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 ϵ0, we prove the existence of solutions known as semiclassical wavepackets which are asymptotic up to “Ehrenfest time” tln1/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.

In this work we study the non-dimensionalized time-dependent Schrödinger equation for ψϵ(x,t):d×[0,),

iϵtψϵ=12ϵ2Δxψϵ+V(xϵ)ψϵ+W(x)ψϵ,ψϵ(x,0)=ψ0ϵ(x),
(1.1)

where ϵ is a positive real parameter which we assume to be small: ϵ1. 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 allvΛ,zd,
(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 ϵ0, the mean position and momentum of the semi-classical wavepacket evolve along the classical trajectories associated with the “Bloch band” Hamiltonian Hn:=En(p)+W(q), where pEn(p) is the dispersion relation associated with the nth spectral (Bloch) band of the periodic Schrödinger operator 12Δz+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 Qϵ(t), mean momentum Pϵ(t), and wave-amplitude profile aϵ(y,t) is a closed system of Hamiltonian type (Theorem 1.2). When d = 3, this system takes the form

Q˙ϵ(t)P˙ϵ(t)==PϵHn(Qϵ(t),Pϵ(t))QϵHn(Qϵ(t),Pϵ(t))Dynamics generated by“Bloch band” HamiltonianHn:=En(Pϵ)+W(Qϵ)++ϵ{ϵ{P˙ϵ(t)×Pϵ×An(Pϵ(t))Anomalous velocity due toBerry curvature+C1[aϵ](t)+C2[aϵ](t)“Particle-field”coupling toenvelopeaϵ}},itaϵ=Hnϵ(t)aϵ;Hnϵ(t):=12yDPϵ2En(Pϵ(t))y+12yDQϵ2W(Qϵ(t))yQuantum harmonic oscillator Hamiltonianwith parametric forcing throughQϵ(t),Pϵ(t),
(1.3)

where DPϵ2En,DQϵ2W denote the Hessian matrices and An is the Berry connection (1.26). For the explicit forms of C1[a], C2[a] and the generalization of (1.3) to arbitrary dimensions d1, 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” tln1/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 Joye7,8 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(xϵ,x), where U is periodic in its first argument,

U(z+v,x)=U(z,x)for allz,xd,vΛ
(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 (V0) 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 

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

itψ=22mΔxψ+V(x)ψ+W(x)ψ,
(1.5)

where 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=ml2.
(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)=ml2τ2V(xl),W(x)=mL2T2W(xL).
(1.7)

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

x:=xL,t:=tT,ψ(x,t):=ψ(x,t),
(1.8)

(1.5) becomes

iTtψ=22mL2Δxψ+ml2τ2V(Lxl)ψ+mL2T2W(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:=TmL2,ϵ:=lL.
(1.10)

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

ihtψh,ϵ=h22Δxψh,ϵ+h2ϵ2V(xϵ)ψh,ϵ+W(x)ψh,ϵ,
(1.11)

where we have written ψh,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 ϵ0 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 ϵ0 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.

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

H:=12Δz+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 pB,

H(p)χ(z;p)=E(p)χ(z;p),
χ(z+v;p)=χ(z;p)for all zd,vΛ,
(1.13)
H(p):=12(piz)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,

E1(p)E2(p)En(p).
(1.14)

For fixed p, the associated normalized eigenfunctions χn(z;p) are a basis of the space,

Lper2:={fLloc2:f(z+v)=f(z)for all vΛ,zd}.
(1.15)

Varying p over the Brillouin zone, the maps pEn(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 L2(d) can be written as a superposition of Bloch waves,

{Φn(z;p)=eipzχn(z;p):n,pB},
(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 En(p),

σ(H)L2(d)=n{En(p):pB}.
(1.17)

The map pEn(p) extends to a map on d which is periodic with respect to the reciprocal lattice Λ*,

for anybΛ*,En(p+b)=En(p).
(1.18)

If the eigenvalue En(p) is simple, then (up to a constant phase shift) χn(z;p+b)=eibzχn(z;p). A more detailed account of the Floquet-Bloch theory which we require, in particular results on the regularity of the maps pEn(p),χn(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 (q0,p0)d×d are such that the flow generated by the classical Hamiltonian Hn(q,p):=En(p)+W(q),

q˙(t)=pEn(p(t)),
p˙(t)=qW(q(t)),
(1.19)
q(0),p(0)=q0,p0

has a unique smooth solution (q(t),p(t))d×d,t0, and that there exists a constant M>0 such that

infmn|Em(p(t))En(p(t))|M for all t0.
(1.20)

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

Assumption 1.2.|α|=1,2,3,4|xαW(x)|L(d).

Remark 1.2. An example of W satisfying Assumption 1.2 is the “Stark” potential W(x)=Ex for any constant vector Ed. 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)=12xMx, where M is any real positive definite d×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 t0:|α|=1,2,3,4|xαW(x,t)|C.

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 a0(y),b0(y)S(d). Let S(t) denote the classical action along the path (q(t),p(t)),

S(t)=0tp(t)pEn(p(t))En(p(t))W(q(t))dt.
(1.21)

Let a(y, t) satisfy

ita(y,t)=H(t)a(y,t),a(y,0)=a0(y),
(1.22)

where

H(t):=12yDp2En(p(t))y+12yDq2W(q(t))y.
(1.23)

And let b(y, t) satisfy

itb(y,t)=H(t)b(y,t)+I(t)a(y,t),b(y,0)=b0(y),
(1.24)

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

I(t):=16p[yDp2En(p(t))y](iy)+16q[yDq2W(q(t))y]y+p[qW(q(t))An(p(t))](iy)+q[qW(q(t))An(p(t))]y,
(1.25)

where An(p) denotes the nth band Berry connection,

An(p):=iχn(;p))|pχn(;p)L2(Ω),
(1.26)

where Ω denotes a fundamental period cell of the lattice Λ. Let ϕB(t) be the Berry phase associated with transport of χn along the path p(t)B given by

ϕB(t)=0tp˙(t)An(p(t))dt=p0p(t)An(p)dp.
(1.27)

Then, there exists a constant ϵ0>0 such that for all 0<ϵϵ0 the following holds. Let ψϵ(x,t) be the unique solution of the initial value problem (1.1) with “Bloch wavepacket” initial data,

iϵtψϵ=12ϵ2Δxψϵ+V(xϵ)ψϵ+W(x)ψϵψϵ(x,0)=ϵd/4eip0(xq0)/ϵ{a0(xq0ϵ1/2)χn(xϵ;p0)+ϵ1/2[(iy)a0(xq0ϵ1/2)pχn(xϵ;p0)+b0(xq0ϵ1/2)χn(xϵ;p0)]}.
(1.28)

Then, for all t0 the solution evolves as a modulated “Bloch wavepacket” plus a corrector ηϵ(x,t),

ψϵ(x,t)=ϵd/4eiS(t)/ϵeip(t)(xq(t))/ϵeiϕB(t){a(xq(t)ϵ1/2,t)χn(xϵ;p(t))+ϵ1/2[(iy)a(xq(t)ϵ1/2,t)pχn(xϵ;p(t))+b(xq(t)ϵ1/2,t)χn(xϵ;p(t))]}+ηϵ(x,t),
(1.29)

where the corrector ηϵ satisfies the following estimate:

ηϵ(,t)L2(d)Cϵect,
(1.30)

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

supt[0,Cln1/ϵ]ηϵ(,t)L2(d)=o(ϵ1/2),
(1.31)

where C is any constant satisfying C<12c.

Remark 1.4. We include the pre-factorϵd/4throughout so that the Lx2(d) norm of ψϵ(x,t) is of order 1 as ϵ0.

Remark 1.5. We have improved the error bound of Carles and Sparber66from Cϵ1/2ect to Cϵectby including correction terms in the asymptotic expansion proportional to ϵ1/2. Note that we must also assume that the initial data are well-prepared up to terms proportional toϵ1/2(1.28). By keeping more terms in the expansion we may produce approximations where the corrector can be bounded byCϵN/2ectfor any positive integer N. The only changes in the proof are that we include corrections to the initial data proportional toϵN/2, and that Assumption 1.2 is replaced by |α|=1,2,,N+2|xαW(x)|L(d).

Remark 1.6. Keeping terms proportional to ϵ1/2 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), tln1/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 ϵ1/2) 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/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,

a0(y)=N[detA0]1/2exp(i12yB0A01y),
(1.32)

where N is an arbitrary non-zero constant, and A0, B0 are d×d complex matrices satisfying

A0TB0B0TA0=0,A0T¯B0B0T¯A0=2iI.
(1.33)

Remark 1.8. The conditions (1.33) imply that

  1. the matrices B0, A0are invertible,

  2. the matrix B0A01 is complex symmetric: (B0A01)T=B0A01,

  3. the imaginary part of the matrix B0A01 is symmetric, positive definite, and satisfiesImB0A01=(A0A0T¯)1,

and are equivalent to the condition that the matrix

Y:=(ReA0ImA0ReB0ImB0)is symplectic:YTJY=J,whereJ:=(0II0).
(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 |a0(y)|Cec|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 t0,

a(y,t)=N[detA(t)]1/2exp(i12yB(t)A1(t)y).
(1.35)

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

A˙(t)=Dp2En(p(t))B(t),B˙(t)=Dq2W(q(t))A(t),A(0)=A0,B(0)=B0.
(1.36)

Moreover, for allt0, 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)|C(t)ec(t)|y|2whereC(t)>0,c(t)>0for all t0. More generally, we may construct a basis of L2(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.

We now deduce consequences for the physical observables associated with the solution ψϵ(x,t) of (1.28), using the asymptotic form (1.29). Denote the solution of (1.28) through order ϵ1/2 by

ψϵ(y,z,t):=ϵd/4eiS(t)/ϵeip(t)y/ϵ1/2eiϕB(t){a(y,t)χn(z;p(t))+ϵ1/2[(iy)a(y,t)pχn(z;p(t))+b(y,t)χn(z;p(t))]}.
(1.37)

Thus,

ψϵ(x,t)=ψϵ(y,z,t)|z=xϵ,y=xq(t)ϵ1/2+ηϵ(x,t),
(1.38)

where ηϵ is the corrector which satisfies the bound (1.30). Define the physical observables

Qϵ(t):=1Nϵ(t)dx|ψϵ(y,z,t)|z=xϵ,y=xq(t)ϵ1/22dx,Pϵ(t):=1Nϵ(t)dψϵ(y,z,t)¯(iϵy1/2)ψϵ(y,z,t)|z=xϵ,y=xq(t)ϵ1/2dx,
(1.39)

where Nϵ(t) is the normalization factor,

Nϵ(t)=d|ψϵ(y,z,t)|z=xϵ,y=xq(t)ϵ1/22dx.
(1.40)

We will refer to Qϵ(t),Pϵ(t) as the center of mass and average quasi-momentum of the wavepacket. We will see (Theorem 1.2) that Qϵ(t)=q(t)+o(1),Pϵ(t)=p(t)+o(1) up to “Ehrenfest time” tln1/ϵ.

Remark 1.10. In the uniform background case V = 0, solutions of (1.13) are independent of z: χn(z;p)=1for all pB. The asymptotic solution (1.37) obtained in this case is therefore independent of z, and our definition of Pϵ(t) reduces to the usually defined momentum observable,

dψϵ(x,t)¯(iϵx)ψϵ(x,t)dx.
(1.41)

In the periodic background case V0, Pϵ(1.39) corresponds to the quasi-momentum and may be measured in experiments.3434Let aϵ(y,t) satisfy the equation of a quantum harmonic oscillator, with parametric forcing defined by (Qϵ(t),Pϵ(t)),

itaϵ=Hϵ(t)aϵ,Hϵ(t):=12yDPϵ2En(Pϵ(t))y+12yDQϵ2W(Qϵ(t))y,aϵ(y,0)=a0(y).
(1.42)

Note that we have replaced dependence on (q(t),p(t)) in equation (1.22) with dependence on Qϵ(t),Pϵ(t).

For simplicity of presentation of the following theorem we assume that

a0(y)|ya0(y)Ly2(d)=a0(y)|(iy)a0(y)Ly2(d)=0,a0(y)Ly2(d)=1.
(1.43)

The result holds for general a0(y)S(d); see Section IV.

Theorem 1.2.Letψϵ(y,z,t)denote the asymptotic solution (1.37) including corrections proportional to ϵ1/2. Let Qϵ(t),Pϵ(t)denote the observables (1.39). Then, there exists an ϵ0>0 such that for all0<ϵϵ0, and for all t[0,Cln1/ϵ] whereC>0is a constant independent oft,ϵ.

  1. Qϵ(t),Pϵ(t)satisfy
    Qϵ(t)=q(t)+ϵ[b(y,t)|ya(y,t)Ly2(d)+a(y,t)|yb(y,t)Ly2(d)]+ϵAn(p(t))+o(ϵ),Pϵ(t)=p(t)+ϵ[b(y,t)|(iy)a(y,t)Ly2(d)+a(y,t)|(iy)b(y,t)Ly2(d)]+o(ϵ),
    (1.44)
    where An(p) is the nth band Berry connection (1.26), and a(y, t),b(y, t) satisfy (1.22) and (1.24), respectively.
  2. Letaϵ(y,t)satisfy (1.42). Then,
    Q˙αϵ(t)=PαϵEn(Pϵ(t))ϵP˙βϵ(t)Fn,αβ(Pϵ(t))+ϵ12Pαϵyaϵ(y,t)|DPϵ2En(Pϵ(t))yaϵ(y,t)Ly2(d)+o(ϵ)P˙αϵ(t)=QαϵW(Qϵ(t))ϵ12Qαϵyaϵ(y,t)|DQϵ2W(Qϵ(t))yaϵ(y,t)Ly2(d)+o(ϵ),
    (1.45)
    where,
    Fn,αβ(Pϵ(t))
    denotes the Berry curvature of the n-th band,:
    Fn,αβ(Pϵ):=An,βPαϵ(Pϵ)An,αPβϵ(Pϵ),
    (1.46)
    where An(Pϵ) is the nth band Berry connection (1.26). When d = 3 the anomalous velocity P˙βϵ(t)Fn,αβ(Pϵ(t)) may be re-written using the cross product as in (1.3); see Remark 1.11.
  3. After dropping the terms of o(ϵ) in (1.45), Equations (1.45) and (1.42) form a closed, coupled “particle-field” system for Qϵ(t),Pϵ(t),aϵ(y,t).

  4. Let
    Qϵ(t):=Qϵ(t)ϵAn(Pϵ(t)),Pϵ(t):=Pϵ(t).
    (1.47)

Let 𝔞ϵ(y,t) denote the solution of (1.42) with co-efficients evaluated at Qϵ(t),Pϵ(t) rather thanQϵ(t),Pϵ(t).

Then, after dropping terms of o(ϵ), Qϵ(t),Pϵ(t),𝔞ϵ(y,t)satisfy a closed, coupled “particle-field” system which is expressible as an ϵ-dependent Hamiltonian system,

Q˙ϵ=PϵHϵ,P˙ϵ=QϵHϵ,it𝔞ϵ=δHϵδ𝔞ϵ¯,
(1.48)

with Hamiltonian

Hϵ(Pϵ,Qϵ,𝔞ϵ¯,𝔞ϵ):=En(Pϵ)+W(Qϵ)+ϵQϵW(Qϵ)An(Pϵ)+ϵ12y𝔞ϵ|DPϵ2En(Pϵ)y𝔞ϵLy2(d)+ϵ12y𝔞ϵ|DQϵ2W(Qϵ)y𝔞ϵLy2(d).
(1.49)

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

P˙βϵ(t)Fn,αβ(Pϵ(t))=P˙βϵ(t)(PαϵAn,β(Pϵ(t))PβϵAn,α(Pϵ(t)))=(δαγδβϕδαϕδβγ)P˙βϵ(t)PγϵAn,ϕ(Pϵ(t))=ϵηαβϵηγϕP˙βϵ(t)PγϵAn,ϕ(Pϵ(t))=(P˙ϵ(t)×Pϵ×An(Pϵ(t)))α,
(1.50)

whereϵ and δare 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 connectionPϵ×An(Pϵ)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 envelopeaϵ. 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.

  1. Choose initial data a0(y) of the form (1.32)–(1.33) with N=πd/4 so that a0(y)Ly2=1. Then, aϵ(y,t), the solution of the initial value problem (1.42), is given by
    aϵ(y,t)=N[detAϵ(t)]1/2exp(i12yBϵ(t)Aϵ1(t)y),
    (1.51)
    where Aϵ(t),Bϵ(t) satisfy
    A˙ϵ(t)=DPϵ2En(Pϵ(t))Bϵ(t),B˙ϵ(t)=DQϵ2W(Qϵ(t))Aϵ(t),Aϵ(0)=A0,Bϵ(0)=B0.
    (1.52)
  2. After the change of variables (1.47), the full coupled system governing (Qϵ,Pϵ,Aϵ(t),Bϵ(t)) governed by (1.45) (with o(ϵ)terms dropped) and (1.52), is expressible as a Hamiltonian system
    Q˙ϵ=HϵPϵ,ϵ=Hϵ\mathcalyQϵ,A˙ϵ(t)=4HϵBϵ¯,B˙ϵ(t)=4HϵAϵ¯,
    (1.53)
    with Hamiltonian
    Hϵ(Pϵ,Qϵ,Aϵ¯,Aϵ,Bϵ¯,Bϵ):=En(Pϵ)+W(Qϵ)+ϵQϵW(Qϵ)An(Pϵ)+ϵ14Tr[(Bϵ¯)TDPϵ2En(Pϵ)Bϵ]+ϵ14Tr[(Aϵ¯)TDQϵ2W(Qϵ)Aϵ].
    (1.54)

Remark 1.13. In the special case where the periodic background potential V = 0 the Bloch band dispersion function En(Pϵ) reduces to the “free” dispersion relation 12(Pϵ)2 and the Hamiltonian (1.54) takes on the simple form

Hϵ(Pϵ,Qϵ,Aϵ¯,Aϵ,Bϵ¯,Bϵ):=12(Pϵ)2+W(Qϵ)+ϵ14Tr[(Aϵ¯)TDQϵ2W(Qϵ)Aϵ]+ϵ14Tr[(Bϵ¯)TBϵ].
(1.55)

The system (1.53), with Hamiltonian Hϵ 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 Qϵ,Pϵ proportional toϵdue to “field-particle” coupling to theAϵ,Bϵsystem 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” tln1/ϵ, and that under further assumptions on the classical dynamics we expect that this time scale may be extended up to t=o(1/ϵ). Note that the Berry curvature terms and the new “field-particle” coupling terms occur at the same order inϵ. It is an interesting question to determine their impact on the dynamics for t greater than the “Ehrenfest time.”

The ϵ0 limit of (1.1) has been studied by other methods: for example, by space-adiabatic perturbation theory15,35–37 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(xϵ,x), where U is periodic in its first argument,

U(z+v,x)=U(z,x)for allz,xd,vΛ.
(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,

H(p,x)χn(z;p,x)=En(p,x)χn(z;p,x),χn(z+v;p,x)=χn(z;p,x)for allz,xd,vΛ,H(p,x):=12(piz)2+U(z,x).
(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.

  • 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
    pαpβf(p)(iyα)(iyβ)g(y),
    (1.58)
    it is understood that we are summing over α{1,,d},β{1,,d}.
  • Where there is no danger of confusion, we will use the standard conventions
    vβwβ=vw,vβvβ=vv.
    (1.59)
  • Δx=x2 denotes the d-dimensional Laplacian.

  • Dx2 denotes the d-dimensional Hessian matrix with respect to x.

  • We will adopt multi-index notation where appropriate so that
    |α|=l|xαf(x)|L(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,
    Σl(d):={fL2(d):fΣl:=|α|+|β|lyα(iy)βf(y)Ly2<,}.
    (1.61)
  • The space of Schwartz functions S(d) is the space of functions defined as
    S(d):=lΣl(d).
    (1.62)
  • We will refer throughout to the space of L2-integrable functions which are periodic on the lattice Λ,
    Lper2:={fLloc2(d):for allzd,vΛ,f(z+v)=f(z)}.
    (1.63)
  • We will write a fundamental period cell in d of the lattice Λ as Ω.

  • We will make use of the Sobolev norms on a fundamental period cell Ω for integers s0,
    f(z)Hzs:=|j|s(z)jf(z)Lz2.
    (1.64)
  • It will also be useful to introduce the “shifted” Sobolev norms, for arbitrary pd,
    f(z)Hz,ps:=|j|s(piz)jf(z)Lz2.
    (1.65)
  • Define the dual lattice to Λ,
    Λ*:={bd:vΛ:bv=2πn,n}.
    (1.66)
  • We will refer to a fundamental cell in pd of the dual lattice Λ* as the Brillouin zone, or B

  • We make the standard convention for the L2-inner product,
    f|gL2(D):=Df(x)¯g(x)dx.
    (1.67)
  • We will make the conventions,
    fϵ(x,t)=O(ϵKect)c>0,C>0,independent of t,ϵsuch thatfϵ(x,t)Lx2CϵKect,gϵ(t)=O(ϵKect)c>0,C>0,independent of t,ϵsuch that|gϵ(t)|CϵKect.
    (1.68)
  • Let A be a complex matrix. Then we will write AT for its transpose, A¯ for its complex conjugate, and TrA for its trace. Using index notation,
    (Aαβ)T:=Aβα,(A¯)αβ:=Aαβ¯,TrA:=Aαα.
    (1.69)
  • The Kronecker delta δαβ is defined,
    δαβ={+1whenα=β0whenαβ.
    (1.70)
  • In dimension d = 3, the Levi-Civita symbol ϵαβγ is defined,
    ϵαβγ:={+1when(α,β,γ){(1,2,3),(2,3,1),(3,1,2)}1when(α,β,γ){(1,3,2),(2,3,1),(3,2,1)}0whenα=β,β=γ,orγ=α
    (1.71)
    and satisfies the identities
    εαβγ=εβγα=εγαβ,εϕαβεϕγϕ=δαγδβϕδβγδαϕ.
    (1.72)
    The cross product of 3-vectors v,w may then be written as
    (v×w)α=ϵαβγvβwγ.
    (1.73)

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

H:=12Δz+V(z),
(2.1)

where V is periodic with respect to the lattice Λ.29,30 For pd, define the spaces of p-pseudo-periodic L2 functions as follows:

Lp2:={fLloc2:f(z+v)=eipvf(z)for allzd,vΛ}.
(2.2)

Let Λ* denote the lattice dual to Λ,

Λ*:={bd:vΛ:vb=2πn,n},
(2.3)

and since the p–pseudo-periodic boundary condition is invariant under pp+b where bΛ*, 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 pB,

HΦ(z;p)=E(p)Φ(z;p),Φ(z+v;p)=eipvΦ(z;p)for allzd,vΛ.
(2.4)

We can also define the space Lloc2 functions which are periodic with respect to the lattice,

Lper2:={f(z)Lloc2(d):vΛ,f(z+v)=f(z)}.
(2.5)

Then solving the eigenvalue problem (2.4) is equivalent via Φ(z;p)=eipzχ(z;p) to solving the family of eigenvalue problems,

H(p)χ(z;p)=E(p)χ(z;p),χ(z+v;p)=χ(z;p)for allzd,vΛ,H(p)=12(piz)2+V(z)
(2.6)

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

E1(p)E2(p)En1(p)En(p)En+1(p),
(2.7)

and the set of normalized eigenfunctions {χn(z;p):n} is complete in Lper2. The set of Floquet-Bloch waves {Φn(z;p)=eipzχn(z;p):n,pB} are complete in L2(d),

gL2(d)g(x)=n1𝔹gn(p)Φn(x;p)dp,wheregn(p):=Φn(;p)|g()L2(d),
(2.8)

where the sum converges in L2. The L2(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)L2(d)=n{En(p):pB}¯.
(2.9)

Our results require sufficient regularity of the maps,

En:B,  pEn(p),
(2.10)
χn:BLper2,  pχn(z;p).
(2.11)

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

infmn|Em(p)En(p)|>0.
(2.12)

We have in this case the following.

Theorem 2.1. (Smoothness of isolated bands). Let En(p),χn(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. LetEm(p),En(p):m,n,mnbe eigenvalue bands of the eigenvalue problem (2.6). Ifp*Bis such that

Em(p*)=En(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 En(p), Em(p) are only Lipschitz continuous, and the eigenfunction maps pχn(z;p),χm(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 pEn(p),χn(z;p) to maps on all of d. Let pB, and let bΛ* denote a reciprocal lattice vector. Then we have that

H(p+b)(eibzχn(z;p))=eibzH(p)χn(z;p)=eibzEn(p)χn(z;p)=En(p)(eibzχn(z;p)),for all vΛ,eib(z+v)χn(z+v;p)=eibveibzχn(z;p)=eibzχn(z;p),
(2.14)

so that if χn(z;p) satisfies (2.6) with eigenvalue En(p), then eibzχn(z;p) satisfies (2.6) with p replaced by p + b, with the same eigenvalue. It then follows that the map pEn(p) extends to a periodic function with respect to the reciprocal lattice Λ*. If the eigenvalue En(p) is simple, then (up to a constant phase shift) χn(z;p+b)=eibzχn(z;p).

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

ψϵ(x,t)=ϵd/4eiS(t)/ϵeip(t)y/ϵ1/2fϵ(y,z,t)|z=xϵ,y=xq(t)ϵ1/2+ηϵ(x,t).
(3.1)

Substituting (3.1) into (1.1) gives an inhomogeneous time-dependent Schrödinger equation for ηϵ(x,t), with a source term rϵ(x,t) which depends on S(t), q(t), p(t), and fϵ(y,z,t),

iϵtηϵ(x,t)=[ϵ22Δx+V(xϵ)+W(x)]ηϵ(x,t)+rϵ[S,q,p,fϵ](x,t),ηϵ(x,0)=η0ϵ[S,q,p,fϵ](x)=ψϵ(x,0)ϵd/4eiS(0)/ϵeip(0)y/ϵ1/2fϵ(z,y,0)|z=xϵ,y=xq(0)ϵ1/2.
(3.2)

The idea behind the proof of Theorem 1.1 is to choose the functions S(t), q(t), p(t), and fϵ(y,z,t) so that

rϵ[S,q,p,fϵ](x,t)=O(ϵ2),η0ϵ[S,q,p,fϵ](x)=O(ϵ).
(3.3)

We will derive S(t),q(t),p(t),fϵ(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 ηϵ(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 rϵ(x,t) has the explicit form

rϵ(x,t)=ϵd/4eiS(t)/ϵeip(t)y/ϵ1/2{ϵ[12(iy)2it]+ϵ1/2[(p(t)iz)(iy)q˙(t)(iy)+p˙(t)y]+[S˙(t)q˙(t)p(t)+12(p(t)iz)2+V(z)+W(q(t)+ϵ1/2y)]}fϵ(y,z,t)|z=xϵ,y=xq(t)ϵ1/2.
(3.4)

Since W is assumed smooth, we can replace W(q(t)+ϵ1/2y) by its Taylor series expansion in ϵ1/2y,

W(q(t)+ϵ1/2y)=W(q(t))+ϵq1/2W(q(t))y+ϵ12qαqβW(q(t))yαyβ+ϵ3/216qαqβqγW(q(t))yαyβyγ+ϵ201(τ1)44!qαqβqγqδW(q(t)+τϵ1/2y)dτyαyβyγyδ.
(3.5)

We expand fϵ(y,z,t) as a formal power series,

fϵ(y,z,t)=f0(y,z,t)+ϵ1/2f1(y,z,t)+
(3.6)

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

for allvΛ,fj(y,z+v,t)=fj(y,z,t),fj(y,z,t)ΣyRj(d).
(3.7)

The Σl-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):=12(piz)2+V(z).
(3.8)

Substituting (3.5) and (3.6) then gives

rϵ(x,t)=ϵd/4eiS(t)/ϵeip(t)y/ϵ1/2{ϵ2[01(τ1)44!qαqβqγqδW(q(t)+τϵ1/2y)dτyαyβyγyδ]+ϵ3/2[16qαqβqγW(q(t))yαyβyγ]+ϵ[12(iy)2+12qαqβW(q(t))yαyβit]+ϵ1/2[(p(t)iz)(iy)+qW(q(t))yq˙(t)(iy)+p˙(t)y]+[S˙(t)q˙(t)p(t)+H(p(t))]}{f0(y,z,t)+ϵ1/2f1(y,z,t)+}|z=xϵ,y=xq(t)ϵ1/2.
(3.9)

In order to prove Theorem 1.1 it will be sufficient to choose the fj(y,z,t),j{0,,3} so that terms of orders ϵj/2,j{0,,3} vanish. With this choice of fj,j{0,,3} we will then prove rigorously in Section III B that rϵ(x,t) can be bounded by Cϵ2ect for constants c>0,C>0 independent of ϵ,t. There will then be no loss of accuracy in the approximation by taking fj(y,z,t)=0,j4.

1. Analysis of leading order terms

Recall that we assume each fj,j{0,1,2,} 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:

H(p(t))f0(y,z,t)=[S˙(t)+q˙(t)p(t)]f0(y,z,t),for allvΛ,f0(y,z+v,t)=f0(y,z,t),f0(y,z,t)ΣyR(d).
(3.10)

Under Assumption 1.1, En(p(t)) is a simple eigenvalue with eigenfunction χn(z;p(t)) for all t0. Projecting equation (3.10) onto the subspace of

Lper2:={fLloc2(d):vΛ,f(z+v)=f(z)},
(3.11)

spanned by χn(z;p(t)), implies

S˙(t)=q˙(t)p(t)En(p(t))
(3.12)

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

[H(p(t))En(p(t))]f0(y,z,t)=0,for allvΛ,f0(y,z+v,t)=f0(y,z,t),f0(y,z,t)ΣyR(d),
(3.13)

which has the general solution

f0(y,z,t)=a0(y,t)χn(z;p(t)),
(3.14)

where a0(y, t) is an arbitrary function in ΣyR(d), to be fixed at higher order in the expansion.

2. Analysis of order ϵ1/2 terms

Collecting terms of order ϵ1/2 in (3.9), substituting the form of S˙(t) (3.12), and setting equal to zero gives the following inhomogeneous self-adjoint elliptic equation in z for f1(y, z, t),

[H(p(t))En(p(t))]f1(y,z,t)=ξ1(y,z,t),for allzΛ,f1(y,z+v,t)=f1(y,z,t),f1(y,z,t)ΣyR1(d),ξ1:=[(p(t)iz)(iy)+qW(q(t))yq˙(t)(iy)+p˙(t)y]f0(y,z,t).
(3.15)

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

Remark 3.1. Collecting terms of ordersϵj/2for eachj{1,2,}and setting equal to zero, we obtain inhomogeneous self-adjoint elliptic equations of the form

[H(p(t))En(p(t))]fj(y,z,t)=ξj[f0,f1,,fj1](y,z,t),forallzΛ,fj(y,z+v,t)=fj(y,z,t),fj(y,z,t)ΣyRj(d).
(3.16)

Our strategy for solving (3.16) will be the same for each j. Under Assumption 1.1, the eigenvalue En(p(t)) is simple with eigenfunctionχn(z;p(t))for allt0. By the Fredholm alternative, equation (3.16) is solvable if and only if

forallt0,χn(z;p(t))|ξj(y,z,t)Lz2(Ω)=0.
(3.17)

We will first use identities derived in Appendix Afrom the eigenvalue equation [H(p)En(p)]χn(z;p)=0 to writeξj(y,z,t)as a sum,

ξj(y,z,t)=ξj(y,z,t)+[H(p(t))En(p(t))]uj(y,z,t).
(3.18)

Note that by self-adjointness of H(p(t)) − En(p(t)), condition (3.17) is equivalent to the same condition withξj(y,z,t)replaced byξj(y,z,t),

forallt0,χn(z;p(t))|ξj(y,z,t)Lz2(Ω)=0.
(3.19)

ForfLper2, define

Pn(p)f(z):=f(z)χn(z;p)|f(z)Lz2(Ω)χn(z;p)
(3.20)

to be the projection onto the orthogonal complement of the subspace ofLper2spanned byχn(z;p(t)). Then, assuming (3.19) is satisfied, the general solution of (3.16) is

fj(y,z,t)=aj(y,t)χn(z;p(t))+uj(y,z,t)+[H(p(t))En(p(t))]1Pn(p(t))ξj(y,z,t).
(3.21)

Note that we have again made use of Assumption 1.1 to ensure that the operator[H(p(t))En(p(t))]1Pn(p(t)):Lper2Lper2is bounded for allt0. When j=1, condition (3.19) may be enforced by choosing q˙(t),p˙(t) to satisfy (1.19). Forj2, enforcing the constraint (3.19) leads to evolution equations for aj−2(y, t).

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

Lemma 3.1.ξ1(y,z,t), defined in (3.15), satisfies

ξ1(y,z,t)=ξ1(y,z,t)+[H(p(t))En(p(t))]u1(y,z,t),
(3.22)

where

ξ1(y,z,t):=[(pEn(p(t))q˙(t))(iy)a0(y,t)+(qW(q(t))+p˙(t))ya0(y,t)]χn(z;p(t)),u1(y,z,t):=(iy)a0(y,t)pχn(z;p(t)).
(3.23)

The solvability condition of (3.15), given by (3.19) with j = 1 on ξ1(y,z,t) (3.23), is then equivalent to

(pEn(p(t))q˙(t))(iy)a0(y,t)+(qW(q(t))+p˙(t))ya0(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 Hn(q,p)=En(p)+W(q),

q˙(t)=pEn(p(t)),p˙(t)=qW(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 u1,ξ1 are given by (3.23). With the choice (3.25) for q˙(t),p˙(t) we have that ξ1=0 for all t0 so that the general solution reduces to

f1(y,z,t)=a1(y,t)χn(z;p(t))+(iy)a0(y,t)pχn(z;p(t)),
(3.26)

where a1(y, t) is an arbitrary function in ΣyR1(d) to be fixed at higher order in the expansion. Note that since a0(y,t)ΣyR(d), this ensures that f1(y,z,t)ΣyR1(d) as required.

Proof of Lemma 3.1. By Assumption 1.1, En(p) is smooth in a neighborhood of p(t). By adding and subtracting pEn(p(t))(iy)f0(y,z,t), ξ1(y,z,t) is equal to

ξ1(y,z,t)=[((p(t)iz)pEn(p(t)))(iy)]f0(y,z,t)[(pEn(p(t))q˙(t))(iy)]f0(y,z,t)[(qW(q(t))+p˙(t))y]f0(y,z,t).
(3.27)

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

ξ1(y,z,t)=(iy)a0(y,t)[(p(t)iz)pEn(p(t))]χn(z;p(t))(iy)a0(y,t)[pEn(p(t))q˙(t)]χn(z;p(t))ya0(y,t)[qW(q(t))+p˙(t)]χn(z;p(t)).
(3.28)

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

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

It is possible to continue the procedure outlined in Remark 3.1 to any order in ϵ1/2. 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 ϵ3/2. In particular, we derive the evolution equations of the amplitudes a0(y, t), a1(y, t) and show that

a0(y,t)=a(y,t)eiϕB(t),a1(y,t)=b(y,t)eiϕB(t),
(3.29)

where a(y,t),b(y,t),ϕB(t) satisfy Equations (1.22), (1.24), and (1.27), respectively.

Let

f3ϵ(y,z,t):=f0(y,z,t)+ϵ1/2f1(y,z,t)+ϵf2(y,z,t)+ϵ3/2f3(y,z,t),
(3.30)

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

ψ3ϵ(x,t):=ϵd/4eiS(t)/ϵeip(t)y/ϵ1/2f3ϵ(y,z,t)|z=xϵ,y=xq(t)ϵ1/2.
(3.31)

Let ψϵ(x,t) denote the exact solution of the initial value problem (1.28). From the manipulations of Sec. III A, we have that η3ϵ(x,t):=ψϵ(x,t)ψ3ϵ(x,t) satisfies

iϵtη3ϵ(x,t)=[ϵ22Δx+V(xϵ)+W(x)]η3ϵ(x,t)+r3ϵ(x,t),η3ϵ(x,0)=η3,0ϵ(x),
(3.32)

where r3ϵ(x,t) is given by

r3ϵ(x,t)=ϵd/4eiS(t)/ϵeip(t)y/ϵ1/2{ϵ201(τ1)44!qαqβqγqδW(q(t)+τϵ1/2y)dτyαyβyγyδ(f0(y,z,t)+ϵ1/2f1(y,z,t)+ϵf2(y,z,t)+ϵ3/2f3(y,z,t))+ϵ2[16qαqβqγW(q(t))yαyβyγ](f1(y,z,t)+ϵ1/2f2(y,z,t)+ϵf3(y,z,t))+ϵ2[it+12(iy)2+12qαqβW(q(t))yαyβ](f2(y,z,t)+ϵ1/2f3(y,z,t))+ϵ2[((p(t)iz)pEn(p(t)))(iy)]f3(y,z,t)}|z=xϵ,y=xq(t)ϵ1/2
(3.33)

And η3,0ϵ(x) is given by

η3,0ϵ(x)=ϵd/4eip0y/ϵ1/2{ϵ[f2(z,y,0)+ϵ1/2f3(z,y,0)]}|z=xϵ,y=xq0ϵ1/2.
(3.34)

Since the fj(y,z,t),j{0,,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,

r3ϵ(x,t)Lx2=ϵ201(τ1)44!qαqβqγqδW(q(t)+τϵ1/2y)dτyαyβyγyδ(f0(y,z,t)+ϵ1/2f1(y,z,t)+ϵf2(y,z,t)+ϵ3/2f3(y,z,t))+ϵ2[16qαqβqγW(q(t))yαyβyγ](f1(y,z,t)+ϵ1/2f2(y,z,t)+ϵf3(y,z,t))+ϵ2[it+12(iy)2+12qαqβW(q(t))yαyβ](f2(y,z,t)+ϵ1/2f3(y,z,t))+ϵ2[((p(t)iz)pEn(p(t)))(iy)]f3(y,z,t)|Lz,Ly2η3ϵ(x,0)Lx2ϵ[f2(z,y,0)+ϵ1/2f3(z,y,0)]Lz,Ly2,
(3.35)

where we have used the fact that

ϵd/4f(xq(t)ϵ1/2)Lx2=f(y)Ly2.
(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 Lz; see  Appendix D. Let

I(t):=ϵ201(τ1)44!qαqβqγqδW(q(t)+τϵ1/2y)dτyαyβyγyδf0(y,z,t)Lz,Ly2,
(3.37)

where f0(y,z,t)=a0(y,t)χn(z;p(t)) (3.14). By Assumption 1.2, |α|=4|xαW(x)|L(d),

I(t)ϵ214!|α|=4|xαW(x)|L(d)|α|=4yαa(y,t)χn(z;p(t))Lz,Ly2.
(3.38)

Recall Assumption 1.1. Define

Sn:={pd:infmn|Em(p)En(p)|M},
(3.39)

so that for all t[0,),p(t)Sn. For each fixed pSn, by elliptic regularity, χn(z;p) is smooth in z so that χn(z;p)Lz<. Using compactness of the Brillouin zone B and smoothness of χn(z;p) for pSn we have that

suppBSnχn(z;p)Lz<.
(3.40)

Recall that for any reciprocal lattice vector bΛ*, χn(z;p+b)=eibzχn(z;p). It then follows that

for all bΛ*,χn(z;p+b)Lz=χn(z;p)Lz.
(3.41)

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

suppSnχn(z;p)Lz<.
(3.42)

In  Appendix D we show how to bound all z-dependence in r3ϵ(x,t) (3.33) uniformly in pSn in a similar way.

We have therefore that (3.38)

I(t)ϵ214!|α|=4|xαW(x)|L(d)suppSnχn(z;p)Lz|α|=4yαa(y,t)Ly2.
(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,

ita(y,t)=12pαpβEn(p(t))(iyα)(iyβ)a(y,t)+12qαqβW(q(t))yαyβa(y,t),a(y,0)=a0(y).
(3.44)

Following the work of Carles and Sparber6 we first define, for any l, the spaces

Σl(d):={fL2(d):fΣl:=|α|+|β|lyα(iy)βf(y)Ly2<}.
(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). Letu0L2(d), and ηαβ(t),ζαβ(t)be real-valued, symmetric, continuous, and uniformly bounded in t. Then the equation

itu=12ηαβ(t)(iyα)(iyβ)u+12ζαβ(t)yαyβu,u(y,0)=u0(y)
(3.46)

has a unique solution uC([0,);L2(d)). It satisfies

u(,t)L2(d)=u0()L2(d).
(3.47)

Moreover, if u0Σl(d), then uC([0,);Σl(d)).

We seek quantitative bounds on u(,t)Σl(d) for l1. For simplicity, we consider in detail the case l = 1. Recall (1.23),

H(t):=12ηαβ(t)(iyα)(iyβ)+12ζαβ(t)yαyβ,
(3.48)

and let u0(y)S(d) so that l0, the solution of (3.46), u(y,t)C([0,);Σl(d)). Then (iyα)u(y,t)S() solves

it(iyα)u=H(t)(iyα)u+[(iyα),H(t)]u,(iyα)u(y,0)=(iyα)u0(y).
(3.49)

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

(iyα)u(y,t)Ly2(iyα)u(y,0)Ly2+0t[(iyα),H(s)]u(y,s)Ly2ds.
(3.50)

Since ζαβ(t) is symmetric, the commutator is given explicitly by

[(iyα),H(s)]=(i)ζαβ(t)yβ,
(3.51)

so that

(iyα)u(y,t)Ly2(iyα)u0(y)Ly2+0t|ζαβ(t)|yβu(y,s)Ly2ds.
(3.52)

By an identical reasoning, we can derive a similar bound on yαu(y,t),

yαu(y,t)Ly2yαu0(y)Ly2+0t|ηαβ(s)|(iyβ)u(y,s)Ly2ds.
(3.53)

Adding inequalities (3.52) and (3.53) gives

u(,t)Σ1u0()Σ1+20tmaxα,β{1,,d}{|ηαβ(s)|,|ζαβ(s)|}u(,s)Σ1ds,
(3.54)

using the following version of Gronwall’s inequality.

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

v(t)a(t)+0tb(s)v(s)ds,
(3.55)

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

v(t)a(t)exp(0tb(s)ds).
(3.56)

We have that

u(,t)Σ1u0()Σ1e20tmaxα,β{1,,d}{|ηαβ(s)|,|ζαβ(s)|}ds.
(3.57)

More generally, we have for any l0 that there exists a constant Cl>0 such that

u(,t)Σlu0()ΣleCl0tmaxα,β{1,,d}{|ηαβ(s)|,|ζαβ(s)|}ds.
(3.58)

We have proved the following.

Lemma 3.4. (Bound on solutions of (3.46) in the spaces Σl(d)). Let the time-dependent co-efficientsηαβ(t),ζαβ(t)be real-valued, symmetric, continuous, and uniformly bounded in t. Let u0(y)Σl(d). Then, by Lemma 3.2, there exists a unique solutionu(y,t)C([0,);Σl(d)). For each integerl0, there exists a constantCl>0such that this solution satisfies

u(,t)Σl(d)u0()ΣleCl0tmaxα,β{1,,d}{|ηαβ(s)|,|ζαβ(s)|}ds
(3.59)

Since the map pEn(p) is B-periodic and smooth for all pSn, we have that under Assumption 1.1, supmaxα,β{1,,d}t[0,)|pαpβEn(p(t))|<. Under Assumption 1.2 we have that supmaxα,β{1,,d}t[0,)|qαqβW(q(t))|<. Since pαpβEn(p(t)),qαqβW(q(t)) are clearly real-valued, symmetric, and continuous in t, we have that Lemma 3.4 applies to solutions of (3.44). Since a0(y)S(d) by assumption we have that for any integer l0,

a(y,t)Σl(d)a0(y)Σl(d)exp(Cmaxα,β{1,,d}lsups[0,){|pαpβEn(p(s))|,|qαqβW(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

ϵ201(τ1)44!qαqβqγqδW(q(t)+τϵ1/2y)dτyαyβyγyδa(y,t)χn(z;p(t))Lz,Ly2c1ϵ2ec2t,
(3.61)

where

c1:=14!|α|=4|xαW(x)|Lx(d)suppSnχn(z;p)Lza0(y)Σy4(d),c2:=Cmaxα,β{1,,d}lsups[0,){|pαpβEn(p(s))|,|qαqβW(q(s))|}
(3.62)

are constants independent of t,ϵ.

We conclude that there exist constants C1,C2,C3>0, independent of t,ϵ such that

η3,0ϵ(x)Lx2C1ϵ,r3ϵ(x,t)Lx2C2eC3tϵ2.
(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. Letψ(x,t)be the unique solution of

itψ=Hψ+f,ψ(x,0)=ψ0(x),
(3.64)

where H is a self-adjoint operator. Then,

ψ(,t)L2(d)ψ0()L2(d)+0tf(,t)L2(d)dt,
(3.65)

and when f = 0, we have

ψ(,t)L2(d)=ψ0()L2(d).
(3.66)

Applying Lemma 3.5 to Equation (3.32) then gives the bound on η3ϵ(x,t),

η3ϵ(x,t)Lx2η3ϵ(x,0)Lx2+1ϵ0tr3ϵ(x,t)Lx2dtC1ϵ+1ϵ0tC2eC3tϵ2dtCeCtϵ,
(3.67)

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

Let ψϵ(x,t) be the solution of (1.28). By Theorem 1.1 we have that this solution has the form

ψϵ(x,t)=ψϵ(y,z,t)|y=xq(t)ϵ1/2,z=xϵ+ηϵ(x,t),
(4.1)

where

ψϵ(y,z,t):=ϵd/4eiS(t)/ϵeip(t)y/ϵ1/2eiϕB(t){a(y,t)χn(z;p(t))+ϵ1/2[(iy)a(y,t)pχn(z;p(t))+b(y,t)χn(z;p(t))]}.
(4.2)

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

Qϵ(t):=1Nϵ(t)dx|ψϵ(y,z,t)|z=xϵ,y=xq(t)ϵ1/22dx,Pϵ(t):=1Nϵ(t)dψϵ(y,z,t)¯(iϵy1/2)ψϵ(y,z,t)|z=xϵ,y=xq(t)ϵ1/2dx,
(4.3)

where

Nϵ(t)=d|ψϵ(y,z,t)|z=xϵ,y=xq(t)ϵ1/22dx.
(4.4)

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

fϵ(x,t)=O(ϵKect)c>0,C>0independentoft,ϵsuchthatfϵ(x,t)Lx2CϵKect,gϵ(t)=O(ϵKect)c>0,C>0independentoft,ϵsuchthat|gϵ(t)|CϵKect.
(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. LetfS(d), g smooth and periodic with respect to the lattice Λ, sa constant, andδ>0an arbitrary positive parameter. Then for any positive integerN>0,

df(x)g(xδ+sδ2)dx=(df(x)dx)(Ωg(z)dz)+O(δN).
(4.6)

For the proof, see  Appendix E.

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

Nϵ(t)=ϵd/2d|ψϵ(y,z,t)|z=yϵ1/2+q(t)ϵ2dy.
(4.7)

Substituting (4.2) into (4.7) gives

=d{a(y,t)χn(z;p(t))+ϵ1/2[(iy)a(y,t)pχn(z;p(t))+b(y,t)χn(z;p(t))]¯}{a(y,t)χn(z;p(t))+ϵ1/2[(iy)a(y,t)pχn(z;p(t))+b(y,t)χn(z;p(t))]}|z=yϵ1/2+q(t)ϵdy.
(4.8)

We expand the product in the integral and apply Lemma 4.1 term by term with s=q(t),δ=ϵ1/2. Since the χn are assumed normalized, for all t[0,)χn(;p(t))L2(Ω)=1, we have

Nϵ(t)=a(y,t)Ly2(d)2+ϵ1/2[(iy)a(y,t)|a(y,t)Ly2(d)pχn(z;p(t))|χn(z;p(t))Lz2(Ω)+a(y,t)|(iy)a(y,t)Ly2(d)χn(z;p(t))|pχn(z;p(t))Lz2(Ω)+b(y,t)|a(y,t)Ly2(d)+a(y,t)|b(y,t)Ly2(d)]+O(ϵect).
(4.9)

Remark 4.2. In (4.9) we have made explicit all terms through orderϵ1/2. To justify the error bound, consider that the remaining terms may be bounded byC(t)ϵwhereC(t)depends onΣyl1-norms of a(y, t), b(y, t) andLz2-norms ofpl2χn(z;p(t))where l1, l2 are positive integers. By an identical reasoning to that given in Section III B we have thatC(t)may be bounded by Cect where c>0,C>0 are constants independent ofϵ,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)B, the mapping pχn(z;p) is smooth. Hence, we may differentiate the normalization condition: χn(;p)L2(Ω)2=1 with respect to p and evaluate along the curve p(t) to obtain the identity,

χn(z;p(t))|pχn(z;p(t))Lz2(Ω)+pχn(z;p(t))|χn(z;p(t))Lz2(Ω)=0.
(4.10)

It follows from this, and the fact that (iy) is symmetric with respect to the Ly2-inner product, that

(iy)a(y,t)|a(y,t)Ly2(d)pχn(z;p(t))|χn(z;p(t))Lz2(Ω)+a(y,t)|(iy)a(y,t)Ly2(d)χn(z;p(t))|pχn(z;p(t))Lz2(Ω)=0
(4.11)

so that (4.9) reduces to

Nϵ(t)=a(y,t)Ly2(d)2+ϵ1/2[b(y,t)|a(y,t)Ly2(d)+a(y,t)|b(y,t)Ly2(d)]+O(ϵect).
(4.12)

From L2-norm conservation for solutions of (1.22), we have that a(y,t)Ly2=a0(y)Ly2. In  Appendix F we calculate (F17)

ddt[b(y,t)|a(y,t)Ly2(d)+a(y,t)|b(y,t)Ly2(d)]=0,
(4.13)

so that

N˙ϵ(t)=O(ϵeCt).
(4.14)

Integrating in time then gives

Nϵ(t)=Nϵ(0)+O(ϵect)=a0(y)Ly2(d)2+ϵ1/2[b0(y)|a0(y)Ly2(d)+a0(y)|b0(y)Ly2(d)]+O(ϵect).
(4.15)

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

x=q(t)+ϵ1/2y|y=xq(t)ϵ1/2,
(4.16)

we have

Qϵ(t)=q(t)+ϵ1/2+d/21Nϵ(t)dy|ψϵ(y,z,t)|z=q(t)ϵ+yϵ1/22dy,Pϵ(t)=ϵ1/2+d/21Nϵ(t)dψϵ(y,z,t)¯(iy)ψϵ(y,z,t)|z=q(t)ϵ+yϵ1/2dy.
(4.17)

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

Qαϵ(t)=qα(t)+ϵ1/21Nϵ(t)dyα|a(y,t)χn(z;p(t))+ϵ1/2[(iyβ)a(y,t)pβχn(z;p(t))+b(y,t)χn(z;p(t))]|z=q(t)ϵ+yϵ1/22dy,Pαϵ(t)=pα(t)+ϵ1/21Nϵ(t)×da(y,t)χn(z;p(t))+ϵ1/2[(iyβ)a(y,t)pβχn(z;p(t))+b(y,t)χn(z;p(t))]¯(iyα)(a(y,t)χn(z;p(t))+ϵ1/2[(iyβ)a(y,t)pβχn(z;p(t))+b(y,t)χn(z;p(t))])|z=q(t)ϵ+yϵ1/2dy.
(4.18)

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

Qαϵ(t)=qα(t)+ϵ1/21Nϵ(t)a(y,t)|yαa(y,t)Ly2(d)+ϵ1Nϵ(t)[b(y,t)|yαa(y,t)Ly2(d)+a(y,t)|yαb(y,t)Ly2(d)+(iyβ)a(y,t)|yαa(y,t)Ly2(d)pβχn(z;p(t))|χn(z;p(t))Lz2(Ω)+yαa(y,t)|(iyβ)a(y,t)Ly2(d)χn(z;p(t))|pβχn(z;p(t))Lz2(Ω)]+O(ϵ3/2ect)Pαϵ(t)=pα(t)+ϵ1/21Nϵ(t)a(y,t)|(iyα)a(y,t)Ly2(d)+ϵ1Nϵ(t)[b(y,t)|(iyα)a(y,t)Ly2(d)+a(y,t)|(iyα)b(y,t)Ly2(d)]+O(ϵ3/2ect).
(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 (iy) is self-adjoint, we have that

(iyβ)a(y,t)|yαa(y,t)Ly2(d)pβχn(z;p(t))|χn(z;p(t))Lz2(Ω)+yαa(y,t)|(iyβ)a(y,t)Ly2(d)χn(z;p(t))|pβχn(z;p(t))Lz2(Ω)=a(y,t)|[yα,(iyβ)]a(y,t)Ly2(d)χn(z;p(t))|pβχn(z;p(t))Lz2(Ω),
(4.20)

where [yα,(iyβ)]:=yα(iyβ)(iyβ)yα is the commutator. Since [yα,(iyβ)]=iδαβ, we have that

a(y,t)|[yα,(iyβ)]a(y,t)Ly2(d)χn(z;p(t))|pβχn(z;p(t))Lz2(Ω)=ia(y,t)Ly2(d)2χn(z;p(t))|pαχn(z;p(t))Lz2(Ω).
(4.21)

Using L2-norm conservation for solutions of (1.22), we have that for all t0, a(y,t)Ly2(d)2=a0(y)Ly2(d)2. Using (4.15), we have that a0(y)Ly2(d)2=Nϵ(t)+O(ϵ1/2ect) (4.12). We have proved that

ia(y,t)Ly2(d)2χn(z;p(t))|pχn(z;p(t))Lz2(Ω)=Nϵ(t)An(p(t))+O(ϵ1/2ect),
(4.22)

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

Qϵ(t)=q(t)+ϵ1/21Nϵ(t)a(y,t)|ya(y,t)Ly2(d)+ϵ1Nϵ(t)[b(y,t)|ya(y,t)Ly2(d)+a(y,t)|yb(y,t)Ly2(d)]+ϵAn(p(t))+O(ϵ3/2ect)Pϵ(t)=p(t)+ϵ1/21Nϵ(t)a(y,t)|(iy)a(y,t)Ly2(d)+ϵ1Nϵ(t)[b(y,t)|(iy)a(y,t)Ly2(d)+a(y,t)|(iy)b(y,t)Ly2(d)]+O(ϵ3/2ect).
(4.23)

Differentiating both sides of (4.23) with respect to time and using N˙ϵ(t)=O(ϵect) (4.14) gives

Q˙αϵ(t)=q˙α(t)+ϵ1/21Nϵ(t)ddta(y,t)|yαa(y,t)Ly2(d)+ϵ1Nϵ(t)ddt[b(y,t)|yαa(y,t)Ly2(d)+a(y,t)|yαb(y,t)Ly2(d)]+ϵp˙β(t)An,αpβ(p(t))+O(ϵ3/2ect)P˙αϵ(t)=p˙α(t)+ϵ1/21Nϵ(t)ddta(y,t)|(iyα)a(y,t)Ly2(d)+ϵ1Nϵ(t)ddt[<b(y,t)|(iyα)a(y,t)>Ly2(d)+<a(y,t)|(iyα)b(y,t)>Ly2(d)]+O(ϵ3/2ect).
(4.24)

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

ddta(y,t)|yαa(y,t)Ly2(d)=pαpβEn(p(t))<a(y,t)|(iyβ)a(y,t)>Ly2(d),ddt<a(y,t)|(iyα)a(y,t)>Ly2(d)=qαqβW(q(t))<a(y,t)|yβa(y,t)>Ly2(d),
(4.25)

and (F18),

ddt[b(y,t)|yαa(y,t)Ly2(d)+a(y,t)|yαb(y,t)Ly2(d)]=pαpβEn(p(t))[<b(y,t)|(iyβ)a(y,t)>Ly2(d)+<a(y,t)|(iyβ)a(y,t)>Ly2(d)]+12pαpβpγEn(p(t))<a(y,t)|(iyβ)(iyγ)a(y,t)>Ly2(d)+qβW(q(t))An,βpα(p(t))a(y,t)Ly2(d)2ddt[<b(y,t)|(iyα)a(y,t)>Ly2(d)+<a(y,t)|(iyα)b(y,t)>Ly2(d)]=qαqβW(q(t))[<b(y,t)|yβa(y,t)>Ly2(d)+<a(y,t)|yβa(y,t)>Ly2(d)]12qαqβqγW(q(t))<a(y,t)|yβyγa(y,t)>Ly2(d)qαqβW(q(t))An,β(p(t))a(y,t)Ly2(d)2.
(4.26)

Substituting these expressions into (4.24) and using a(y,t)Ly2(d)2=Nϵ(t)+O(ϵ1/2ect) (4.9), we have

Q˙αϵ(t)=pαEn(p(t))+ϵ1/21Nϵ(t)pαpβEn(p(t))<a(y,t)|(iyβ)a(y,t)>Ly2(d)+ϵ1Nϵ(t)pαpβEn(p(t))[<b(y,t)|(iyβ)a(y,t)>Ly2(d)+<a(y,t)|(iyβ)a(y,t)>Ly2(d)]+ϵ121Nϵ(t)pαpβpγEn(p(t))<a(y,t)|(iyβ)(iyγ)a(y,t)>Ly2(d)+ϵqβW(q(t))An,βpα(p(t))ϵqβW(q(t))An,αpβ(p(t))+O(ϵ3/2ect)P˙αϵ(t)=qαW(q(t))ϵ1/21Nϵ(t)qαqβW(q(t))<a(y,t)|yβa(y,t)>Ly2(d)ϵ1Nϵ(t)qαqβW(q(t))[<b(y,t)|yβa(y,t)>Ly2(d)+<b(y,t)|yβa(y,t)>Ly2(d)]ϵ121Nϵ(t)qαqβqγW(q(t))<a(y,t)|yβyγa(y,t)>Ly2(d)ϵqαqβW(q(t))An,β(p(t))+O(ϵ3/2ect).
(4.27)

Equation (4.23) gives expressions for q(t), p(t) in terms of Qϵ(t),Pϵ(t),

qα(t)=Qαϵ(t)ϵ1/21Nϵ(t)a(y,t)|yαa(y,t)Ly2(d)ϵ1Nϵ(t)[b(y,t)|yαa(y,t)Ly2(d)+a(y,t)|yαb(y,t)Ly2(d)]ϵAn,α(p(t))+O(ϵ3/2ect)pα(t)=Pαϵ(t)ϵ1/21Nϵ(t)<a(y,t)|(iyα)a(y,t)>Ly2(d)ϵ1Nϵ(t)[<b(y,t)|(iyα)a(y,t)>Ly2(d)+<a(y,t)|(iyα)b(y,t)>Ly2(d)]+O(ϵ3/2ect).
(4.28)

Substituting these expressions into (4.27), Taylor-expanding in ϵ1/2, and again using Nϵ(t)=a0(y)Ly2(d)2+O(ϵ1/2ect) (4.9) then gives

Q˙αϵ(t)=PαϵEn(Pϵ(t))+ϵ12PαϵPβϵPγϵEn(Pϵ(t))[1a0(y)Ly2(d)2<a(y,t)|(iyβ)(iyγ)a(y,t)>Ly2(d)1a0(y)Ly2(d)4<a(y,t)|(iyβ)a(y,t)>Ly2(d)<a(y,t)|(iyγ)a(y,t)>Ly2(d)]+ϵQγϵW(Qϵ(t))Fn,αγ(Pϵ(t))+O(ϵ3/2ect),P˙αϵ(t)=QαϵW(Qϵ(t))ϵ12QαϵQβϵQγϵW(Qϵ(t))[1a0(y)Ly2(d)2a(y,t)|yβyγa(y,t)Ly2(d)1a0(y)Ly2(d)4<a(y,t)|yβa(y,t)>Ly2(d)<a(y,t)|yγa(y,t)>Ly2(d)]+O(ϵ3/2ect),
(4.29)

where Fn,αγ(Pϵ):=An,γPαϵ(Pϵ)An,αPβϵ(Pϵ) 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 aϵ(y,t) (1.42) as the solution of (1.22) with co-efficients evaluated at Qϵ(t),Pϵ(t),

itaϵ(y,t)=12PαϵPβϵEn(Pϵ(t))(iyα)(iyβ)aϵ(y,t)+12QαϵQβϵW(Qϵ(t))yαyβaϵ(y,t),aϵ(y,0)=a0(y),
(4.30)

Recall the definition of the Σl norms (3.45). If we can show that aϵ(y,t)a(y,t)Σyl(d)=O(ϵ1/2ect) for each positive integer l, then we may replace a(y,t) by aϵ(y,t) everywhere in (4.29) and, after dropping error terms, we will have obtained a closed system for Qϵ(t),Pϵ(t),aϵ(y,t). Let

Hϵ(t):=12PαϵPβϵEn(Pϵ(t))(iyα)(iyβ)+12QαϵQβϵW(q(t))yαyβ,H(t):=12pαpβEn(p(t))(iyα)(iyβ)+12qαqβW(q(t))yαyβ,
(4.31)

then aϵ(y,t)a(y,t) satisfies

it(aϵ(y,t)a(y,t))=Hϵ(t)aϵ(y,t)H(t)a(y,t)=Hϵ(t)(aϵ(y,t)a(y,t))+(Hϵ(t)H(t))a(y,t),aϵ(y,0)a(y,0)=0.
(4.32)

Using the fact that Hϵ(t) is self-adjoint on Ly2(d) for each t, it follows from (4.32) that

ddtaϵ(y,t)a(y,t)Ly2(d)2=i(Hϵ(t)H(t))a(y,t)|aϵ(y,t)a(y,t)Ly2(d)iaϵ(y,t)a(y,t)|×(Hϵ(t)H(t))a(y,t)Ly2(d).
(4.33)

By the Cauchy-Schwarz inequality,

2aϵ(y,t)a(y,t)Ly2(d)(Hϵ(t)H(t))a(y,t)Ly2(d),
(4.34)

it then follows that

aϵ(y,t)a(y,t)Ly2(d)0t(Hϵ(s)H(s))a(y,s)Ly2(d)ds.
(4.35)

Using the precise forms of Hϵ(t),H(t), we have

aϵ(y,t)a(y,t)Ly2(d)0tsups[0,),α,β{1,,d}(|PαϵPβϵEn(Pϵ(s))pαpβEn(p(s))|+|QαϵQβϵW(Qϵ(s))qαqβW(q(s))|)a(y,s)Σy2(d)ds,
(4.36)

where Σy2(d) is the norm defined in (3.45). Recall that |Qϵ(t)q(t)|+|Pϵ(t)p(t)|=O(ϵ1/2ect) (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 En(p), W(q) for all p along the line segments connecting p(t) and Pϵ(t), and all q along the line segments connecting q(t) and Qϵ(t). We may therefore conclude from the mean - value theorem that there exist constants c>0,C>0 independent of ϵ,t such that

aϵ(y,t)a(y,t)Ly2(d)Cϵ1/20tecsa(y,s)Σy2(d)ds.
(4.37)

We now use the a priori bounds on the Σyl(d)-norms of a(y, t) for each l (Lemma 3.4) to see that

aϵ(y,t)a(y,t)Ly2(d)ϵ1/2Cect
(4.38)

for some constants c>0,C>0 independent of ϵ,t. By a similar argument, we see that for any integer l0 there exist a constants cl>0,Cl>0 such that

aϵ(y,t)a(y,t)Σyl(d)ϵ1/2Cleclt.
(4.39)

It then follows that we may replace a(y, t) by aϵ(y,t) everywhere in (4.29), generating further errors which are O(ϵ3/2ect) to derive

Q˙αϵ(t)=PαϵEn(Pϵ(t))+ϵ12PαϵPβϵPγϵEn(Pϵ(t))[1a0(y)Ly2(d)2<aϵ(y,t)|(iyβ)(iyγ)aϵ(y,t)>Ly2(d)1a0(y)Ly2(d)4<aϵ(y,t)|(iyβ)aϵ(y,t)>Ly2(d)<aϵ(y,t)|(iyγ)aϵ(y,t)>Ly2(d)]+ϵQγϵW(Qϵ(t))Fn,αγ(Pϵ(t))+O(ϵ3/2ect),P˙αϵ(t)=QαϵW(Qϵ(t))ϵ12QαϵQβϵQγϵW(Qϵ(t))[1a0(y)Ly2(d)2<aϵ(y,t)|yβyγaϵ(y,t)>Ly2(d)1a0(y)Ly2(d)4<aϵ(y,t)|yβaϵ(y,t)>Ly2(d)<aϵ(y,t)|yγaϵ(y,t)>Ly2(d)]+O(ϵ3/2ect).
(4.40)

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

Qϵ(t):=Qϵ(t)ϵAn(Pϵ(t)),Pϵ(t):=Pϵ(t).
(4.41)

Let 𝔞ϵ(y,t) denote the solution of (1.42) with co-efficients evaluated at Qϵ(t),Pϵ(t) rather than Qϵ(t),Pϵ(t), with initial data normalized in Ly2(d),

it𝔞ϵ(y,t)=12PαϵPβϵEn(\mathcalyPϵ(t))(iyα)(iyβ)𝔞ϵ(y,t)+12QαϵQβϵW(Qϵ(t))yαyβ𝔞ϵ(y,t),𝔞ϵ(y,0)=a0(y)a0(y)Ly2(d).
(4.42)

Since Qϵ(t)Qϵ(t)=O(ϵect), Pϵ(t)=Pϵ(t), by a similar argument to that given in Sec. IV C we have that for each integer l0,

aϵ(y,t)a0(y)Ly2(d)𝔞ϵ(y,t)Σyl(d)=O(ϵect).
(4.43)

Differentiating (4.41), using Equations (4.40) for Q˙ϵ(t),P˙ϵ(t), and using (4.43) to replace aϵ(y,t)a0(y)Ly2(d) with 𝔞ϵ(y,t) everywhere, we obtain

Q˙αϵ(t)=PαϵEn(Pϵ(t))+ϵ12PαϵPβPγϵEn(Pϵ(t))[𝔞ϵ(y,t)|(iyβ)(iyγ)𝔞ϵ(y,t)Ly2(d)𝔞ϵ(y,t)|(iyβ)𝔞ϵ(y,t)Ly2(d)𝔞ϵ(y,t)|(iyγ)𝔞ϵ(y,t)Ly2(d)]+QβϵW(Qϵ(t))An,αPβϵ(Pϵ(t))+O(ϵ3/2ect),P˙αϵ(t)=QαϵW(Qϵ(t))ϵ12QαϵQβϵQγϵW(Qϵ(t))[𝔞ϵ(y,t)|yβyγ𝔞ϵ(y,t)Ly2(d)𝔞ϵ(y,t)|yβ𝔞ϵ(y,t)Ly2(d)𝔞ϵ(y,t)|yγ𝔞ϵ(y,t)Ly2(d)]QαϵQβϵW(Qϵ(t))An,β(Pϵ(t))+O(ϵ3/2ect).
(4.44)

Note that, up to error terms, Equations (4.44) and (4.42) constitute a closed system for Qϵ(t),Pϵ(t),𝔞ϵ(y,t).

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

μϵ(t):=𝔞ϵ(y,t)|y𝔞ϵ(y,t)Ly2(d)λϵ(t):=𝔞ϵ(y,t)|(iy)𝔞ϵ(y,t)Ly2(d).
(4.45)

Then, we may write (4.44) as

Q˙αϵ(t)=PαϵEn(Pϵ(t))+ϵ12PαϵPβPγϵEn(Pϵ(t))[𝔞ϵ(y,t)|(iyβ)(iyγ)𝔞ϵ(y,t)Ly2(d)λβϵ(t)λγϵ(t)]+ϵQβϵW(Qϵ(t))PβϵAn,α(Pϵ(t))+O(ϵ3/2ect),P˙αϵ(t)=QαϵW(Qϵ(t))ϵ12QαϵQβϵQγϵW(Qϵ(t))[𝔞ϵ(y,t)|yβyγ𝔞ϵ(y,t)Ly2(d)μαϵ(t)μβϵ(t)]ϵQαϵQβϵW(Qϵ(t))An,β(Pϵ(t))+O(ϵ3/2ect).
(4.46)

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

μ˙αϵ(t)=\mathcalhPαϵ\mathcalhPβϵEn(\mathcalhPϵ(t))λβϵ(t),λ˙αϵ(t)=\mathcalhQαϵ\mathcalhQβϵW(\mathcalyQϵ(t))μβϵ(t).
(4.47)

Let

Hϵ(Qϵ,Pϵ,𝔞ϵ¯,𝔞ϵ,μϵ,λϵ):=En(Pϵ)+ϵW(Qϵ)+ϵQϵW(Qϵ)An(Pϵ)+ϵ12PαϵPβϵEn(Pϵ)[yαaϵ|yβaϵLy2(d)λαϵλβϵ]+ϵ12QαϵQβϵW(Qϵ)[yαaϵ|yβaϵLy2(d)μαϵμβϵ].
(4.48)

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

Q˙ϵ=HϵPϵ(Pϵ),P˙ϵ=HϵQϵ(Qϵ),it𝔞ϵ=δHϵδ𝔞ϵ¯,μ˙ϵ(t)=Hϵλϵ,λ˙ϵ(t)=Hϵμϵ.
(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 ϵ3/2C1ec1t,ϵ3/2C2ec2t,ϵ3/2C3ec3t for positive constants cj,Cj,j{1,2,3}. Define c:=maxj{1,2,3}cj,C:=maxj{1,2,3}Cj. Then all of these errors may be bounded by ϵ3/2Cect. It follows that these terms are o(ϵ) for all t[0,Cln1/ϵ] where C is any constant such that C<12c. Next, in  Appendix F (F13) and (F14) we show that a0(y)|ya0(y)Ly2(d)=a0(y)|(iy)a0(y)Ly2(d)=0 implies that for all t0a(y,t)|ya(y,t)Ly2(d)=a(y,t)|(iy)a(y,t)Ly2(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 λϵ,μϵ degrees of freedom in (4.48) and (4.49) since for all t0, λϵ(t)=μϵ(t)=0. In this way we obtain the simplified Hamiltonian system (1.48) and (1.49).

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.).

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

[H(p)E(p)]χ(z;p)=0,H(p)=12(piz)2+V(z)
(A1)

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

[(piz)pEn(p)]χn(z;p)+[H(p)En(p)]pχn(z;p)=0.
(A2)

Taking two derivatives with respect to p of the equation gives

[δαβpαpβEn(p)]χn(z;p)+[(piz)αpαEn(p)]pβχn(z;p)+[(piz)βpβEn(p)]pαχn(z;p)+[H(p)En(p)]pαpβχn(z;p)=0,
(A3)

where δαβ is the Kronecker delta. Taking the derivative with respect to pγ of (A3) gives

[pαpβpγEn(p)]χn(z;p)+[δαβpαpβEn(p)]pγχn(z;p)+[δαγpαpγEn(p)]pβχn(z;p)+[δβγpβpγEn(p)]pαχn(z;p)+[(piz)αpαEn(p)]pβpγχn(z;p)+[(piz)βpβEn(p)]pαpγχn(z;p)+[(piz)γpγEn(p)]pαpβχn(z;p)+[H(p)En(p)]pαpβpγχn(z;p)=0.
(A4)

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

[H(p(t))En(p(t))]f2(y,z,t)=ξ2(y,z,t)for allvΛ,f2(y,z+v,t)=f2(y,z,t);f2(y,z,t)ΣyR2(d)ξ2:=[12(iy)2+12qαqβW(q(t))yαyβit]f0(y,z,t)[((p(t)iz)pEn(p(t)))(iy)]f1(y,z,t).
(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.ξ2(y,z,t)defined in (B1) satisfies

ξ2(y,z,t)=ξ2(y,z,t)+[H(p(t))En(p(t))]u2(y,z,t),
(B2)

where

ξ2(y,z,t)=[ita0(y,t)12pαpβEn(p(t))(iyα)(iyβ)a0(y,t)12qαqβW(q(t))yαyβa0(y,t)qW(q(t))An(p(t))]χn(z;p(t))+Pn(p(t))[ia0(y,t)qW(q(t))pχn(z;p(t))]u2(y,z,t)=(iy)a1(y,t)pχn(z;p(t))+12(iyα)(iyβ)a0(y,t)pαpβχn(z;p(t)).
(B3)

Here, An(p(t)) is the Berry connection (1.26) andPn(p(t))is the orthogonal projection operator away from the subspace of Lper2 spanned byχn(z;p(t)) (3.20).

Imposing the solvability condition of Equation (B1), given by (3.19) with j = 2 and ξ2(y,z,t) given by (B3), gives the following evolution equation for a0(y, t):

ita0(y,t)=12pαpβEn(p(t))(iyα)(iyβ)a0(y,t)+12qαqβW(q(t))yαyβa0(y,t)+qW(q(t))An(p(t))a0(y,t).
(B4)

Taking a0(y,t)=a(y,t)eiϕB(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,

f2(y,z,t)=a2(y,t)χn(z;p(t))+(iy)a1(y,t)pχn(z;p(t))+12(iyα)(iyβ)a0(y,t)pαpβχn(z;p(t))[H(p(t))En(p(t))]1Pn(p(t))[iqW(q(t))a0(y,t)pχn(z;p(t))],
(B5)

where a2(y, t) is an arbitrary function in ΣyR2(d) to be fixed at higher order in the expansion.

Proof of Lemma B.1. Adding and subtracting terms, using smoothness of the band En(p) in a neighborhood of p(t) (Assumption 1.1), we can re-write ξ2 (B1) as

ξ2(y,z,t)=[12pαpβEn(p(t))(iyα)(iyβ)+12qαqβW(q(t))yαyβit]f0(y,z,t)[12(δαβpαpβEn(p(t)))(iyα)(iyβ)]f0(y,z,t)[((p(t)iz)pEn(p(t)))(iy)]f1(y,z,t).
(B6)

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

ξ2(y,z,t)=[12pαpβEn(p(t))(iyα)(iyβ)a0(y,t)+12qαqβW(q(t))yαyβa0(y,t)ita0(y,t)]χn(z;p(t))+ia0(y,t)p˙(t)pχn(z;p(t))(iyα)(iyβ)a0(y,t)12(δαβpαpβEn(p(t)))χn(z;p(t))(iyα)(iyβ)a0(y,t)((p(t)iz)αpαEn(p(t)))pβχn(z;p(t))(iy)a1(y,t)((p(t)iz)pEn(p(t)))χn(z;p(t)).
(B7)

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

(iy)a1(y,t)((p(t)iz)pEn(p(t)))χn(z;p(t))=(iy)a1(y,t)[H(p(t))En(p(t))]pχn(z;p(t)).
(B8)

Using (A3), and the symmetry: (iyα)(iyβ)a0(y,t)=(iyβ)(iyα)a0(y,t), we can simplify the terms,

(iyα)(iyβ)a0(y,t)12(δαβpαpβEn(p(t)))χn(z;p(t))(iyα)(iyβ)a0(y,t)((p(t)iz)αpαEn(p(t)))pβχn(z;p(t))=12(iyα)(iyβ)a0(y,t)[H(p(t))En(p(t))]pαpβχn(z;p(t)).
(B9)

Recall that p˙(t)=qW(q(t)) (1.19). Substituting this, (B8), and (B9) into (B1) and adding and subtracting a0(y,t)qW(q(t))An(p(t))χn(z;p(t)) gives

ξ2(y,z,t)=[ita0(y,t)12pαpβEn(p(t))(iyα)(iyβ)a0(y,t)12qαqβW(q(t))yαyβa0(y,t)qW(q(t))An(p(t))]χn(z;p(t))+Pn(p(t))[ia0(y,t)qW(q(t))pχn(z;p(t))]+[H(p(t))En(p(t))][12(iyα)(iyβ)a0(y,t)pαpβχn(z;p(t))+(iy)a1(y,t)pχn(z;p(t))],
(B10)

where An(p(t)) is the Berry connection (1.26) and Pn(p(t)) is the orthogonal projection in Lper2 away from the subspace spanned by χn(z;p(t)).

Collecting terms of order ϵ3/2 in the expansion (3.9), using Equations (1.21) for S˙(t) and (1.19) for q˙(t),p˙(t), and setting equal to zero gives the following inhomogeneous self-adjoint elliptic equation in z for f3(y, z, t),

[H(p(t))En(p(t))]f3(y,z,t)=ξ3(y,z,t),for allvΛ,f3(y,z+v,t)=f3(y,z,t),f3(y,z,t)ΣyR3(d),ξ3(y,z,t):=[16qαqβqγW(q(t))yαyβyγ]f0(y,z,t)[12(iy)2+12qαqβW(q(t))yαyβit]f1(y,z,t)[[(p(t)iz)pEn(p(t))](iy)]f2(y,z,t).
(C1)

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

Lemma C.1.ξ3(y,z,t), as defined in (C1), satisfies

ξ3(y,z,t)=ξ3(y,z,t)+[H(p(t))En(p(t))]u3(y,z,t),
(C2)

where ξ3 is given explicitly by (C24) and

u3(y,z,t):=(iy)a2(y,t)pχn(z;p(t))+12(iyα)(iyβ)a1(y,t)pαpβχn(z;p(t))+16(iyα)(iyβ)(iyγ)a0(y,t)pαpβpγχn(z;p(t)).
(C3)

Lemma C.2. The solvability condition for (C1), given by (3.19) with j = 3 andξ3(y,z,t)given by (C24), is equivalent to the following evolution equation for a1(y, t):

ita1(y,t)=12pαpβEn(p(t))(iyα)(iyβ)a1(y,t)+12qαqβW(q(t))yαyβa1(y,t)+qW(q(t))An(p(t))a1(y,t)+16pαpβpγEn(p(t))(iyα)(iyβ)(iyγ)a0(y,t)+16qαqβqγW(q(t))yαyβyγa0(y,t)+qβW(q(t))An,βpγ(p(t))(iyγ)a0(y,t)+qβqγW(q(t))An,β(p(t))yγa0(y,t).
(C4)

Here, An(p(t)) is the Berry connection (1.26).

Taking a1(y,t)=b(y,t)eiϕB(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),

f3(y,z,t)=a3(y,t)χn(z;p(t))+u3(y,z,t)+[H(p(t))En(p(t))]1Pn(p(t))ξ3(y,z,t),
(C5)

where ξ3(y,z,t) is given by (C24) and u3(y, z, t) by (C3). a3(y, t) is an arbitrary function in ΣyR3(d) to be fixed at higher order in the expansion. Note that all manipulations so far are valid as long as R3.

Proof of Lemma C.1. Adding and subtracting terms using smoothness of the band En(p) in a neighborhood of p(t) (Assumption 1.1), we can re-write ξ3(y,z,t) (C1) as

ξ3(y,z,t)=[16pαpβpγEn(p(t))(iyα)(iyβ)(iyγ)+16qαqβqγW(q(t))yαyβyγ]f0(y,z,t)[16pαpβpγEn(p(t))(iyα)(iyβ)(iyγ)]f0(y,z,t)[12pαpβEn(p(t))(iyα)(iyβ)+12qαqβW(q(t))yαyβit]f1(y,z,t)[12(δαβpαpβEn(p(t)))(iyα)(iyβ)]f1(y,z,t)[((p(t)iz)pEn(p(t)))(iy)]f2(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),

(iy)a2(y,t)((p(t)iz)pEn(p(t)))χn(z;p(t)),
(C7)

which can be simplified using (A2),

(iy)a2(y,t)((p(t)iz)pEn(p(t)))χn(z;p(t))=[H(p(t))En(p(t))][(iy)a2(y,t)pχn(z;p(t))].
(C8)
[12pαpβEn(p(t))(iyα)(iyβ)a1(y,t)+12qαqβW(q(t))yαyβa1(y,t)ita1(y,t)]χn(z;p(t))+ip˙(t)pχn(z;p(t))a1(y,t)(iyα)(iyβ)a1(y,t)12(δαβpαpβEn(p(t)))χn(z;p(t))(iyα)(iyβ)a1(y,t)((p(t)iz)αpEn(p(t))α)pβχn(z;p(t)).
(C9)

Note that these terms have an identical form to the terms depending on a0(y, t) in expression (B7) for ξ2(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

=[ita1(y,t)12pαpβEn(p(t))(iyα)(iyβ)a1(y,t)12qαqβW(q(t))yαyβa1(y,t)qW(q(t))An(p(t))]χn(z;p(t))+Pn(p(t))[ia1(y,t)qW(q(t))pχn(z;p(t))]+[H(p(t))En(p(t))][12(iyα)(iyβ)a1(y,t)pαpβχn(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

T1:=[16pαpβpγEn(p(t))(iyα)(iyβ)(iyγ)a0(y,t)+16qαqβqγW(q(t))yαyβyγa0(y,t)]χn(z;p(t)),
(C11)
T2:=(iyα)(iyβ)(iyγ)a0(y,t)[16pαpβpγEn(p(t))χn(z;p(t))](iyα)(iyβ)(iyγ)a0(y,t)[12(δαβpαpβEn(p(t)))pγχn(z;p(t))](iyα)(iyβ)(iyγ)a0(y,t)[12((p(t)iz)αpαEn(p(t)))pβpγχn(z;p(t))],
(C12)
T3:=[12pαpβEn(p(t))(iyα)(iyβ)+12qαqβW(q(t))yαyβit](iyγ)a0(y,t)pγχn(z;p(t)),
(C13)
T4:=i((p(t)iz)βpβEn(p(t)))qγW(q(t))(iyβ)a0(y,t)[H(p(t))En(p(t))]1Pn(p(t))pγχn(z;p(t)).
(C14)

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

T2=[H(p(t))En(p(t))][16(iyα)(iyβ)(iyγ)a0(y,t)pαpβpγχn(z;p(t))].
(C15)

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

T3=[12pαpβEn(p(t))(iyα)(iyβ)+12qαqβW(q(t))yαyβ](iyγ)a0(y,t)pγχn(z;p(t))+(iyγ)[ita0(y,t)]pγχn(z;p(t))+(iyγ)a0(y,t)ip˙β(t)pβpγχn(z;p(t))=12qαqβW(q(t))yαyβ(iyγ)a0(y,t)pγχn(z;p(t))+12qαqβW(q(t))(iyγ)yαyβa0(y,t)×pγχn(z;p(t))+qβW(q(t))An,β(p(t))(iyγ)a0(y,t)pγχn(z;p(t))iqβW(q(t))(iyγ)a0(y,t)×pβpγχn(z;p(t)).
(C16)

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

T3,1:=12qαqβW(q(t))yαyβ(iyγ)a0(y,t)pγχn(z;p(t))+12qαqβW(q(t))(iyγ)yαyβa0(y,t)pγχn(z;p(t)),
(C17)
T3,2:=qβW(q(t))An,β(p(t))(iyγ)a0(y,t)pγχn(z;p(t))iqβW(q(t))×(iyγ)a0(y,t)pβpγχn(z;p(t)).
(C18)

We can simplify T3,1 as follows. We first re-arrange (C17),

T3,1=((iyγ)yαyβyαyβ(iyγ))12qαqβW(q(t))a0(y,t)pγχn(z;p(t)).
(C19)

Using the identity (iyα)yβyβ(iyα)=iδαβ twice, we have that

(iyγ)yαyβyαyβ(iyγ)=(iδαγ)yβ+(iδβγ)yα.
(C20)