Commun.
math. Phys.
28,
251—257
(1972)
© by
Springer-Verlag
1972
The
Finite Group Velocity
of
Quantum Spin Systems
Elliott
H.
Lieb*
Dept.
of
Mathematics, Massachusetts Institute
of
Technology
Cambridge, Massachusetts,
USA
Derek
W.
Robinson**
Dept.
of
Physics, Univ.
Aix-Marseille
II,
Marseille-Luminy, France
Received
May 15,
1972
Abstract.
It is
shown that
if Φ is a finite
range interaction
of a
quantum spin
system,
τf the
associated group
of
time translations,
τ
x
the
group
of
space translations,
and
A,
B
local observables, then
lim
||[τ?τ
3e
(A),B]||e"
(l
'
)t
=0
\*\>v\t\
whenever
v is
sufficiently
large
(v >
V
φ
)
where
μ(υ)
> 0. The
physical content
of the
statement
is
that
information
can
propagate
in the
system only with
a finite
group velocity.
1.
Introduction
In
[2] it was
demonstrated that
for a
large class
of
translationally
invariant
interactions,
time translations
of
quantum spin systems
can be
defined
as
automorphisms
of a
C*-algebra,
j/,
of
quasi-local observables,
i.e.
the
abstract
algebra
generated
by the
spin operators. This should
allow
one to
discuss features
of the
dynamical propagation
of
physical
effects
in an
algebraic manner independent
of the
state
of the
system, i.e.
independent
of the
kinematical data.
It is
expected that this propagation
has
many features
in
common with
the
propagation
of
waves
in
con-
tinuous
matter
and the
point
of
this
paper
is to
demonstrate such
a
feature,
namely
a
finite
bound
for the
group velocity
of a
system with
finite
range interaction. This result
is
obtained
by a
simple estimation
derived
from
the
equations
of
motion
and it is
possible that more
detailed estimations would give more precise information
of the
form
of
spin-wave propagation.
We
briefly
discuss this possibility
at the end of
Section
3.
*
Work supported
by
National Science Foundation Grant
N°:
GP-31674X.
**
Work supported
by
National Science Foundation Grants
N°:
GP-31239X
and
GP-30819X.
252 E. H.
Lieb
and D. W.
Robinson
:
2.
Basic
Notation
We
use the
formalism introduced
in [1] and
[2].
For
completeness
we
recall
the
basic
definitions which
will
be
used
in the
sequel.
The
kinematics
of a
quantum spin system constrained
to a
v-dimen-
sional cubic lattice,
Z
v
,
are
introduced
by
associating with each point
x
E
Έ
v
an
Λf-dimensional
vector space
3?
x
and
with each
finite
set A C
Έ
v
the
direct product space
The
algebra
of
strictly local observables,
jtf
Λ
,
of the
subsystem
A, is
defined
to be the
algebra
of all
matrices acting
on
J^
Λ
.
If
Λ
1
CΛ
2
,
the
algebra
X
4l
acting
on
jj?
Λl
can be
identified
with
the
algebra
jtf
Λί
(x)
^A
2
\Ai
acting
on
J^
2
(^
Λ2
\
Λl
is the
identity operator
on
Jtf
Λ2
\
Λl
)
and
with this
identification
jtf
Λl
C&f
Λ2
Due
to
^is
isotony relation
the set
theoretic
union
of all
s$
A
,
with
A C
Έ
finite
is a
normed *-algebra
and we
define
the
completion
of
this algebra
to be the
C*-algebra
$0
of
(quasi-)
local
kinematical observables
of the
spin system.
The
group
Έ
v
of
space
translations
is a
subgroup
of the
automorphism group
of
j/,
and we
denote
the
action
of
this
group
by A e
#ί
Λ
-+τ
x
A
e
^
Λ+x
for
xεΈ
v
.
To
define
the
dynamics
of our
system
we
introduce
an
interaction
Φ
as a
function
from
the
finite
sets
X
cΈ
v
to
elements
Φ(X)
C
<$?
x
.
In
con-
trast
to [2] we
will only consider
finite
range interactions
in the
sequel.
Thus
we
demand
that
Φ
satisfies:
1.
Φ(X)
is
Hermitian
for X C
Έ\
2.
Φ(X
+ a)
=
τ
a
Φ(X)
for X C
Έ
v
and a e
Έ
v
.
3.
The
union
#
φ
of all X
such that
X
9
0 and
Φ(X)
φ
0 is a
finite
subset
of
Έ
v
.
[Physically only particles situated
at the
points
x e
R
φ
have
a
non-
zero sinteraction with
a
particle
at the
origin.]
The
Hamiltonian
of a
finite
system
A
with interaction
Φ is
defined
by:
H
Φ
(A)=
Σ
Φ(X)
XcΛ
In
[2] it was
established that each interaction
Φ
defines
a
strongly
continuous, one-parameter group
of
automorphisms
τf of
s$.
Explicitly
we
have
for
each
A e
,$/
and t e R an
element
τf
(A)
e
s/
such
that:
lim
\\τ*(A)-e
itH
*
(Λ}
Ae-
itH
*
(Λ}
\\
=0
Λ-+OO
\im\\τf(A)-A\\=0
ί-»0
τf(AB)
=
τf(A)τf(B),
etc .....
Quantum
Spin Systems
253
In
fact
t
->
τf(A)
with
A
E
s/
Λ
is
analytic
in a
strip
|
Imt
| <
a
φ
with
a
φ
> 0
and
further:
1
**
ffί
,
,N
^
lim
Λ->
oo
For
details
see
[2].
3.
Local
Commutativity
Our aim is to
discuss
the
behaviour
of
commutators
for
A and B
strictly local,
i.e.
contained
in
some
<$/
Λ
.
We
wish
to
examine
the
magnitude
of
these commutators
for
large
x and t and to
show that
information
propagates with
a finite
group velocity
V
φ
.
More precisely,
we
have
the
following.
Theorem.
For
each
finite
range
interaction
Φ
there
exists
a finite
group
velocity
V
φ
and a
strictly
positive
increasing
function
μ
such that
for
v>V
φ
lim
e
μ(v
^
||
[τf
τ
x
(A),
B]\\
=0
|f|->oo
\*\>v\t\
for all
strictly
local
A and B.
Proof.
First
we
note that
it is
sufficient
to
prove
the
theorem
for
A,
B
E
j/
{0}
.
This
is
because each strictly local
A,Bes/
Λ
can be
written
as a
polynomial
in
elements
of
j/
w
with
x
E
A.
Hence
the
norm
of the
general commutator
can be
bounded
above
by a finite sum of
norms
of
similar
commutators
but
with
A E
s/
{x}
,
B E
s/^
y}
and x, y E A.
Using
translation invariance each
of
these commutators
can be
reduced
to a
commutator with
A,BE
j/
{0}
.
As
J^Q
is
finite
dimensional
we can
choose
a
finite
basis
α
l9
...,
α
N2
of
j^
{0}
closed under multiplication with
||
α
{
\\
= 1 and
such that every
A E
^
{
Q
}
has a
unique decomposition
of the
form
N
2
Further,
if
Ae
<s/
(xι
>>t<
jXΠ}
,
then
A has a
unique decomposition
as a
polynomial
N
2
N
2
n
Σ
e(i^...,i
n
\A)l\τ
x
.(a
i
),
in=l
254 E. H.
Lieb
and D. W.
Robinson:
Next,
with
B e
£/
{0}
fixed,
consider
and
From
the
definition
of the
time translation automorphisms
and
their
properties cited
in the
previous section,
one
obtains
the
differential
equations
,
—
C
i
(x,t)
=
i Σ
\τ?τ
x
([_Φ(X),a3),B\.
For
each
X in
this
sum,
the
corresponding
Φ(X)
can be
written
as
a
polynomial
in the set of
elements
τ
y
(a^
yeX,
j=l,2,
...,N
2
.
The
commutator
D
t
(X)
=
[Φ(X\
a^\
is
then
a
polynomial
of the
same kind.
Each monomial,
M,
in
Dι(X)
is of the
form
TT
yeX
Hence
τf
τχ
(M)=γ\τfτ
x+y
(a
J(y)
).
yeX
The
commutator
[τf
τ
x
(M\
B~]
will have
N(X)
terms
(the
number
of
points
in
X),
each obtained
by
taking
the
commutator
[τf
τ
Λ+y
(α
ί(y)
),
B~]
and
leaving
the
other elements
in
τfτ
x
(M)
as
coefficients.
Each
of
these
coefficients
has
norm
one
(the
a^
have norm
one and
automorphisms
preserve
the
norm). Hence
and
N
2
^
v—\
v~~ι
^
Σ
Σ<
yeR
φ
j=l
N
2
^
v^
v^
^
Σ
Σ«
yeΛ
φ
j=l
where
dμ(Φ\y)
is a
non-negative
coefficient
that depends
on the
inter-
action
Φ
and the
point
v,
whilst
d
ti
(Φ)
= max
d
n
(Φ;
y).
Using
the
triangle
yeflφ
inequality
it is
easy
to
verify
that
||
dί
where
2f
t
denotes
the
upper derivative,
i.e.
Quantum
Spin Systems
255
Thus
FI
satisfies
the
differential
inequality
and the
initial data
F
i
(
where
ω
[For
the
sequel note that
we
could have used
the
analyticity
of
τf
,
etc.
to
deduce
the
partial
difference
inequality
for
h
small, where
K is a
positive constant. This would have avoided
the
introduction
of the
upper derivative
to
circumvent
the
possible non-
differentiability
of
FJ.]
Next, consider
the
Green's
function
G*(x,
ί)
defined
by the
differential
equations
and
(&
is the
adjoint
of
Subsequently
we
shall
use the
non-negativity
of the
coefficients
dij(Φ)
to
conclude that
the
G\
are
non-negative,
for all
£>0,
x,
f
and fc,
that they
are
real analytic
in
£,
and
that there exists
a
constant
F
φ
> 0,
and
an
increasing
function
μ(υ)
> 0
such that
lim
e
μ(f;)ί
G?(x,ί)
=
0,
t;>F
φ
.
ί->oo
\x\>vi
For the
moment
we
accept these properties
and
reconsider
the
dif-
ferential
inequalities
satisfied
by the
F
t
.
We
deduce
from
the
non-
negativity
of the Gf
that
X
ΣvJdί
i=l
xeΈ
0
^Σ
i=l
xeΈ
0
Upon integrating
by
parts
on the
left
hand side,
a
process
which
is
legitimized
by the
real analyticity
of
G\
in t and the
interpretation
of the
differential
inequality
as a
partial
difference
inequality,
we find
that