (z; p (t)) | {\tilde{ξ}}^{j} (y, z, t) ⟩}_{L_{z}^{2} (Ω)} = 0.

(3.19)

For $f \in L_{p e r}^{2}$ ⁠, define

P_{n}^{⊥} (p) f (z) := f (z) - {⟨ χ_{n} (z; p) | f (z) ⟩}_{L_{z}^{2} (Ω)} χ_{n} (z; p)

(3.20)

to be the projection onto the orthogonal complement of the subspace of $L_{p e r}^{2}$ spanned by $χ_{n} (z; p (t))$ ⁠. Then, assuming (3.19) is satisfied, the general solution of (3.16) is

f^{j} (y, z, t) = a^{j} (y, t) χ_{n} (z; p (t)) + u^{j} (y, z, t) + {[H (p (t)) - E_{n} (p (t))]}^{- 1} P_{n}^{⊥} (p (t)) {\tilde{ξ}}^{j} (y, z, t) .

(3.21)

Note that we have again made use of Assumption 1.1 to ensure that the operator ${[H (p (t)) - E_{n} (p (t))]}^{- 1} P_{n}^{⊥} (p (t)) : L_{p e r}^{2} \to L_{p e r}^{2}$ is bounded for all $t \geq 0$ ⁠. When j=1, condition (3.19) may be enforced by choosing $\dot{q} (t), \dot{p} (t)$ to satisfy (1.19). For $j \geq 2$ ⁠, enforcing the constraint (3.19) leads to evolution equations for a^j−2(y, t).

We will give the proof of the following lemma at the end of this section.

Lemma 3.1. $ξ^{1} (y, z, t)$ ⁠, defined in (3.15), satisfies

ξ^{1} (y, z, t) = {\tilde{ξ}}^{1} (y, z, t) + [H (p (t)) - E_{n} (p (t))] u^{1} (y, z, t),

(3.22)

where

\begin{matrix} {\tilde{ξ}}^{1} (y, z, t) := \\ - [(\nabla_{p} E_{n} (p (t)) - \dot{q} (t)) \cdot (- i \nabla_{y}) a^{0} (y, t) + (\nabla_{q} W (q (t)) + \dot{p} (t)) \cdot y a^{0} (y, t)] χ_{n} (z; p (t)), \\ u^{1} (y, z, t) := (- i \nabla_{y}) a^{0} (y, t) \cdot \nabla_{p} χ_{n} (z; p (t)) . \end{matrix}

(3.23)

The solvability condition of (3.15), given by (3.19) with j = 1 on ${\tilde{ξ}}^{1} (y, z, t)$ (3.23), is then equivalent to

(\nabla_{p} E_{n} (p (t)) - \dot{q} (t)) \cdot (- i \nabla_{y}) a^{0} (y, t) + (\nabla_{q} W (q (t)) + \dot{p} (t)) \cdot y a^{0} (y, t) = 0,

(3.24)

which we can satisfy by choosing (q(t), p(t)) to evolve as the Hamiltonian flow of the nth Bloch band Hamiltonian $H_{n} (q, p) = E_{n} (p) + W (q)$ ⁠,

\dot{q} (t) = \nabla_{p} E_{n} (p (t)), \dot{p} (t) = - \nabla_{q} W (q (t)) .

(3.25)

Taking q(0), p(0) = q₀,p₀ to match with the initial data (1.28) implies (1.19).

The general solution of (3.15) is given by taking j = 1 in (3.21), where $u^{1}, {\tilde{ξ}}^{1}$ are given by (3.23). With the choice (3.25) for $\dot{q} (t), \dot{p} (t)$ we have that ${\tilde{ξ}}^{1} = 0$ for all $t \geq 0$ so that the general solution reduces to

f^{1} (y, z, t) = a^{1} (y, t) χ_{n} (z; p (t)) + (- i \nabla_{y}) a^{0} (y, t) \cdot \nabla_{p} χ_{n} (z; p (t)),

(3.26)

where a¹(y, t) is an arbitrary function in $Σ_{y}^{R - 1} (ℝ^{d})$ to be fixed at higher order in the expansion. Note that since $a^{0} (y, t) \in Σ_{y}^{R} (ℝ^{d})$ ⁠, this ensures that $f^{1} (y, z, t) \in Σ_{y}^{R - 1} (ℝ^{d})$ as required.

Proof of Lemma 3.1. By Assumption 1.1, E_n(p) is smooth in a neighborhood of p(t). By adding and subtracting $\nabla_{p} E_{n} (p (t)) \cdot (- i \nabla_{y}) f^{0} (y, z, t)$ ⁠, $ξ^{1} (y, z, t)$ is equal to

ξ^{1} (y, z, t) = - [((p (t) - i \nabla_{z}) - \nabla_{p} E_{n} (p (t))) \cdot (- i \nabla_{y})] f^{0} (y, z, t) - [(\nabla_{p} E_{n} (p (t)) - \dot{q} (t)) \cdot (- i \nabla_{y})] f^{0} (y, z, t) - [(\nabla_{q} W (q (t)) + \dot{p} (t)) \cdot y] f^{0} (y, z, t) .

(3.27)

Substituting the explicit form of f⁰(y, z, t) (3.14) into (3.27), we have

ξ^{1} (y, z, t) = - (- i \nabla_{y}) a^{0} (y, t) \cdot [(p (t) - i \nabla_{z}) - \nabla_{p} E_{n} (p (t))] χ_{n} (z; p (t)) - (- i \nabla_{y}) a^{0} (y, t) \cdot [\nabla_{p} E_{n} (p (t)) - \dot{q} (t)] χ_{n} (z; p (t)) - y a^{0} (y, t) \cdot [\nabla_{q} W (q (t)) + \dot{p} (t)] χ_{n} (z; p (t)) .

(3.28)

(3.23) then follows immediately from identity (A2).

3. Analysis of order $ϵ$ and $ϵ^{3 / 2}$ terms (summary)

It is possible to continue the procedure outlined in Remark 3.1 to any order in $ϵ^{1 / 2}$ ⁠. In Appendices B and C we show the details of how to continue the procedure in order to cancel terms in the expansion of orders $ϵ$ and $ϵ^{3 / 2}$ ⁠. In particular, we derive the evolution equations of the amplitudes a⁰(y, t), a¹(y, t) and show that

a^{0} (y, t) = a (y, t) e^{i ϕ_{B} (t)}, a^{1} (y, t) = b (y, t) e^{i ϕ_{B} (t)},

(3.29)

where $a (y, t), b (y, t), ϕ_{B} (t)$ satisfy Equations (1.22), (1.24), and (1.27), respectively.

B. Proof of estimate (1.30) for the corrector $η$

Let

f_{3}^{ϵ} (y, z, t) := f^{0} (y, z, t) + ϵ^{1 / 2} f^{1} (y, z, t) + ϵ f^{2} (y, z, t) + ϵ^{3 / 2} f^{3} (y, z, t),

(3.30)

where the f⁰, f¹, f², f³ are given by (3.14), (3.26), (B5), and (C5), respectively, and define

ψ_{3}^{ϵ} (x, t) := {ϵ^{- d / 4} e^{i S (t) / ϵ} e^{i p (t) \cdot y / ϵ^{1 / 2}} f_{3}^{ϵ} (y, z, t) |}_{z = \frac{x}{ϵ}, y = \frac{x - q (t)}{ϵ^{1 / 2}}} .

(3.31)

Let $ψ^{ϵ} (x, t)$ denote the exact solution of the initial value problem (1.28). From the manipulations of Sec. III A, we have that $η_{3}^{ϵ} (x, t) := ψ^{ϵ} (x, t) - ψ_{3}^{ϵ} (x, t)$ satisfies

\begin{matrix} i ϵ \partial_{t} η_{3}^{ϵ} (x, t) = [- \frac{ϵ^{2}}{2} Δ_{x} + V (\frac{x}{ϵ}) + W (x)] η_{3}^{ϵ} (x, t) + r_{3}^{ϵ} (x, t), \\ η_{3}^{ϵ} (x, 0) = η_{3,0}^{ϵ} (x), \end{matrix}

(3.32)

where $r_{3}^{ϵ} (x, t)$ is given by

r_{3}^{ϵ} (x, t) = {ϵ^{- d / 4} e^{i S (t) / ϵ} e^{i p (t) \cdot y / ϵ^{1 / 2}} {ϵ^{2} \int_{0}^{1} \frac{{(τ - 1)}^{4}}{4!} \partial_{q_{α}} \partial_{q_{β}} \partial_{q_{γ}} \partial_{q_{δ}} W (q (t) + τ ϵ^{1 / 2} y) d τ y_{α} y_{β} y_{γ} y_{δ} \cdot (f^{0} (y, z, t) + ϵ^{1 / 2} f^{1} (y, z, t) + ϵ f^{2} (y, z, t) + ϵ^{3 / 2} f^{3} (y, z, t)) + ϵ^{2} [\frac{1}{6} \partial_{q_{α}} \partial_{q_{β}} \partial_{q_{γ}} W (q (t)) y_{α} y_{β} y_{γ}] (f^{1} (y, z, t) + ϵ^{1 / 2} f^{2} (y, z, t) + ϵ f^{3} (y, z, t)) + ϵ^{2} [- i \partial_{t} + \frac{1}{2} {(- i \nabla_{y})}^{2} + \frac{1}{2} \partial_{q_{α}} \partial_{q_{β}} W (q (t)) y_{α} y_{β}] (f^{2} (y, z, t) + ϵ^{1 / 2} f^{3} (y, z, t)) + ϵ^{2} [((p (t) - i \nabla_{z}) - \nabla_{p} E_{n} (p (t))) \cdot (- i \nabla_{y})] f^{3} (y, z, t)} |}_{z = \frac{x}{ϵ}, y = \frac{x - q (t)}{ϵ^{1 / 2}}}

(3.33)

And $η_{3,0}^{ϵ} (x)$ is given by

η_{3,0}^{ϵ} (x) = - {ϵ^{- d / 4} e^{i p_{0} \cdot y / ϵ^{1 / 2}} {ϵ [f^{2} (z, y, 0) + ϵ^{1 / 2} f^{3} (z, y, 0)]} |}_{z = \frac{x}{ϵ}, y = \frac{x - q_{0}}{ϵ^{1 / 2}}} .

(3.34)

Since the $f^{j} (y, z, t), j \in {0, \dots, 3}$ are periodic with respect to the lattice $Λ$ ⁠, we will follow the work of Carles and Sparber⁶ and bound the above expressions in the uniform norm in z and the L² norm in y,

\begin{matrix} {∥ r_{3}^{ϵ} (x, t) ∥}_{L_{x}^{2}} \leq \\ = ∥ ϵ^{2} \int_{0}^{1} \frac{{(τ - 1)}^{4}}{4!} \partial_{q_{α}} \partial_{q_{β}} \partial_{q_{γ}} \partial_{q_{δ}} W (q (t) + τ ϵ^{1 / 2} y) d τ y_{α} y_{β} y_{γ} y_{δ} \\ \cdot (f^{0} (y, z, t) + ϵ^{1 / 2} f^{1} (y, z, t) + ϵ f^{2} (y, z, t) + ϵ^{3 / 2} f^{3} (y, z, t)) \\ + ϵ^{2} [\frac{1}{6} \partial_{q_{α}} \partial_{q_{β}} \partial_{q_{γ}} W (q (t)) y_{α} y_{β} y_{γ}] (f^{1} (y, z, t) + ϵ^{1 / 2} f^{2} (y, z, t) + ϵ f^{3} (y, z, t)) \\ + ϵ^{2} [- i \partial_{t} + \frac{1}{2} {(- i \nabla_{y})}^{2} + \frac{1}{2} \partial_{q_{α}} \partial_{q_{β}} W (q (t)) y_{α} y_{β}] (f^{2} (y, z, t) + ϵ^{1 / 2} f^{3} (y, z, t)) \\ + {ϵ^{2} [((p (t) - i \nabla_{z}) - \nabla_{p} E_{n} (p (t))) \cdot (- i \nabla_{y})] f^{3} (y, z, t) |}_{L_{z}^{\infty}, L_{y}^{2}} \\ {∥ η_{3}^{ϵ} (x, 0) ∥}_{L_{x}^{2}} \leq {∥ ϵ [f^{2} (z, y, 0) + ϵ^{1 / 2} f^{3} (z, y, 0)] ∥}_{L_{z}^{\infty}, L_{y}^{2}}, \end{matrix}

(3.35)

where we have used the fact that

{∥ ϵ^{- d / 4} f (\frac{x - q (t)}{ϵ^{1 / 2}}) ∥}_{L_{x}^{2}} = {∥ f (y) ∥}_{L_{y}^{2}} .

(3.36)

We show how to bound the first term in (3.35). Bounding the other terms is similar although care must be taken in bounding terms in $L_{z}^{\infty}$ ⁠; see Appendix D. Let

I (t) := ϵ^{2} {∥ \int_{0}^{1} \frac{{(τ - 1)}^{4}}{4!} \partial_{q_{α}} \partial_{q_{β}} \partial_{q_{γ}} \partial_{q_{δ}} W (q (t) + τ ϵ^{1 / 2} y) d τ y_{α} y_{β} y_{γ} y_{δ} f^{0} (y, z, t) ∥}_{L_{z}^{\infty}, L_{y}^{2}},

(3.37)

where $f^{0} (y, z, t) = a^{0} (y, t) χ_{n} (z; p (t))$ (3.14). By Assumption 1.2, $\sum_{| α | = 4} | \partial_{x}^{α} W (x) | \in L^{\infty} (ℝ^{d})$ ⁠,

I (t) \leq ϵ^{2} \frac{1}{4!} {∥ \sum_{| α | = 4} | \partial_{x}^{α} W (x) | ∥}_{L^{\infty} (ℝ^{d})} \sum_{| α | = 4} {∥ y^{α} a (y, t) χ_{n} (z; p (t)) ∥}_{L_{z}^{\infty}, L_{y}^{2}} .

(3.38)

Recall Assumption 1.1. Define

S_{n} := {p \in ℝ^{d} : \underset{m \neq n}{i n f} | E_{m} (p) - E_{n} (p) | \geq M},

(3.39)

so that for all $t \in [0, \infty), p (t) \in S_{n}$ ⁠. For each fixed $p \in S_{n}$ ⁠, by elliptic regularity, $χ_{n} (z; p)$ is smooth in z so that ${∥ χ_{n} (z; p) ∥}_{L_{z}^{\infty}} < \infty$ ⁠. Using compactness of the Brillouin zone $B$ and smoothness of $χ_{n} (z; p)$ for $p \in S_{n}$ we have that

\sup_{p \in B \cap S_{n}} {∥ χ_{n} (z; p) ∥}_{L_{z}^{\infty}} < \infty .

(3.40)

Recall that for any reciprocal lattice vector $b \in Λ^{*}$ ⁠, $χ_{n} (z; p + b) = e^{- i b \cdot z} χ_{n} (z; p)$ ⁠. It then follows that

for all b \in Λ^{*}, {∥ χ_{n} (z; p + b) ∥}_{L_{z}^{\infty}} = {∥ χ_{n} (z; p) ∥}_{L_{z}^{\infty}} .

(3.41)

It then follows from combining (3.40) and (3.41) that

\sup_{p \in S_{n}} {∥ χ_{n} (z; p) ∥}_{L_{z}^{\infty}} < \infty .

(3.42)

In Appendix D we show how to bound all z-dependence in $r_{3}^{ϵ} (x, t)$ (3.33) uniformly in $p \in S_{n}$ in a similar way.

We have therefore that (3.38)

I (t) \leq ϵ^{2} \frac{1}{4!} {∥ \sum_{| α | = 4} | \partial_{x}^{α} W (x) | ∥}_{L^{\infty} (ℝ^{d})} \sup_{p \in S_{n}} {∥ χ_{n} (z; p) ∥}_{L_{z}^{\infty}} \sum_{| α | = 4} {∥ y^{α} a (y, t) ∥}_{L_{y}^{2}} .

(3.43)

We see that to complete the bound, we require a bound on the 4th moments of a(y, t), which solves the Schrödinger equation with time-dependent co-efficients,

\begin{matrix} i \partial_{t} a (y, t) = \frac{1}{2} \partial_{p_{α}} \partial_{p_{β}} E_{n} (p (t)) (- i \partial_{y_{α}}) (- i \partial_{y_{β}}) a (y, t) + \frac{1}{2} \partial_{q_{α}} \partial_{q_{β}} W (q (t)) y_{α} y_{β} a (y, t), \\ a (y, 0) = a_{0} (y) . \end{matrix}

(3.44)

Following the work of Carles and Sparber⁶ we first define, for any $l \in ℕ$ ⁠, the spaces

Σ^{l} (ℝ^{d}) := {f \in L^{2} (ℝ^{d}) : {∥ f ∥}_{Σ^{l}} := \sum_{| α | + | β | \leq l} {∥ y^{α} {(- i \partial_{y})}^{β} f (y) ∥}_{L_{y}^{2}} < \infty} .

(3.45)

We then require the following lemma due to the work of Kitada.⁵⁰

Lemma 3.2. (Existence of unitary solution operator for the envelope equation). Let $u_{0} \in L^{2} (ℝ^{d})$ ⁠, and $η_{α β} (t), ζ_{α β} (t)$ be real-valued, symmetric, continuous, and uniformly bounded in t. Then the equation

\begin{matrix} i \partial_{t} u = \frac{1}{2} η_{α β} (t) (- i \partial_{y_{α}}) (- i \partial_{y_{β}}) u + \frac{1}{2} ζ_{α β} (t) y_{α} y_{β} u, \\ u (y, 0) = u_{0} (y) \end{matrix}

(3.46)

has a unique solution $u \in C ([0, \infty); L^{2} (ℝ^{d}))$ . It satisfies

{∥ u (\cdot, t) ∥}_{L^{2} (ℝ^{d})} = {∥ u_{0} (\cdot) ∥}_{L^{2} (ℝ^{d})} .

(3.47)

Moreover, if $u_{0} \in Σ^{l} (ℝ^{d})$ , then $u \in C ([0, \infty); Σ^{l} (ℝ^{d}))$ ⁠.

We seek quantitative bounds on ${∥ u (\cdot, t) ∥}_{Σ^{l} (ℝ^{d})}$ for $l \geq 1$ ⁠. For simplicity, we consider in detail the case l = 1. Recall (1.23),

H (t) := \frac{1}{2} η_{α β} (t) (- i \partial_{y_{α}}) (- i \partial_{y_{β}}) + \frac{1}{2} ζ_{α β} (t) y_{α} y_{β},

(3.48)

and let $u_{0} (y) \in S (ℝ^{d})$ so that $\forall l \geq 0$ ⁠, the solution of (3.46), $u (y, t) \in C ([0, \infty); Σ^{l} (ℝ^{d}))$ ⁠. Then $(- i \partial_{y_{α}}) u (y, t) \in S (ℝ)$ solves

\begin{matrix} i \partial_{t} (- i \partial_{y_{α}}) u = H (t) (- i \partial_{y_{α}}) u + [(- i \partial_{y_{α}}), H (t)] u, \\ (- i \partial_{y_{α}}) u (y, 0) = (- i \partial_{y_{α}}) u_{0} (y) . \end{matrix}

(3.49)

We can solve this equation using Duhamel’s formula and the solution operator of Equation (3.46). It follows that

{∥ (- i \partial_{y_{α}}) u (y, t) ∥}_{L_{y}^{2}} \leq {∥ (- i \partial_{y_{α}}) u (y, 0) ∥}_{L_{y}^{2}} + \int_{0}^{t} {∥ [(- i \partial_{y_{α}}), H (s)] u (y, s) ∥}_{L_{y}^{2}} d s .

(3.50)

Since $ζ_{α β} (t)$ is symmetric, the commutator is given explicitly by

[(- i \partial_{y_{α}}), H (s)] = (- i) ζ_{α β} (t) y_{β},

(3.51)

so that

\begin{matrix} {∥ (- i \partial_{y_{α}}) u (y, t) ∥}_{L_{y}^{2}} \leq {∥ (- i \partial_{y_{α}}) u_{0} (y) ∥}_{L_{y}^{2}} + \int_{0}^{t} | ζ_{α β} (t) | {∥ y_{β} u (y, s) ∥}_{L_{y}^{2}} d s . \end{matrix}

(3.52)

By an identical reasoning, we can derive a similar bound on $y_{α} u (y, t)$ ⁠,

{∥ y_{α} u (y, t) ∥}_{L_{y}^{2}} \leq {∥ y_{α} u_{0} (y) ∥}_{L_{y}^{2}} + \int_{0}^{t} | η_{α β} (s) | {∥ (- i \partial_{y_{β}}) u (y, s) ∥}_{L_{y}^{2}} d s .

(3.53)

Adding inequalities (3.52) and (3.53) gives

{∥ u (\cdot, t) ∥}_{Σ^{1}} \leq {∥ u_{0} (\cdot) ∥}_{Σ^{1}} + 2 \int_{0}^{t} max_{α, β \in {1, \dots, d}} {| η_{α β} (s) |, | ζ_{α β} (s) |} {∥ u (\cdot, s) ∥}_{Σ^{1}} d s,

(3.54)

using the following version of Gronwall’s inequality.

Lemma 3.3. (Gronwall’s inequality). Let $υ$ (t) satisfy the inequality,

v (t) \leq a (t) + \int_{0}^{t} b (s) v (s) d s,

(3.55)

where b(t) is non-negative and a(t) is non-decreasing. Then,

v (t) \leq a (t) \exp (\int_{0}^{t} b (s) d s) .

(3.56)

We have that

{∥ u (\cdot, t) ∥}_{Σ^{1}} \leq {∥ u_{0} (\cdot) ∥}_{Σ^{1}} e^{2 \int_{0}^{t} \max_{α, β \in {1, \dots, d}} {| η_{α β} (s) |, | ζ_{α β} (s) |} d s} .

(3.57)

More generally, we have for any $l \geq 0$ that there exists a constant $C_{l} > 0$ such that

{∥ u (\cdot, t) ∥}_{Σ^{l}} \leq {∥ u_{0} (\cdot) ∥}_{Σ^{l}} e^{C_{l} \int_{0}^{t} \max_{α, β \in {1, \dots, d}} {| η_{α β} (s) |, | ζ_{α β} (s) |} d s} .

(3.58)

We have proved the following.

Lemma 3.4. (Bound on solutions of (3.46) in the spaces $Σ^{l} (ℝ^{d})$ ⁠). Let the time-dependent co-efficients $η_{α β} (t), ζ_{α β} (t)$ be real-valued, symmetric, continuous, and uniformly bounded in t. Let $u_{0} (y) \in Σ^{l} (ℝ^{d})$ . Then, by Lemma 3.2, there exists a unique solution $u (y, t) \in C ([0, \infty); Σ^{l} (ℝ^{d}))$ ⁠. For each integer $l \geq 0$ ⁠, there exists a constant $C_{l} > 0$ such that this solution satisfies

{∥ u (\cdot, t) ∥}_{Σ^{l} (ℝ^{d})} \leq {∥ u_{0} (\cdot) ∥}_{Σ^{l}} e^{C_{l} \int_{0}^{t} \max_{α, β \in {1, \dots, d}} {| η_{α β} (s) |, | ζ_{α β} (s) |} d s}

(3.59)

Since the map $p \mapsto E_{n} (p)$ is $B$ -periodic and smooth for all $p \in S_{n}$ ⁠, we have that under Assumption 1.1, $s u p {}_{t \in [0, \infty)}\max_{α, β \in {1, \dots, d}} | \partial_{p_{α}} \partial_{p_{β}} E_{n} (p (t)) | < \infty$ ⁠. Under Assumption 1.2 we have that $s u p {}_{t \in [0, \infty)}\max_{α, β \in {1, \dots, d}} | \partial_{q_{α}} \partial_{q_{β}} W (q (t)) | < \infty$ ⁠. Since $\partial_{p_{α}} \partial_{p_{β}} E_{n} (p (t)), \partial_{q_{α}} \partial_{q_{β}} W (q (t))$ are clearly real-valued, symmetric, and continuous in t, we have that Lemma 3.4 applies to solutions of (3.44). Since $a_{0} (y) \in S (ℝ^{d})$ by assumption we have that for any integer $l \geq 0$ ⁠,

{∥ a (y, t) ∥}_{Σ^{l} (ℝ^{d})} \leq {∥ a_{0} (y) ∥}_{Σ^{l} (ℝ^{d})} \cdot \exp (C {}_{l}{\max_{α, β \in {1, \dots, d}}} \sup_{s \in [0, \infty)} {| \partial_{p_{α}} \partial_{p_{β}} E_{n} (p (s)) |, | \partial_{q_{α}} \partial_{q_{β}} W (q (s)) |} t) .

(3.60)

Remark 3.2. Terms which depend on b(y, t) rather than a(y, t) may be dealt with similarly, by an application of Duhamel’s formula and a Gronwall inequality.

We have therefore that

ϵ^{2} {∥ \int_{0}^{1} \frac{{(τ - 1)}^{4}}{4!} \partial_{q_{α}} \partial_{q_{β}} \partial_{q_{γ}} \partial_{q_{δ}} W (q (t) + τ ϵ^{1 / 2} y) d τ y_{α} y_{β} y_{γ} y_{δ} a (y, t) χ_{n} (z; p (t)) ∥}_{L_{z}^{\infty}, L_{y}^{2}} \leq c_{1} ϵ^{2} e^{c_{2} t},

(3.61)

where

\begin{matrix} c_{1} := \frac{1}{4!} {∥ \sum_{| α | = 4} | \partial_{x}^{α} W (x) | ∥}_{L_{x}^{\infty} (ℝ^{d})} \sup_{p \in S_{n}} {∥ χ_{n} (z; p) ∥}_{L_{z}^{\infty}} {∥ a_{0} (y) ∥}_{Σ_{y}^{4} (ℝ^{d})}, \\ c_{2} := C {}_{l}{max_{α, β \in {1, \dots, d}}} \sup_{s \in [0, \infty)} {| \partial_{p_{α}} \partial_{p_{β}} E_{n} (p (s)) |, | \partial_{q_{α}} \partial_{q_{β}} W (q (s)) |} \end{matrix}

(3.62)

are constants independent of $t, ϵ$ ⁠.

We conclude that there exist constants $C_{1}, C_{2}, C_{3} > 0$ ⁠, independent of $t, ϵ$ such that

\begin{matrix} {∥ η_{3,0}^{ϵ} (x) ∥}_{L_{x}^{2}} \leq C_{1} ϵ, \\ {∥ r_{3}^{ϵ} (x, t) ∥}_{L_{x}^{2}} \leq C_{2} e^{C_{3} t} ϵ^{2} . \end{matrix}

(3.63)

The bound (1.30) then follows from the basic a priori L² bound for solutions of the linear time-dependent Schrödinger equation:

Lemma 3.5. Let $ψ (x, t)$ be the unique solution of

\begin{matrix} i \partial_{t} ψ = H ψ + f, \\ ψ (x, 0) = ψ_{0} (x), \end{matrix}

(3.64)

where H is a self-adjoint operator. Then,

{∥ ψ (\cdot, t) ∥}_{L^{2} (ℝ^{d})} \leq {∥ ψ_{0} (\cdot) ∥}_{L^{2} (ℝ^{d})} + \int_{0}^{t} {∥ f (\cdot, t^{'}) ∥}_{L^{2} (ℝ^{d})} d t^{'},

(3.65)

and when f = 0, we have

{∥ ψ (\cdot, t) ∥}_{L^{2} (ℝ^{d})} = {∥ ψ_{0} (\cdot) ∥}_{L^{2} (ℝ^{d})} .

(3.66)

Applying Lemma 3.5 to Equation (3.32) then gives the bound on $η_{3}^{ϵ} (x, t)$ ⁠,

{∥ η_{3}^{ϵ} (x, t) ∥}_{L_{x}^{2}} \leq {∥ η_{3}^{ϵ} (x, 0) ∥}_{L_{x}^{2}} + \frac{1}{ϵ} \int_{0}^{t} {∥ r_{3}^{ϵ} (x, t^{'}) ∥}_{L_{x}^{2}} d t^{'} \leq C_{1} ϵ + \frac{1}{ϵ} \int_{0}^{t} C_{2} e^{C_{3} t^{'}} ϵ^{2} d t^{'} \leq C e^{C t} ϵ,

(3.67)

where C is a constant independent of $ϵ, t$ ⁠. This completes the proof of Theorem 1.1.

IV. PROOF OF THEOREM 1.2 ON DYNAMICS OF PHYSICAL OBSERVABLES

Let $ψ^{ϵ} (x, t)$ be the solution of (1.28). By Theorem 1.1 we have that this solution has the form

ψ^{ϵ} (x, t) = {{\tilde{ψ}}^{ϵ} (y, z, t) |}_{y = \frac{x - q (t)}{ϵ^{1 / 2}}, z = \frac{x}{ϵ}} + η^{ϵ} (x, t),

(4.1)

where

{\tilde{ψ}}^{ϵ} (y, z, t) := ϵ^{- d / 4} e^{i S (t) / ϵ} e^{i p (t) \cdot y / ϵ^{1 / 2}} e^{i ϕ_{B} (t)} {a (y, t) χ_{n} (z; p (t)) + ϵ^{1 / 2} [(- i \nabla_{y}) a (y, t) \cdot \nabla_{p} χ_{n} (z; p (t)) + b (y, t) χ_{n} (z; p (t))]} .

(4.2)

In this section we compute the dynamics of the physical observables,

\begin{matrix} Q^{ϵ} (t) := \frac{1}{N^{ϵ} (t)} \int_{ℝ^{d}} x {| {\tilde{ψ}}^{ϵ} (y, z, t) |}_{z = \frac{x}{ϵ}, y = \frac{x - q (t)}{ϵ^{1 / 2}}}^{2} d x, \\ P^{ϵ} (t) := \frac{1}{N^{ϵ} (t)} \int_{ℝ^{d}} \bar{{\tilde{ψ}}^{ϵ} (y, z, t)} (- i ϵ {}^{1 / 2}\nabla_{y}) {\tilde{ψ}}^{ϵ} (y, z, t) |_{z = \frac{x}{ϵ}, y = \frac{x - q (t)}{ϵ^{1 / 2}}} d x, \end{matrix}

(4.3)

where

N^{ϵ} (t) = \int_{ℝ^{d}} {| {\tilde{ψ}}^{ϵ} (y, z, t) |}_{z = \frac{x}{ϵ}, y = \frac{x - q (t)}{ϵ^{1 / 2}}}^{2} d x .

(4.4)

Remark 4.1. Throughout this section we will employ a short-hand notation,

\begin{matrix} f^{ϵ} (x, t) = O (ϵ^{K} e^{c t}) ⟺ \exists c > 0, C > 0 i n d e p e n d e n t o f t, ϵ s u c h t h a t {∥ f^{ϵ} (x, t) ∥}_{L_{x}^{2}} \leq C ϵ^{K} e^{c t}, \\ g^{ϵ} (t) = O (ϵ^{K} e^{c t}) ⟺ \exists c > 0, C > 0 i n d e p e n d e n t o f t, ϵ s u c h t h a t | g^{ϵ} (t) | \leq C ϵ^{K} e^{c t} . \end{matrix}

(4.5)

We will use the following lemma which is a mild generalization of that found in Ref. 24 (as Lemma 4.2):

Lemma 4.1. Let $f \in S (ℝ^{d})$ ⁠, g smooth and periodic with respect to the lattice $Λ$ ⁠, $s \in ℝ$ a constant, and $δ > 0$ an arbitrary positive parameter. Then for any positive integer $N > 0$ ⁠,

\int_{ℝ^{d}} f (x) g (\frac{x}{δ} + \frac{s}{δ^{2}}) d x = (\int_{ℝ^{d}} f (x) d x) (\int_{Ω} g (z) d z) + O (δ^{N}) .

(4.6)

For the proof, see Appendix E.

A. Asymptotic expansion and dynamics of $N^{ϵ} (t)$

By changing variables in the integral (4.4), we have that

N^{ϵ} (t) = ϵ^{d / 2} \int_{ℝ^{d}} {| {\tilde{ψ}}^{ϵ} (y, z, t) |}_{z = \frac{y}{ϵ^{1 / 2}} + \frac{q (t)}{ϵ}}^{2} d y .

(4.7)

Substituting (4.2) into (4.7) gives

\begin{matrix} = \int_{ℝ^{d}} {\bar{a (y, t) χ_{n} (z; p (t)) + ϵ^{1 / 2} [(- i \nabla_{y}) a (y, t) \cdot \nabla_{p} χ_{n} (z; p (t)) + b (y, t) χ_{n} (z; p (t))]}} \\ {{a (y, t) χ_{n} (z; p (t)) + ϵ^{1 / 2} [(- i \nabla_{y}) a (y, t) \cdot \nabla_{p} χ_{n} (z; p (t)) + b (y, t) χ_{n} (z; p (t))]} |}_{z = \frac{y}{ϵ^{1 / 2}} + \frac{q (t)}{ϵ}} d y . \end{matrix}

(4.8)

We expand the product in the integral and apply Lemma 4.1 term by term with $s = q (t), δ = ϵ^{1 / 2}$ ⁠. Since the $χ_{n}$ are assumed normalized, for all $t \in [0, \infty)$ ${∥ χ_{n} (\cdot; p (t)) ∥}_{L^{2} (Ω)} = 1$ ⁠, we have

N^{ϵ} (t) = {∥ a (y, t) ∥}_{L_{y}^{2} (ℝ^{d})}^{2} + ϵ^{1 / 2} [{⟨ (- i \nabla_{y}) a (y, t) | a (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})} \cdot {⟨ \nabla_{p} χ_{n} (z; p (t)) | χ_{n} (z; p (t)) ⟩}_{L_{z}^{2} (Ω)} + {⟨ a (y, t) | (- i \nabla_{y}) a (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})} \cdot {⟨ χ_{n} (z; p (t)) | \nabla_{p} χ_{n} (z; p (t)) ⟩}_{L_{z}^{2} (Ω)} + {⟨ b (y, t) | a (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})} + {⟨ a (y, t) | b (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})}] + O (ϵ e^{c t}) .

(4.9)

Remark 4.2. In (4.9) we have made explicit all terms through order $ϵ^{1 / 2}$ ⁠. To justify the error bound, consider that the remaining terms may be bounded by $C (t) ϵ$ where $C (t)$ depends on $Σ_{y}^{l_{1}}$ -norms of a(y, t), b(y, t) and $L_{z}^{2}$ -norms of $\partial_{p}^{l_{2}} χ_{n} (z; p (t))$ where l1, l2 are positive integers. By an identical reasoning to that given in Section III B we have that $C (t)$ may be bounded by Ce^ct where $c > 0, C > 0$ are constants independent of $ϵ, t$ ⁠. Error terms of this type will arise throughout the following discussion and will be treated similarly.

Under Assumption 1.1, in a neighborhood of the curve $p (t) \in B$ ⁠, the mapping $p \mapsto χ_{n} (z; p)$ is smooth. Hence, we may differentiate the normalization condition: ${∥ χ_{n} (\cdot; p) ∥}_{L^{2} (Ω)}^{2} = 1$ with respect to p and evaluate along the curve p(t) to obtain the identity,

{⟨ χ_{n} (z; p (t)) | \nabla_{p} χ_{n} (z; p (t)) ⟩}_{L_{z}^{2} (Ω)} + {⟨ \nabla_{p} χ_{n} (z; p (t)) | χ_{n} (z; p (t)) ⟩}_{L_{z}^{2} (Ω)} = 0.

(4.10)

It follows from this, and the fact that $(- i \nabla_{y})$ is symmetric with respect to the $L_{y}^{2}$ -inner product, that

{⟨ (- i \nabla_{y}) a (y, t) | a (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})} \cdot {⟨ \nabla_{p} χ_{n} (z; p (t)) | χ_{n} (z; p (t)) ⟩}_{L_{z}^{2} (Ω)} + {⟨ a (y, t) | (- i \nabla_{y}) a (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})} \cdot {⟨ χ_{n} (z; p (t)) | \nabla_{p} χ_{n} (z; p (t)) ⟩}_{L_{z}^{2} (Ω)} = 0

(4.11)

so that (4.9) reduces to

N^{ϵ} (t) = {∥ a (y, t) ∥}_{L_{y}^{2} (ℝ^{d})}^{2} + ϵ^{1 / 2} [{⟨ b (y, t) | a (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})} + {⟨ a (y, t) | b (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})}] + O (ϵ e^{c t}) .

(4.12)

From L²-norm conservation for solutions of (1.22), we have that ${∥ a (y, t) ∥}_{L_{y}^{2}} = {∥ a_{0} (y) ∥}_{L_{y}^{2}}$ ⁠. In Appendix F we calculate (F17)

\frac{d}{d t} [{⟨ b (y, t) | a (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})} + {⟨ a (y, t) | b (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})}] = 0,

(4.13)

so that

{\dot{N}}^{ϵ} (t) = O (ϵ e^{C t}) .

(4.14)

Integrating in time then gives

\begin{matrix} N^{ϵ} (t) & = N^{ϵ} (0) + O (ϵ e^{c t}) \\ = {∥ a_{0} (y) ∥}_{L_{y}^{2} (ℝ^{d})}^{2} + ϵ^{1 / 2} [{⟨ b_{0} (y) | a_{0} (y) ⟩}_{L_{y}^{2} (ℝ^{d})} + {⟨ a_{0} (y) | b_{0} (y) ⟩}_{L_{y}^{2} (ℝ^{d})}] + O (ϵ e^{c t}) . \end{matrix}

(4.15)

B. Asymptotic expansion of $Q^{ϵ} (t), P^{ϵ} (t)$ ⁠, proof of assertion (1) of theorem 1.2

Changing variables in the integrals (4.3) and using the identity,

x = q (t) + {ϵ^{1 / 2} y |}_{y = \frac{x - q (t)}{ϵ^{1 / 2}}},

(4.16)

we have

\begin{matrix} Q^{ϵ} (t) = q (t) + ϵ^{1 / 2 + d / 2} \frac{1}{N^{ϵ} (t)} \int_{ℝ^{d}} y {| {\tilde{ψ}}^{ϵ} (y, z, t) |}_{z = \frac{q (t)}{ϵ} + \frac{y}{ϵ^{1 / 2}}}^{2} d y, \\ P^{ϵ} (t) = {ϵ^{1 / 2 + d / 2} \frac{1}{N^{ϵ} (t)} \int_{ℝ^{d}} \bar{{\tilde{ψ}}^{ϵ} (y, z, t)} (- i \nabla_{y}) {\tilde{ψ}}^{ϵ} (y, z, t) |}_{z = \frac{q (t)}{ϵ} + \frac{y}{ϵ^{1 / 2}}} d y . \end{matrix}

(4.17)

Substituting (4.2) into (4.17) we have, for each $α \in {1, \dots, d}$ ⁠,

\begin{matrix} Q_{α}^{ϵ} (t) = q_{α} (t) + ϵ^{1 / 2} \frac{1}{N^{ϵ} (t)} \int_{ℝ^{d}} y_{α} | a (y, t) χ_{n} (z; p (t)) \\ + {ϵ^{1 / 2} [(- i \partial_{y_{β}}) a (y, t) \partial_{p_{β}} χ_{n} (z; p (t)) + b (y, t) χ_{n} (z; p (t))] |}_{z = \frac{q (t)}{ϵ} + \frac{y}{ϵ^{1 / 2}}}^{2} d y, \\ P_{α}^{ϵ} (t) = p_{α} (t) + ϵ^{1 / 2} \frac{1}{N^{ϵ} (t)} \\ \times \int_{ℝ^{d}} \bar{a (y, t) χ_{n} (z; p (t)) + ϵ^{1 / 2} [(- i \partial_{y_{β}}) a (y, t) \partial_{p_{β}} χ_{n} (z; p (t)) + b (y, t) χ_{n} (z; p (t))]} \\ {(- i \partial_{y_{α}}) (a (y, t) χ_{n} (z; p (t)) + ϵ^{1 / 2} [(- i \partial_{y_{β}}) a (y, t) \partial_{p_{β}} χ_{n} (z; p (t)) + b (y, t) χ_{n} (z; p (t))]) |}_{z = \frac{q (t)}{ϵ} + \frac{y}{ϵ^{1 / 2}}} d y . \end{matrix}

(4.18)

Expanding all products and applying Lemma 4.1 term by term in (4.18), we obtain

\begin{matrix} Q_{α}^{ϵ} (t) = q_{α} (t) + ϵ^{1 / 2} \frac{1}{N^{ϵ} (t)} {⟨ a (y, t) | y_{α} a (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})} \\ + ϵ \frac{1}{N^{ϵ} (t)} [{⟨ b (y, t) | y_{α} a (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})} + {⟨ a (y, t) | y_{α} b (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})} \\ + {⟨ (- i \partial_{y_{β}}) a (y, t) | y_{α} a (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})} {⟨ \partial_{p_{β}} χ_{n} (z; p (t)) | χ_{n} (z; p (t)) ⟩}_{L_{z}^{2} (Ω)} \\ + {⟨ y_{α} a (y, t) | (- i \partial_{y_{β}}) a (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})} {⟨ χ_{n} (z; p (t)) | \partial_{p_{β}} χ_{n} (z; p (t)) ⟩}_{L_{z}^{2} (Ω)}] \\ + O (ϵ^{3 / 2} e^{c t}) \\ P_{α}^{ϵ} (t) = p_{α} (t) + ϵ^{1 / 2} \frac{1}{N^{ϵ} (t)} {⟨ a (y, t) | (- i \partial_{y_{α}}) a (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})} \\ + ϵ \frac{1}{N^{ϵ} (t)} [{⟨ b (y, t) | (- i \partial_{y_{α}}) a (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})} + {⟨ a (y, t) | (- i \partial_{y_{α}}) b (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})}] \\ + O (ϵ^{3 / 2} e^{c t}) . \end{matrix}

(4.19)

Here, terms of higher order than $ϵ$ are bounded by a similar reasoning to that given in Section III B (Remark 4.2). Using the identity (4.10) and the fact that $(- i \nabla_{y})$ is self-adjoint, we have that

{⟨ (- i \partial_{y_{β}}) a (y, t) | y_{α} a (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})} {⟨ \partial_{p_{β}} χ_{n} (z; p (t)) | χ_{n} (z; p (t)) ⟩}_{L_{z}^{2} (Ω)} + {⟨ y_{α} a (y, t) | (- i \partial_{y_{β}}) a (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})} {⟨ χ_{n} (z; p (t)) | \partial_{p_{β}} χ_{n} (z; p (t)) ⟩}_{L_{z}^{2} (Ω)} = {⟨ a (y, t) | [y_{α}, (- i \partial_{y_{β}})] a (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})} {⟨ χ_{n} (z; p (t)) | \partial_{p_{β}} χ_{n} (z; p (t)) ⟩}_{L_{z}^{2} (Ω)},

(4.20)

where $[y_{α}, (- i \partial_{y_{β}})] := y_{α} (- i \partial_{y_{β}}) - (- i \partial_{y_{β}}) y_{α}$ is the commutator. Since $[y_{α}, (- i \partial_{y_{β}})] = i δ_{α β}$ ⁠, we have that

{⟨ a (y, t) | [y_{α}, (- i \partial_{y_{β}})] a (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})} {⟨ χ_{n} (z; p (t)) | \partial_{p_{β}} χ_{n} (z; p (t)) ⟩}_{L_{z}^{2} (Ω)} = i {∥ a (y, t) ∥}_{L_{y}^{2} (ℝ^{d})}^{2} {⟨ χ_{n} (z; p (t)) | \partial_{p_{α}} χ_{n} (z; p (t)) ⟩}_{L_{z}^{2} (Ω)} .

(4.21)

Using L²-norm conservation for solutions of (1.22), we have that for all $t \geq 0$ ⁠, ${∥ a (y, t) ∥}_{L_{y}^{2} (ℝ^{d})}^{2} = {∥ a_{0} (y) ∥}_{L_{y}^{2} (ℝ^{d})}^{2}$ ⁠. Using (4.15), we have that ${∥ a_{0} (y) ∥}_{L_{y}^{2} (ℝ^{d})}^{2} = N^{ϵ} (t) + O (ϵ^{1 / 2} e^{c t})$ (4.12). We have proved that

i {∥ a (y, t) ∥}_{L_{y}^{2} (ℝ^{d})}^{2} {⟨ χ_{n} (z; p (t)) | \nabla_{p} χ_{n} (z; p (t)) ⟩}_{L_{z}^{2} (Ω)} = N^{ϵ} (t) A_{n} (p (t)) + O (ϵ^{1 / 2} e^{c t}),

(4.22)

where the last equality holds by the definition of the nth band Berry connection (1.26). We have proved that

\begin{matrix} Q^{ϵ} (t) = q (t) + ϵ^{1 / 2} \frac{1}{N^{ϵ} (t)} {⟨ a (y, t) | y a (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})} \\ + ϵ \frac{1}{N^{ϵ} (t)} [{⟨ b (y, t) | y a (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})} + {⟨ a (y, t) | y b (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})}] \\ + ϵ A_{n} (p (t)) + O (ϵ^{3 / 2} e^{c t}) \\ P^{ϵ} (t) = p (t) + ϵ^{1 / 2} \frac{1}{N^{ϵ} (t)} {⟨ a (y, t) | (- i \nabla_{y}) a (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})} \\ + ϵ \frac{1}{N^{ϵ} (t)} [{⟨ b (y, t) | (- i \nabla_{y}) a (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})} + {⟨ a (y, t) | (- i \nabla_{y}) b (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})}] \\ + O (ϵ^{3 / 2} e^{c t}) . \end{matrix}

(4.23)

C. Computation of dynamics of $Q^{ϵ} (t), P^{ϵ} (t)$ ⁠, proof of assertion (2) of Theorem 1.2

Differentiating both sides of (4.23) with respect to time and using ${\dot{N}}^{ϵ} (t) = O (ϵ e^{c t})$ (4.14) gives

\begin{matrix} {\dot{Q}}_{α}^{ϵ} (t) = {\dot{q}}_{α} (t) + ϵ^{1 / 2} \frac{1}{N^{ϵ} (t)} \frac{d}{d t} {⟨ a (y, t) | y_{α} a (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})} \\ + ϵ \frac{1}{N^{ϵ} (t)} \frac{d}{d t} [{⟨ b (y, t) | y_{α} a (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})} + {⟨ a (y, t) | y_{α} b (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})}] \\ + ϵ {\dot{p}}_{β} (t) \partial {}_{p_{β}}A_{n, α} (p (t)) + O (ϵ^{3 / 2} e^{c t}) \\ {\dot{P}}_{α}^{ϵ} (t) = {\dot{p}}_{α} (t) + ϵ^{1 / 2} \frac{1}{N^{ϵ} (t)} \frac{d}{d t} {⟨ a (y, t) | (- i \partial_{y_{α}}) a (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})} \\ + ϵ \frac{1}{N^{ϵ} (t)} \frac{d}{d t} [{< b (y, t) | (- i \partial_{y_{α}}) a (y, t) >}_{L_{y}^{2} (ℝ^{d})} + {< a (y, t) | (- i \partial_{y_{α}}) b (y, t) >}_{L_{y}^{2} (ℝ^{d})}] \\ + O (ϵ^{3 / 2} e^{c t}) . \end{matrix}

(4.24)

Recall that (q(t), p(t)) satisfy the classical system (1.19). In Appendix F, we calculate (F13),

\begin{matrix} \frac{d}{d t} {⟨ a (y, t) | y_{α} a (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})} = \partial_{p_{α}} \partial_{p_{β}} E_{n} (p (t)) {< a (y, t) | (- i \partial_{y_{β}}) a (y, t) >}_{L_{y}^{2} (ℝ^{d})}, \\ \frac{d}{d t} {< a (y, t) | (- i \partial_{y_{α}}) a (y, t) >}_{L_{y}^{2} (ℝ^{d})} = - \partial_{q_{α}} \partial_{q_{β}} W (q (t)) {< a (y, t) | y_{β} a (y, t) >}_{L_{y}^{2} (ℝ^{d})}, \end{matrix}

(4.25)

and (F18),

\begin{matrix} \frac{d}{d t} [{⟨ b (y, t) | y_{α} a (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})} + {⟨ a (y, t) | y_{α} b (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})}] \\ = \partial_{p_{α}} \partial_{p_{β}} E_{n} (p (t)) [{< b (y, t) | (- i \partial_{y_{β}}) a (y, t) >}_{L_{y}^{2} (ℝ^{d})} + {< a (y, t) | (- i \partial_{y_{β}}) a (y, t) >}_{L_{y}^{2} (ℝ^{d})}] \\ + \frac{1}{2} \partial_{p_{α}} \partial_{p_{β}} \partial_{p_{γ}} E_{n} (p (t)) {< a (y, t) | (- i \partial_{y_{β}}) (- i \partial_{y_{γ}}) a (y, t) >}_{L_{y}^{2} (ℝ^{d})} + \partial_{q_{β}} W (q (t)) \partial {}_{p_{α}}A_{n, β} (p (t)) {∥ a (y, t) ∥}_{L_{y}^{2} (ℝ^{d})}^{2} \\ \frac{d}{d t} [{< b (y, t) | (- i \partial_{y_{α}}) a (y, t) >}_{L_{y}^{2} (ℝ^{d})} + {< a (y, t) | (- i \partial_{y_{α}}) b (y, t) >}_{L_{y}^{2} (ℝ^{d})}] \\ = - \partial_{q_{α}} \partial_{q_{β}} W (q (t)) [{< b (y, t) | y_{β} a (y, t) >}_{L_{y}^{2} (ℝ^{d})} + {< a (y, t) | y_{β} a (y, t) >}_{L_{y}^{2} (ℝ^{d})}] \\ - \frac{1}{2} \partial_{q_{α}} \partial_{q_{β}} \partial_{q_{γ}} W (q (t)) {< a (y, t) | y_{β} y_{γ} a (y, t) >}_{L_{y}^{2} (ℝ^{d})} - \partial_{q_{α}} \partial_{q_{β}} W (q (t)) A_{n, β} (p (t)) {∥ a (y, t) ∥}_{L_{y}^{2} (ℝ^{d})}^{2} . \end{matrix}

(4.26)

Substituting these expressions into (4.24) and using ${∥ a (y, t) ∥}_{L_{y}^{2} (ℝ^{d})}^{2} = N^{ϵ} (t) + O (ϵ^{1 / 2} e^{c t})$ (4.9), we have

\begin{matrix} {\dot{Q}}_{α}^{ϵ} (t) = \partial_{p_{α}} E_{n} (p (t)) + ϵ^{1 / 2} \frac{1}{N^{ϵ} (t)} \partial_{p_{α}} \partial_{p_{β}} E_{n} (p (t)) {< a (y, t) | (- i \partial_{y_{β}}) a (y, t) >}_{L_{y}^{2} (ℝ^{d})} \\ + ϵ \frac{1}{N^{ϵ} (t)} \partial_{p_{α}} \partial_{p_{β}} E_{n} (p (t)) [{< b (y, t) | (- i \partial_{y_{β}}) a (y, t) >}_{L_{y}^{2} (ℝ^{d})} + {< a (y, t) | (- i \partial_{y_{β}}) a (y, t) >}_{L_{y}^{2} (ℝ^{d})}] \\ + ϵ \frac{1}{2} \frac{1}{N^{ϵ} (t)} \partial_{p_{α}} \partial_{p_{β}} \partial_{p_{γ}} E_{n} (p (t)) {< a (y, t) | (- i \partial_{y_{β}}) (- i \partial_{y_{γ}}) a (y, t) >}_{L_{y}^{2} (ℝ^{d})} \\ + ϵ \partial_{q_{β}} W (q (t)) \partial {}_{p_{α}}A_{n, β} (p (t)) - ϵ \partial_{q_{β}} W (q (t)) \partial {}_{p_{β}}A_{n, α} (p (t)) + O (ϵ^{3 / 2} e^{c t}) \\ {\dot{P}}_{α}^{ϵ} (t) = - \partial_{q_{α}} W (q (t)) - ϵ^{1 / 2} \frac{1}{N^{ϵ} (t)} \partial_{q_{α}} \partial_{q_{β}} W (q (t)) {< a (y, t) | y_{β} a (y, t) >}_{L_{y}^{2} (ℝ^{d})} \\ - ϵ \frac{1}{N^{ϵ} (t)} \partial_{q_{α}} \partial_{q_{β}} W (q (t)) [{< b (y, t) | y_{β} a (y, t) >}_{L_{y}^{2} (ℝ^{d})} + {< b (y, t) | y_{β} a (y, t) >}_{L_{y}^{2} (ℝ^{d})}] \\ - ϵ \frac{1}{2} \frac{1}{N^{ϵ} (t)} \partial_{q_{α}} \partial_{q_{β}} \partial_{q_{γ}} W (q (t)) {< a (y, t) | y_{β} y_{γ} a (y, t) >}_{L_{y}^{2} (ℝ^{d})} \\ - ϵ \partial_{q_{α}} \partial_{q_{β}} W (q (t)) A_{n, β} (p (t)) + O (ϵ^{3 / 2} e^{c t}) . \end{matrix}

(4.27)

Equation (4.23) gives expressions for q(t), p(t) in terms of $Q^{ϵ} (t), P^{ϵ} (t)$ ⁠,

\begin{matrix} q_{α} (t) = Q_{α}^{ϵ} (t) - ϵ^{1 / 2} \frac{1}{N^{ϵ} (t)} {⟨ a (y, t) | y_{α} a (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})} \\ - ϵ \frac{1}{N^{ϵ} (t)} [{⟨ b (y, t) | y_{α} a (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})} + {⟨ a (y, t) | y_{α} b (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})}] - ϵ A_{n, α} (p (t)) + O (ϵ^{3 / 2} e^{c t}) \\ p_{α} (t) = P_{α}^{ϵ} (t) - ϵ^{1 / 2} \frac{1}{N^{ϵ} (t)} {< a (y, t) | (- i \partial_{y_{α}}) a (y, t) >}_{L_{y}^{2} (ℝ^{d})} \\ - ϵ \frac{1}{N^{ϵ} (t)} [{< b (y, t) | (- i \partial_{y_{α}}) a (y, t) >}_{L_{y}^{2} (ℝ^{d})} + {< a (y, t) | (- i \partial_{y_{α}}) b (y, t) >}_{L_{y}^{2} (ℝ^{d})}] + O (ϵ^{3 / 2} e^{c t}) . \end{matrix}

(4.28)

Substituting these expressions into (4.27), Taylor-expanding in $ϵ^{1 / 2}$ ⁠, and again using $N^{ϵ} (t) = {∥ a_{0} (y) ∥}_{L_{y}^{2} (ℝ^{d})}^{2} + O (ϵ^{1 / 2} e^{c t})$ (4.9) then gives

\begin{matrix} {\dot{Q}}_{α}^{ϵ} (t) = \partial_{P_{α}^{ϵ}} E_{n} (P^{ϵ} (t)) + ϵ \frac{1}{2} \partial_{P_{α}^{ϵ}} \partial_{P_{β}^{ϵ}} \partial_{P_{γ}^{ϵ}} E_{n} (P^{ϵ} (t)) [\frac{1}{{∥ a_{0} (y) ∥}_{L_{y}^{2} (ℝ^{d})}^{2}} < a (y, t) | (- i \partial_{y_{β}}) (- i \partial_{y_{γ}}) a (y, t) >_{L_{y}^{2} (ℝ^{d})} \\ - \frac{1}{{∥ a_{0} (y) ∥}_{L_{y}^{2} (ℝ^{d})}^{4}} < a (y, t) | (- i \partial_{y_{β}}) a (y, t) >_{L_{y}^{2} (ℝ^{d})} < a (y, t) | (- i \partial_{y_{γ}}) a (y, t) >_{L_{y}^{2} (ℝ^{d})}] \\ + ϵ \partial_{Q_{γ}^{ϵ}} W (Q^{ϵ} (t)) F_{n, α γ} (P^{ϵ} (t)) + O (ϵ^{3 / 2} e^{c t}), \\ {\dot{P}}_{α}^{ϵ} (t) = - \partial_{Q_{α}^{ϵ}} W (Q^{ϵ} (t)) - ϵ \frac{1}{2} \partial_{Q_{α}^{ϵ}} \partial_{Q_{β}^{ϵ}} \partial_{Q_{γ}^{ϵ}} W (Q^{ϵ} (t)) [\frac{1}{{∥ a_{0} (y) ∥}_{L_{y}^{2} (ℝ^{d})}^{2}} {⟨ a (y, t) | y_{β} y_{γ} a (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})} \\ - \frac{1}{{∥ a_{0} (y) ∥}_{L_{y}^{2} (ℝ^{d})}^{4}} < a (y, t) | y_{β} a (y, t) >_{L_{y}^{2} (ℝ^{d})} < a (y, t) | y_{γ} a (y, t) >_{L_{y}^{2} (ℝ^{d})}] + O (ϵ^{3 / 2} e^{c t}), \end{matrix}

(4.29)

where $F_{n, α γ} (P^{ϵ}) := \partial {}_{P_{α}^{ϵ}}A_{n, γ} (P^{ϵ}) - \partial {}_{P_{β}^{ϵ}}A_{n, α} (P^{ϵ})$ is the nth band Berry curvature (1.46). Note that the system (4.29) is not closed: a(y, t) satisfies an equation parametrically forced by q(t), p(t) (1.22). Recall the definition of $a^{ϵ} (y, t)$ (1.42) as the solution of (1.22) with co-efficients evaluated at $Q^{ϵ} (t), P^{ϵ} (t)$ ⁠,

\begin{matrix} i \partial_{t} a^{ϵ} (y, t) = \frac{1}{2} \partial_{P_{α}^{ϵ}} \partial_{P_{β}^{ϵ}} E_{n} (P^{ϵ} (t)) (- i \partial_{y_{α}}) (- i \partial_{y_{β}}) a^{ϵ} (y, t) + \frac{1}{2} \partial_{Q_{α}^{ϵ}} \partial_{Q_{β}^{ϵ}} W (Q^{ϵ} (t)) y_{α} y_{β} a^{ϵ} (y, t), \\ a^{ϵ} (y, 0) = a_{0} (y), \end{matrix}

(4.30)

Recall the definition of the $Σ^{l}$ norms (3.45). If we can show that ${∥ a^{ϵ} (y, t) - a (y, t) ∥}_{Σ_{y}^{l} (ℝ^{d})} = O (ϵ^{1 / 2} e^{c t})$ for each positive integer l, then we may replace a(y,t) by $a^{ϵ} (y, t)$ everywhere in (4.29) and, after dropping error terms, we will have obtained a closed system for $Q^{ϵ} (t), P^{ϵ} (t), a^{ϵ} (y, t)$ ⁠. Let

\begin{matrix} H^{ϵ} (t) := \frac{1}{2} \partial_{P_{α}^{ϵ}} \partial_{P_{β}^{ϵ}} E_{n} (P^{ϵ} (t)) (- i \partial_{y_{α}}) (- i \partial_{y_{β}}) + \frac{1}{2} \partial_{Q_{α}^{ϵ}} \partial_{Q_{β}^{ϵ}} W (q (t)) y_{α} y_{β}, \\ H (t) := \frac{1}{2} \partial_{p_{α}} \partial_{p_{β}} E_{n} (p (t)) (- i \partial_{y_{α}}) (- i \partial_{y_{β}}) + \frac{1}{2} \partial_{q_{α}} \partial_{q_{β}} W (q (t)) y_{α} y_{β}, \end{matrix}

(4.31)

then $a^{ϵ} (y, t) - a (y, t)$ satisfies

\begin{matrix} i \partial_{t} (a^{ϵ} (y, t) - a (y, t)) = H^{ϵ} (t) a^{ϵ} (y, t) - H (t) a (y, t) \\ = H^{ϵ} (t) (a^{ϵ} (y, t) - a (y, t)) + (H^{ϵ} (t) - H (t)) a (y, t), \\ a^{ϵ} (y, 0) - a (y, 0) = 0. \end{matrix}

(4.32)

Using the fact that $H^{ϵ} (t)$ is self-adjoint on $L_{y}^{2} (ℝ^{d})$ for each t, it follows from (4.32) that

\frac{d}{d t} {∥ a^{ϵ} (y, t) - a (y, t) ∥}_{L_{y}^{2} (ℝ^{d})}^{2} = i {⟨ (H^{ϵ} (t) - H (t)) a (y, t) | a^{ϵ} (y, t) - a (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})} - i {⟨ a^{ϵ} (y, t) - a (y, t) | \times (H^{ϵ} (t) - H (t)) a (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})} .

(4.33)

By the Cauchy-Schwarz inequality,

\leq 2 {∥ a^{ϵ} (y, t) - a (y, t) ∥}_{L_{y}^{2} (ℝ^{d})} {∥ (H^{ϵ} (t) - H (t)) a (y, t) ∥}_{L_{y}^{2} (ℝ^{d})},

(4.34)

it then follows that

{∥ a^{ϵ} (y, t) - a (y, t) ∥}_{L_{y}^{2} (ℝ^{d})} \leq \int_{0}^{t} {∥ (H^{ϵ} (s) - H (s)) a (y, s) ∥}_{L_{y}^{2} (ℝ^{d})} d s .

(4.35)

Using the precise forms of $H^{ϵ} (t), H (t)$ ⁠, we have

{∥ a^{ϵ} (y, t) - a (y, t) ∥}_{L_{y}^{2} (ℝ^{d})} \leq \int_{0}^{t} \sup_{s \in [0, \infty), α, β \in {1, \dots, d}} (| \partial_{P_{α}^{ϵ}} \partial_{P_{β}^{ϵ}} E_{n} (P^{ϵ} (s)) - \partial_{p_{α}} \partial_{p_{β}} E_{n} (p (s)) | + | \partial_{Q_{α}^{ϵ}} \partial_{Q_{β}^{ϵ}} W (Q^{ϵ} (s)) - \partial_{q_{α}} \partial_{q_{β}} W (q (s)) |) {∥ a (y, s) ∥}_{Σ_{y}^{2} (ℝ^{d})} d s,

(4.36)

where $Σ_{y}^{2} (ℝ^{d})$ is the norm defined in (3.45). Recall that $| Q^{ϵ} (t) - q (t) | + | P^{ϵ} (t) - p (t) | = O (ϵ^{1 / 2} e^{c t})$ (4.23). It follows from compactness of the Brillouin zone and Assumptions 1.1 and 1.2 that there exists a uniform bound in t on third derivatives of E_n(p), W(q) for all p along the line segments connecting p(t) and $P^{ϵ} (t)$ ⁠, and all q along the line segments connecting q(t) and $Q^{ϵ} (t)$ ⁠. We may therefore conclude from the mean - value theorem that there exist constants $c > 0, C > 0$ independent of $ϵ, t$ such that

{∥ a^{ϵ} (y, t) - a (y, t) ∥}_{L_{y}^{2} (ℝ^{d})} \leq C ϵ^{1 / 2} \int_{0}^{t} e^{c s} {∥ a (y, s) ∥}_{Σ_{y}^{2} (ℝ^{d})} d s .

(4.37)

We now use the a priori bounds on the $Σ_{y}^{l} (ℝ^{d})$ -norms of a(y, t) for each $l \in ℕ$ (Lemma 3.4) to see that

{∥ a^{ϵ} (y, t) - a (y, t) ∥}_{L_{y}^{2} (ℝ^{d})} \leq ϵ^{1 / 2} C^{'} e^{c^{'} t}

(4.38)

for some constants $c^{'} > 0, C^{'} > 0$ independent of $ϵ, t$ ⁠. By a similar argument, we see that for any integer $l \geq 0$ there exist a constants $c^{'}_{l} > 0, C^{'}_{l} > 0$ such that

{∥ a^{ϵ} (y, t) - a (y, t) ∥}_{Σ_{y}^{l} (ℝ^{d})} \leq ϵ^{1 / 2} C^{'}_{l} e^{c^{'}_{l} t} .

(4.39)

It then follows that we may replace a(y, t) by $a^{ϵ} (y, t)$ everywhere in (4.29), generating further errors which are $O (ϵ^{3 / 2} e^{c t})$ to derive

\begin{matrix} {\dot{Q}}_{α}^{ϵ} (t) = \partial_{P_{α}^{ϵ}} E_{n} (P^{ϵ} (t)) + ϵ \frac{1}{2} \partial_{P_{α}^{ϵ}} \partial_{P_{β}^{ϵ}} \partial_{P_{γ}^{ϵ}} E_{n} (P^{ϵ} (t)) [\frac{1}{{∥ a_{0} (y) ∥}_{L_{y}^{2} (ℝ^{d})}^{2}} < a^{ϵ} (y, t) | (- i \partial_{y_{β}}) (- i \partial_{y_{γ}}) a^{ϵ} (y, t) >_{L_{y}^{2} (ℝ^{d})} \\ - \frac{1}{{∥ a_{0} (y) ∥}_{L_{y}^{2} (ℝ^{d})}^{4}} < a^{ϵ} (y, t) | (- i \partial_{y_{β}}) a^{ϵ} (y, t) >_{L_{y}^{2} (ℝ^{d})} < a^{ϵ} (y, t) | (- i \partial_{y_{γ}}) a^{ϵ} (y, t) >_{L_{y}^{2} (ℝ^{d})}] \\ + ϵ \partial_{Q_{γ}^{ϵ}} W (Q^{ϵ} (t)) F_{n, α γ} (P^{ϵ} (t)) + O (ϵ^{3 / 2} e^{c t}), \\ {\dot{P}}_{α}^{ϵ} (t) = - \partial_{Q_{α}^{ϵ}} W (Q^{ϵ} (t)) - ϵ \frac{1}{2} \partial_{Q_{α}^{ϵ}} \partial_{Q_{β}^{ϵ}} \partial_{Q_{γ}^{ϵ}} W (Q^{ϵ} (t)) [\frac{1}{{∥ a_{0} (y) ∥}_{L_{y}^{2} (ℝ^{d})}^{2}} < a^{ϵ} (y, t) | y_{β} y_{γ} a^{ϵ} (y, t) >_{L_{y}^{2} (ℝ^{d})} \\ - \frac{1}{{∥ a_{0} (y) ∥}_{L_{y}^{2} (ℝ^{d})}^{4}} < a^{ϵ} (y, t) | y_{β} a^{ϵ} (y, t) >_{L_{y}^{2} (ℝ^{d})} < a^{ϵ} (y, t) | y_{γ} a^{ϵ} (y, t) >_{L_{y}^{2} (ℝ^{d})}] + O (ϵ^{3 / 2} e^{c t}) . \end{matrix}

(4.40)

D. Hamiltonian structure of dynamics of $Q^{ϵ} (t), P^{ϵ} (t)$ ⁠, proof of assertion (3) of Theorem 1.2

Following Ref. 16, we introduce the new variables (1.47),

\begin{matrix} Q^{ϵ} (t) := Q^{ϵ} (t) - ϵ A_{n} (P^{ϵ} (t)), \\ P^{ϵ} (t) := P^{ϵ} (t) . \end{matrix}

(4.41)

Let $𝔞^{ϵ} (y, t)$ denote the solution of (1.42) with co-efficients evaluated at $Q^{ϵ} (t), P^{ϵ} (t)$ rather than $Q^{ϵ} (t), P^{ϵ} (t)$ ⁠, with initial data normalized in $L_{y}^{2} (ℝ^{d})$ ⁠,

\begin{matrix} i \partial_{t} 𝔞^{ϵ} (y, t) = \frac{1}{2} \partial_{P_{α}^{ϵ}} \partial_{P_{β}^{ϵ}} E_{n} (\mathcaly P^{ϵ} (t)) (- i \partial_{y_{α}}) (- i \partial_{y_{β}}) 𝔞^{ϵ} (y, t) + \frac{1}{2} \partial_{Q_{α}^{ϵ}} \partial_{Q_{β}^{ϵ}} W (Q^{ϵ} (t)) y_{α} y_{β} 𝔞^{ϵ} (y, t), \\ 𝔞^{ϵ} (y, 0) = \frac{a_{0} (y)}{{∥ a_{0} (y) ∥}_{L_{y}^{2} (ℝ^{d})}} . \end{matrix}

(4.42)

Since $Q^{ϵ} (t) - Q^{ϵ} (t) = O (ϵ e^{c t})$ ⁠, $P^{ϵ} (t) = P^{ϵ} (t)$ ⁠, by a similar argument to that given in Sec. IV C we have that for each integer $l \geq 0$ ⁠,

{∥ \frac{a^{ϵ} (y, t)}{{∥ a_{0} (y) ∥}_{L_{y}^{2} (ℝ^{d})}} - 𝔞^{ϵ} (y, t) ∥}_{Σ_{y}^{l} (ℝ^{d})} = O (ϵ e^{c t}) .

(4.43)

Differentiating (4.41), using Equations (4.40) for ${\dot{Q}}^{ϵ} (t), {\dot{P}}^{ϵ} (t)$ ⁠, and using (4.43) to replace $\frac{a^{ϵ} (y, t)}{{∥ a_{0} (y) ∥}_{L_{y}^{2} (ℝ^{d})}}$ with $𝔞^{ϵ} (y, t)$ everywhere, we obtain

\begin{matrix} {\dot{Q}}_{α}^{ϵ} (t) = \partial_{P_{α}^{ϵ}} E_{n} (P^{ϵ} (t)) + ϵ \frac{1}{2} \partial_{P_{α}^{ϵ}} \partial_{P_{β}} \partial_{P_{γ}^{ϵ}} E_{n} (P^{ϵ} (t)) [{⟨ 𝔞^{ϵ} (y, t) | (- i \partial_{y_{β}}) (- i \partial_{y_{γ}}) 𝔞^{ϵ} (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})} \\ - {⟨ 𝔞^{ϵ} (y, t) | (- i \partial_{y_{β}}) 𝔞^{ϵ} (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})} {⟨ 𝔞^{ϵ} (y, t) | (- i \partial_{y_{γ}}) 𝔞^{ϵ} (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})}] \\ + \partial_{Q_{β}^{ϵ}} W (Q^{ϵ} (t)) \partial {}_{P_{β}^{ϵ}}A_{n, α} (P^{ϵ} (t)) + O (ϵ^{3 / 2} e^{c t}), \\ {\dot{P}}_{α}^{ϵ} (t) = - \partial_{Q_{α}^{ϵ}} W (Q^{ϵ} (t)) - ϵ \frac{1}{2} \partial_{Q_{α}^{ϵ}} \partial_{Q_{β}^{ϵ}} \partial_{Q_{γ}^{ϵ}} W (Q^{ϵ} (t)) [{⟨ 𝔞^{ϵ} (y, t) | y_{β} y_{γ} 𝔞^{ϵ} (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})} \\ - {⟨ 𝔞^{ϵ} (y, t) | y_{β} 𝔞^{ϵ} (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})} {⟨ 𝔞^{ϵ} (y, t) | y_{γ} 𝔞^{ϵ} (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})}] \\ - \partial_{Q_{α}^{ϵ}} \partial_{Q_{β}^{ϵ}} W (Q^{ϵ} (t)) A_{n, β} (P^{ϵ} (t)) + O (ϵ^{3 / 2} e^{c t}) . \end{matrix}

(4.44)

Note that, up to error terms, Equations (4.44) and (4.42) constitute a closed system for $Q^{ϵ} (t), P^{ϵ} (t), 𝔞^{ϵ} (y, t)$ ⁠.

We now show that this system may be derived from a Hamiltonian. Let

\begin{matrix} μ^{ϵ} (t) := {⟨ 𝔞^{ϵ} (y, t) | y 𝔞^{ϵ} (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})} \\ λ^{ϵ} (t) := {⟨ 𝔞^{ϵ} (y, t) | (- i \nabla_{y}) 𝔞^{ϵ} (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})} . \end{matrix}

(4.45)

Then, we may write (4.44) as

\begin{matrix} {\dot{Q}}_{α}^{ϵ} (t) = \partial_{P_{α}^{ϵ}} E_{n} (P^{ϵ} (t)) + ϵ \frac{1}{2} \partial_{P_{α}^{ϵ}} \partial_{P_{β}} \partial_{P_{γ}^{ϵ}} E_{n} (P^{ϵ} (t)) [{⟨ 𝔞^{ϵ} (y, t) | (- i \partial_{y_{β}}) (- i \partial_{y_{γ}}) 𝔞^{ϵ} (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})} \\ - λ_{β}^{ϵ} (t) λ_{γ}^{ϵ} (t)] + ϵ \partial_{Q_{β}^{ϵ}} W (Q^{ϵ} (t)) \partial_{P_{β}^{ϵ}} A_{n, α} (P^{ϵ} (t)) + O (ϵ^{3 / 2} e^{c t}), \\ {\dot{P}}_{α}^{ϵ} (t) = - \partial_{Q_{α}^{ϵ}} W (Q^{ϵ} (t)) - ϵ \frac{1}{2} \partial_{Q_{α}^{ϵ}} \partial_{Q_{β}^{ϵ}} \partial_{Q_{γ}^{ϵ}} W (Q^{ϵ} (t)) [{⟨ 𝔞^{ϵ} (y, t) | y_{β} y_{γ} 𝔞^{ϵ} (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})} - μ_{α}^{ϵ} (t) μ_{β}^{ϵ} (t)] \\ - ϵ \partial_{Q_{α}^{ϵ}} \partial_{Q_{β}^{ϵ}} W (Q^{ϵ} (t)) A_{n, β} (P^{ϵ} (t)) + O (ϵ^{3 / 2} e^{c t}) . \end{matrix}

(4.46)

By an identical calculation to that given in Appendix F (F13) and (F14), we have that

\begin{matrix} {\dot{μ}}_{α}^{ϵ} (t) = \partial_{\mathcalh P_{α}^{ϵ}} \partial_{\mathcalh P_{β}^{ϵ}} E_{n} (\mathcalh P^{ϵ} (t)) λ_{β}^{ϵ} (t), \\ {\dot{λ}}_{α}^{ϵ} (t) = - \partial_{\mathcalh Q_{α}^{ϵ}} \partial_{\mathcalh Q_{β}^{ϵ}} W (\mathcaly Q^{ϵ} (t)) μ_{β}^{ϵ} (t) . \end{matrix}

(4.47)

Let

H^{ϵ} (Q^{ϵ}, P^{ϵ}, \bar{𝔞^{ϵ}}, 𝔞^{ϵ}, μ^{ϵ}, λ^{ϵ}) := E_{n} (P^{ϵ}) + ϵ W (Q^{ϵ}) + ϵ \nabla_{Q^{ϵ}} W (Q^{ϵ}) \cdot A_{n} (P^{ϵ}) + ϵ \frac{1}{2} \partial_{P_{α}^{ϵ}} \partial_{P_{β}^{ϵ}} E_{n} (P^{ϵ}) [{⟨ \partial_{y_{α}} a^{ϵ} | \partial_{y_{β}} a^{ϵ} ⟩}_{L_{y}^{2} (ℝ^{d})} - λ_{α}^{ϵ} λ_{β}^{ϵ}] + ϵ \frac{1}{2} \partial_{Q_{α}^{ϵ}} \partial_{Q_{β}^{ϵ}} W (Q^{ϵ}) [{⟨ y_{α} a^{ϵ} | y_{β} a^{ϵ} ⟩}_{L_{y}^{2} (ℝ^{d})} - μ_{α}^{ϵ} μ_{β}^{ϵ}] .

(4.48)

Then we may write the closed system (4.42), (4.47), and (4.46) as

\begin{matrix} {\dot{Q}}^{ϵ} = \nabla {}_{P^{ϵ}}H^{ϵ} (P^{ϵ}), {\dot{P}}^{ϵ} = - \nabla {}_{Q^{ϵ}}H^{ϵ} (Q^{ϵ}), \\ i \partial_{t} 𝔞^{ϵ} = \frac{δ H^{ϵ}}{δ \bar{𝔞^{ϵ}}}, {\dot{μ}}^{ϵ} (t) = - \nabla {}_{λ^{ϵ}}H^{ϵ}, {\dot{λ}}^{ϵ} (t) = \nabla {}_{μ^{ϵ}}H^{ϵ} . \end{matrix}

(4.49)

The precise statements (1), (2), (3) of Theorem 1.2 follow from the following observations. The errors in Equations (4.23), (4.40), and (4.46) may each be bounded by $ϵ^{3 / 2} C_{1} e^{c_{1} t}, ϵ^{3 / 2} C_{2} e^{c_{2} t}, ϵ^{3 / 2} C_{3} e^{c_{3} t}$ for positive constants $c_{j}, C_{j}, j \in {1,2,3}$ ⁠. Define $c^{'} := \max_{j \in {1,2,3}} c_{j}, C^{'} := \max_{j \in {1,2,3}} C_{j}$ ⁠. Then all of these errors may be bounded by $ϵ^{3 / 2} C^{'} e^{c^{'} t}$ ⁠. It follows that these terms are $o (ϵ)$ for all $t \in [0, {\tilde{C}}^{'} \ln 1 / ϵ]$ where ${\tilde{C}}^{'}$ is any constant such that ${\tilde{C}}^{'} < \frac{1}{2 c^{'}}$ ⁠. Next, in Appendix F (F13) and (F14) we show that ${⟨ a_{0} (y) | y a_{0} (y) ⟩}_{L_{y}^{2} (ℝ^{d})} = {⟨ a_{0} (y) | (- i \nabla_{y}) a_{0} (y) ⟩}_{L_{y}^{2} (ℝ^{d})} = 0$ implies that for all $t \geq 0$ ${⟨ a (y, t) | y a (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})} = {⟨ a (y, t) | (- i \nabla_{y}) a (y, t) ⟩}_{L_{y}^{2} (ℝ^{d})} = 0$ ⁠. Imposing the constraints (1.43), then the simplified expressions (1.44) and (1.45) follow from (4.23) and (4.40), respectively. We are also justified in ignoring the $λ^{ϵ}, μ^{ϵ}$ degrees of freedom in (4.48) and (4.49) since for all $t \geq 0$ ⁠, $λ^{ϵ} (t) = μ^{ϵ} (t) = 0$ ⁠. In this way we obtain the simplified Hamiltonian system (1.48) and (1.49).

ACKNOWLEDGMENTS

The authors wish to thank George Hagedorn, Christof Sparber, and Tomoki Ohsawa for stimulating discussions. This research was supported in part by National Science Foundation Grant Nos. DMS-1412560 (A.W. and M.I.W.) and DMS-1454939 (J.L.), and Simons Foundation Math + X Investigator Award No. 376319 (M.I.W.).

APPENDIX A: USEFUL IDENTITIES INVOLVING $E_{n}, χ_{n}$

Let $E (p), χ (z; p)$ satisfy the eigenvalue problem,

\begin{matrix} [H (p) - E (p)] χ (z; p) = 0, \\ H (p) = \frac{1}{2} {(p - i \nabla_{z})}^{2} + V (z) \end{matrix}

(A1)

and assume that $E (p), χ (z; p)$ are smooth functions of p. Taking the gradient with respect to p gives

[(p - i \nabla_{z}) - \nabla_{p} E_{n} (p)] χ_{n} (z; p) + [H (p) - E_{n} (p)] \nabla_{p} χ_{n} (z; p) = 0.

(A2)

Taking two derivatives with respect to p of the equation gives

[δ_{α β} - \partial_{p_{α}} \partial_{p_{β}} E_{n} (p)] χ_{n} (z; p) + [{(p - i \partial_{z})}_{α} - \partial_{p_{α}} E_{n} (p)] \partial_{p_{β}} χ_{n} (z; p) + [{(p - i \partial_{z})}_{β} - \partial_{p_{β}} E_{n} (p)] \partial_{p_{α}} χ_{n} (z; p) + [H (p) - E_{n} (p)] \partial_{p_{α}} \partial_{p_{β}} χ_{n} (z; p) = 0,

(A3)

where $δ_{α β}$ is the Kronecker delta. Taking the derivative with respect to $p_{γ}$ of (A3) gives

[- \partial_{p_{α}} \partial_{p_{β}} \partial_{p_{γ}} E_{n} (p)] χ_{n} (z; p) + [δ_{α β} - \partial_{p_{α}} \partial_{p_{β}} E_{n} (p)] \partial_{p_{γ}} χ_{n} (z; p) + [δ_{α γ} - \partial_{p_{α}} \partial_{p_{γ}} E_{n} (p)] \partial_{p_{β}} χ_{n} (z; p) + [δ_{β γ} - \partial_{p_{β}} \partial_{p_{γ}} E_{n} (p)] \partial_{p_{α}} χ_{n} (z; p) + [{(p - i \partial_{z})}_{α} - \partial_{p_{α}} E_{n} (p)] \partial_{p_{β}} \partial_{p_{γ}} χ_{n} (z; p) + [{(p - i \partial_{z})}_{β} - \partial_{p_{β}} E_{n} (p)] \partial_{p_{α}} \partial_{p_{γ}} χ_{n} (z; p) + [{(p - i \partial_{z})}_{γ} - \partial_{p_{γ}} E_{n} (p)] \partial_{p_{α}} \partial_{p_{β}} χ_{n} (z; p) + [H (p) - E_{n} (p)] \partial_{p_{α}} \partial_{p_{β}} \partial_{p_{γ}} χ_{n} (z; p) = 0.

(A4)

APPENDIX B: DERIVATION OF LEADING-ORDER ENVELOPE EQUATION

Collecting terms of order $ϵ$ in the expansion (3.9), using Equations (1.21) for $\dot{S} (t)$ and (1.19) for $\dot{q} (t), \dot{p} (t)$ ⁠, and setting equal to zero gives the following inhomogeneous self-adjoint elliptic equation in z for f²(y, z, t),

\begin{matrix} [H (p (t)) - E_{n} (p (t))] f^{2} (y, z, t) = ξ^{2} (y, z, t) \\ for all v \in Λ, f^{2} (y, z + v, t) = f^{2} (y, z, t); f^{2} (y, z, t) \in Σ_{y}^{R - 2} (ℝ^{d}) \\ ξ^{2} := - [\frac{1}{2} {(- i \nabla_{y})}^{2} + \frac{1}{2} \partial_{q_{α}} \partial_{q_{β}} W (q (t)) y_{α} y_{β} - i \partial_{t}] f^{0} (y, z, t) \\ - [((p (t) - i \nabla_{z}) - \nabla_{p} E_{n} (p (t))) \cdot (- i \nabla_{y})] f^{1} (y, z, t) . \end{matrix}

(B1)

We follow the strategy outlined in Remark 4.1. The proof of the following lemma will be given at the end of this section.

Lemma B.1. $ξ^{2} (y, z, t)$ defined in (B1) satisfies

ξ^{2} (y, z, t) = {\tilde{ξ}}^{2} (y, z, t) + [H (p (t)) - E_{n} (p (t))] u^{2} (y, z, t),

(B2)

where

\begin{matrix} {\tilde{ξ}}^{2} (y, z, t) = [i \partial_{t} a^{0} (y, t) - \frac{1}{2} \partial_{p_{α}} \partial_{p_{β}} E_{n} (p (t)) (- i \partial_{y_{α}}) (- i \partial_{y_{β}}) a^{0} (y, t) - \frac{1}{2} \partial_{q_{α}} \partial_{q_{β}} W (q (t)) y_{α} y_{β} a^{0} (y, t) \\ - \nabla_{q} W (q (t)) \cdot A_{n} (p (t))] χ_{n} (z; p (t)) \\ + P_{n}^{⊥} (p (t)) [- i a^{0} (y, t) \nabla_{q} W (q (t)) \cdot \nabla_{p} χ_{n} (z; p (t))] \\ u^{2} (y, z, t) = (- i \nabla_{y}) a^{1} (y, t) \cdot \nabla_{p} χ_{n} (z; p (t)) + \frac{1}{2} (- i \partial_{y_{α}}) (- i \partial_{y_{β}}) a^{0} (y, t) \partial_{p_{α}} \partial_{p_{β}} χ_{n} (z; p (t)) . \end{matrix}

(B3)

Here, $A_{n} (p (t))$ is the Berry connection (1.26) and $P_{n}^{⊥} (p (t))$ is the orthogonal projection operator away from the subspace of $L_{p e r}^{2}$ spanned by $χ_{n} (z; p (t))$ (3.20).

Imposing the solvability condition of Equation (B1), given by (3.19) with j = 2 and ${\tilde{ξ}}^{2} (y, z, t)$ given by (B3), gives the following evolution equation for a⁰(y, t):

i \partial_{t} a^{0} (y, t) = \frac{1}{2} \partial_{p_{α}} \partial_{p_{β}} E_{n} (p (t)) (- i \partial_{y_{α}}) (- i \partial_{y_{β}}) a^{0} (y, t) + \frac{1}{2} \partial_{q_{α}} \partial_{q_{β}} W (q (t)) y_{α} y_{β} a^{0} (y, t) + \nabla_{q} W (q (t)) \cdot A_{n} (p (t)) a^{0} (y, t) .

(B4)

Taking $a^{0} (y, t) = a (y, t) e^{i ϕ_{B} (t)}$ and matching with the initial data implies Equations (1.27) and (1.22). The general solution of (B1) is given by (3.21) with j = 2,

f^{2} (y, z, t) = a^{2} (y, t) χ_{n} (z; p (t)) + (- i \nabla_{y}) a^{1} (y, t) \cdot \nabla_{p} χ_{n} (z; p (t)) + \frac{1}{2} (- i \partial_{y_{α}}) (- i \partial_{y_{β}}) a^{0} (y, t) \partial_{p_{α}} \partial_{p_{β}} χ_{n} (z; p (t)) {[H (p (t)) - E_{n} (p (t))]}^{- 1} P_{n}^{⊥} (p (t)) [- i \nabla_{q} W (q (t)) a^{0} (y, t) \cdot \nabla_{p} χ_{n} (z; p (t))],

(B5)

where a²(y, t) is an arbitrary function in $Σ_{y}^{R - 2} (ℝ^{d})$ to be fixed at higher order in the expansion.

Proof of Lemma B.1. Adding and subtracting terms, using smoothness of the band E_n(p) in a neighborhood of p(t) (Assumption 1.1), we can re-write $ξ^{2}$ (B1) as

ξ^{2} (y, z, t) = - [\frac{1}{2} \partial_{p_{α}} \partial_{p_{β}} E_{n} (p (t)) (- i \partial_{y_{α}}) (- i \partial_{y_{β}}) + \frac{1}{2} \partial_{q_{α}} \partial_{q_{β}} W (q (t)) y_{α} y_{β} - i \partial_{t}] f^{0} (y, z, t) - [\frac{1}{2} (δ_{α β} - \partial_{p_{α}} \partial_{p_{β}} E_{n} (p (t))) (- i \partial_{y_{α}}) (- i \partial_{y_{β}})] f^{0} (y, z, t) - [((p (t) - i \nabla_{z}) - \nabla_{p} E_{n} (p (t))) \cdot (- i \nabla_{y})] f^{1} (y, z, t) .

(B6)

Substituting the forms of f⁰(y, z, t) (3.14) and f¹(y, z, t) (3.26) gives

ξ^{2} (y, z, t) = - [\frac{1}{2} \partial_{p_{α}} \partial_{p_{β}} E_{n} (p (t)) (- i \partial_{y_{α}}) (- i \partial_{y_{β}}) a^{0} (y, t) + \frac{1}{2} \partial_{q_{α}} \partial_{q_{β}} W (q (t)) y_{α} y_{β} a^{0} (y, t) - i \partial_{t} a^{0} (y, t)] χ_{n} (z; p (t)) + i a^{0} (y, t) \dot{p} (t) \cdot \nabla_{p} χ_{n} (z; p (t)) - (- i \partial_{y_{α}}) (- i \partial_{y_{β}}) a^{0} (y, t) \frac{1}{2} (δ_{α β} - \partial_{p_{α}} \partial_{p_{β}} E_{n} (p (t))) χ_{n} (z; p (t)) - (- i \partial_{y_{α}}) (- i \partial_{y_{β}}) a^{0} (y, t) ({(p (t) - i \partial_{z})}_{α} - \partial_{p_{α}} E_{n} (p (t))) \partial_{p_{β}} χ_{n} (z; p (t)) - (- i \nabla_{y}) a^{1} (y, t) \cdot ((p (t) - i \nabla_{z}) - \nabla_{p} E_{n} (p (t))) χ_{n} (z; p (t)) .

(B7)

Using (A2) we can simplify the term involving a¹,

- (- i \nabla_{y}) a^{1} (y, t) \cdot ((p (t) - i \nabla_{z}) - \nabla_{p} E_{n} (p (t))) χ_{n} (z; p (t)) = (- i \nabla_{y}) a^{1} (y, t) \cdot [H (p (t)) - E_{n} (p (t))] \nabla_{p} χ_{n} (z; p (t)) .

(B8)

Using (A3), and the symmetry: $(- i \partial_{y_{α}}) (- i \partial_{y_{β}}) a^{0} (y, t) = (- i \partial_{y_{β}}) (- i \partial_{y_{α}}) a^{0} (y, t)$ ⁠, we can simplify the terms,

- (- i \partial_{y_{α}}) (- i \partial_{y_{β}}) a^{0} (y, t) \frac{1}{2} (δ_{α β} - \partial_{p_{α}} \partial_{p_{β}} E_{n} (p (t))) χ_{n} (z; p (t)) - (- i \partial_{y_{α}}) (- i \partial_{y_{β}}) a^{0} (y, t) ({(p (t) - i \partial_{z})}_{α} - \partial_{p_{α}} E_{n} (p (t))) \partial_{p_{β}} χ_{n} (z; p (t)) = \frac{1}{2} (- i \partial_{y_{α}}) (- i \partial_{y_{β}}) a^{0} (y, t) [H (p (t)) - E_{n} (p (t))] \partial_{p_{α}} \partial_{p_{β}} χ_{n} (z; p (t)) .

(B9)

Recall that $\dot{p} (t) = - \nabla_{q} W (q (t))$ (1.19). Substituting this, (B8), and (B9) into (B1) and adding and subtracting $a^{0} (y, t) \nabla_{q} W (q (t)) \cdot A_{n} (p (t)) χ_{n} (z; p (t))$ gives

ξ^{2} (y, z, t) = [i \partial_{t} a^{0} (y, t) - \frac{1}{2} \partial_{p_{α}} \partial_{p_{β}} E_{n} (p (t)) (- i \partial_{y_{α}}) (- i \partial_{y_{β}}) a^{0} (y, t) - \frac{1}{2} \partial_{q_{α}} \partial_{q_{β}} W (q (t)) y_{α} y_{β} a^{0} (y, t) - \nabla_{q} W (q (t)) \cdot A_{n} (p (t))] χ_{n} (z; p (t)) + P_{n}^{⊥} (p (t)) [- i a^{0} (y, t) \nabla_{q} W (q (t)) \cdot \nabla_{p} χ_{n} (z; p (t))] + [H (p (t)) - E_{n} (p (t))] [\frac{1}{2} (- i \partial_{y_{α}}) (- i \partial_{y_{β}}) a^{0} (y, t) \partial_{p_{α}} \partial_{p_{β}} χ_{n} (z; p (t)) + (- i \nabla_{y}) a^{1} (y, t) \cdot \nabla_{p} χ_{n} (z; p (t))],

(B10)

where $A_{n} (p (t))$ is the Berry connection (1.26) and $P_{n}^{⊥} (p (t))$ is the orthogonal projection in $L_{p e r}^{2}$ away from the subspace spanned by $χ_{n} (z; p (t))$ ⁠.

APPENDIX C: DERIVATION OF FIRST-ORDER ENVELOPE EQUATION

Collecting terms of order $ϵ^{3 / 2}$ in the expansion (3.9), using Equations (1.21) for $\dot{S} (t)$ and (1.19) for $\dot{q} (t), \dot{p} (t)$ ⁠, and setting equal to zero gives the following inhomogeneous self-adjoint elliptic equation in z for f³(y, z, t),

\begin{matrix} [H (p (t)) - E_{n} (p (t))] f^{3} (y, z, t) = ξ^{3} (y, z, t), \\ for all v \in Λ, f^{3} (y, z + v, t) = f^{3} (y, z, t), f^{3} (y, z, t) \in Σ_{y}^{R - 3} (ℝ^{d}), \\ ξ^{3} (y, z, t) := - [\frac{1}{6} \partial_{q_{α}} \partial_{q_{β}} \partial_{q_{γ}} W (q (t)) y_{α} y_{β} y_{γ}] f^{0} (y, z, t) \\ - [\frac{1}{2} {(- i \nabla_{y})}^{2} + \frac{1}{2} \partial_{q_{α}} \partial_{q_{β}} W (q (t)) y_{α} y_{β} - i \partial_{t}] f^{1} (y, z, t) \\ - [[(p (t) - i \nabla_{z}) - \nabla_{p} E_{n} (p (t))] \cdot (- i \nabla_{y})] f^{2} (y, z, t) . \end{matrix}

(C1)

We claim the following lemmas, the proofs of which will be given at the end of this section.

Lemma C.1. $ξ^{3} (y, z, t)$ ⁠, as defined in (C1), satisfies

ξ^{3} (y, z, t) = {\tilde{ξ}}^{3} (y, z, t) + [H (p (t)) - E_{n} (p (t))] u^{3} (y, z, t),

(C2)

where ${\tilde{ξ}}^{3}$ is given explicitly by (C24) and

u^{3} (y, z, t) := (- i \nabla_{y}) a^{2} (y, t) \cdot \nabla_{p} χ_{n} (z; p (t)) + \frac{1}{2} (- i \partial_{y_{α}}) (- i \partial_{y_{β}}) a^{1} (y, t) \partial_{p_{α}} \partial_{p_{β}} χ_{n} (z; p (t)) + \frac{1}{6} (- i \partial_{y_{α}}) (- i \partial_{y_{β}}) (- i \partial_{y_{γ}}) a^{0} (y, t) \partial_{p_{α}} \partial_{p_{β}} \partial_{p_{γ}} χ_{n} (z; p (t)) .

(C3)

Lemma C.2. The solvability condition for (C1), given by (3.19) with j = 3 and ${\tilde{ξ}}^{3} (y, z, t)$ given by (C24), is equivalent to the following evolution equation for a¹(y, t):

i \partial_{t} a^{1} (y, t) = \frac{1}{2} \partial_{p_{α}} \partial_{p_{β}} E_{n} (p (t)) (- i \partial_{y_{α}}) (- i \partial_{y_{β}}) a^{1} (y, t) + \frac{1}{2} \partial_{q_{α}} \partial_{q_{β}} W (q (t)) y_{α} y_{β} a^{1} (y, t) + \nabla_{q} W (q (t)) \cdot A_{n} (p (t)) a^{1} (y, t) + \frac{1}{6} \partial_{p_{α}} \partial_{p_{β}} \partial_{p_{γ}} E_{n} (p (t)) (- i \partial_{y_{α}}) (- i \partial_{y_{β}}) (- i \partial_{y_{γ}}) a^{0} (y, t) + \frac{1}{6} \partial_{q_{α}} \partial_{q_{β}} \partial_{q_{γ}} W (q (t)) y_{α} y_{β} y_{γ} a^{0} (y, t) + \partial_{q_{β}} W (q (t)) \partial {}_{p_{γ}}A_{n, β} (p (t)) (- i \partial_{y_{γ}}) a^{0} (y, t) + \partial_{q_{β}} \partial_{q_{γ}} W (q (t)) A_{n, β} (p (t)) y_{γ} a^{0} (y, t) .

(C4)

Here, $A_{n} (p (t))$ is the Berry connection (1.26).

Taking $a^{1} (y, t) = b (y, t) e^{i ϕ_{B} (t)}$ and matching with the initial data implies Equation (1.24) for b(y, t). The solution of (C1) is then given by (3.21),

f^{3} (y, z, t) = a^{3} (y, t) χ_{n} (z; p (t)) + u^{3} (y, z, t) + {[H (p (t)) - E_{n} (p (t))]}^{- 1} P_{n}^{⊥} (p (t)) {\tilde{ξ}}^{3} (y, z, t),

(C5)

where ${\tilde{ξ}}^{3} (y, z, t)$ is given by (C24) and u³(y, z, t) by (C3). a³(y, t) is an arbitrary function in $Σ_{y}^{R - 3} (ℝ^{d})$ to be fixed at higher order in the expansion. Note that all manipulations so far are valid as long as $R \geq 3$ ⁠.

Proof of Lemma C.1. Adding and subtracting terms using smoothness of the band E_n(p) in a neighborhood of p(t) (Assumption 1.1), we can re-write $ξ^{3} (y, z, t)$ (C1) as

ξ^{3} (y, z, t) = - [\frac{1}{6} \partial_{p_{α}} \partial_{p_{β}} \partial_{p_{γ}} E_{n} (p (t)) (- i \partial_{y_{α}}) (- i \partial_{y_{β}}) (- i \partial_{y_{γ}}) + \frac{1}{6} \partial_{q_{α}} \partial_{q_{β}} \partial_{q_{γ}} W (q (t)) y_{α} y_{β} y_{γ}] f^{0} (y, z, t) - [- \frac{1}{6} \partial_{p_{α}} \partial_{p_{β}} \partial_{p_{γ}} E_{n} (p (t)) (- i \partial_{y_{α}}) (- i \partial_{y_{β}}) (- i \partial_{y_{γ}})] f^{0} (y, z, t) - [\frac{1}{2} \partial_{p_{α}} \partial_{p_{β}} E_{n} (p (t)) (- i \partial_{y_{α}}) (- i \partial_{y_{β}}) + \frac{1}{2} \partial_{q_{α}} \partial_{q_{β}} W (q (t)) y_{α} y_{β} - i \partial_{t}] f^{1} (y, z, t) - [\frac{1}{2} (δ_{α β} - \partial_{p_{α}} \partial_{p_{β}} E_{n} (p (t))) (- i \partial_{y_{α}}) (- i \partial_{y_{β}})] f^{1} (y, z, t) - [((p (t) - i \nabla_{z}) - \nabla_{p} E_{n} (p (t))) \cdot (- i \nabla_{y})] f^{2} (y, z, t) .

(C6)

Substituting the forms of f⁰(y, z, t) (3.14), f¹(y, z, t) (3.26), and f²(y, z, t) (B5) gives a very long expression on the right-hand side. We simplify this expression by treating terms which depend on a²(y, t),a¹(y, t),a⁰(y, t) in turn.Contributions to (C6) depending on a²(y, t). There is one term which depends on a²(y, t),

- (- i \nabla_{y}) a^{2} (y, t) \cdot ((p (t) - i \nabla_{z}) - \nabla_{p} E_{n} (p (t))) χ_{n} (z; p (t)),

(C7)

which can be simplified using (A2),

- (- i \nabla_{y}) a^{2} (y, t) \cdot ((p (t) - i \nabla_{z}) - \nabla_{p} E_{n} (p (t))) χ_{n} (z; p (t)) = [H (p (t)) - E_{n} (p (t))] [(- i \nabla_{y}) a^{2} (y, t) \cdot \nabla_{p} χ_{n} (z; p (t))] .

(C8)

- [\frac{1}{2} \partial_{p_{α}} \partial_{p_{β}} E_{n} (p (t)) (- i \partial_{y_{α}}) (- i \partial_{y_{β}}) a^{1} (y, t) + \frac{1}{2} \partial_{q_{α}} \partial_{q_{β}} W (q (t)) y_{α} y_{β} a^{1} (y, t) - i \partial_{t} a^{1} (y, t)] χ_{n} (z; p (t)) + i \dot{p} (t) \cdot \nabla_{p} χ_{n} (z; p (t)) a^{1} (y, t) - (- i \partial_{y_{α}}) (- i \partial_{y_{β}}) a^{1} (y, t) \frac{1}{2} (δ_{α β} - \partial_{p_{α}} \partial_{p_{β}} E_{n} (p (t))) χ_{n} (z; p (t)) - (- i \partial_{y_{α}}) (- i \partial_{y_{β}}) a^{1} (y, t) ({(p (t) - i \partial_{z})}_{α} - \partial_{p} E_{n} {(p (t))}_{α}) \partial_{p_{β}} χ_{n} (z; p (t)) .

(C9)

Note that these terms have an identical form to the terms depending on a⁰(y, t) in expression (B7) for $ξ^{2} (y, z, t)$ which were simplified to the form (B10). We may therefore manipulate these terms in an identical way (specifically, using (1.19) and (A3)) into the form

= [i \partial_{t} a^{1} (y, t) - \frac{1}{2} \partial_{p_{α}} \partial_{p_{β}} E_{n} (p (t)) (- i \partial_{y_{α}}) (- i \partial_{y_{β}}) a^{1} (y, t) - \frac{1}{2} \partial_{q_{α}} \partial_{q_{β}} W (q (t)) y_{α} y_{β} a^{1} (y, t) - \nabla_{q} W (q (t)) \cdot A_{n} (p (t))] χ_{n} (z; p (t)) + P_{n}^{⊥} (p (t)) [- i a^{1} (y, t) \nabla_{q} W (q (t)) \cdot \nabla_{p} χ_{n} (z; p (t))] + [H (p (t)) - E_{n} (p (t))] [\frac{1}{2} (- i \partial_{y_{α}}) (- i \partial_{y_{β}}) a^{1} (y, t) \partial_{p_{α}} \partial_{p_{β}} χ_{n} (z; p (t))] .

(C10)

Contributions to(C6)depending on a⁰(y, t). The terms which depend on a⁰(y, t) may be written as T₁ + T₂ + T₃ + T₄, where

T_{1} := - [\frac{1}{6} \partial_{p_{α}} \partial_{p_{β}} \partial_{p_{γ}} E_{n} (p (t)) (- i \partial_{y_{α}}) (- i \partial_{y_{β}}) (- i \partial_{y_{γ}}) a^{0} (y, t) + \frac{1}{6} \partial_{q_{α}} \partial_{q_{β}} \partial_{q_{γ}} W (q (t)) y_{α} y_{β} y_{γ} a^{0} (y, t)] χ_{n} (z; p (t)),

(C11)

T_{2} := - (- i \partial_{y_{α}}) (- i \partial_{y_{β}}) (- i \partial_{y_{γ}}) a^{0} (y, t) [- \frac{1}{6} \partial_{p_{α}} \partial_{p_{β}} \partial_{p_{γ}} E_{n} (p (t)) χ_{n} (z; p (t))] - (- i \partial_{y_{α}}) (- i \partial_{y_{β}}) (- i \partial_{y_{γ}}) a^{0} (y, t) [\frac{1}{2} (δ_{α β} - \partial_{p_{α}} \partial_{p_{β}} E_{n} (p (t))) \partial_{p_{γ}} χ_{n} (z; p (t))] - (- i \partial_{y_{α}}) (- i \partial_{y_{β}}) (- i \partial_{y_{γ}}) a^{0} (y, t) [\frac{1}{2} ({(p (t) - i \partial_{z})}_{α} - \partial_{p_{α}} E_{n} (p (t))) \partial_{p_{β}} \partial_{p_{γ}} χ_{n} (z; p (t))],

(C12)

T_{3} := - [\frac{1}{2} \partial_{p_{α}} \partial_{p_{β}} E_{n} (p (t)) (- i \partial_{y_{α}}) (- i \partial_{y_{β}}) + \frac{1}{2} \partial_{q_{α}} \partial_{q_{β}} W (q (t)) y_{α} y_{β} - i \partial_{t}] (- i \partial_{y_{γ}}) a^{0} (y, t) \partial_{p_{γ}} χ_{n} (z; p (t)),

(C13)

T_{4} := i ({(p (t) - i \partial_{z})}_{β} - \partial_{p_{β}} E_{n} (p (t))) \partial_{q_{γ}} W (q (t)) (- i \partial_{y_{β}}) a^{0} (y, t) {[H (p (t)) - E_{n} (p (t))]}^{- 1} P_{n}^{⊥} (p (t)) \partial_{p_{γ}} χ_{n} (z; p (t)) .

(C14)

Using (A4) and the equality of mixed partial derivatives, we can simplify T₂,

T_{2} = [H (p (t)) - E_{n} (p (t))] [\frac{1}{6} (- i \partial_{y_{α}}) (- i \partial_{y_{β}}) (- i \partial_{y_{γ}}) a^{0} (y, t) \partial_{p_{α}} \partial_{p_{β}} \partial_{p_{γ}} χ_{n} (z; p (t))] .

(C15)

We can simplify T₃ using the evolution equation for a⁰(y, t) (B4),

\begin{matrix} T_{3} = - [\frac{1}{2} \partial_{p_{α}} \partial_{p_{β}} E_{n} (p (t)) (- i \partial_{y_{α}}) (- i \partial_{y_{β}}) + \frac{1}{2} \partial_{q_{α}} \partial_{q_{β}} W (q (t)) y_{α} y_{β}] (- i \partial_{y_{γ}}) a^{0} (y, t) \partial_{p_{γ}} χ_{n} (z; p (t)) \\ + (- i \partial_{y_{γ}}) [i \partial_{t} a^{0} (y, t)] \partial_{p_{γ}} χ_{n} (z; p (t)) + (- i \partial_{y_{γ}}) a^{0} (y, t) i {\dot{p}}_{β} (t) \partial_{p_{β}} \partial_{p_{γ}} χ_{n} (z; p (t)) \\ = - \frac{1}{2} \partial_{q_{α}} \partial_{q_{β}} W (q (t)) y_{α} y_{β} (- i \partial_{y_{γ}}) a^{0} (y, t) \partial_{p_{γ}} χ_{n} (z; p (t)) + \frac{1}{2} \partial_{q_{α}} \partial_{q_{β}} W (q (t)) (- i \partial_{y_{γ}}) y_{α} y_{β} a^{0} (y, t) \\ \times \partial_{p_{γ}} χ_{n} (z; p (t)) + \partial_{q_{β}} W (q (t)) A_{n, β} (p (t)) (- i \partial_{y_{γ}}) a^{0} (y, t) \partial_{p_{γ}} χ_{n} (z; p (t)) - i \partial_{q_{β}} W (q (t)) (- i \partial_{y_{γ}}) a^{0} (y, t) \\ \times \partial_{p_{β}} \partial_{p_{γ}} χ_{n} (z; p (t)) . \end{matrix}

(C16)

We now write T₃ = T_3,1 + T_3,2, where

T_{3,1} := - \frac{1}{2} \partial_{q_{α}} \partial_{q_{β}} W (q (t)) y_{α} y_{β} (- i \partial_{y_{γ}}) a^{0} (y, t) \partial_{p_{γ}} χ_{n} (z; p (t)) + \frac{1}{2} \partial_{q_{α}} \partial_{q_{β}} W (q (t)) (- i \partial_{y_{γ}}) y_{α} y_{β} a^{0} (y, t) \partial_{p_{γ}} χ_{n} (z; p (t)),

(C17)

T_{3,2} := \partial_{q_{β}} W (q (t)) A_{n, β} (p (t)) (- i \partial_{y_{γ}}) a^{0} (y, t) \partial_{p_{γ}} χ_{n} (z; p (t)) - i \partial_{q_{β}} W (q (t)) \times (- i \partial_{y_{γ}}) a^{0} (y, t) \partial_{p_{β}} \partial_{p_{γ}} χ_{n} (z; p (t)) .

(C18)

We can simplify T_3,1 as follows. We first re-arrange (C17),

T_{3,1} = ((- i \partial_{y_{γ}}) y_{α} y_{β} - y_{α} y_{β} (- i \partial_{y_{γ}})) \frac{1}{2} \partial_{q_{α}} \partial_{q_{β}} W (q (t)) a^{0} (y, t) \partial_{p_{γ}} χ_{n} (z; p (t)) .

(C19)

Using the identity $(- i \partial_{y_{α}}) y_{β} - y_{β} (- i \partial_{y_{α}}) = - i δ_{α β}$ twice, we have that

(- i \partial_{y_{γ}}) y_{α} y_{β} - y_{α} y_{β} (- i \partial_{y_{γ}}) = (- i δ_{α γ}) y_{β} + (- i δ_{β γ}) y_{α} .

(C20)

Using the symmetry $\partial_{q_{α}}$

Wavepackets in inhomogeneous periodic media: Effective particle-field dynamics and Berry curvature

I. INTRODUCTION

A. Dimensional analysis, derivation of equation (1.1)

B. Statement of results

C. Dynamics of semiclassical wavepackets in a periodic background

D. Dynamics of observables associated with the asymptotic solution

E. Discussion of results, relation to previous work

F. Notation and conventions

II. SUMMARY OF RELEVANT FLOQUET-BLOCH THEORY

III. PROOF OF THEOREM 1.1 BY MULTISCALE ANALYSIS

A. Derivation of asymptotic solution (1.29) via multiscale expansion

1. Analysis of leading order terms

2. Analysis of order ϵ1/2 terms

3. Analysis of order ϵ and ϵ3/2 terms (summary)

B. Proof of estimate (1.30) for the corrector η

IV. PROOF OF THEOREM 1.2 ON DYNAMICS OF PHYSICAL OBSERVABLES

A. Asymptotic expansion and dynamics of Nϵ(t)

B. Asymptotic expansion of Qϵ(t),Pϵ(t)⁠, proof of assertion (1) of theorem 1.2

C. Computation of dynamics of Qϵ(t),Pϵ(t)⁠, proof of assertion (2) of Theorem 1.2

D. Hamiltonian structure of dynamics of Qϵ(t),Pϵ(t)⁠, proof of assertion (3) of Theorem 1.2

ACKNOWLEDGMENTS

APPENDIX A: USEFUL IDENTITIES INVOLVING En,χn

APPENDIX B: DERIVATION OF LEADING-ORDER ENVELOPE EQUATION

APPENDIX C: DERIVATION OF FIRST-ORDER ENVELOPE EQUATION

2. Analysis of order $ϵ^{1 / 2}$ terms

3. Analysis of order $ϵ$ and $ϵ^{3 / 2}$ terms (summary)

B. Proof of estimate (1.30) for the corrector $η$

A. Asymptotic expansion and dynamics of $N^{ϵ} (t)$

B. Asymptotic expansion of $Q^{ϵ} (t), P^{ϵ} (t)$ ⁠, proof of assertion (1) of theorem 1.2

C. Computation of dynamics of $Q^{ϵ} (t), P^{ϵ} (t)$ ⁠, proof of assertion (2) of Theorem 1.2

D. Hamiltonian structure of dynamics of $Q^{ϵ} (t), P^{ϵ} (t)$ ⁠, proof of assertion (3) of Theorem 1.2

APPENDIX A: USEFUL IDENTITIES INVOLVING $E_{n}, χ_{n}$