Download Spontaneously Broken Symmetries

ESI
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Institute for Mathematical Physics
Boltzmanngasse 9
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Spontaneously Broken Symmetries
H. Narnhofer
W. Thirring
Vienna, Preprint ESI 490 (1997)
Supported by Federal Ministry of Science and Research, Austria
Available via http://www.esi.ac.at
September 25, 1997
UWThPh-1996-56
ESI 490
Spontaneously Broken Symmetries
H. Narnhofer and W. Thirring
Institut fur Theoretische Physik
Universitat Wien
Boltzmanngasse 5, A-1090 Wien
Abstract
The possibility of broken symmetry can be decided in the xpoint algebra of the
symmetry group by solving an appropriate eigenvalue problem. The method is applied
for the fermi algebra and spin systems.
1
1 Introduction
Spontaneous symmetry breaking is one of the main themes of physics in the past decades
and crucial eects in elementary particle physics and condensed matter physics are attributed to it [GHK]. Since this phenomenon was mainly studied within the context of
quantum eld theory it became considered as typical for innite quantum systems. We
shall study it in this paper in the context of general dynamical systems and we will realize that the mechanism of symmetry breaking can be found also in some nite systems.
For nite systems one is used to have a unique Hamiltonian determining the dynamics,
and the degeneracy of its pure point spectrum is responsible for spontaneous symmetry
breaking. If the system allows inequivalent representations (which is always the case for
classical systems and for innite quantum systems) then the symmetry breaking is related to the unitary implementability of automorphisms. But this implementability has
dierent consequences if one is in a pure state or in a mixed state. We shall consider
in detail only the simplest case of a discrete symmetry group and there it is possible
to decide whether spontaneous symmetry breaking can occur by examining whether an
appropriate operator has eigenvalue one.
We will use this strategy to show e.g. that translation invariant states on the fermi
elds have to be even and that symmetry breaking in the X-Y model is only possible for
appropriate parameters in the ground state but not in faithful states like temperature
states. In particular relativistic QFT with a Poincare-invariant vacuum cannot give a
nonvanishing vacuum expectation value of a fermi (i.e. anticommuting) eld irrespective
of whether the vacuum is degenerated or not as long as it is translation invariant. This
is surprising in supersymmetric theories where bose and fermi elds can be mixed and
bose elds can certainly have nonvanishing vacuum expectation values.
In Section 2 we shall give the general denitions and recapitulate in various examples
where symmetry breaking does occur or not. We shall see that the interesting examples
have the form of a crossed product and in Section 3 we shall derive the main theorem
which states when for such a structure spontaneous symmetry breaking can occur. Our
results are not limited to vacuum or equilibrium states but we need some properties of
the time evolution of the algebra invariant under the symmetry. This will shed some light
on the question why the innite spin chain and the fermionic chain behave so dierently
in this respect though for nite chains they are isomorphic. Finally we shall illustrate the
power of this theorem by applying it to the rotation algebra.
2 Denitions and Examples
A dynamical system (A; ) consists of a C*-algebra A (the \observables") and a one
parameter group t of automorphisms of A, t 2 R or Z (the \time evolution"). A state
! : A+ ! R+ is called invariant if ! = ! and extremal invariant if it is not a convex
combination of dierent invariant states. A symmetry 2 AutA is said to be dynamically
broken if [; ] 6= 0 and spontaneously broken by an invariant state ! if [; ] = 0 and
! 6= !. We are interested only in the latter case.
2
For the question whether a symmetry can be broken spontaneously we can restrict
ourselves to study extremal invariant states because if they respect a symmetry so does
a mixture of them. Since R and Z are amenable groups there always exists at least one
invariant state ! by taking the mean over t of ! t, ! any state, and one invariant state
with respect to and by averaging ! n over n. Thus the only remaining question is
whether there are extremal -invariant states which break .
Examples (2.1)
1. Classical mechanics in one dimension.
A = ff (x; p)g, given by the ow of the Hamiltonian H = p2 + V (x), : (x; p) !
(?x; ?p) (parity) is dynamically unbroken if V (x) = V (?x). Invariant states are
given by probability distributions (H ).
(a) V = x2. The states c (H ? E ), E > 0 are extremal invariant and is never
broken spontaneously.
(b) V = x4 ? x2. The states c (H ? E ), E > 0 are extremal invariant and do
not break spontaneously. For ?3=16 < E < 0 (H ? E )(x) is extremal
invariant and is broken spontaneously. Of course, generally if ! is invariant so
is 21 (! + ! ) and does not break but is not necessarily extremal -invariant.
(c) V = x2 +1=jxj. Here all invariant states c (H ? E ) are not extremal invariant
and its components c (H ? E )(x) break .
Thus the extremal invariant states may (a) never, (b) sometimes, (c) always break
but there are always invariant states where remains unbroken.
2. Finite quantum systems where the Hamiltonian H has pure point spectrum.
A = B(H), t(A) = eiHtAe?iHt, A 2 A. A symmetry is given by any unitary V 2
A, (A) = V AV ?1, [V; H ] = 0. The extremal invariant states are !i (A) = hijAjii,
H jii = Eijii. We see that if Ei 6= Ej 8 i 6= j all symmetries of the dynamical system
(A; ) are of the form V (H ) and cannot be broken spontaneously. If for some i 6= j
we have Ei = Ej there is some V 6= 1 in the degeneracy space [V; H ] = 0 and V
is not a function of H . The corresponding symmetry is broken spontaneously by
!i but not by 21 (!i + !j ). Thus there is spontaneous symmetry breaking i there is
degeneracy.
3. Innite quantum systems
Here new features appear since may not be implemented by some eiHt and if it
is in some representations H may have continuous spectrum and may not belong
to A. Then V (H ) = eif (H ) will generally not generate an automorphism of A. This
new situation arises already for
3
(a) Free fermions
The CAR algebra A is generated by the a(f ), f 2 L2, [a(f ); a(g)]+ = 0,
[a(f ); a(g)]+ = hf jgi, t(a(f )) = a(e?ihtf ), h = h 2 L(L2), (a(f )) =
?a(f ).
Theorem (2.2)
If h has purely absolutely continuous spectrum cannot be broken spontaneously.
Remark. The theorem appears absurd since the following seems an easy
counterexample.
Take a translation (= x) invariant Hamiltonian H and add to
R
it " dx(a(x)+a (x)). Assume this creates another time evolution " 6= ?1" ,
["; x] = 0. This " will have a KMS state ! which we may assume to be invariant, if it is not we can average !x over x. Then ! is KMS with respect
to ?1" and thus ! 6= ! since a state cannot be KMS with respect to the
two dierent automorphisms " and ?1" . Thus ! is -translation invariant
but not invariant and we have a contradiction to the theorem in its general
form since the generator of translations has a purely absolutely continuous
spectrum. The solution to this puzzle is that " does not exist since the formal
expression H" does not generate an automorphism of A.
Because the theorem is surprising we shall give its easy proof.
Proof: We have to show that ! = ! implies !(A) = 0 whenever A is
an odd polynomial in the a(f ) and a(f ). Since h has absolutely continuous
spectrum 8 f; g, " > 0 9 T such that
k[a(f ); ta(g)]+k = jhf jeihtgij < " 8 t T:
Therefore also for any odd polynomial A 9 T such that k[A; tA]+k < "
8 t > T . Thus since j!(A)j2 21 !([A; A]+) we see
0 N
1
X
2
N
X
1
1
j!(A)j2 = N !( Tk A) 2N 2 ! @ [ TkA; Tj (A)]+A
k=1
1j;k=1
0 N
X
= 2N1 2 ! @ [A; (j?k)T A]+A 21N + N (2NN?2 1) " if kAk 1:
j;k=1
Since this holds for all " > 0 and N we conclude !(A) = 0.
Remarks
1. We see that the proof is more general, need not to be quasifree but only
asymptotically anticommutative.
4
2. The result does not extend to the bigger gauge group (a) = ia. By
a translation invariant Bogoliubov transformation one can construct a
translation invariant state ! with ! (a(f )a(g)) 6= 0.
(b) The spin chain (X-Y model)
A = Nj2Z M2(j) generated by f~ig. Let be determined by the (formal) Hamiltonian
X
H = ?J [(1 + c)jxjx+1 + (1 ? c)jy jy+1 + 2jz ]; c 2 (0; 1); 2 R
j
and
Here we have the
jx;y = ?jx;y :
Theorem (2.3) [AM]
(a) jj 1 : 9! ground state. For each 9! KMS state.
(b) jj < 1 : 9 2 ground states. For each 9! KMS state.
Corollary. Since ! has the same KMS or ground state properties as !,
no symmetry is broken in KMS states but is broken in the ground states i
jj < 1 since they are not invariant.
4. The quantum CAT (= quantized torus)
Here A is the subalgebra of the Weyl algebra linearly generated by W (~n) =
ei(n1x+n2 p) when ~n is restricted to the lattice Z 2. It is characterized by the relations
W (~n)W (m
~ ) = ei(~n;~m)W (~n + m
~)
where ~n; m
~ 2 Z 2. (~n; m
~ ) is the symplectic form n1m2 ? n2m1 and 2 (?1; 1]
plays the r^ole of Planck's constant. Linear symplectic transformations of ~n give
automorphisms of the Weyl algebra and they leave A invariant if they have integer
coecients. Thus we consider a discrete time evolution T (W (~n)) = W (T~n) with
T 2 Sp (2; Z ) = SL(2; Z ) or explicitly
!
a
b
T = c d ; a; b; c; d 2 Z and ad ? bc = 1:
The shifts ~k W (~n) = eih~kj~niW (~n) are symmetries of (A; T ) i h~k ? T ~kj~ni = 2g,
g 2 Z 8 ~n, which holds if (1 ? T ) ~k 2 2Z 2. Nevertheless the spontaneous
symmetry breaking in this seemingly nite quantum system is severely restricted
by the
5
Theorem (2.4) [NT1, N1]
For any T 2 SL(2; Z ) nf1g with probability 1 in there is only one invariant state
of (A ; T ), namely the trace !0(W (~n)) = ~n;0.
Corollary. With probability one in there is no spontaneous breaking of any
symmetry of (A; T ).
Proof: !0 is also T invariant, !0 T = !0 T = !0 and the latter
must equal !0 since it is also tracial and the trace is unique.
Remarks (2.5)
1. Probability is with respect to the Lebesgue measure of (?1; 1). The statement
does not hold for = rational and a countable subset of irrationals but it
holds nevertheless for almost all .
2. In physics it is quite unusual that there is only one invariant state over a local
C*-algebra, one expects at least KMS states for all temperatures. T can be
extended to a continuous dilation automorphism over the whole Weyl algebra
but this automorphism does not have KMS states [NT2].
3. For rational it is shown in [N1] that spontaneous symmetry breaking is
possible by some states not normal to the trace-representation. In the classical
case = 0 T has some periodic orbits and taking the average of the functions
on such orbits gives other invariant states which may break a symmetry. For
= 1=2 A1=2 is generated by 2 anticommuting unitaries u; v, uv = ?vu and
it is easy to nd other invariant states, for instance, if
!
m
1
2
T = 0 1 ) T (u) = uv2; T (v) = v; !(vnum) = 1 + (2?)
is T invariant. T commutes with the symmetries 1(u; v) = (?u; v), 2(u; v) =
(u; ?v) and ! does not break 1 but it does 2. In the next section we shall
develop a systematic way of constructing such states.
Our ndings so far are actually more general since the symmetry breaking properties
of 's which are conjugated with automorphisms respecting are the same
Lemma (2.6)
Let 2 Aut A commute with and e = ?1 . Then (A; e) can break spontaneously
if and only if (A; ) does.
Proof: Let !e e = !e 6= !e . Then !e ?1 is -invariant, !e ?1 = !e e ?1 = !e ?1
but breaks : !e ?1 = !e ?1 =
6 !e ?1. Since is invertible the conclusion goes
in both directions.
6
Remarks (2.7)
1. For nite quantum systems where all automorphisms are unitarily implementable,
= ad eiB := eiB e?iB , = ad G, [G; B ] = 0, and H has a pure point spectrum
then the Lemma simply says that H and (H ) have the same spectrum such that
the degeneracy criterium works the same way in both cases.
2. Let a(f ) ! a(Mf ) + a(Nf ) be a Bogoliubov transformation in example 3(a). It
respects and thus the Lemma says that if a Hamiltonian can be diagonalized by
a Bogoliubov transformation and then exhibits an absolutely continuous spectrum
cannot be broken spontaneously.
3. If ; e 2 Aut A both respect and approach each other asymptotically such that
the Mller automorphism limt!1 ?1et = 2 Aut A, limt!1 e?t t = ?1, exists
then the spontaneous symmetry breaking properties of (A; ) and (A; e) are the
same.
3 Crossed products
There is a striking dierence between the fermionic and the spin chain, though for nite
chains they are isomorphic: translation invariant states for fermions are even whereas for
the spin chain any constant magnetic eld not in the z-axis can be used to construct
as equilibrium states states which are translation invariant and break the symmetry
x;y = ?x;y. Yet the two systems have many things in common:
3.1 Relation of Af and As
The algebra Ae of xpoints of of the two algebras Af and As are isomorphic
As Ae = fpolynomials in ~k where #x + #y is eveng
Af Ae = fpolynomials in ak ; a` where #ak + #a` is eveng.
They are identied via
jz = aj aj ? aj aj
kY
?1
jxkx = (aj + aj ) (a` a` ? a`a` )(ak + ak ):
`=j
Remark. For nite chains also Af Qand
As can be identied via the above Klein transk
formation. But for the innite chain `=?1 (a` a` ? a`a` ) is ill dened, whereas for Ae it
reduces to a well dened nite product.
7
3.2 Af and As as crossed products
It is possible to pass from Ae to Af resp. to As by adding a missing element, e.g. (a0 + a0)
for Af and 0x for As. These elements are unitary and implement an automorphism resp. on Ae, satisfying 2 = 2 = id. Thus
As = Ae Z (2); 2 = 2 = id:
Z (2) is the group with two elements. Ae is embedded in Af resp. Ae as xedpoint algebra
of the gauge group, A = A, A 2 Ae. Gauge invariant automorphisms , = on Af
resp. As are also automorphisms on Ae and can be regained from in a unique way.
States on Ae with ! = ! can also be enlarged to states on Af resp. As and whether
Af = Ae Z (2);
this extension is unique (no symmetry breaking) or not can be decided on the level of
Ae.
In the following we will demonstrate these statements.
Proposition (3.3)
Let Ae be a C*-algebra,
2 Aut Ae, 6= ad V , V 2 Ae, 2 = ad W , W = W , W 2 Ae.
Let Ab = Ae Z (2). Then
!
A
B
(i) Ab = (B )W (A) is a C*-subalgebra of Ae M2.
(ii) can be extended to Ab and there it is inner, b = ad
!
0 1 .
W 0
!
1
0
2
b
(iii) There is a grading on A, i.e. an automorphism , = id, = ad 0 ?1 2
Aut Ab. need not be inner, b = b and Ae is the xpoint algebra of Ab with
respect to .
(iv) Automorphisms that are equivalent to modulo inner automorphisms, i.e. x =
ad x , x 2 Ae, lead with Wx = x (x)W to algebras Abx Ae M2, which are
unitarily related
!
!
1
0
1
0
b
b
Ax = 0 x A 0 x such that is respected.
The various claims are veried by straightforward calculation and do not warrant an
ingenious proof.
8
Remarks (3.4)
1. If 2 = 1, W = 1, Ab is isomorphic to the crossed product Ae Z (2) [BR, T]. Thus,
if W can be written as W = (x)x then Ab is unitarily equivalent to a crossed
product.
2. Conversely, given an algebra Ab with a grading 2 Aut Ab, 2 = 1, and given an
odd unitary element u = ?u 2 Ab then Ab can be written as sum of its even and
odd part
C = A + Bu; A = A; B = B; A = 12 (C + C ):
With = ad u, W = u2 the multiplication law is the same as the one of the
matrices in (3.3.i). Subsequently we shall concentrate on the special case 2 = 1.
The general case only makes the notation a little heavier. It is considered e.g. in
[N1].
3. If Ae has trivial center, 2 = ad W implies already W = W .
Proof: 3 = ad W = ad W = ad W and therefore W W = c1
together with 2W = W this implies c = 1.
Proposition (3.5)
Let be an automorphism of Ae. It can be extended to b 2 Aut Ab, if and only if
V 2 Ae :
Then V ?1 = V = (V ). For algebras with trivial center b is unique
up to b ! b . If
!
U
0
is implemented by U , then b is implemented by 0 V U .
1 ?1?1 = ad V;
Proof: Ab is generated algebraically by Ae which is isomorphic to the subalgebra of
elements of the form
!
A 0 ;
0 A
A 2 Ae ;
and
!
0 1 :
1 0
!
0
1
Thus b is determined by its action on 1 0 . Since it maps even elements into even
elements as bijection it has to map odd elements into odd elements. Therefore there must
be a V 2 Ae such that
!
!
0
1
0
V
b 1 0 = V 0 :
9
!
!2
!
!2
0
1
0
1
1
0
0
V
Unitarity of 1 0 demands V = V , 1 0 = 0 1 = V 0 implies
V V = V V = V V = 1:
!
0
1
Since 1 0 implements b we have
!
!
!
!
!
0
1
A
0
0
1
A
0
V
AV
0
b 1 0
0 A
1 0 = 0 A =
0
(V AV )
which proves ?1?1 = ad V . V is xed up to unitaries z of the center Z . For Z = C 1
with (V ) = V only z = 1 is allowed, which corresponds to the freedom b or b .
Conversely, if ?1?1 = ad V , then ?1 = ad V = ad V and V can be used
for the extension b. If b exists, [b; ] = 0 and = ad U , then (3.5) holds with V = b(U ),
U 2 Ae .
Proposition (3.6)
If ! is a unique invariant state on Ae there exists a b invariant state !b with !b = !b ,
!b jA = !. Furthermore any extension !b 1 of ! is dominated by 2!b .
e
Proof: In the GNS representation ! of ! we have !(A) = h
j! (A)j
i. We dene
A) ! (B )
!b (A + Bu) = 0 !((B
) ! (A)
!
*
+
:
0
This state satises obviously all requirements. The second claim follows from !b = 21 (!b1 +
!b1 ) 21 !b1.
Proposition
(3.7)
+
, 0 =
6 2 H! extends ! i can be unitarily implemented in the representation,
i.e. (A) = WAW , W = unitary 2 B(H! ) for = W (we write A instead of ! (A)).
+
Proof: gives extension of ! , h j(A)j i = h
jAj
i. If (A) = WAW take
j i = W j
i. Conversely, if h j(A)j i = h
jAj
i, dene WAj
i = (A)j i 8 A 2
A. W is densely dened, unitary, and h
jB W AWC j
i = h j(B ?1 (A)C )j i =
h
jB ?1(A)C j
i ) W implements . W is not uniquely xed, B 0W with B 0 a unitary
from ! (Ae)0 is just as good.
If is cyclic and separating we shall normalize W by requiring W = JWJ =: W 0
where J generates the canonical conjugation ! (Ae)0 = J!(Ae)J [BR]. If ! (Ae) is a
factor together with !(Ae )0 it generates all of B(H! ). Since W 2 generates the identity
10
transformation on both in this case we have W 2 = 1 or W = W ?1 = W .
However, this extension is not necessarily b-invariant. There remains the problem
whether !b is the only b invariant extension of ! and if not, whether the other extensions
break the symmetry. As we have seen it is necessary for symmetry breaking that !b can
be decomposed into other b invariant states. We use the following well established fact
[BR]
Proposition (3.8)
If !b1 is dominated by a multiple of !b , then there exists a unique positive operator T 2
!b(Ab)0 such that
!b1(A) = h
!b j!b(A)T j
!bi:
b Ub = T , where Ub implements b and Ub j
!b i = j
!bi. !
!b1 is b invariant, i bT := UT
A0 W ,
One nds easily that the commutant b! (Ab)0 are operators of the form W
A0
where A0 2 ! (Ae)0 and W! (A) = ! (A)W 8 A 2 Ae. >From 2 = 1, (A) = (A)
we infer that W 2; W W; WW 2 ! (Ae)0. Such a W will not exist if the representations
! and ! are disjoint. In this case there are sequences An * 0, (An) 6* 0 which
contradicts WAn = (An )W for W 6= 0.
Proposition (3.9)
Assume that ! and ! are disjoint. Then !b cannot be decomposed into gauge breaking
invariant states.
Proof: In this case
b! (Ab)0 =
(
!
A0 0 ;
0 A0
A0 2 !(Ae )0
)
:
!b can only be decomposed by elements with A0 = A0, which just correspond to decompositions of ! and do not mix between the even and odd part.
Corollary (3.10)
If ! is extremal invariant for A and ! and ! are disjoint, then !b is b extremal
invariant for Ab and cannot be broken by !b .
Theorem (3.11)
Let ! denote a -invariant state over Ae faithful over ! (Ae)00 and fU g0 \ ! (Ae)0 = 1
where is implemented by U and extends to B(H!). U is chosen in the standard way
U = JUJ where J is the modular conjugation of !. Let be unitarily implemented.
Then ! has a breaking extension i one of the four following equivalent conditions
hold.
11
(i)
(ii)
(iii)
(iv)
!
1
W
There exists W implementing such that T = W 1 2 b! (Ab)0 is -invariant,
V = (W )W = (W )W (V from (3.5) and the second equality is implied by
(V ) = V from the rst),
W can be written as B 00W0, B 00 2 ! (Ae)00, unitary, W0 2 B(H!), [W0; U ] = 0,
V = (B 00)B 00 = W0 (B 00)B 00W0,
9 j i 2 H! , U j i = V j i
Proof:
breaking , (i): A0 in (3.8) has to be a c number, the choice c = 1 makes T positive.
(i) , (ii):
U 0
0 V U
!
! !
!
!
0 W
U 0 =
0
U W U V = 0 W
W 0
0 U V
V U WU
0
W 0
is equivalent to (ii).
(iii) ) (ii):
(W )W = UB 00W0U W0B 00 = (B 00)B 00 = V
(W )W = UW0B 00UB 00W0 = W0 (B 00)B 00W0 = V:
(ii) ) (iii): With the standard W = W = JWJ (3.7) we have UWU W = V V 0,
V 0 = JV J 2 ! (A0e) and we can write W = B 0W , B 0 2 ! (Ae)0. Then
V = (W )W = UB 0WU WB 0 = (B 0)V V 0B 0 ) V 0 = (B 0)B 0 ) V = (B 00)B 00
with B 00 = JB 0J since J commutes with . Now dene W0 = B 00W , then
W0 (W0) = W B 00 (B 00) (W ) = W V V W = 1 ) (W0) = W0:
Finally
V = (W )W = (W0B 00)B 00W0 = W0 (B 00)B 00W0:
(ii) ) (iv): j i = W j
i.
(iv) ) (iii): Since ! is faithful for Ae and
AW j
i = 0 , h
j(AA)j
i = 0 , A = 0
we see that W j
i is separating for Ae and thus cyclic for ! (Ae)0. Since we assumed
that ! is the only invariant state in the folium of ! we have h jAj i = !(A)
12
8 A 2 (Ae)00. Hence we may write j i = B 0W j
i, B 0 2 ! (Ae)0, B 0B 0 = 1. If
B 0W j
i = V UB 0WU j
i we have
0
0 0
j
i = WB 0V |UB{z0U } UWU
| {z W} W j
i = (B (B )V )j
i
(B 0 )
VV0
by extending to ! (Ae )0. Since j
i is separating for !(Ae )0 we conclude
V 0 = (B 0)B 0 ) V = (B 00)B 00:
At this point we can identify B 0W with W from (ii) and proceed as in (ii) ) (iii).
Remark (3.12)
If ! is a pure state (for instance, a ground state) then the symmetry can only be
broken if is inner in ! (Ae)00 since it is all of B(H!). If W 2 !(Ae )00 the algebra
generated by A + BW 2 ! (Ae)00 with A; B 2 ! (Ae) is isomorphic to ! (Ae )00. Thus
in this case the \fermionic operators" (odd elements) can be obtained as strong limits of
\bosonic operators" (even elements). On the other hand, if is inner in !(Ae )00, (iii) is
satised with an appropriate choice for the sign of V which then determines b. That the
symmetry is actually broken follows since h
j! (B )j i = 0 8 B 2 Ae implies = 0
since j
i is cyclic for Ae.
3.3 Cluster properties
The uniqueness of the invariant state ! as assumed in (3.11) is guaranteed if (Ae; ; !) is
weakly asymptotically abelian in its folium and ! is extremally invariant. The dynamical
system (Ae ; ; !) is clustering which is equivalent to saying that U has one eigenvalue 1
(eigenvector j
i) and otherwise a continuous spectrum. We shall now investigate whether
these properties carry over to (Ab; b; !b ).
a) No spontaneous symmetry breaking
!
U
0
We have to establish the spectrum of Ub = 0 V U . By assumption V U
does not have an eigenvalue 1 and we shall rst show that it has no other proper
eigenvalues. From V j i = eiU j i we would conclude as in the proof of (3.11),
(iv) ) (iii) that V = ei (B 00)B 00. We shall show that for 6= 0; this is incompatible with (V ) = V : ei(V ) = e2i (B 00)B 00 = WB 00 (B 00)W where we
have again W = B 0B 00W0. Thus the equation continues = W0B 0 (B 00)B 0B 00W0 =
(W0B 00W0)W0B 00W0 or (B 00W0B 00W0) = e2iB 00W0B 00W0. But our assumptions on (Ae; ; !) imply that ! (Ae)00 does not have an element 6= c1 with periodic
time+dependence thus e2i = 1, B 00W0B 00W0 = c1. Thus Ub has only the eigenvector
if there is no -breaking. (ei = ?1 corresponds to our previous ambiguity
0
13
V ! ?V .) For a good criterium to exclude the singular continuous spectrum further knowledge on the spectrum of U and the relation between and would be
necessary.
Proposition (3.13)
If there is no spontaneous -breaking then V U has only continuous spectrum and
(Ab; b; !b ) is -abelian (i.e. invariant means of operators are c numbers in ! ). If
the spectrum of V U is absolutely continuous the system is weakly asymptotically
abelian and extremal invariant states are clustering.
b) Spontaneous -breaking
Since we can expect clustering only for extremally invariant states we rst have to
decompose !b with -invariant projections from !b(Ab)0. This is easily done since
we will learn from a) that W = B 00W0 is actually hermitian. We know that
W is hermitian, W = B 0B 00W0 = W = W0B 00B 0 and from a) we get with
an appropriate choice of the phase B 00 = W0B 00W0. Combining the two we see
B 0B 00W0 = W0B 0W0B 00W0. Thus with J J of the rst relation we have B 0 =
W0B 0W0 = W0B 0W0 ) W0 = W0 and W = B 00W0 = W0B 00 = (B 00W0) = W .
!
1
W
1
Thus P = 2 W 1
are -invariant projectors 2 !b(Ab)0 and the states
!b () = !b (P ) are extremally invariant. They correspond to the vectors
+
1
b
b
b
j
i = P j
i=kP j
ik = p2 W :
For them we have indeed clustering and weak asymptotic abelianness since Ub n
converges weakly to the projector onto the Ub -invariant subspace. With V U =
W we have
WU
!
!
bU n = U n 0 n * j
ih
j 0 = j
b + ih
b + j + j
b ? ih
b ? j:
0 W j
ih
jW
0 (WU W )
Further
j
ih
j !
j
ih
j
W
n
b
w- lim P U P = j
ih
jW W j
ih
jW
is a one-dimensional projection.
Proposition (3.14)
For spontaneous symmetry breaking weak asymptotic abelianness carries over to
the extremal invariant states over Ab without further assumptions.
14
4 Examples
4.1 A baby model
To illustrate that it is not so much the innity of the system that opens the possibility
of symmetry breaking but on the contrary the existence of inequivalent representations
which can hinder symmetry breaking we consider the 1 spin system:
Ab = f1;~g;
x;y = ?x;y ;
Ae = f1; z g;
x
z
z
= ad ;
= ? ;
= id:
Therefore
!
!
z)
(
0
0
1
!
x
b!
b! ( ) = 1 0 :
0 ?!(z ) ;
(i) If ! is pure, i.e. ! (z ) = 1 = (z ), then are inequivalent representations
and b! is irreducible. The extension !b (z ) = 1 is by (3.9) even, generally !(~) = ~s,
k~sk 1 and j!(z )j = 1 implies !(x;y ) = 0.
(ii) If ! is mixed, i.e. !(z ) = cos2 ? sin2 , then H! is two dimensional and we
span B(H!) by Pauli matrices f1;~g. It contains x 2= Ae that implements . Since
= b = id, (3.11.iii)
is trivially satised with V = 1. If we represent ! by the
cos +
vector = sin , then we take j i = xj
i and
! +
* 1
0
1
1 0 = 2 cos sin !b (x) = 2
(z ) =
breaks the symmetry and Ab is irreducibly represented.
(iii) If we take instead b = ad (z )t (notice that Ae is commutative, therefore the
extension is not unique) and if we write again ! (z ) = z , Ubt = (z )t (z )t then
(3.11.iii) with W = x cannot be satised and the symmetry (which equals b)
remains unbroken. Thus our intuition is correct that for a state invariant under
rotations around the z-axis !(~) has to point in the z-direction.
This baby may be named SUSY since it is the simplest realization of N = 2
supersymmetry. Ab is generated by two fermionic charges (Q1; Q2) = (x; y ),
[Qi; Qj ]+ = Hij , H = 2. The Witten parity iH1 [Q1; Q2] = z generates Ae . Thus
our crossed product realization coincides with the 2 2 matrix representation of
supersymmetry which is nothing but the -symmetry. H > 0 says that it can be
broeken by a ground state [J]. Our construction makes the two diagonal element of
H isomorphic and thus does not cover the case when their zero eigenspaces have
dierent dimensions.
15
4.2 The spin chain
Here Ab is the innite tensor product M2 of 2 2 matrices which we generate
by the usual Pauli matrices. The grading is iz = iz , ix;y = ?ix;y 8 i. Ae =
fpolynomials in ~; #x + #y eveng and ad 0x can be used to create Ab out of Ae . For
we consider the shift b(~i) = ~i+1. It satises bb?1 ?1 = ad 1x0x and 1x0x 2 Ae, so
that the construction of (3.5) works.
Assume !e is an extremal translation invariant state and therefore clustering and
!e (0x) = a 6= 0. Then
!2
N
X
1
x
x
x
lim h
b j N b (n 0 ) ? ab (0 ) j
b i =
N !1
n=1
N
N
X
X
1
2
a
x
x
b
b
= Nlim
h
j
(nk ) ? N b (nx) + a2j
b i = 0
!1 N 2
n=1
n;k=1
and therefore
N
X
xx
b(0x) = st-lim 1
n
0
Na
n=1
00
2 (Ae) . Again
where
2 Ae , so that
(3.11.iii) is trivially satised and
e
e
the -symmetry is broken by !. We can pass from ! to ! over Ae and the gauge invariant
extension !b of ! can be decomposed into 12 (!e + !e ) according to (3.11.i).
nx0x
b (0x) = W
4.3 The fermion chain
We consider the CAR algebra for simplicity over `2 , i.e. Ab is generated by ai; ak , i; k 2 Z ,
[ai; ak ]+ = 0 and [ai; ak ]+ = ik . The grading ai = ?ai 8 i denes the even algebra Ae.
For we take the shift, but any quasifree automorphism as in (2.3a) would work in the
same way. We want to recover the result that any b invariant state !b has to be even,
!b = !b .
For we take ad (a0 + a0) which evidently maps Ae into Ae. Then
V = (a1 + a1)(a0 + a0)
or
Notice that
(4:3:1)
n ?n ?1 = ad Vn = ad (an + an)(a0 + a0):
Vn = ?Vn
(4:3:2)
which corresponds to anticommutativity, but expressed in Ae. In order to produce broken
symmetry we have to nd a solution to
V U j i = j i;
U n j i = Vn j i:
(4:3:3)
16
Now with P = w- limn!1 U n = j
ih
j
n
nlim
!1h jU j i = h jP j i = nlim
!1h jVn j i = nlim
!1hVn j i
(3:5)
?n
j i = ? nlim
!1hVn j i = ? nlim
!1h jU j i = ?h jP j i:
Therefore h jP j i = 0 hence h
j i = 0 for any solution of (4.3.3). With such a we
=
calculate
nlim
!1hVn
! +
0 1 = 2h
j i = 0:
1 0 The same argument works if we recall (3.3.iv) and replace by ad B , B 2 Ae, such
that now st-limn!1 (Vn + Vn ) = 0,
!e (b!eB (a0 + a0)) = 0;
and therefore !e does not break the symmetry.
To show explicitly that (3.11.iii) cannot be met take an even polynomial g0(a#?k : : : a#k )
and n > 2k +1. Now multiply the relation W U n W = VnU n = (a0 + a0)(an + an)U n with
g0 g0. If w-limn!1 U n = j
ih
j we see with gn = U n g0U ?n
0 jh
jWg0j
ij2 h
jg0W U n Wg0j
i
= h
jg0(a0 + a0)(an + an)gn j
i = ?h
j(an + an)g0gn (a0 + a0)j
i
= ?h
j(an + an)gn g0(a0 + a0)j
i = ?h
j((g0 ))n(an + an)(a0 + a0)(g0)j
i
= ?h
j(g0)(a0 + a0)U n (a0 + a0)(g0)j
i
= ?h
j(g0)W U n W(g0)j
i ! ?jh
jW(g0)j
ij2 0
) jh
jWg0j
ij2 = 0:
Since k was arbitrary the g0 j
i are dense in H! we conclude W j
i = 0 which contradicts
the unitarity of W .
!e (be! (a0 + a0)) =
*
4.4 The rotation algebra
So far (3.11) has not given more than (2.2). But next we will apply (3.11) to systems
which are not anticommutative. We return to the quantized torus (2.4) and choose the
rotation parameter = 1=2. With the grading
(W (n1; n2)) = ein1 W (n1; n2)
the even algebra Ae built by W (2n1; n2) is abelian. It is isomorphic to the functions on
the torus T 2 which we shall represent by W (2; 0) = u2 = e2ix and W (0; 1) = v = e2iy ,
x; y 2 [0; 1). We write A1=2 as the crossed product of Ae with the automorphism =
ad W (1; 0) = ad u, u2 = u2, v = ?v. Thus we represent A1=2
!
n um
v
0
n
m
if m = even (v u ) =
0 (?)nvnum ;
num?1 !
0
v
n
m
:
if m = odd (v u ) = (?)nvnum+1
0
17
!
3
2
As automorphism we consider T with T = 1 1 . Extremal states on Ae are point
measures on T 2. Among them the xed points of give also invariant states. They have
to satisfy
(x; y) = (x + g1; y + g1); (g1; g2) 2 Z 2;
i.e., a) x = y = 0 and b) x = 0, y = 1=4), c) x = 0, y = 3=4 are solutions.
To enlarge the corresponding states on A1=2 we calculate
?1 ?1 ! u2v = V:
Since the representation on Ae is pure, U = 1. Therefore we have to solve V =
+ +
(a) V = 1. We have symmetry breaking. The vectors 11 and ?11 dene invariant
states.
(b),(c) V = i. The unique extension is even.
One can also consider periodic orbits, that is to say, states that are given by xed points
of N , N 2 Z +. The U is then an N N matrix and so is V . Again UV has eigenvalue
1 in some (but not all) examples and symmetry breaking is possible.
4.5 The X-Y model
We have already stated the relation of the spin chain and the fermion chain (3.1). The
passage from Ae to Af resp. As uses f = ad (a0 + a0) or s = ad 0x. The two are related
via
s f = ? ;
for n 0
"n = ?11 for
?(an) = "nan; ? (nx) = "nnx; ? (nz ) = nz ;
n < 0:
This relation is the main input in the analysis of the X-Y model in [AM] of (2.2). Whereas
space translations are suciently explicit for an analysis (norm asymptotic abelian) we
have to extract the structure of time invariant states with respect to H in 2.b) from our
knowledge of the time evolution on Ae where it reduces to a quasifree time evolution with
absolutely continuous one particle spectrum after an appropriate Bogoliubov transformation. In order to be in a Fock space the Bogoliubov transformation has to be adjusted
such that the one particle spectrum is positive. Now f is a Bogoliubov transformation
that turns exactly one particle into an antiparticle, which means that it has index 1 (or
odd). In (2.3) the index of the Bogoliubov transformation ? is evaluated for the dierent
parameters in the Hamiltonian, e.g. for jj < 1, 6= 0, therefore for a suciently weak
magnetic eld in the z-direction and a suciently strong attraction of the spins if they
point in the x-direction, it is odd. Combined with f we get a Bogoliubov transformation
of even index together with a scattering transformation. This scattering transformation
18
is inner in the Fock representation, and so is the even Bogoliubov transformation. Therefore in the Fock representation we satisfy the demands of (3.12.i) and there is symmetry
breaking. But the scattering transformation is not inner in Ae and in general, especially
in faithful quasifree representations it is either not implementable or at least not inner
in ! (Ae)00 [N2]. This corresponds to the results (2.3) that for nite temperatures there
is no symmetry breaking.
If, on the other hand, the Bogoliubov transformation is even (as it happens for suciently strong magnetic elds) then the odd Bogoliubov transformation f hinders broken
symmetry as for the fermi system.
Finally we have to consider the spectral properties of V U . We consider quasifree
states. There U is a quasifree evolution of the CCR algebra built by creation and annihilation operators of the algebra and its commutant whose spectrum is absolutely continuous. V corresponds to a perturbation on the one particle level by an operator of
nite rank which thus can only change the pure point spectrum and not the absolutely
continuous spectrum. This guarantees that for all extremally invariant states the system
is clustering and otherwise the evolution is weakly asymptotically abelian.
References
[GHK] G. Guralnik, C. Hagen, T. Kibble, Adv. in Part. Phys., Vol. 2, 567 (1968)
[J]
G. Junker, Supersymmetric Methods in Quantum and Satistical Physics. Texts
and Monographs in Physics, Springer (1996)
[AM] H. Araki, T. Matsui, Commun. Math, Phys. 101, 213 (1985)
[BR] O. Bratteli, D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics, Springer, New York, 1981
[NT1] H. Narnhofer, W. Thirring, Lett. Math. Phys. 35, 145 (1995)
[NT2] H. Narnhofer, W. Thirring, Lett. Math. Phys. 27, 133 (1993)
[N1] H. Narnhofer, Vienna preprint UWThPh-1996-43, to be published in Rep. Math.
Phys.
[N2] H. Narnhofer, Rep. Math. Phys. 16, 1 (1979)
[T]
M. Takesaki, Theory of Operator Algebras, Springer, New York, 1979