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\hbox{}
\vskip 1truein\centerline{{\bf ADIABATIC CURVATURE AND THE $S$-MATRIX}}
\vs.2
\centerline{by}
\vs.2
\centerline{Lorenzo Sadun${}^1$ and Joseph E. Avron${}^2$}
\footnote{}{1\ Department of Mathematics, University of Texas, Austin, TX 78712, USA. Email: sadun@math.utexas.edu.
Research supported in part by an NSF Mathematical Sciences
Postdoctoral Fellowship and Texas ARP Grant 003658-037 \hfil}
\footnote{}{2\ Department of Physics, Israel Institute of Technology,
Haifa, Israel. \hfill \break
Email: avron@physics.technion.ac.il.
Research supported in part by GIF, DFG and the
Fund for Promotion of Research at the Technion. \hfil}
\vs.5 \centerline{{\bf Abstract}}
\vs.1 \nd
We study the relation of the adiabatic
curvature associated to scattering states
and the scattering matrix. We show that there cannot be any formula
relating the two locally. However, the first Chern number,
which is proportional to the integral of the curvature,
{\it can} be computed by
integrating a 3-form constructed from the $S$-matrix.
Similar formulas relate higher Chern classes to integrals of higher degree
forms constructed from scattering data.
We show that level crossings of the
on-shell $S$-matrix can be assigned an index so that the first
Chern number of the scattering states is the sum of the indices. We
construct an example which
is the natural scattering analog of Berry's spin 1/2
Hamiltonian.
\vs1
\centerline{December 1995}
\centerline{AMS 1995 Subject Classification:\ \ 81U20, 58Z05, also
14L30, 34L40, 81Q30}
\centerline{1994 PACS numbers:\ \ 03.65.Bz, also 03.80+r, 02.40Re}
\vfill\eject
\nd {\bf I. Motivation}
In the quantum Hall effect, the conductance is related to a
Chern number --- the integral of the adiabatic curvature
[\tknn,\berry]. In many systems, conductance admits a description in terms of
scattering data via Landauer type formulas [\landauer]. This seems to suggest
that there may be a direct and general link between adiabatic curvature and/or
Chern numbers, on the one hand, and scattering data on the other. The study of
this relation is the central theme of the work presented here. As we shall
see, there is no such (pointwise) link between scattering data and adiabatic
curvature, but there {\it is} such a link between scattering data and Chern
numbers.
We shall consider local deformations of quantum Hamiltonians that
are associated with a scattering situation and have a band of absolutely
continuous spectrum. We study the adiabatic curvature associated with
this band. We shall not consider deformations
that ``act at infinity". This limits the applications of our
results to the usual transport theory since transport of
charge to infinity typically involves deformations at infinity.
For potential scattering in one dimension the scattering matrix alone does not
determine the scattering potential; one needs to know certain norming
constants associated with bound states [\faddeev]. Because of this,
it may not be too surprising that scattering can sometime
fail to know about the
adiabatic curvature due to local deformations. From this point of view it is
perhaps more remarkable that the $S$-matrix nevertheless determines the Chern
numbers.
There is also a mathematical motivation for studying the geometry and topology
of vector bundles from the perspective of scattering theory. The classical
studies of vector bundles are concerned with finite dimensional fibers.
Scattering situations give rise to bundles with infinite dimensional fibers
which arise from the consideration of the scattering states that lie in a band
of energies. The geometry comes about by studying how these infinite
dimensional subspaces of a fixed Hilbert space rotate. Our results can be
phrased as stating that the scattering data determine the topology of such
bundles, but not their local curvature.
We shall see that for a class of tight-binding models the first Chern class of
the scattering bundle is determined by an explicit 3-form constructed from
scattering data:
$$
s_3(y) = {1\over 4\pi^2}\ \sum_\a d\theta_\a
(y)\wedge \Omega(P_\a^S). \eqno(\3form)$$
%
Here $P_\a^S$ is the $\a$-th
spectral projection of the on-shell scattering matrix $S(y)$, and
$\theta_\a(y)$ is half the corresponding phase shift, i.e.,
$$S(y)P_\a^S(y)= \exp i\theta_\a (y) P_\a^S(y). \eqno(1.2)$$
$y_0 =k$ parameterizes the energy and $\{y_j\}, j=1,\dots,\ell$ are additional
parameters that parameterize the deformation of the Hamiltonian. $\Omega (P)$
is the trace of the usual adiabatic curvature associated to
the projection $P$; see
Eq.~(\curvature) below. If $S(y)$ is a non-degenerate $n\times n$ matrix,
then $\alpha =1,\dots, n$ and $P_\a^S$ is a rank-1 orthogonal projection in $\complex^n$.
Integrating $s_3(y)$ over the band of scattering energies parameterized by
$y_0$ gives a closed 2-form on the space of parameters governing the
deformation of the system. We shall see that the cohomology class of this
2-form is the first Chern class of the infinite-dimensional vector bundle in
question.
In general, only unitary invariant properties of the on-shell $S$-matrix have
physical significance [\birman]. The phase shifts are, of course, unitary
invariant. $\Omega (P_\a^S)$ is not. Rather, under the
transformation $S \to U^\dagger S U$, $P_\a^S \to U^\dagger P_\a^S U$ and
$$ \Omega (P_\a^S) \to \Omega (P_\a^S) +d\ Tr(P_\a^S U^\dagger dU). \
\eqno(\unit1)
$$
It follows that the 3-form $s_3(y)$ is not
invariant under $y$-dependent unitary transformations,
but its cohomology class {\it is} invariant.
First Chern numbers,
which are periods of this three form, are
invariant, as they should be.
A similar construction gives the higher Chern classes of the infinite
dimensional bundle associated to $P$. Let $F(P)$ be the adiabatic
curvature operator whose trace is $\Omega(P)$; see Eq.~(2.1). The
$2n$-form ${1 \over n! (2\pi)^n}Tr(F(P)^n)$ is closely related to the $n$-th
Chern class of the bundle of scattering states defined by $P$. ($P$ is infinite
dimensional). This form is cohomologous to the integral over $y_0$ of the
$2n+1$
form $$ s_{2n+1}(y) = {1\over n! (2\pi)^{n+1}}\ \sum_\a d\theta_\a (y)\wedge
Tr(F(P_\a^S)^n), \eqno(1.4) $$ which is computed from scattering data (and
where
$P_\a^S$ are finite dimensional).
In section II we review the theory of adiabatic curvature for finite
dimensional projections and its extension to infinite dimensions.
In section III we give a family of examples that show that the adiabatic
curvature cannot, in general, be computed pointwise from the $S$-matrix.
In these examples there is a nonzero curvature associated to two parameters,
but the $S$-matrix is independent of one of the parameters.
In section IV we state precise hypotheses under which the 3-form
$s_3(y)$ computes first Chern numbers. We also show how these Chern numbers
are related to numerical indices associated to level crossings of the
$S$-matrix. In Section V we work a key example, the natural scattering
analog of Berry's spin Hamiltonian. Section VI is the proof of the
main theorem, as stated in Section IV. In Section VII we consider
some exceptional cases and generalize our results to cover higher
Chern classes. Finally, we include an appendix
reviewing scattering in tight binding models.
\vfill\eject
\nd {\bf II. Preliminaries--Geometry of Projections}
Let $X$ be a space of parameters with local coordinates $y=(y_1,\dots,y_\ell)$.
Let $P(y)$ be a family of orthogonal finite dimensional projections that
depends smoothly on $y\in X$.
$Range(P)$ is a vector bundle over $X$ with a natural connection.
We are interested in the trace of the resulting curvature:
$$\eqalign{\Omega(P)&= -i\ Tr \ (P dP \wedge dP P)\cr
&=-i\ \sum_{1\le i0$ and $P$ finite dimensional, the adiabatic
curvature is a positive, increasing function of $P$. If $P$ has finite
codimension, then
Eq.~(\flat ) says that the curvature is a negative increasing function of $P$.
This is peculiar. Finite codimensional projections are clearly ``larger''
than finite dimensional projections,
and the adiabatic curvature increases with
$P$, so how can the curvature be positive for finite dimensional projections
and negative for finite codimensional projections?
Perhaps a useful analogy, where something similar happens, is negative
temperatures in canonical ensembles; energy is an increasing function of
temperature, but ensembles with negative temperature have more energy than
those with positive temperature.
An important fact about curvatures is that their integrals over closed regions
of parameter space are quantized. For finite dimensional bundles this is
a standard result of Chern-Weil theory. See e.g. [\MS, \DFN] for the theory of
Chern classes for finite dimensional bundles and [\segal, \freed] for its
extension to infinite dimensional bundles.
For infinite dimensional bundles, the existence of Chern classes
depends on the
structure group. The group $U(H)$ of unitary operators on the infinite
dimensional Hilbert space $H$ is too big. One must reduce
the structure group to a small enough subgroup of $U(H)$, where Chern classes
are defined and can be expressed by curvature formulas.
Let $U_c(H)$ be the space of unitary operators $U$ with $U-I$ compact.
For each integer $p$ let $U_p(H)$ be the subspace of $U_c(H)$
such that $U-I$ is in $L^p$. In particular,
$p=1$ means trace class and $p=2$ means Hilbert-Schmidt.
As long as the structure group can
be reduced to $U_c(H)$, Chern classes are well defined topologically.
Freed [\freed] showed that, when the structure group is $U_p$,
the Chern-Weil formulas for $c_i$ in terms of curvature hold for all
$i \ge p$.
The situation where $dP$ is Hilbert-Schmidt is intermediate between
$U_1$ and $U_2$.
For any path $\gamma$, the operator $U^\gamma$ that gives parallel
transport along the path may be obtained by integrating the equation
$ dU = [dP, P] U$ along the path, with initial condition $U=1$. Since
the right hand side is Hilbert-Schmidt, every path
$\gamma$ has $U^\gamma-I$ Hilbert-Schmidt, so our structure group reduces to
$U_2$. However, we have more. The curvature is trace class
and holonomies along closed
null-homotopic loops are actually in $U_1$. Pressley and Segal [\segal]
showed how to construct the determinant bundle of $P$, from which one
can show that the first Chern class
is represented by $\Omega/2\pi$. In short, we have the following
\nd {\bf Proposition 2}
Let $P$ be a family of orthogonal projections such that $dP$ is
Hilbert-Schmidt, and let $\Sigma$ be a (smooth) closed 2-surface in parameter
space. Then
%
$$c_1(\Sigma,P)= {1\over 2\pi}
\int_\Sigma\Omega(P) \eqno(\chern )$$
%
is an integer.
\vfill\eject
{\bf \nd III. Curvature Is Not Computable from Scattering Data}
Here we construct examples that show that curvature is not
necessarily detected by the $S$-matrix.
In these examples the curvature associated to two parameters may be
nonzero, but the $S$-matrix is independent of one of the parameters.
Let $V$ be any reasonable perturbation of the Laplacian in
one dimension so that there is a good scattering theory
and one or more bound states. For example, let $V$
be a short range potential on the line. Consider the family of Hamiltonians
%
$$H(a,b)= U(a,b)\Big(-{d^2\over dx^2} +V\Big)U^\dagger(a,b)
=\left(-i{d\over dx} - b\Lambda' (x-a)\right)^2 +V(x-a), \eqno(3.1) $$
%
where $U$ is as in Eq.~(\weyl ). Let $y=(k,a,b)$ and let
$\psi_y$ be a solution of the
differential equation $\Big(H(a,b)-k^2\Big)\psi_y=0$.
Since $\psi_y=U(a,b) \psi_{k,0,0}$ we have, in the limit
$|x|\to\infty$,
$$\psi_y(x)=e^{ib\Lambda(\pm\infty)}\psi_{k,0,0}(x-a). \eqno(3.2) $$
>From this and the definition of the on-shell $S$-matrix (see
appendix), we see that
$$S(k,a,b)=
\left(\matrix{r_R(k)\ e^{2ika}&t_L(k)\ e^{ib\Delta\Lambda}\cr
t_R(k)\ e^{-ib\Delta\Lambda}&r_L(k)\ e^{-2ika}\cr}\right), \eqno(3.3)
$$
where $\Delta\Lambda = \Lambda (\infty)- \Lambda (-\infty)$. In particular,
if $\Lambda(\infty )=\Lambda(-\infty )$, the $S$-matrix is independent of $b$.
Since curvature is a property of pairs of variables and only one parameter
affects $S$, $S$ cannot see any curvature.
Now let $Q$ be
the projection on the (finite dimensional)
subspace of bound states of $H$, and suppose that
$Tr Q(-\Delta+\Lambda ^2) Q <\infty$. Let $P=1-Q$. $P$ is the projection on
the (positive energy) scattering states. From proposition 1 and example 1 we
have that
%
$$\Omega(P) = -\Omega(Q) = - da \wedge db \ Tr \ Q \Lambda ' Q .
\eqno(3.4) $$
%
Since $\Lambda$ can be chosen independently of $V$, and hence of $Q$, we
can easily arrange for $Tr \ Q \Lambda ' Q $ to be nonzero. For example,
we can take
%
$$
\Lambda '(x) = \cases{1 & $|x|1$ which
becomes a threshold state when
$v=\pm 1$. At the edge of the continuous spectrum ($z=\pm 1$),
$S(k,v)=-1$ for all $v\neq \pm 1$.
{\nd \bf Theorem 1:} {\it Assume the above hypotheses. The integral of the
3-form $s_3(y)$ over the spectral interval $I=(0,\pi)$ is a closed
2-form on $X$. The cohomology class of this 2-form is the first Chern
class of the bundle of scattering states in the spectral interval $I$.
In particular, if $\Sigma$ is an oriented
surface in $X$, then the first Chern number of the bundle defined by
$P$ over $\Sigma$ is }
%
$$
c_1(\Sigma,P ) = \int_{I\times \Sigma} s_3(y). \eqno(\main )$$
%
{\nd \bf Remark 4:}
The tight-binding assumption is made to make the
entire continuous spectrum be a single band, and the proof is tailored
to this case. It should be straightforward to prove similar results for
compact perturbations of periodic potentials (either on the continuum
or on a lattice) subject to similar conditions on the $S$-matrix.
In addition, the formula Eq.~(\main) is the
simplest in a class of formulas for
the Chern number. More complicated formulas hold if Eq.~(\limit) is replaced
by other limiting values for the $S$-matrix that occur when threshold states
exist at $y_0=0$ or $y_0=\pi$. These formulas are discussed in section VII.
{\nd \bf Remark 5:} If the system is made of disconnected pieces, it
is possible to arrange for embedded eigenvalues, associated with a
compact part of the system, to lie in the energy interval associated
to the scattering states. These embedded eigenstates can carry Chern numbers,
but are irrelevant to the problem we study here. The caveat ``bundle of scattering states" in the theorem indicates that we are not including
these eigenstates.
{\nd \bf Remark 6:} Since $|y|^{-2}$ is integrable in three
dimensions, the integral in Eq.~(\main) is absolutely convergent.
Let $Y= I \times X$. The eigenvalues and eigenvectors of the on-shell
$S$-matrix have a consistent labeling in $Y/\{crossings\}$. By the assumptions
about $A(y)$ in Eq.~(\limit), there is a consistent labeling of the the
spectrum on the section $y_0\times X$ with $y_0$ near zero. Since codimension
3 crossings do not destroy simple connectivity, this labeling propagates to
$Y/\{crossings\}$.
When $X$ is 2-dimensional (or when we are studying a 2-dimensional surface
$\Sigma$ in $X$) the generic level crossings occur at isolated points.
We associate numerical indices to these level crossings, as follows:
{\nd \bf Index:} Let $z_j$ be a crossing point for the $\a$ and $\b$
eigenvalues of the $S$-matrix with $01/2$. In this example the $S$-matrix
turns out to be essentially
the Hilbert transform of Berry's spin Hamiltonian, so it may be viewed as
playing the analogous role in scattering situations.
Let $h(\vec B)=\vec B\cdot \vec\sigma +| B|$,
and consider the following tight binding Hamiltonian $H(\vec B)$ on the
non-negative integers:
%
$$(H(\vec B)\psi)(n) = \psi(n+1) + \psi (n-1) + \delta_{n0}\ h(\vec B)\psi(n),
\eqno(5.1) $$
%
where $\psi(n)\in \complex^2$ and $\psi (-1)=0$.
The absolutely continuous spectrum is the interval $[-2,2]$. It is free of
embedded eigenvalues and is all of the spectrum if $| B|<1/2 $.
For $|B|>1/2$ the spectrum also has one bound state with energy
$2|B|+{1\over 2|B|}$.
The bound state for $|\vec B|>1/2 $
has an exponentially localized wave function
%
$$
\psi_0(y) =(2|B|)^{-n}\ \vec B\cdot\vec\sigma \left (
\matrix {1 \cr 0} \right ). \eqno(5.2)
$$
%
Let $P$ be the projection onto the scattering states. $P_\perp=1-P$
is the projection onto the bound states.
Since $P_\perp$ is smooth and finite rank,
we can use Eq.~(\flat) and Eq.~(2.2) to compute the
adiabatic curvature of $P$:
%
$$\Omega(P)=-\Omega(P_\perp)=\cases{0&if $|\vec B|<1/2 $;\cr
-\omega (\vec B)&if $|\vec B|>1/2 $,} \eqno(5.3) $$
%
where
%
$$\omega (\vec B)=
{1\over2|B|^3}\left(B_1 dB_2\wedge dB_3 +B_2 dB_3\wedge dB_1 +
B_3 dB_1\wedge dB_2\right)\eqno(\angle)$$
%
is half the spherical angle 2-form. Integrating this we find that
for a 2-sphere $S^2$ enclosing the origin
in the 3-dimensional space of magnetic fields,
$$c_1(S^2,P)=\cases{0,\ &if \
$|B|<1/2$;\cr -1 & if\ $|B|>1/2$.} \eqno(5.5) $$
Now we recompute this Chern number from theorem 1.
The scattering matrix is (see Appendix)
%
$$S(k,\vec B)= -{h(\vec B) -z\over h(\vec B) -1/z}, \eqno(5.6) $$
%
where $ \ \ z=\exp ik,\ \ 0\le k\le\pi.$
Since the spectrum of $h(\vec B)$ is $\{0, 2|B|\}$, $S$ is smooth
for $k$ real and $\vec B$ away from the sphere $|\vec B|=1/2$.
$S$ satisfies the basic hypotheses in paricular Eq.~(\limit) holds with
$A(\vec B) =2/(h(\vec B) -1)$ and $T(\vec B) =2/(h(\vec B) +1)$.
The eigenvalues and adiabatic curvature for the corresponding
spectral projections of $S$ are:
%
$$\exp i\theta_0(k,\vec B) = -z^2,\ \exp i\theta_1(k,\vec B) =
-{(2|\vec B|-z)/(2|\vec B| -1/z)}, \eqno(5.7) $$
%
$$\Omega(P^S_0) =
-\Omega(P^S_1)=-\omega (\vec B). \eqno(5.8) $$
%
The windings of $\theta$ are thus
%
$$\ell_0=1,\ \ell_1=
\cases{1 & for $|\vec B|<1/2 $;\cr 0 &for $|\vec B|>1/2$.} \eqno(5.9) $$
%
It follows that the 3-form of Eq.~(\3form) is
%
$$ s_3(y)= {1\over 4\pi^2} (d\theta_1-d\theta_0)\wedge \omega(\vec B).
\eqno(5.10) $$
%
Integrating $s_3$ over $I \times S^2$, or using proposition 3,
we confirm that the Chern
number of the scattering states is given by Eq.~(5.6).
\nd {\bf VI. Derivation of Eq.~(\main)}
This section is a proof of formula Eq.~(\main) in theorem 1
given the hypotheses in section IV. By assumption our scattering
potential is compactly supported, hence supported on a disk of radius $M$
for some integer $M$.
Now pick an integer $L >> M$ and apply a Dirichlet condition at $L$.
If $L$ is chosen large enough, this causes only a small change in
the wavefunctions of the bound states, and therefore does not change
the Chern classes of these states. Thus it also does not change
the Chern class of the complementary part of the spectrum, corresponding to
the energy interval $[-2,2]$.
The cutoff at $L$ breaks the system up into two noninteracting subsystems.
The exterior states, supported on $\{ x>L \}$, have absolutely
continuous spectrum and are completely independent of $\{y_i\}$. These states
contribute nothing to the curvature and are henceforth ignored.
The interior states, supported on $\{ x|x|>M$, the wavefunctions take the form
%
$$\psi_{m,\a}(x) = \zeta_\a(k,y) \bigg(\exp(-ik|x|) +
\exp(i(\theta_\a + k |x|))\bigg), \eqno(6.1) $$
%
where $\psi$ takes values in $\complex^n$,
$\zeta_\a(k,\{y_i\})$ is an eigenvector
of $S(k,\{y_i\})$ with eigenvalue $\exp(i\theta_\a(y))$,
and the ``energy bands", $k_{m\a}(\{y_i\})$, solve
%
$$\theta_\a(k,\{y_i\}) + 2kL = (2m-1)\pi,\ \ \ m=1,\dots, L+\ell_\a-1.
\eqno(\exa) $$
%
Eq.~(\exa) is equivalent to the Dirichlet condition $\psi(L)=0$. Although $k=0$
is a solution to Eq.~(\exa) with $m=0$, $\psi_{0,\a}$ is identically zero, so
this solution is not counted. Similarly $k=\pi$ solves Eq.~(\exa) for
$m=L+\ell$, but this also generates the zero wavefunction.
We temporarily suppress the $\a$ index and
as before write $y_0=k$.
Taking derivatives we find that, for
fixed $m$, $\partial k/\partial y_j = - (\partial\theta/\partial y_j)/2L$.
We also define a density-of-states
function
$$ \rho(k) = (2L + d\theta/dk)/2\pi. \eqno(\exb) $$
Of course, $1/\rho(k)$ is not precisely the spacing between levels.
Rather,
$$ \rho(k_m) (k_{m+1} - k_{m-1})/2 = 1 + O(L^{-3}). \eqno (6.4) $$
Each energy level $k_{m}(y)$ satisfying Eq.~(\exa) is associated to two line
bundles. One is the sub-bundle of the trivial
Hilbert space bundle $\Sigma \times \ell_2$ spanned
by $\psi_{m}$. The other is the sub-bundle of $\Sigma \times \complex^n$
spanned by $\zeta(k,y)$. These two bundles are isomorphic, as the
limiting behavior of $\psi$ defines $\zeta$, and as each $\zeta$,
together with a solution to Eq.~(\exa), defines an eigenfunction $\psi$.
Isomorphic bundles have the same Chern classes, so we may compute
the Chern class of the $\psi$ bundle by integrating the curvature
of the $\zeta$ bundle. This is just the restriction to the
surface $k_{m} (y)$ of the 2-form $\Omega(P^S)$ on $I \times X$.
Two tangents to the surface $k_{m} (\{y_j\})$ are
$(-\partial_1 \theta /2L, 1, 0)$ and
$(-\partial_2 \theta /2L, 0, 1)$.
Applying $\Omega$ to these two vectors, we find that $\Omega$, restricted
to the surface $k_m(\{y_j\})$, equals $f(k_{m}(y), y_1, y_2)
dy^1 \wedge dy^2$, where
%
$$f(y) =\Omega_{12} +
{\Omega_{20} \partial_1 \theta +
\Omega_{01} \partial_2\theta \over 2L}.\eqno(\F)$$
%
So we can write
%
$$c_1(\Sigma,P) = {1 \over 2\pi} \sum_{\a=1}^n\sum_{m=1}^{L+\ell_\a-1}
\int_{\Sigma} f_{\a}(k_{\a,m}(y), y_1, y_2)
dy_1 \wedge dy_2. \eqno(6.6) $$
%
Next we replace the sum over $m$ with an integral over $k_0$, using
the fact that
$$ f(k_m, y_1, y_2) = \int_{(k_{m-1} + k_m)/2}^{(k_{m} + k_{m+1})/2}
f(k, y_1, y_2)\rho(k) dk
+ O(L^{-2}). \eqno(6.7)
$$
Note that $f(y)$ is defined by Eq.~(\F) for all $y_0$,
not just for $y_0=k_m(y)$.
Some care is required for $f(k_1)$ and $f(k_{L+\ell-1})$.
Eq.~(6.7) still applies, as long as we take $k_0=0$ and $k_{L+\ell}=\pi$.
We also have that
$$ \eqalign{\int_{0}^{k_1/2} f(y) \rho(y) dk = & f(0, y_1, y_2)/2
+ O(L^{-1}) \cr
\int_{(\pi+k_{L+\ell-1})/2}^{\pi} f(y) \rho(y) dk = &
f(\pi, y_1, y_2)/2 +
O(L^{-1}).} \eqno(6.8)
$$
Plugging (6.7) and (6.8) into (6.6) we find
$$ c_1(\Sigma,P) = {1 \over 2\pi} \sum_{\a=1}^n \int_{I\times\Sigma}
f_\a(y) \rho_\a(y) - {1 \over 4\pi} \int_\Sigma \sum_\a \left (
f_\a(0,y_1,y_2) \! + \! f_\a(\pi,y_1,y_2) \right )
+ O(L^{-1}). \eqno(6.9)
$$
By Eq.~(\sumrule ) and Eq.~(\flat ), $\sum_\a \Omega(P_\a^S)$ is
identically zero, so $\sum_\a f_\a(0,y_1,y_2)+
\sum_\a f_\a(\pi,y_1,y_2) = O(L^{-1})$.
We are
thus left with the triple integral of $\sum_{\a} f_\a(y) \rho_\a(y)$.
But
$$ f(y) \rho(y) = {L \over \pi} \Omega_{12} + {1 \over 2\pi}
(\Omega_{20} \partial_1 \theta + \Omega_{01} \partial_2 \theta
+ \Omega_{12} \partial_0 \theta) + O(L^{-1}). \eqno(6.10)
$$
Summing over $\a$ eliminates the $O(L)$ term, as $\sum_\a \Omega(P^S_\a) =0$.
The $O(1)$ terms of Eq.~(6.10), summed over $\a$, are
precisely $2 \pi s_3(y)$. This shows that
%
$$c_1(\Sigma,P) = \int_{I\times\Sigma} s(y) + O(L^{-1}). \eqno(6.11) $$
Since $c_1(\Sigma,P)$ and $\int_{I\times\Sigma} s(y)$ are independent of $L$,
the $O(L^{-1})$ correction must in fact be zero. This establishes
theorem 1.
\nd {\bf VII. Threshold States and Higher Chern Classes}
In this section we prove two extensions of theorem 1. The
first extension is to allow
threshold states to exist at $k=0$ and $k=\pi$. The second extension
is to compute the higher Chern classes of the bundle $Range(P)$ in
term of scattering data.
For the first extension the hypotheses are as in theorem 1, except
that the eigenvalues of the $S$ matrix do not all have to approach
$-1$ as $k \to 0$ or $k \to \pi$. Rather, some $+1$ eigenvalues may
occur, corresponding to threshold states. Specifically, we assume that
$$
S(y_0, \{y_j\})+1=\cases{2 B(\{y_j\}) + i\ y_0\ A (\{y_j\})+O(y_0^2)
\phantom{O((y_0-\pi)^2)} \hbox{ near $y_0=0$;} \cr
2 R(\{y_j\}) + i\ (y_0-\pi)\ T(\{y_j\})+O((y_0-\pi)^2) \hbox{ near $y_0=\pi$},}
\eqno(7.1)
$$
with $A(\{y_j\})$, $T(\{y_j\})$ smooth, Hermitian, matrix-valued functions that
have no level crossings in $X$, and with $B(y_i)$, $R(y_i)$
orthogonal projections on $\complex^n$ that depend smoothly on $\{ y_i \}$.
This implies that $Range(B)$ and $Range(R)$ are finite-dimensional
bundles over $X$ with well-defined Chern numbers.
{\nd \bf Theorem 2:} {\it Assume the above hypotheses.
If $\Sigma$ is an oriented
surface in $X$, then the first Chern number of the bundle defined by
$P$ over $\Sigma$ is}
%
$$ c_1(\Sigma,P) = {c_1(\Sigma,B) + c_1(\Sigma,R) \over 2} + \int_{I\times
\Sigma} s_3(y). \eqno(7.2)$$
%
The proof of Eq.~(7.2) is almost identical to that of Eq.~(\main ).
The only difference is that, for the states in $Range(B)$, $\theta(0)=0$
instead of $-\pi$, and as a result $k_1 = \pi/L + O(L^{-2})$,
not $2 \pi/L + O(L^{-2})$. Replacing the sum over $m$ with an
integral over $k$ gives an
integral with lower limit $k=0$, not $k=k_1/2$. This, and similar
considerations at $k=\pi$, cause Eq.~(6.9) to be replaced by
$$ \eqalign{ c_1(\Sigma,P) = & {1 \over 2\pi} \sum_{\a=1}^n
\int_{I\times\Sigma} f_\a(y) \rho_\a(y) d^3y \cr &
- {1 \over 4\pi} \int_\Sigma \sum_{\b \not\in Range(B)}\! \! f_\b(0,y_1,y_2)
- {1 \over 4\pi} \int_\Sigma \sum_{\gamma \not \in Range(R)}
\! \! f_\gamma (\pi,y_1,y_2) + O(L^{-1}).} \eqno(7.3)
$$
Since $\sum_\a f_\a(0,y_1,y_2)= O(L^{-1})$, a negative sum over
$\b \not \in
Range(B)$ can be replaced with a positive sum over $\b \in Range(B)$, with
a similar substitution for $\gamma$. As a result,
$$
\eqalign{ c_1(\Sigma,P) = & {1 \over 2\pi} \sum_{\a=1}^n \int_{I\times\Sigma}
f_\a(y) \rho_\a(y) d^3y \cr & + {1 \over 4\pi} \int_\Sigma \sum_{\b \in
Range(B)} f_\b(0,y_1,y_2) + {1 \over 4\pi} \int_\Sigma \sum_{\gamma \in
Range(R)} f_\gamma (\pi,y_1,y_2) \cr = & \int_{I \times \Sigma} s_3(y) +
c_1(\Sigma,B)/2 + c_1(\Sigma,R)/2.\phantom{ \int_\Sigma \sum_{\gamma \in
Range(R)} f_\gamma(\pi,y_1)}\qed} \eqno(7.4) $$
The example of section V, with $|B|=1/2$, illustrates this theorem.
There is no bound state, but there is a threshold at $k=\pi$
whose Chern number is $+1$.
As $k$ goes from $0$ to $\pi$, $\theta_0$ goes from
$-\pi$ to $\pi$, as before, but $\theta_1$ goes from $-\pi$ to 0. From
Eq.~(5.10) we see that the integral of $s_3$ is $-1/2$.
Adding this to half the
Chern number of the threshold state gives 0. This is
indeed the Chern number of
the scattering states, since, in the absence of bound
states, the projection
$P=1$.
We next turn our attention to higher Chern classes. For any family
of projections $P$, let $F(P)$ be
the operator-valued 2-form $-i P dP \wedge dP P$.
The cohomology class of the $2n$-form
$$ \omega_n = {1 \over n! (2\pi)^n} Tr(F(P)^n) \eqno(7.5) $$
is a topological invariant, a linear combination of the n-th Chern
class and products of lower Chern classes.
For example, if the first Chern class is zero, then
$\omega_2$ gives minus the 2nd Chern class.
{\nd \bf Theorem 3:} {\it Assume a tight-binding model with a
finite number of scattering channels and a compactly supported
scattering potential that depends on a $2k$-dimensional compact
oriented parameter space $X$ with
local coordinates $\{ y_i \}$.
Assume the $S$-matrix $S(y)$ depends smoothly on all variables and has
level crossings at a finite number of points in $I \times X$,
with $F(P^S_\a) = O(d^{-2})$ near the level
crossings, where $d$ is the distance to the crossing.
Assume that $S$ has the limiting behavior given in Eq.~(4.1). Then
$$
s_{2k+1}(y) = {1\over k! (2\pi)^{k+1}}\ \sum_\a d\theta_\a
(y)\wedge Tr(F(P_\a)^k) \eqno(7.6)
$$
is defined almost everywhere, and}
$$ \int_X \omega_k = \int_{I \times X} s_{2k+1}. \eqno(7.7)
$$
The proof is almost identical to that of theorem 1, and so is only sketched
here. The form $s_{2k+1}(y)$ is defined everywhere except at level
crossings. Near level crossings $s_{2k+1}=O(d^{-2k})$, which is integrable
in dimension $2k+1$.
As before, we apply a cutoff at a large distance $L$ and examine
the finite number of interior states. Eigenstates of the Hamiltonian
are in 1-1 correspondence with eigenstates of the $S$-matrix on
the energy bands (6.2). Since the integral of $\omega_{k}$ is topological,
we can use the curvature of the eigenbundles of $S$, restricted to the
energy bands, to compute the topological class of the eigenbundles of $H$.
The form $\omega_{k}(P_\a^S)$, restricted to the surface $k_m(y)$, takes
the form $f(y) dy^1 \wedge \cdots \wedge dy^{2k}$, where
$$
f(y) = \omega_{1\ldots 2k} + {1 \over 2L} \sum_{i=1}^{2k}
(-1)^i \partial_i \theta \omega_{0,\ldots, i-1, \hat i, i+1, \ldots, 2k},
\eqno(7.8)
$$
where $\hat i$ denotes that the subscript $i$ is not included.
We replace the sum over $m$ with an integral over $k$. As before, this
involves multiplying $f(y)$ by the density of states:
$$
f(y) \rho(y) = {L \over \pi} \omega_{1,\ldots,2k} + {1 \over 2\pi}
(d\theta \wedge \omega_{k})_{0,1,\ldots,2k}. \eqno(7.9)
$$
Summing over $\a$ and integrating eliminates the $O(L)$ term, since the
trivial $\complex^n$ bundle has zero invariants. What remains is the
integral of $s_{2k+1}(y)$.\hfil\qed
To construct an example for theorem 3, we recall the non-Abelian
analog of Berry's spin-1/2 example. (For details, see [\asss]).
Let $X=\real^5$
be the space of real, symmetric, traceless $3 \times 3$ matrices $Q$.
Consider a spin-3/2 particle with the Hamiltonian
$$
h(Q) = |Q| + \sum_{i,j=1}^3 Q_{ij} J_i J_j, \eqno(7.10)
$$
where $|Q|^2 = {3 \over 2} Tr(Q^2)$ and $\{ J_i \}$ are the usual angular
momentum operators. The spectrum of $h$ is $\{0, 2|Q| \}$, and
each eigenvalue is doubly degenerate. If we restrict ourselves to
a 4-sphere $S^4$ enclosing the origin in $X$, then the upper eigenbundle
has 2nd Chern number $c_2=+1$, while the lower eigenbundle has $c_2=-1$.
We can now duplicate the construction of Section V.
Consider a spin-3/2 particle on
a semi-infinite chain with potential $V(n)=h(Q) \delta_{n0}$.
Our parameter space is a 4-sphere of the form $|Q|=$constant. As before,
there is continuous spectrum over the energy range $[-2,2]$. If $|Q|>1/2$
there is a bound state with energy $2|Q| + {1 \over 2|Q|}$.
This bound state
is doubly degenerate and has (2nd) Chern number $+1$. The scattering
states therefore have (2nd) Chern number 0 if $|Q|<1/2$ and $-1$ if
$Q > 1/2$. The 5-form $s_5(y)$ is proportional to $d\theta_0 - d\theta_1$
times the area form on $S^4$, and integrating $s_5$ over $I \times S^4$
correctly computes the Chern number in all cases.
\nd {\bf VIII. Appendix: Scattering and Tight Binding Models}
Here we recall some basic facts from scattering theory and
tight-binding models on graphs.
\nd {\bf Discrete Schr\"odinger Operators for Graphs: }
For an arbitrary graph there is a conventional notion of a discrete
Schr\"odinger operator associated with the graph. For each vertex $v$
we have a real potential $V(v)$, and for each pair of adjacent
vertices we have a hopping amplitude $t_{v v'}$ satisfying
$t_{vv'}=\bar t_{v'v}$. We consider the Hermitian operator $H$ defined
by
%
$$(H\psi)(v) = \sum t_{vv'} \psi (v') + V(v)\psi(v).
\eqno(\schrod)
$$
Consider a graph with n strands going to infinity, which we label by
$\a\in1,\dots,n$. Let $x$ be an integer label of the vertices along a given
strand (so that the point at infinity corresponds to $x=\infty$).
The function
$\exp \pm ikx$, with $0\le k\le\pi$, is a plane wave.
We henceforth assume that $t_{vv'}=1$ and $V(v)=0$ for all but
a finite number of vertices.
These assumptions guarantee absolutely
continuous spectrum, with multiplicity $n$, for the momentum interval
$[0,\pi]$, corresponding to the energy interval $[-2,2]$.
{\bf Bound States:} Bound states with exponentially decaying solutions
behave at infinity like $(\pm 1)^n e^{-\kappa n}$ with $\kappa>0$
and have energies $\pm
2\cosh \kappa$. These are always outside the continuous spectrum $[-2,2]$.
Eigenvalues embedded in the continuous spectrum $[-2,2]$, if they exist, are
associated with compactly supported eigenfunctions.
Complex hermitian Hamiltonians with embedded eigenvalues
are of codimension $2n$ while real symmetric Hamiltonians with
embedded eigenvalues are of codimension $n$.
(The condition that $\psi$ vanishes at a vertex is codimension 2 in
the complex case and codimension 1 in the real case. This has to
occur on all strands simultaneously.) For a connected system, our hypotheses in
section IV preclude embedded eigenvalues, as these would conflict with the
unitarity of S on $\complex^n$. In the real case this is the generic setting
if at least four strands go to infinity. In the complex case there need be at
least two strands. The example in section V is of this type since one strand
with spin is equivalent to two strands without spin.
{\bf Scattering States:} Let $\psi$ be a solution
of the difference equation $(H-2\cos k)\psi=0$ with $ k\in[0,\pi]$.
To each such $\psi$ we can associate two vectors in $\complex^n$
so that $\psi(x) \to
\zeta_{out} e^{ikx} + \zeta_{in} e^{-ikx}$ as $x \to \infty$.
The on-shell $S$-matrix is the
defined by
%
$$\zeta_{out} = S(k)\zeta_{in}.\eqno(A.2) $$
%
Since there is no distinguished
basis in $\complex^n$, one focuses on unitary invariant properties of $S(k)$.
There is a class of graphs for which there is a simple formula for the on-shell
$S$-matrix. Consider first a compact graph, and let $h$ be the tight binding
Hamiltonian for the graph (either with or without spin). Now attach an infinite
strand (with free evolution on the strand) to each vertex of the graph. The
scattering states are determined by the solutions of
%
$$(h-E)(\psi_{in}+\psi_{out})= -(\psi_{in}/z + z \psi_{out}).
\eqno(A.4) $$
%
It follows that
%
$$S=-{h-z\over h-1/z}, \eqno(A.5) $$
%
where $z=\exp ik$ and the energy is $E=z+1/z$.
At the edges of the continuous spectrum, where $z=\pm 1$, the
phase shifts are $\exp i\theta_\alpha = - 1$, except in the special
case where
$h$ has eigenvalues $\pm 1$. If $h\mp 1$ is invertible, the
$S$-matrix at the edges of the spectrum is $-1$ and
Eq.~(\limit) holds with $A=2/(h-1)$, $T=2/(h+1)$.
The condition $Ker (h\mp1) =0$ is a codimension 1 condition, (since $h$ is a
hermitian matrix). It follows that a generic matrix family $h(y)$ may violate the conditions in the hypotheses in section IV. This is not surprising, as
the hypotheses do not allow bound states to appear or disappear. If two
Hamiltonians have different numbers of bound states, then any path between
them must contain a point where the hypotheses are violated. On the
other hand, it is easy to construct examples where the nature of the
spectrum does not change, and where the hypotheses are
satisfied. For example, take $h(y) = u(y) h_0 u^\dagger (y)$, with
$h_0$ a fixed Hermitian matrix (whose spectrum does not contain $\pm 1$),
and with $u(y)$ a family of unitary matrices.
\vfill\eject
\nd {\bf Acknowledgment.}
The authors gratefully acknowledge the hospitality of the Erwin Schr\"odinger
Institute and to the ITP, Technion, where part of this work was done. We also
thank Dan Freed for helpful discussions on the topology of infinite dimensional
bundles.
\vs.1
\nd {\bf References}
\vs.1
\nd
[\tknn] D.~J.~Thouless, M.~Kohmoto,~P.~Nightingale and M.
den Nijs, {\it Quantum Hall conductance in a two dimensional periodic
potential}, Phys.~Rev.~Lett. {\bf 49}, 40, (1982).
\par\nd
[\berry] B.~Simon, {\it Holonomy, the quantum adiabatic theorem and Berry's
phase,} \hfil \break Phys. Rev. Lett.~{\bf 51}, 2167 (1983);
M.V.~Berry, Proc.~Roy.~Soc.~A {\bf 392},45--57,(1984)\par
\nd[\landauer] R. Landauer,{\it IBM J. of Research and
Development} {\bf 32} (1988)\par
\nd [\faddeev] V.A. Marchenko, {\it Sturm
Liouville operators and their applications}, Birkhauser, (1986).\par
\nd [\birman] M.~SH. Birman and D.R.~Yafaev, {\it Spectral properties of the
scattering matrix}, St.~Petersburg Math. J. {\bf 4}, 1055--1079, (1993);
{\it The spectral shift function, the work of M.G. Krein and its further
developments}, {\it ibid} 833--870.\par
\nd [\MS] J. Milnor and J. Stasheff, {\it Characteristic Classes},
Princeton University Press and University of Tokyo Press, Princeton, 1974.\par
\nd [\DFN] B.A. Dubrovin, A.T. Fomenko and S.P. Novikov, {\it Modern
Geometry--- Methods and Applications}, Vol. II, Springer, (1984)
\nd [\segal] A. Pressley and G. Segal, {\it Loop Groups}, Oxford University
Press, (1988). \par
\nd [\freed] D. Freed, {\it An Index Theorem for Families of Fredholm
Operators Parameterized by a Group}, Topology {\bf 27} (1988) 279--300.\par
\nd [\simon] B. Simon, {\it Trace Ideals and their Applications}, Cambridge
University Press, (1979)\par
\nd[\kato] A. Jensen and T. Kato. {\it Spectral Properties of Schr\"odinger
Operators and Time-Decay of the Wave Functions}, Duke.~Math.~J.
{\bf 46} 583--611 (1979) \par
\nd [\thirring] W. Thirring, {\it
Quantum Mechanics of Atoms and Molecules}, Springer, (1979)\par
\nd [\asss] J.E. Avron, L. Sadun, J. Segert and B. Simon, {\it
Chern Numbers, Quaternions, and Berry's Phases in Fermi Systems},
Commun. Math. Phys {\bf 124}, 595--627 (1989)
\vfill\eject
\bye