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 , we prove the existence of solutions known as semiclassical wavepackets which are asymptotic up to “Ehrenfest time” . 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 ,
where is a positive real parameter which we assume to be small: . We assume throughout that the function V is smooth and periodic with respect to a d-dimensional lattice Λ so that
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 , the mean position and momentum of the semi-classical wavepacket evolve along the classical trajectories associated with the “Bloch band” Hamiltonian , where is the dispersion relation associated with the nth spectral (Bloch) band of the periodic Schrödinger operator . The order 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 , mean momentum , and wave-amplitude profile is a closed system of Hamiltonian type (Theorem 1.2). When d = 3, this system takes the form
where denote the Hessian matrices and is the Berry connection (1.26). For the explicit forms of C1[a], C2[a] and the generalization of (1.3) to arbitrary dimensions , 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” , 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 (see Assumption 1.1). We do not attempt a description of wavepacket dynamics when this distance (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 , where U is periodic in its first argument,
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 () 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,
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 denote the quantum time scale,
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,”
After re-scaling x, t by the macroscopic length and time scales,
We now identify two dimensionless parameters. Let h denote a scaled Planck’s constant, and the ratio of the lattice constant to the macroscopic scale,
Writing (1.9) in terms of and dropping the tildes we arrive at
where we have written to emphasize the dependence of the solution on both parameters. We obtain the problem depending only on (1.1) by setting . Therefore, the limit 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 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,
where V is periodic with respect to a d-dimensional lattice Λ.29,30 Let denote the dual lattice to and define the first Brillouin zone to be a fundamental period cell. Consider the family of self-adjoint eigenvalue problems parameterized by ,
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,
For fixed p, the associated normalized eigenfunctions are a basis of the space,
Varying p over the Brillouin zone, the maps 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 is called the band structure of H (1.12). Any function in can be written as a superposition of Bloch waves,
The map extends to a map on which is periodic with respect to the reciprocal lattice ,
If the eigenvalue En(p) is simple, then (up to a constant phase shift) . A more detailed account of the Floquet-Bloch theory which we require, in particular results on the regularity of the maps , 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 are such that the flow generated by the classical Hamiltonian ,
has a unique smooth solution , and that there exists a constant such that
That is, the nth spectral band is uniformly isolated along the classical trajectory (q(t), p(t)).
Assumption 1.2. .
Remark 1.2. An example of W satisfying Assumption 1.2 is the “Stark” potential for any constant vector . 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 , where M is any real positive definite 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 such that for all .
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 . Let S(t) denote the classical action along the path (q(t),p(t)),
Let a(y, t) satisfy
And let b(y, t) satisfy
where is as in (1.23) and
where denotes the nth band Berry connection,
where denotes a fundamental period cell of the lattice . Let be the Berry phase associated with transport of along the path given by
Then, there exists a constant such that for all the following holds. Let be the unique solution of the initial value problem (1.1) with “Bloch wavepacket” initial data,
Then, for all the solution evolves as a modulated “Bloch wavepacket” plus a corrector ,
where the corrector satisfies the following estimate:
where are constants independent of . It follows that
where is any constant satisfying .
Remark 1.4. We include the pre-factor throughout so that the norm of is of order 1 as .
Remark 1.5. We have improved the error bound of Carles and Sparber66 from to by including correction terms in the asymptotic expansion proportional to . Note that we must also assume that the initial data are well-prepared up to terms proportional to (1.28). By keeping more terms in the expansion we may produce approximations where the corrector can be bounded by for any positive integer N. The only changes in the proof are that we include corrections to the initial data proportional to , and that Assumption 1.2 is replaced by .
Remark 1.6. Keeping terms proportional to in the expansion will allow us to calculate corrections to the dynamics of physical observables proportional to ; see Theorem 1.2 and Section IV.
Remark 1.7. The time scale of validity of the approximation (1.29), , 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 ) 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 ; see Refs. 5 and 31 for further discussion.
where is an arbitrary non-zero constant, and A0, B0 are complex matrices satisfying
Remark 1.8. The conditions (1.33) imply that
the matrices B0, A0 are invertible,
the matrix is complex symmetric: ,
the imaginary part of the matrix is symmetric, positive definite, and satisfies ,
and are equivalent to the condition that the matrix
Note that it follows from assertion (3) of Remark 1.8 that a0(y) (1.32) satisfies for constants . We have then the following.
Here, the complex matrices A(t), B(t) satisfy
Moreover, for all, 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), wherefor all . More generally, we may construct a basis of of solutions of the envelope equation (1.22), consisting of products of Gaussians with polynomials, known as the “semiclassical wavepacket” basis.11,32,33
D. Dynamics of observables associated with the asymptotic solution
where is the corrector which satisfies the bound (1.30). Define the physical observables
where is the normalization factor,
We will refer to as the center of mass and average quasi-momentum of the wavepacket. We will see (Theorem 1.2) that up to “Ehrenfest time” .
Remark 1.10. In the uniform background case V = 0, solutions of (1.13) are independent of z: for all . The asymptotic solution (1.37) obtained in this case is therefore independent of z, and our definition of reduces to the usually defined momentum observable,
In the periodic background case , (1.39) corresponds to the quasi-momentum and may be measured in experiments.3434Let satisfy the equation of a quantum harmonic oscillator, with parametric forcing defined by ,
Note that we have replaced dependence on (q(t),p(t)) in equation (1.22) with dependence on .
For simplicity of presentation of the following theorem we assume that
The result holds for general ; see Section IV.
Theorem 1.2. Let denote the asymptotic solution (1.37) including corrections proportional to . Let denote the observables (1.39). Then, there exists an such that for all , and for all where is a constant independent of .
Let denote the solution of (1.42) with co-efficients evaluated at rather than .
Then, after dropping terms of , satisfy a closed, coupled “particle-field” system which is expressible as an -dependent Hamiltonian system,
Remark 1.11. In three spatial dimensions (d = 3), the anomalous velocity may be re-written using the cross product
where and 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 connectionis 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 . 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).
Remark 1.13. In the special case where the periodic background potential V = 0 the Bloch band dispersion function reduces to the “free” dispersion relation and the Hamiltonian (1.54) takes on the simple form
The system (1.53), with Hamiltonian 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 proportional todue to “field-particle” coupling to thesystem can lead to qualitatively different dynamical behavior. In particular, the coupling may destabilize periodic orbits of the unperturbed () 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” , and that under further assumptions on the classical dynamics we expect that this time scale may be extended up to . Note that the Berry curvature terms and the new “field-particle” coupling terms occur at the same order in . 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 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 , where U is periodic in its first argument,
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,
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 expressionit is understood that we are summing over .(1.58)
- Where there is no danger of confusion, we will use the standard conventions(1.59)
denotes the d-dimensional Laplacian.
denotes the d-dimensional Hessian matrix with respect to x.
- We will adopt multi-index notation where appropriate so thatmeans all derivatives of order l of f(x) are uniformly bounded.(1.60)
- It will be useful to introduce the energy spaces for every ,(1.61)
- The space of Schwartz functions is the space of functions defined as(1.62)
- We will refer throughout to the space of L2-integrable functions which are periodic on the lattice ,(1.63)
We will write a fundamental period cell in of the lattice as .
- We will make use of the Sobolev norms on a fundamental period cell for integers ,(1.64)
- It will also be useful to introduce the “shifted” Sobolev norms, for arbitrary ,(1.65)
- Define the dual lattice to ,(1.66)
We will refer to a fundamental cell in of the dual lattice as the Brillouin zone, or
- We make the standard convention for the L2-inner product,(1.67)
- We will make the conventions,(1.68)
- Let A be a complex matrix. Then we will write AT for its transpose, for its complex conjugate, and for its trace. Using index notation,(1.69)
- The Kronecker delta is defined,(1.70)
- In dimension d = 3, the Levi-Civita symbol is defined,and satisfies the identities(1.71)The cross product of 3-vectors may then be written as(1.72)(1.73)
II. SUMMARY OF RELEVANT FLOQUET-BLOCH THEORY
In this section we recall the spectral theory of the operator,
where V is periodic with respect to the lattice .29,30 For , define the spaces of p-pseudo-periodic L2 functions as follows:
Let denote the lattice dual to ,
and since the p–pseudo-periodic boundary condition is invariant under where , the dual lattice to , it is natural to restrict to a fundamental cell, .
We now consider the family of eigenvalue problems depending on the parameter ,
We can also define the space functions which are periodic with respect to the lattice,
Then solving the eigenvalue problem (2.4) is equivalent via to solving the family of eigenvalue problems,
For fixed p, the operator H(p) with periodic boundary conditions is self-adjoint and has compact resolvent. So, for each , there exists an eigenpair . The eigenvalues are real and can be ordered with multiplicity,
and the set of normalized eigenfunctions is complete in . The set of Floquet-Bloch waves are complete in ,
where the sum converges in L2. The 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 ,
Our results require sufficient regularity of the maps,
Definition 2.1. We will call an eigenvalue band En(p) of the problem (2.6) isolated at a point if
We have in this case the following.
Theorem 2.1. (Smoothness of isolated bands). Let 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 be eigenvalue bands of the eigenvalue problem (2.6). If is such that
we will say that the bands En(p) and Em(p) have a band crossing at .
In a neighborhood of a crossing, the band functions En(p), Em(p) are only Lipschitz continuous, and the eigenfunction maps 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 to maps on all of . Let , and let denote a reciprocal lattice vector. Then we have that
III. PROOF OF THEOREM 1.1 BY MULTISCALE ANALYSIS
A. Derivation of asymptotic solution (1.29) via multiscale expansion
The idea behind the proof of Theorem 1.1 is to choose the functions S(t), q(t), p(t), and so that
We will derive 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 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 has the explicit form
Since W is assumed smooth, we can replace by its Taylor series expansion in ,
We expand as a formal power series,
and assume that for all the fj(y, z, t) are periodic with respect to the lattice in z and have sufficient smoothness and decay in y,
The -spaces are defined in (1.61). is a fixed positive integer which we will take as large as required. Recall the notation
In order to prove Theorem 1.1 it will be sufficient to choose the so that terms of orders vanish. With this choice of we will then prove rigorously in Section III B that can be bounded by for constants independent of . There will then be no loss of accuracy in the approximation by taking .
1. Analysis of leading order terms
Recall that we assume each to be periodic with respect to the lattice 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:
Under Assumption 1.1, En(p(t)) is a simple eigenvalue with eigenfunction for all . Projecting equation (3.10) onto the subspace of
spanned by , implies
which has the general solution
where a0(y, t) is an arbitrary function in , to be fixed at higher order in the expansion.
2. Analysis of order terms
Before solving (3.15) we remark on our general strategy for solving equations of this type.
Remark 3.1. Collecting terms of orders for each and setting equal to zero, we obtain inhomogeneous self-adjoint elliptic equations of the form
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 eigenfunctionfor all. By the Fredholm alternative, equation (3.16) is solvable if and only if
We will first use identities derived in Appendix Afrom the eigenvalue equation to writeas a sum,
Note that by self-adjointness of H(p(t)) − En(p(t)), condition (3.17) is equivalent to the same condition withreplaced by,
Note that we have again made use of Assumption 1.1 to ensure that the operatoris bounded for all. When j=1, condition (3.19) may be enforced by choosing to satisfy (1.19). For, 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. , defined in (3.15), satisfies
which we can satisfy by choosing (q(t), p(t)) to evolve as the Hamiltonian flow of the nth Bloch band Hamiltonian ,
where a1(y, t) is an arbitrary function in to be fixed at higher order in the expansion. Note that since , this ensures that 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 , is equal to
3. Analysis of order and terms (summary)
It is possible to continue the procedure outlined in Remark 3.1 to any order in . 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 and . In particular, we derive the evolution equations of the amplitudes a0(y, t), a1(y, t) and show that
B. Proof of estimate (1.30) for the corrector
where is given by
And is given by
Since the are periodic with respect to the lattice , 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,
where we have used the fact that
where (3.14). By Assumption 1.2, ,
Recall Assumption 1.1. Define
so that for all . For each fixed , by elliptic regularity, is smooth in z so that . Using compactness of the Brillouin zone and smoothness of for we have that
Recall that for any reciprocal lattice vector , . It then follows that
We have therefore that (3.38)
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,
Following the work of Carles and Sparber6 we first define, for any , the spaces
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 , and be real-valued, symmetric, continuous, and uniformly bounded in t. Then the equation
has a unique solution . It satisfies
Moreover, if , then .
We seek quantitative bounds on for . For simplicity, we consider in detail the case l = 1. Recall (1.23),
and let so that , the solution of (3.46), . Then solves
We can solve this equation using Duhamel’s formula and the solution operator of Equation (3.46). It follows that
Since is symmetric, the commutator is given explicitly by
By an identical reasoning, we can derive a similar bound on ,
using the following version of Gronwall’s inequality.
Lemma 3.3. (Gronwall’s inequality). Let (t) satisfy the inequality,
where b(t) is non-negative and a(t) is non-decreasing. Then,
We have that
More generally, we have for any that there exists a constant such that
We have proved the following.
Lemma 3.4. (Bound on solutions of (3.46) in the spaces ). Let the time-dependent co-efficients be real-valued, symmetric, continuous, and uniformly bounded in t. Let . Then, by Lemma 3.2, there exists a unique solution . For each integer , there exists a constant such that this solution satisfies
Since the map is -periodic and smooth for all , we have that under Assumption 1.1, . Under Assumption 1.2 we have that . Since are clearly real-valued, symmetric, and continuous in t, we have that Lemma 3.4 applies to solutions of (3.44). Since by assumption we have that for any integer ,
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
are constants independent of .
We conclude that there exist constants , independent of such that
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 be the unique solution of
where H is a self-adjoint operator. Then,
and when f = 0, we have
Applying Lemma 3.5 to Equation (3.32) then gives the bound on ,
where C is a constant independent of . This completes the proof of Theorem 1.1.
IV. PROOF OF THEOREM 1.2 ON DYNAMICS OF PHYSICAL OBSERVABLES
Let be the solution of (1.28). By Theorem 1.1 we have that this solution has the form
In this section we compute the dynamics of the physical observables,
Remark 4.1. Throughout this section we will employ a short-hand notation,
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 , g smooth and periodic with respect to the lattice , a constant, and an arbitrary positive parameter. Then for any positive integer ,
For the proof, see Appendix E.
A. Asymptotic expansion and dynamics of
By changing variables in the integral (4.4), we have that
We expand the product in the integral and apply Lemma 4.1 term by term with . Since the are assumed normalized, for all , we have
Remark 4.2. In (4.9) we have made explicit all terms through order . To justify the error bound, consider that the remaining terms may be bounded by where depends on -norms of a(y, t), b(y, t) and -norms of where l1, l2 are positive integers. By an identical reasoning to that given in Section III B we have that may be bounded by Cect where are constants independent of . 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 , the mapping is smooth. Hence, we may differentiate the normalization condition: with respect to p and evaluate along the curve p(t) to obtain the identity,
It follows from this, and the fact that is symmetric with respect to the -inner product, that
so that (4.9) reduces to
Integrating in time then gives
B. Asymptotic expansion of , proof of assertion (1) of theorem 1.2
Changing variables in the integrals (4.3) and using the identity,
Expanding all products and applying Lemma 4.1 term by term in (4.18), we obtain
where is the commutator. Since , we have that
where the last equality holds by the definition of the nth band Berry connection (1.26). We have proved that
C. Computation of dynamics of , proof of assertion (2) of Theorem 1.2
Differentiating both sides of (4.23) with respect to time and using (4.14) gives
Equation (4.23) gives expressions for q(t), p(t) in terms of ,
where 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 (1.42) as the solution of (1.22) with co-efficients evaluated at ,
Recall the definition of the norms (3.45). If we can show that for each positive integer l, then we may replace a(y,t) by everywhere in (4.29) and, after dropping error terms, we will have obtained a closed system for . Let
Using the fact that is self-adjoint on for each t, it follows from (4.32) that
By the Cauchy-Schwarz inequality,
it then follows that
Using the precise forms of , we have
where is the norm defined in (3.45). Recall that (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 , and all q along the line segments connecting q(t) and . We may therefore conclude from the mean - value theorem that there exist constants independent of such that
We now use the a priori bounds on the -norms of a(y, t) for each (Lemma 3.4) to see that
for some constants independent of . By a similar argument, we see that for any integer there exist a constants such that
It then follows that we may replace a(y, t) by everywhere in (4.29), generating further errors which are to derive
D. Hamiltonian structure of dynamics of , proof of assertion (3) of Theorem 1.2
Let denote the solution of (1.42) with co-efficients evaluated at rather than , with initial data normalized in ,
Since , , by a similar argument to that given in Sec. IV C we have that for each integer ,
We now show that this system may be derived from a Hamiltonian. Let
Then, we may write (4.44) as
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 for positive constants . Define . Then all of these errors may be bounded by . It follows that these terms are for all where is any constant such that . Next, in Appendix F (F13) and (F14) we show that implies that for all . 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 degrees of freedom in (4.48) and (4.49) since for all , . 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.).
APPENDIX A: USEFUL IDENTITIES INVOLVING
Let satisfy the eigenvalue problem,
and assume that are smooth functions of p. Taking the gradient with respect to p gives
Taking two derivatives with respect to p of the equation gives
where is the Kronecker delta. Taking the derivative with respect to of (A3) gives
APPENDIX B: DERIVATION OF LEADING-ORDER ENVELOPE EQUATION
Collecting terms of order in the expansion (3.9), using Equations (1.21) for and (1.19) for , and setting equal to zero gives the following inhomogeneous self-adjoint elliptic equation in z for f2(y, z, t),
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. defined in (B1) satisfies
where a2(y, t) is an arbitrary function in 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 (B1) as
Using (A2) we can simplify the term involving a1,
Using (A3), and the symmetry: , we can simplify the terms,
where is the Berry connection (1.26) and is the orthogonal projection in away from the subspace spanned by .
APPENDIX C: DERIVATION OF FIRST-ORDER ENVELOPE EQUATION
Collecting terms of order in the expansion (3.9), using Equations (1.21) for and (1.19) for , and setting equal to zero gives the following inhomogeneous self-adjoint elliptic equation in z for f3(y, z, t),
We claim the following lemmas, the proofs of which will be given at the end of this section.
Lemma C.1. , as defined in (C1), satisfies
where is given explicitly by (C24) and
Here, is the Berry connection (1.26).
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 (C1) as
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),
which can be simplified using (A2),
Note that these terms have an identical form to the terms depending on a0(y, t) in expression (B7) for 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
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
Using (A4) and the equality of mixed partial derivatives, we can simplify T2,
We can simplify T3 using the evolution equation for a0(y, t) (B4),
We now write T3 = T3,1 + T3,2, where
We can simplify T3,1 as follows. We first re-arrange (C17),
Using the identity twice, we have that
Using the symmetry