VOLUME
51,
NUMBER 24
PHYSICAL
REVIEW
LETTERS
12 DECEMBER
1/83
Holonomy,
the
Quantum Adiabatic
Theorem,
and
Berry's
Phase
Barry
Simon
Departments
of
Mathematics
and
Physics, California
Institute
of
Technology,
Pasadena,
California
91185
(Received
18
October
1983)
It
is
shown
that
the
"geometrical
phase
factor"
recently
found
by
Berry
in
his
study
of
the
quantum
adiabatic
theorem
is
precisely
the
holonomy
in
a Hermitian
line
bundle
since
the adiabatic theorem
naturally
defines
a
connection
in such
a bundle.
This not
only
takes
the
mystery
out
of
Berry's
phase
factor and
provides calculational
simple formulas, but
makes
a
connection
between
Berry's
work and
that of
Thouless
zf zP.
This
connection
al-
lows the
author
to
use
Berry's
ideas
to
interpret
the
integers
of
Thouless
et gl.
in
terms
of
eigenvalue
degeneracies.
I'ACS
numbers:
03.65.
Db,
02.
40.
+
m
Vector
bundles and their
integral
invariants
(Chem
numbers)
are
already
familiar to
theoret-
ical
physicists
because
of
their occurrence
in
classical
Yang-Mills
theories. Here
I want to
explain
how
they
also
enter
naturally
into
non-
relativistic
quantum
mechanics,
especially
in
problems
connected
with
condensed matter
phys-
ics.
If
one
has a Hermitian
operator
H(x)
depend-
ing
smoothly
on a
parameter
x,
with an
isolated
nondegenerate
eigenvalue
E(x)
depending
continu-
ously
onx,
then
((x,p)
~
H(x)y
=
E(x)p
I
defines
a
line bundle over
the
parameter
space.
l will
show
that
the
twisting
of this line bundle
affects the
phase
of
quantum
mechanical
wave functions.
Berry,
in
a beautiful recent
paper,
'
discovered
a
striking
phenomenon
in
the
quantum
adiabatic
theorem.
'
That theorem
says'
that if
H(t),
0
~
t
~1,
is a
family
of Hermitian
Hamiltonians,
de-
pending
smoothly
on
t,
and if
E(t)
is
a
smooth
function of
t
which is
a
simple
eigenvalue
of
H(t)
isolated
from the rest
of
the
spectrum
of
H(t)
for
each
t,
then the solution
Pr(s)
of
the
time-de-
pendent
Schrodinger equation
where
y(C)
is
an extra
phase
which
Berry
exten-
sively
studies,
and
which he
suggests
could
be
experimentally measured.
'
The
purpose
here
is
first to
advertise
what
Berry
calls a
"remarkable
and
rather
mysterious
result,
"
but
more
basically
to
try
to
take
the mys-
tery
out of
it
by
realizing
that
y
is an
integral of
a
curvature
so
that
Berry's
phenomenon is
es-
sentially
that of
holonomy
which
is
becoming
quite
familiar to
theoretical
physicists.
'
This
realization will
allow us
a
more
compact
formula
than
that
used
principally
by
Berry,
and
one
that
is easier to
compute
with.
Most
importantly,
it
will
give
a
close
mathematical
relationship
be-
tween
his work
and that of
Thouless et
al,
'
so
that
Berry's
interesting
analysis
of
the relation
of
degeneracy
to
y(E)
will
allow
us
to
interpret
the
TEN'
integers
in
a
new and
interesting
way.
To
explain
that
y
is
a
holonomy,
l
begin
by
re-
placing
H(s)
by
H(s)
-E(s)
which
produces
a
trivial, computable
phase change
in
the solution
Pr(s)
of
(1).
Thus,
without
loss,
we
can take
E(s)
=
—
0.
Define
qr
(s)
=
gr
(sT
)
so that
q
solves
idler(s)/ds
=
H(s/T)
gr(s)
i
dg
(
r)/sds
=
TH(s)gr(s),
(3)
is
wrong;
rather,
Berry
finds
p,
=exp[
—
i
f,
E(s/T)
ds]
exp
[iy(C)]
p„
(2)
with
gr(0)
=
p,
where
H(0)
p,
=
E(0)p,
has
the
prop-
erty
that
as
T
-
~,
Pr(T)
approaches
the eigen-
vector
p,
of
H(1)
with
H(1)
p,
=
E(1)p,
.
Berry
asked
the
following
tluestion:
Suppose
that
H(x)
is
a multiple-parameter
family
and that
C(t)
is
a
closed
curve in
parameter
space,
so
that
H(C(t))
=
H(t)
obeys
the
hypo-theses
of the
adiabat-
ic
theorem. Then
that
theorem
says
that
p,
is
just
a
phase
factor times
po
and
Berry
asks,
"%'hat
phase
factor
~"
Surprisingly,
the
obvious"
guess
y,
=exp[-i
f,
E(s/T)
ds]
y,
and
the adiabatic
theorem
says
that
gr(s)
has
a
limit
rt(s)
with
H(s)
q(s)
=0
(4)
[since
we
have
taken
E(s)
=0].
If
now
H(x)
is a
multiparameter
family
of
Hermitian Hamilton-
ians,
so
that
in
some
region
M
of
parameter
space,
0
is an
isolated
nondegenerate
eigenvalue,
then
given
any
curve
C(t)
and
any
choice
r4
of
normalized
zero-energy
eigenvector of
H(C(0))
(i.
e.
,
a
choice
of
phase),
the adiabatic limit
yields
a
way
of
transporting
q,
along
the
curve
C(t),
i.
e.
,
a connection.
In
this
way,
(2)
is
just
an
ex-
pression
of the
holonomy
associated to this
con-
nection.
So
far
this
is
just
fancy words
to
de-
1983 The American
Physicai Society
2167
VOLUME
5
1,
N
UMBER
24
PHYSICAL
REVIEW
LETTERS
12 DEcaMuzR
1983
by
(4).
One
can
give
a rigorous
proof
just
using
the
convergence
of
g
without
worrying
about the
question
of
convergence of
derivatives.
'
Thus
the connection
given
by
the adiabatic theorem
[when
E(s)
=
—
0]
is
precisely
the
conventional
one
for
embedded
Hermitian
bundles
and
y
is the
con-
ventional
integral
of the
curvature
which
is
just
the
Chem
class
of
the
connection.
In
particular,
y
only depends
on
the X„'s,
not
other
aspects
of
H(x).
This means that
one has
a
simple
compact
formula'
for
y.
.
r(c)
=
J,
v,
where
S
is
any
oriented
surface in M
with
9S
=C
and V
can
be defined in
terms
of an
arbitrary
smooth
choice,
'
p(x),
of
unit
vectors
in
X„by
V
=i(dp,
dp),
which is shorthand
for the
two-form"
V
=
g
im(()
y/sx„op/sx,
.
)
dx,
ndx,
.
(5)
written in terms
of
local
coordinates. The
for-
mula
that
Berry
used
has
the
advantage
over
(5)
of
being
manifestly
invariant under
phase changes
of
p(x),
"
but it
appears
to
depend
on details
of
H(x)
and not
just
on
the
spaces
X„.
Moreover,
since it
has a sum over intermediate
states,
it
could
be
difficult
to
compute
in
general,
although
in his
examples
it
is
easy
to
compute,
since the
sum over intermediate
state in these
examples
is
finite.
Even
in
these
examples,
(5)
ean be
very
easy
to
use in
computations.
E(luation
(5)
shows that V
=0
if
one can
choose
the
p(x)
to
be
all
simultaneously real.
Thus,
the
phenomena we discuss
here are
only
present
scribe
Berry's
discovery. However,
there
is a
mathematically
natural
connection
already
long
known in
the situation
of
distinguished
lines in
Hilbert
space.
For
givenx,
let
X„denote
the
zero-energy
eigenspace
for
H(x).
This
yields
a
line bundle
over the
parameter
space
which,
since
it
is
embedded in
M
&
&,
has
a
natural
Hermitian
connection,
studied,
e.
g.
,
by
Bott and
Chem.
'
In
this
connection,
one
transports a
vec-
tor
P,
along
a
curve
C(t
)
so
that
P
(i
)
obeys
(P
(t
+
M), P
(i
))
=1+
O((5t)')
.
I claim that
q(s)
precisely obeys
this
condition;
formally,
one
can
argue
that
(
(
)
dq(s))
).
(
(
)
dq~(s))
=
lim
(q(s),
—
iTH(s)gr(s))
=0
in
magnetic
fields or some
other
condition
pro-
ducing
a nonreal
Hamiltonian.
As
an
example
of
significance below which is
also
considered
by
Berry,
'
let
M=
i(,
"$(0)
and
giv-
enxEM,
let
H(x)
=
x
L
where
L
is a
spin-S
spin
on
C
",
Then
all
eigenvalues
are
nondegenerate
and
we
can,
for each
nz
=
—
S,
—
S
+
1,
.
..
,
S,
com-
pute
a V
(x)
associated to the
eigenvalue
(x(m.
By
rotation
covariance, V
(x)
must
be a function
of
(x(
times
the
area form
A(x)
on
the
sphere
of
radius
x.
Thus,
we
need
only
compute
V
at
x
=
(0,
0,
z).
U
(m)
is
the vector with
L,
(m)
=m(m),
then for x near
(0,
0,
z),
we
can take
p(x)
=exp
i~
—
L,
,
—
—
L„+
O(x'+y')
(m)
~z
'
z
so
that
dy=iz
'[dxL,
—
dyL„](m),
(d
p,
dp)
=
z
dx
hdy
(m(
[L,
,
L„](
m)
=
ia
rndxh
dp,
and
thus,
returning
to
general
x,
V.
(x)
=
m(x(
'W(x}.
(6)
In
particular,
if S
is
any
sphere'
about the
ori-
gin,
(2~)-'J,
V.
(x)
=2m
is
an
integer.
This
is no
coincidence: If
C
is
a
clockwise circuit around
the
equator
of the
sphere
8,
which
breaks
up
S
into
two
hemispheres
S+,
S
with
~S,
=
+
C,
then
exp
[ir(C)]
=exp(i
Js,
v)
=exp(-i
Js
V}
so
that
Jsv
must
be
2))'
times
an
integer.
We
therefore
see the familiar
quantization of
the
inte-
gral
of
the
Chem
class,
V,
of the
bundle,
as a
consistency condition on
the
holonomy,
a standard
fact.
Thouless
eta/.
,
in their
deep
analysis
of the
quantized Hall
effect,
'
considered
a band of
a
two-dimensional
solid in
magnetic
field,
so
that
for
each
k
in
T,
the Brillouin
zone,
the
corre-
sponding
band
energy
is nondegenerate.
If ()()(k)
is
the
corresponding
eigenvector,
then
(i/2))
)
&&
Ir2(d
p
dp)
is
an
integer,
the
TKN'
integer of
the band.
Using
the
source"
analogy
of
Berry,
'
we
can
interpret"
these
integer
s.
Suppose
the
band under
consideration
is the
nth;
and
suppose
an
arbitrary
smooth
interpolation,
H(k),
"
of the
band Hamiltonian
H(k)
is
given
from
the
surface
of
the
torus into the
solid torus
T,
i.
e.
,
H(k)
is
defined for
0
in
T
and
e(luals
H(k)
on the
surface.
2168
VOLUME
51,
NUMBER
24
PHYSICAL
REVIEW
LETTERS
12 DECEMBER
198$
The
Wigner-von
Neumann
theorem"
says
that
generically,
"
the
nth band
is
only
degenerate
for
isolated
points
(P&
j
&,
'
in
T. One can define V
on
T
with
these
points
removed,
and
since
dV=O,
Gauss'
theorem assures us
that the
integral
of V
over the
torus
is
just
the
same as its
integral
over
little
spheres
about the
degeneracies. Each
sphere
has a
charge"
associated with
it
which
is
&q&
with
q&
an
integer,
and
gq,
is the
TKN'
integer.
It is worthwhile
to
expand
slightly
on
this
pic-
tureen'
Consider
a
matrix
family
M(x),
depend-
ing
smoothly
on these
parameters.
If all eigen-
values of
M(x,
)
are
nondegenerate,
we
say
that
xo
is a regular
point.
xo
is a
normal
singular
point
if and
only
if
(i)
only
one
eigenvalue is
de-
generate
and
its
multiplicity
is
2;
(ii)
the degen-
eracy
is
removed
to
first order
for
any
line
through
x,
.
If
0
is
a
normal
singular point
and
P
is
the
projection
onto the degenerate
eigen-
values
of
M(0),
then for
x
near
zero,
PM(x)P
—
PM(0)P
=a
xp+ B
~
x
+
O(x')
where
B is a
vector
of
traceless operators
on P.
Picking
a basis
for
the
range
of
P,
"
we
can write
B
x
=
o Cx
where
C
is
a
3
&
3
matrix and
a
are
the usual
Pauli matrices in the basis. The
condi-
tion
on removal af
degeneracy
says
that
det(C)
c
0. The
Hermiticity
of
M
implies
det(C)
is
real.
I
call
the
sign
of
det(C)
the
signature
o
(x
)
of
the
normal
singular
point.
With
use
of a
deforma-
tion
argument
and the
example
discussed
above
(with
spin
S
=
2),
it
is
not
hard
to
show
that
if
the
nth and
(n+
1)
st
levels
are
degenerate
at
x„and
if
Vz
is
the
Chem
class associated
to the
jth
lev-
el,
then for a small
sphere
S
about
x„we
have
that
(2~)
'fsV„„=
o(x,
);
(2n)
'fs
V„=
o(x,
).
-
If
M
is a
smooth,
regular
matrix
family
on
T'
and
I
is
a smooth interpolation
to
T
with
only
normal
singular
points,
"
then
the nth
TKN
inte-
ger
is
exactly equal
to a
weighted
sum
of singu-
lar
points:
Points
where
the nth level
is
nonde-
generate
get
weight
zero,
those where
it is
de-
generate
with
the next
lower
level
get
weight
o'(P)
where o is the signature
of
P,
and
those where
it
is degenerate
with the next
higher
level
get
weight
-o(P)"
I
conclude
with
a mathematical
remark:
I
have
shown
how the
Chem
integers
associated to
cer-
tain line bundles can be
understood
in
terms
of
singularities
of interpolations
of
the
bundle.
It
would
be interesting
to extend this
picture
to
gen-
eral
vector
bundles.
It is
a
pleasure
to
thank D.
Robinson
and N.
Tru-
dinger
for the
hospitality
of
the
Australian
Na-
tional
University,
where
this
work
was
done,
M.
Berry
for
telling
me
of his
work,
and
B.
Souil-
lard
for
the
remark
(via
M.
Berry)
that there
must be a
connection
between
Berry's
work
and
that of
TKN'.
This
research
was
partially
sup-
ported
by
the
National
Science
Foundation
through
Grant
No.
MCS-81-20833.
'M.
V.
Berry,
to
be
published;
see
also
Proceedings
of the
Corno
Conference
on
Chaos,
1983
(to
be pub-
lished).
28ee
T.
Kato,
J.
Phys.
Soc.
Jpn.
5,
435-439
(1950);
A.
Messiah,
Quantum
Mechanics
(North-Holland,
Am-
sterdam,
1962),
Vol.
2.
3There
are,
in the
infinite-dimensional
case,
also
domain
conditions if
H(It)
is unbounded. We
ignore
these
in our
discussion.
For
purposes
of
this
note,
one can
think
of finite-dimensional
cases.
4Since
f
J'F',
(s/T)ds
=
T
f
&'jE(s)ds
will
be
very large
in
the
limit
T
~
unless
f
0Z(s)ds
=0,
it
may
be difficult
to
set
up
the
experiment
in
such
a
way
that the
"dy-
namic phase"
f
~OR(s/T)ds
does
not
wash out
y(C).
See
T.
Eguchi
etal.
,
Phys.
Rep.
66,
213
(1980);
Y.
Choguet-Bruhat
et
al.
,
Analysis Manifolds
and
Physics
(North-Holland,
Amsterdam,
1983).
6D.
Thouless,
M.
Kohmoto, M.
Nightingale,
and
M.
den
Nijs,
Phys.
Rev. Lett.
49,
405
(1982),
desig-
nated as
TKN2.
B.
Bott and
S.
Chem,
Acta
Math.
114,
71
(1965).
Pick
f
a
smooth function
of
compact
support
and
compute
f(s)
g(s),
ds
=
~~~„
f(s)
gz
(s),
—
Integrate
by
parts
and
get
one
term
(dq~/ds,
q)
which
is
zero
by (3)
and
{4)
and one
term
&"~
ff'(s)(0&,
0)
=
ff'(s)(0,
0)ds=
f
f'(s)de
=0.
While one
gets
(5) by
appealing
to
the abstract
theory,
it
is
quite
easy
to
compute.
If
gP)
=e~&'&pQ),
then
(g(s),
dq/ds)
=0
becomes
(y(t),
dp/dt
)+
i0
0=so
,
that
y(C)
=
$ci(p,
dp)
=
fsi(dy,
dp)
by
Stokes'
theorem.
~OIf
M
has
"holes,
»
it
may
not
be
possible
to choose
y
globally,
but since
(5)
is
phase
invariant, one need
only
make a choice in
a
neighborhood of
any
given
point.
If
S
is
the
image
of a
disk
(as
it
typically
is),
one
can
always
make
a
global
choice.
'This
formula
appears
as
Eq.
(Vb)
in
Ref.
1
but
only
in
passing;
surprisingly,
it is not used
agajn.
For
ex-
ample,
with
it,
the
one-page
calculation in
Ref.
1
that
dt/'=0
is trivial from
d~
=0.
"It
is
an
easy
calculation that
(5)
is invariant under
p(x)
e'
~"&y(g)
since the normalization condition
2169
VOLUME
51,
NUMBEK 24
PHYSICAL
REVIEW
LETTERS
12
DECEMBER
1983
(y,
y)
=1
implies
that
(cp,
dp)
+
(dp,
y)
=0.
Avron,
Seiler,
and Simon
P.
Avron,
H.
Seiler,
and
B.
Simon,
to be
published;
see also
J.
Avron
gtgE.
,
Phys.
Hev.
I
ett,
.
51,
51-53
(1983)]
give
a
simple
manifestly
phase-
invariant form
for
p;
viz.
V
=2i
Tr(dPP
dP)
where
P(x)
is the orthogonal
pro]ection
onto
x'„.
'3I
emphasize
that
(5)
holds
for
the factor
y(C)
in
(2)
even
if
F.
(s)
o
0.
'4Since
d
p
=0
(i.
e.
,
div
&
=
0
if
P
=
&,
dx
Ady+
V„ck
h
dx
+
V
dy
p,
dg),
away
from zY
=0,
(7)
holds for
any
surface
S
surrounding
0.
Since
Berry
is
talking
about
integrating
(e,
lfs)
along
curves
which he makes
analogous
to
a
vector
potential,
he talks about
magnetic monopoles.
Since
we
only
care
about
(de,
de)
whose dual is
divergenceless
away
from
degeneracies,
we do not use the
magnetic monopole
language.
Since
the dual
may
not
have
zero
curl,
elec-
trostatic
language
is
not
appropriate.
Since
the
sourc-
es
have a
sign,
we
still
use
the
phrase
"charge"
for
the
coefficient of the
delta function in
d[(dg,
cQ)]
at
sin-
gularities.
'60ne
can
show
(see
Avron, Seiler,
and
Simon,
and
Avron
eg gg.
,
Ref.
12)
that
such
interpolations
exist;
indeed,
that in the finite-dimensional
case,
the set
of
such
interpolations
is a
dense
open
set in
the
set of
all
interpolations.
J.
von Neumann and
E.
Wigner, Phys.
Z.
30,
467
(1929);
see also
J.
Avron
and
B.
Simon,
Ann.
Phys.
(N.
j('.
)
110,
85-110
(1978).
These
things
will be
discussed further
in
Avron,
Seiler,
and
Simon,
Ref.
12.
'9Changing
basis
multiplies
C
by
a
unitary
and
so
0-(~0)
is
independent
of
basis. It
does
depend
on
an
orientation
of
B (order
of
g,
y,
z)
but
so do the TKN
integers.
Everything
(0,
the TKN
integers,
the
sign
of
spanning
surfaces for
C)
changes
sign
under
change
of orientation.
The
fact that the
sum
of
the
TKN
integers
is
zero
in the finite-dimensional
case is made
particularly
transparent
by
these
relations. The
number
of
points
of
degeneracies of
level
~
and
level
n+
1
is at least
the
absolute value of the
sum of the first
g
TKN
integers.
2170