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Berry's phase, commutators, and the Dirac sea
Nuclear Physics B295 [FS21] (1988) 243-261 North-Holland, Amsterdam
B E R R Y ' S PHASE, C O M M U T A T O R S , AND T H E DIRAC SEA Michael STONE and William E. GOFF University of Illinois at Urbana-Champaign, Department of Physics, Loomis Laboratory, 1110 W Green St., Urbana, IL 61801, USA
Received 22 May 1987 (Revised 2 November 1987)
We discuss the connection between the Berry phase and the Schwinger term in generators of symmetry groups, and stress the relative minus sign between the anomalous commutator of the currents and the anomalous commutator of the full group generators.
1. Introduction Careful computation of the commutators of fermion currents in a quantum field theory reveals that they do not always have the form anticipated from naive manipulations using the canonical commutation relations. Additional terms, usually called Schwinger terms [1-3], were first noticed by Jordan [4] in connection with his neutrino theory of light, and were in fact the key ingredient in this first example of a Fermi-Bose equivalence. Some years later, Tomonaga, in one of his seminal works on q u a n t u m field theory, also used the additional terms to show that a one-dimensional electron gas can be equivalently regarded as a boson system [5]. It is in this p a p e r of T o m o n a g a ' s that one sees clearly stated, for the first time, the equations of what are now called a Kac-Moody algebra. Later Schwinger [1] gave general arguments based on current conservation, the equations of motion and the existence of a ground state to show that additional terms were to be expected in all current algebras. A substantial amount of physics has grown out of these early ideas: the Bose-Fermi equivalences in two dimensions [6], the theory of Kac-Moody algebras and their connection with conformally invariant field theories and strings [7], some current algebra in four dimensions and an enhanced understanding of the origin of anomalies in gauge theories [8-15]. One can obtain these Schwinger terms in many ways, ranging from being careful with the normal ordering of the operators and then using the properties of the set of states to which the operators are to be a p p l i e d - as was done by the early authors - to observing that, in the 1 + 1 dimensional field theories which we will be concerned with here, the short distance current-current correlation function of two 0169-6823/88/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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M. Stone, W.E. Goff / Berry'sphase
fight-handed fermions goes as z-2 and thus, using the complex variable technology recently developed for conformally invariant theories [7, 16], the commutator must contain the derivative of a delta function. Other popular procedures involve separating the points at which the two Fermi operators act or taking a BJL limit [8, 3,17] to extract the commutator from Feynman diagram computations. In the last few years, starting with the work of Faddeev [9], we have begun to appreciate that these extra terms in the algebra of currents, and the implied projective representations of the symmetry groups generated by the currents, have an origin in the complex topology of the configuration spaces of the system on which the groups act [10]. There have been several works connecting the Schwinger terms with the nonabelian anomaly and the Wess-Zumino action via the descent equations [14,15], as well as directly with the Berry phase [18-21] which, in some sense, is the origin of all these effects. Despite these insights actual computations of the anomalous commutators for the full operators generating the symmetries - that is the electric field operators as well as the purely fermionic charges - seem to have caused some difficulty in that different authors report apparently incompatible results. For example there has been extensive work on the problem of anomalous commutators by Jo [8,22] who argues that the electric field operators do not commute in general, namely in 3 + 1 dimensions. In 1 + 1 dimensions, he finds that they do commute but also calculates a zero contribution from the cross-term piece. In ref. [22], Jo shows that the BJL method agrees with that of Faddeev and Shatashvili [23] in 1 + 1 dim, but not in 3 + 1 dim. Kobayashi, Seo, and Sugamoto [17] also use the BJL limit and concur with Jo's result, but no statement is made about the contribution from the various pieces. Niemi and Semenoff [12, 24, 25] take commuting electric fields and find the cross-terms between the fermionic currents and the electric fields to be zero, so the anomalous commutator of the full Gauss-law operator comes entirely from the anomalous commutator of the charges. In this paper we will argue that the cross-terms are non-zero and that the commutator of the full generators differs in sign from that of the fermionic charges. We also find ourselves at variance with the viewpoint expressed by Faddeev and Shatashvili [23] and Hosono [21], that the anomalous commutator arises because of a c-number subtraction of an A-dependent expectation value from the current. This gives the result, which we believe to be erroneous, that, in our notation (see sect. 3), the anomaly arises from 7/([A, B] s) rather than ~/([As, BS]). While we were revising this paper, two papers by Hosono and Seo [26] appeared. They use different methods to arrive a relation similar to ours. Their method is such that 8/8A acts on regulators rather than on the fermionic annihilation and creation operators as we have done. They account for differences between their covariant method and the BJL method by the difference between the consistent and covariant currents (those that give the consistent and covariant anomalies). These currents differ only by a local gauge dependent c-number redefinition. They also hypothesize
M. Stone, W.E. Goff / Berry'sphase
245
that the commutativity or non-commutativity of the electric fields depends on the regularisation procedure used. In their covariant formalism, and assuming that the electric field operators commute, they obtain our 2d relationship. They also obtain the same relationship in 4d, though with some reservations of validity. (A factor of i in one of their definitions may obscure this agreement.) Our intent in this paper is to provide a simple demonstration, in the context of quantum mechanics and in the case of two-dimensional systems, of the commutation of the full gauge generators including a careful treatment of the cross-terms between the fermionic charges and those parts of the symmetry generators which act on the Bose variables. Sect. 2 will review some of the essential geometry, in particular the way that the Berry phase provides a connexion on the bosonic part of the configuration space and, following the general theory introduced by Sonoda [20], show how this connexion influences the commutation relations of symmetry generators. In sect. 3 we will apply this formalism to show how the Berry phase of the vacuum, or Dirac sea, is responsible for the Schwinger term. In the appendix we prove a few useful relations between different formulae for the Berry phase.
2. Berry phase, connexions and commutators In this section we will briefly review the well known ideas of M.V. Berry, as interpreted by Simon in ref. [19], and discuss the significance of these ideas for the action of symmetry groups. Suppose we have a family of hamiltonians H ( h ) labeled by points h ~ ; / / where ~ ' is some manifold. In the systems of interest the parameters h will be some bosonic gauge fields or condensates, such as Higgs fields or charge density waves, and the hamiltonians H(X) will be those of a system of fermions interacting with these bosons. Consider the eigenvalue problem for each H ( h ) H ( h ) q J , = En(X)~b . .
(2.1)
If it does not become degenerate with another state at any point on the manifold, a particular state can be regarded as part of a family of states ~kn(h). Associated with this family of states there is a one-form A = A i d h i on the manifold iAn= - •
(2.2)
This is the Berry phase which would occur in the state q~n if we adiabatically changed the hamiltonian from H ( h ) to H ( h + d h ) and it can be regarded as defining a parallel transport which takes eigenstates of one hamiltonian to the corresponding eigenstate of the slightly changed hamiltonian. In more technical language, A n is a connexion on the bundle of one-dimensional vector spaces (i.e. the multiples of ~kn(X)) over the manifold ;/g.
M. Stone, W.E. Goff / Berry's phase
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As a toy, but essentially generic, example consider the family of hamiltonians corresponding to a spin-½ particle in a magnetic field [27,18] //. = #o -n,
(2.3)
where n is a point in S 2 (the direction of the magnetic field). For each n there are two eigenstates with eigenvalues _+#. If we restrict ourselves to the family of states ~p+ with eigenvalue +/~ we find that we have to be careful about the choice of phases for these states. We notice that
1--( cOS½0) ~P+-
(2.4)
sin ½0 ei~
is a smooth choice of phases for the states in any region not including the south pole (where it is ambiguous since all values of q~ correspond to the same point), while cos ½0 e - i~ ~2+= t
(2.5)
sin½O
is a smooth choice everywhere except at the north pole. Of course,
#1 = ei~+]
(2.6)
everywhere that both phase choices are defined. Eq. (2.5) is the transition function connecting the two patches. This is an example of a non-trivial fibre bundle in that there is no simple way of regarding the combination of base manifold S 2 and the one-dimensional Hilbert spaces at each point as a simple product, except in patches of the base manifold. The transition function sews the patches together in a topologically non-trivial way. There are superficially different A's for each choice:
iAl=-
(~plld~bl+)=- li(-cos0
+ 1) dq~,
(2.7)
iA2+= _ (~2+ [d~pZ+) = _ ½ i ( - c o s 0 - 1) dq~.
(2.8)
while
AI+ and A 2+ differ by a gauge transformation A I _- A +2-
dq~,
(2.9)
which is well-defined except at the poles where dq, is singular. Since these connexions or gauge-fields differ only by a gauge transformation they have the same
M. Stone, W.E. G o l f / Berry's phase
247
curvature two-form F + = d A + = - ½sin0d0 A dq)= - ½d[area].
(2 .lO)
This means that were we to adiabatically rotate the direction of the magnetic field so that it swept out a region on the two-sphere, we would find that the state would return to its original value times a phase equal to minus one-half the area of the region. The integral, divided by 2~r, of the curvature two-form over the base manifold S 2 must be an integer called the first Chern class of the bundle. In this case, since the area of the two-sphere is 4rr we have fs2F = -2~r.
(2.11)
There are possible interesting consequences when the states form a non-trivial bundle over the family of hamiltonians. If the parameters in the family of hamiltonians represent dynamical variables which are being treated in the Born-Oppenheimer approximation, it is well known that the connexion A has an effect on the d y n a m i c s - it gives rise to the Wess-Zumino term in the effective action for the Bose fields. It is also possible that there is an effect on the implementation of symmetry operations on the dynamical system consisting of the internal fermionic (in this case the spin- ~) degrees of freedom and the external Bose (orientation of the magnetic field) variables. A nice account of this has been given by Sonoda [20] and we will follow his argument in the following. Define an operator valued one-form J = J~ dXi by its matrix elements between the states ~k, ( q,,,IJI 4',,,)
-
( q,,,Id i l l +,,,)
n_-/=m
E m - E,
=0
n = m.
(2.12)
Then first-order perturbation theory and the absence of diagonal elements in J tells us that d~b, = & k ,
(2.13)
is the change in the state +n as it is Berry-transported from H to H + dH. Consequently the operator ~ r = Peers
(2.14)
must be diagonal in the ~kn basis and give the total phase change in each state after it has been Berry transported around a closed curve F. For an infinitesimal
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M. Stone, W.E. Goff / Berry'sphase
parallelogram of sides d~ 1 and d~ 2 this reduces to <~p, l P e x p ~ / l a p , ) = 1 + <~n]OIJ2 - 02J 1 -t- [J2, J1]llPn) dXl d2,2 •
(2.15)
In other words, if we define the operator valued two-form F by i F . ~ " 7tF~j 1 • d X ~A
dXj
= 1( O i j j _ O j J i _
=dJ-J
[j/, jj]) dXi A dM
2,
(2.16)
= i~[~b >(~P.lF("),
(2.17)
n it is the curvature two-form. Another expression for the expectation, in energy eigenstates, of the same curvature form is derived in the appendix. It is
(~.lJ21~.) = i F (").
(2.18)
When the variables ~ become dynamical, the wavefunctions of the combined system are q'.(X), the amplitude that the parameters have values ~, and the fermionic "internal" eigenstate is ~b.. If a Lie group f¢ acts on the manifold Jg, if: ,./g ~ ..g,
(2.19)
then, in the absence of "internal" degrees of freedom, the group would act on the wave functions by (gq')(X) = qS(g- aX)
(2.20)
and the generators of infinitesimal transformations (being a trifle unconventional in the omission of factors of i) would be the differential operators on wave functions G , , = - X a, where X a = X ~ 3 i is the right-invariant vector field of infinitesimal displacements (corresponding to multiplying on the left by elements of the group infinitesimally close to the identity). The X a will obey the defining relations of a Lie algebra: [ X o, Xb] = --fab'Xc.
(2.21)
(The minus sign in the structure constants is the one that usually occurs for right-invariant vector fields. It is compensated by the minus sign in the definition of G which itself originates in the g-1 in the action of the group on the wave
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M. Stone, W.E. Goff / Berry'sphase
function.) If we now include the action of the group on the "internal" states ~bn we can do so in a number of ways. The one chosen by Sonoda is to Berry transport the internal state using the operator J. The total group generator is now a sum of two parts G~ = - X a + J ( X a )
=
- -
(2.22)
XtOi-~ JiX~.
The commutator of these Ga generators can be evaluated by taking note that the X ' s also act on the J ' s . We find
[-- Xa + J( Xo), --
+ JC xO] = L;C-- xO +
[J(Xo), J(X )I
-- J([ Xa,
- ( X o J ( X O - X j ( X o ) --J([Xo, =f~aC(-X~+J(X~)) - (dJ-JZ)(x~,
Xb), (2.23)
where we have used d J ( X a , Xb) = X~J( Xb) -- X b J ( X ~ ) - J([ X~, X6] ) .
(2.24)
[Ga, Gb] =fabCGc -- i~aO(~),
(2.25)
O~b ( X ) = 3~ ( X a, Xb; X) = X~ X[F,j( X )
(2.26)
Succinctly,
where the anomalous piece
is made out of the Berry curvature (2.16) and the vector fields. This procedure of Sonoda's does not produce the conventional generators of the symmetry because of its deliberate suppression of the diagonal elements of J in the energy-eigenstate basis. Before we argue that it is a reasonable, if unconventional, procedure let us see how it works in the toy example of the spin-½ particle. Under a rotation, 8o~, H changes by 8 H = 8to A n . o.
(2.27)
We find that (2.28) is the operator " J " corresponding to the rotation. It differs from the more usual operator generating rotations, - ½io, by diagonal terms corresponding to our phase choices. The commutator of the J ' s is in this case diagonal (it is not in general, if there are more than two states) and equal to [Z8,ol, Zs~,~ ] = ½in. (8~ 1 A 8 ~ 2 ) ( n - o ) .
(2.29)
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M. Stone, W.E. Goff / Berry'sphase
We recognize that the diagonal terms are in fact the Berry phases for the states concerned as a consequence of (2.18). The commutator of the total generators Jso, = L • 8~o + Xso, (L being the usual angular momentum operator) is (2.30) and again we see the Berry phase operator occuring as an additional term although with opposite sign compared to the commutator of the spin generators. The additional terms in the commutation relations mean that we have a projective, or ray, representation of the action of the group on the states. However, in the case of our toy example, we know that we can replace our generators by the conventional ones and recover the usual commutation relations. The difference between the conventional and "Berry" generators is just (n. 8~0)n. o which is nowhere singular on the two-sphere. We cannot always do this if the bundle of internal states is non-trivial [11]. To see this, suppose the group acts on the manifold without leaving any point fixed, (this is not so in our toy example) then the orbit of any point on the manifold is a copy of the group manifold. Suppose further, that (as is the case of our simple example) the bundle over this orbit is non-trivial and there is therefore no section of normalized states (that is there is no way to choose representative states smoothly over the orbit), then there can be no conventional group action; if there were, we could just apply a unique group element to a state over some chosen point in the manifold and obtain a definite state at the image of our chosen point. This would be a smooth section in contradiction to our assumption. While there may be no conventional, vector (as opposed to ray) representation of the group, the "Berry" action always provides a valid implementation of the symmetry which is compatible with the topology. Any other implementation of the symmetry must be obtainable from the Berry action via a (globally defined) redefinition of the generators. A good analogy comes from parallel transport on a two-sphere: There are many possible connexions but they must all have curvature that integrates up to 4~r (2~r times the Euler character) because of the Gauss-Bonnet theorem. Having found one all the others can be obtained by the addition of a globally defined vector field to the connexion form. We cannot give a simple criterion foretelling when the Berry prescription can be reduced to a vector representation this way - the toy example shows that it is not just a question of whether the bundle is trivial or not - but there is a well-known formal framework [10,15] for stating the question. Objects like : - ( X , Y; 2~) are antisymmetric functions of elements of the Lie algebra and are functions on the manifold ~¢~'. They are therefore function valued forms on the Lie algebra (which need not always be obtained as pull-backs of forms on J-/). We can define an operation 8 which acts on them and behaves like the ordinary exterior derivative d in that it increases the number of antisymmetric slots by one and gives zero if
M. Stone, W.E.Goff / Berry'sphase
251
applied twice. On e-number one-forms s¢(X; X) we have
&s~C(X,Y,X)=Xssc(Y;X)- Yd(X;?~)-.s¢([X,Y];X)
(2.31)
and on two-forms
&~( X, Y, Z; X) = X.,~(Y, Z; X) + (cyclic permutations) - . ~ ( [ X, Y], Z; X) + (cyclic permutations).
(2.32)
We can define 8 for general objects of this type which we will call members of An(g, ~[). 8 increases the number of antisymmetric slots for elements of the Lie algebra n
n+l
8: A ( N , ~gg) --* A (~q, ~ ' )
(2.33)
and satisfies 62 = 0. If we replace G~ by Ga + ~'~ then o~ changes to o~+ 8~¢. The Jacobi identity for Ga requires 8o~= 0 so ~" is an element of the second cohomology group H2U¢, Jg). Thus the question of whether the additional o~ term can be removed is restated by asking if there exists a globally defined d e AI(~, ~ ' ) such that o~= 8 d . That is, whether o~ is cohomologically trivial*. As the toy case of the spin-½ particle shows the additions to G, need not necessarily be c-numbers. Most of the definitions of generators will differ from Sonoda's by having non-zero, but globally defined, diagonal elements (the off-diagonal ones must coincide with his). This can be taken into account by writing
f = J + ~f(")l~b.)@.l.
(2.34)
n
Here f(") is a globally defined 1-form of diagonal elements on s/g, and J is the Sonoda generator with zero diagonal elements in the energy-eigenstate basis. Then an easy computation shows that
df-f
2= d J - J 2 +
y'df(")p, = ion,
(2.35)
and therefore
[ - X a + f ( X a ) , - - X b + f ( X b ) ] =fabC(-Xc+f(Xc)) -io~(Xa, Xb), (2.36) where, in general, f ' ( X , , X,) is no longer the Berry phase curvature. * Actuallyin a field theorywe require that ~¢ be a localfunctionalof the field so somekind of "local cohomology"is needed. See ref. [10] for details.
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M. Stone, W.E. Goff / Berry'sphase
The Berry phase curvature for any state can still be found from the formula
i ~ = ( n If2ln )
(2.37)
as the diagonal terms do not contribute because of the antisymmetry.
3. 1 + 1 dimensional Fermi systems
In this section we will apply the general formalism to the specific example of 1 + 1 dimensional Fermi systems. These have the advantage that the geometric properties of the anomalous commutators are not obscured by the more complicated renormalizations needed in higher dimensions while retaining all the essential ideas [28]. Consider the fermi field ~b(x) and expand it in terms of the eigenstates of a first quantized hamiltonian Hf(A) which depends on a gauge field A.
~ ( x ) = Ea,.u,.(x),
(3.1)
A )u. = E.u. .
(3.2)
H for a fight-moving particle is, for example,
Hf(A) = - i ( O x + A x ( x ) ) .
(3.3)
There is an ordinary action of a time-independent gauge group ~(~(1) o n Hi(A ) and Un~
u.(x) L (gu.)(x) = EUm(X)(mlgln) = EUm(X)(m, glgln, g),
(3.4)
Hf(A ) L gHfg -1 = Hf( gAg -1 - Oxgg -t) = Hf( Ag).
(3.5)
Here [m) and Ira, g) represent, respectively, the single particle states u,(x) and gun(x) before and after the group action. For mathematical convenience we choose to work on a periodic interval x ~ [0, 2¢r] and g : [0, 2~r] ~ ~, g(0) = g(2~r) = identity ~ N is an element of the "loop group" N(1) of the ordinary Lie group ~¢. The restriction of g to the identity at the endpoints of the interval ensures that the group acts freely (i.e. without fixed points). We want to use this action on the first quantized theory to induce an action of the group on the second quantized theory. That is, we seek to construct operators U
M. Stone, W.E. Goff / Berry'sphase
253
(corresponding at the moment just to the " J " ' s ) which act on ~ as follows:
U(g)+U-l(g) = g - ~ ,
(3.6)
U(g)amU-l(g) = E (mlg-tln)a,,= a,,(g),
(3.7)
n
U(g)atmU-l(g) = E a ~ ( n [ g l m ) - a~(g).
(3.8)
l1
Notice that qJ = EanU n = Ean(g)(gu),. The second-quantized Fermi hamiltonian is constructed from the first-quantized one in the usual way, HF(A ) = +tH,(A) +,
(3.9)
U ( g ) H F ( A ) U - I ( g ) = Hv(gAg -1 - Oxgg -1) = HF(Ag ) .
(3.10)
and we see that
The total hamiltonian includes a hamiltonian HB(il ) for the gauge fields Htot(A ) = H F ( A ) + HB(.4 ) .
(3.11)
There is also a bosonic operator UB which acts on the operator ~f by
u.(g)
fu
1 = g-l
g + g-laxg
(3.12)
so that the total hamiltonian is invariant under the combined effect of both operators
UB(g)U(g)Htot(UB(g)U(g)) -1 = Utot(g)HtotUtotl(g) = Htot .
(3.13)
To study the algebra of the symmetry generators of the group, we construct infinitesimal generators for which an initial guess would be
o(dgg -1) & Y',at,(g)a,,(g)(n, gldgg-llm, g).
(3.14)
Under "naive" manipulations this behaves as expected since, e.g., da¢,(g) = ~,a~(mJdg{n )
= ~_,a*.,(g)(mlg-~dgln) = E a ~ ( g ) ( m , gldgg-lln, g) & [o(dgg-'),a~(g)].
(3.15)
M. Stone, W.E. Goff / Berry'sphase
254
The question marks are inserted because p is ill-defined as it stands since the sum is infinite. It is very easy to make mistakes in these manipulations so for maximal security we will set up an algebraic operation which makes well-defined secondquantized operators from first-quantized ones. We begin with a matrix An,,, which we will always assume to be such that An, . ---, 0 rapidly for n - m ~ oo, and then define a mollified matrix A~,~ by
A~,,,, = ( n l A
Im)L(E.)L(E,,,),
(3.16)
where f~(E,) is a cutoff function with the properties that f~(E,) =f~(lEn[ ) ~ 0 as n ~ oo Vs and f~(En)~ 1 as s ~ 0 Vn. Then, using :A: to denote normal ordering of a and a t with respect to the vacuum of their hamiltonian, we define P(A ~) =
1 "f ET[an, am](nlA[m)fs(E,)f~(Em)
(3.17)
and
. p(AS)='~,.~[an, .1 t am]: (nlAlm)L(En)£(Em),
(3.18)
SO
0(a
= : 0 ( A ' ) - ln(A'),
(3.19)
where ~( A')
=
(3.20)
Esgn( E,)(nlA [n)fsZ( En). n
has a well-defined limit as s ~ 0. With these definitions we have well-defined operators and can compute
[P(A'),P(B~)] =P([A',B'])+!n([A',B'])+i°~(
,
(3.21)
where
ion'(AS,
B s) = - ½ Y'~ (sgn(E.) -
sgn(Em))(nlA [m) (mlB[n)fs2(En)fs2(Em),
n~m
(3.22) by observing that the operators [p(AS), p ( B ' ) ] and p([A s, B q ) are equal up to a (finite) c-number and i.~" is the vacuum expectation of [O(W), p(BS)] and - ~1 / t h e vacuum expectation of p([A ~, Bq). There is so far little profit in these manipulations however, since it is easy to see that ½~([A s, BS]) +
i ~ ( A s, B ~) - O.
(3.23)
M. Stone, W.E. Goff / Berry's phase
255
The problem is that we want o([A, B]S), not p([W, Bq), if we are to make contact with the well-behaved algebra of the first-quantized operators. The cutoff on the sum over the intermediate states in o([A s, B~]) prevents us from preceeding directly. Now
([.4, BI ~- [A ', B']).., =L(e.)L(e.,) E (1 - L2(E,.)) m
×((nlAlm)(mlBln' ) - (nlBlm)(mlAtn'))
(3.24)
has entries only at large values of m, and thus also only at large values of n, n' because of our assumption that matrix elements rapidly vanish if m, n are far apart. (Essentially this means no high Fourier components in dgg -1 - not a great restriction.) If we apply our operators only to "Tomonaga" states [5], for which there exists an Emax and E~n such that all one-particle states are empty if E n > Emax and all full if En < Emin, and normal order the operators, then
:O([W,B'I):-:o([A,B]~):=:O([W,B~I-[A,
BI'): ~O
as s-*O. (3.25)
So,
o([A ~, B']) + -~,~rA', 8'1) = :0([A', 8"1)" 2 \t
--.:o([A,B]S): = o([A, B] s) + ~ ( [ A , B]~),
(3.26)
we see that on Tomonaga states and for small s,
[p(A'),p(B~)I=P([A,B]S)+(i~(AS,
BS)+~n([A,B]')),
(3.27)
and now the two c-numbers do not cancel. For example, if we take A,,, = [shift(+p)] nm ~-- ~n-prn,
(3.28)
B,,, = [shift( +p')] ,,, - 6,_p,,,,
(3.29)
the commutator [A, B] is well-defined (only finite sums in the products) and zero, but i~=pSp_p, and p(A)= lims~0P(A' ) obeys [0 (shift(p)), 0 (shift( p'))] = -pSp_p, as found long ago by Jordan.
(3.30)
M. Stone, W.E. Goff /
256
Berry's phase
Specializing now to the case in hand, we construct our operators which will correspond to the J ' s out of the annihilation and creation operators and matrix elements of the group transformation at the current value of Ag on the orbit, p ( d g g -1) = lira E ~1 [ a ,t ( g ) ,
am(g)](n,gldgg-llm, g)L(E,)L(Em).
s-'~0 t/~m
(3.31)
Notice that the cutoffs on the m and n sums mean that we cannot just replace the g's in a(g) and the In, g) by g = (identity) in this expression although we could perform sums inside the matrix element to rewrite ( n, g [dgg- 1[m, g) = ( n Ig- ld g[ n ) since they converge without a cutoff by our assumption about the matrix elements. Writing the commutator as the product of forms we can express it compactly as
[p(dgg-1)] 2= p((dgg-1) 2) -[-i~-[-
1T/((dgg-1)2).
(3.32)
From this, using the formula for the Berry curvature of an individual state as the expectation value of the commutator,
i~(n) = (nlJ21n),
(3.33)
/
we can read off the curvature of any state
a~l(g)a~2(g)... [0; g) = [nxn2...; g)
(3.34)
as
L~"('''2") = E(ni, gl(dgg
1 2
) [ni, g) + i.~.
(3.35)
i
In words, the Berry curvature of the many-particle state is the sum of each particle's first-quantized curvature, (n i, g](dgg-1)2]ni, ~), plus a contribution equal to the vacuum curvature - and it is latter which appears as the anomalous commutator. To compute the commutator of the total generator, i.e. including the piece that acts on the gauge field, we need dp(dgg -1) - (p(dgg-1)2). 1 t P ( ( d g g - 1 ) s ) = E ~[a.(g),am(g)](n,
gldgg-llm, g)f~(E.)fs(Em)
n,m ---- ~] ~1 [ a .t ( g ) , n,m Therefore, if the cutoff on dp((dgg-~)~) = Z
a,,,(g)](n,elg-ldgtm,e)fs(E,,)fs(Em).
(3.36)
p((dgg-ly ') is much milder than on p ( ( d g g - l y ) ,
([P((dgg-1)¢),½[a:(g),am(g)l](n,elg-'dgl re,e)
n~m 1 ? +-~[a,(g),am(g)](m,el
--
(g-ldg)2ln,e))f~(E.lf,(E.,)
.
(3.37)
257
M. Stone, W.E. Goff / Berry'sphase
The s ~ 0 limit is insensitive to the ordering of the cutoff and gives
dp((dgg-~)s) : [p((dgg-a)"), p((dgg-1)s)] - p((dgg-1) 2) ""~°2(p(dgg-1))2- p((dgg-1)2),
(3.38)
or
do (dgg -1)
-(p(dgg-1))2= (p (dgg-')) 2 -
p((dgg- 1)2)
= ion+ ½~((dgg-1)2). So, writing
(3.39)
p(dgg-l(Xa)) as P(Xa) and similarly for 4, we have
[-Xa + p( Xa), --Xb + p(Xb)] =f~bc(-xc + p( Xc) ) - (i~-( X~, Xb) + ½f~bcn(xc)) (3.40) A simple interpretation is obtained by rewriting this in terms of normal ordered currents. We use the following: d~ ((dgg-t) s) = - ~(((dgg-~)2) s)
(3.41)
and p(dgg-1) =
:p(dgg-1) "- ~im½~/((dgg 1)s),
(3.42)
to find dp (dgg -1) = d: p (dgg -1): - ½d~/(dgg -1) = d:p(dgg-1): + ½*/((dgg-1)2),
(3.43)
and finally
d:p(dgg-1):-(:o(dgg
1);) 2--" i ~ .
(3.44)
This expression is a c-number, and equal to the vacuum phase, because :p(dgg-1): has diagonal elements, by which it differs from the Sonoda expression: :p(dgg -1) -- :p(dgg-1)Sonoda + ~
:af~(g)a,(g):(n, gldgg-lln, g). (3.45)
The last term is just a sum, of the form used in (2.35), of projection operators onto the states. Using (2.35), we see that this is just what is needed to remove the operator part of the anomalous commutator occuring in the Sonoda approach due
M. Stone, W.E. Goff / Berry's phase
258
tO the Berry phases of the one-particle states. The operator term is cohomologically trivial since there exists a normal ~(1) representation on the one particle states. The normal-ordered currents therefore have c-number Schwinger terms. Thus [ - X a + :p(Xa):,--Xb+ :p(Xb): ] =f~bc(--Xc+ :p(Xc): )
-i~'(Xa, Xb)
(3.46)
while
[:p(Xo):,
=Lbc:o( xc): +
xo,
(3.47)
This relative minus sign between the anomalous commutator of the total generators and the fermionic ones is an inevitable consequence of the Sonoda formalism but does not seem to have been widely appreciated.
4. Conclusion We have seen that, despite the extra analytic care necessitated by the infinite Dirac sea, the computations of the anomalous commutators are entirely consistent with the general geometric formulation of the problem by Sonoda. In particular we can see that the Berry phases of the states other than the vacuum are not in general zero - although as long as there is a way to implement the group action on the first-quantized states they are cohomologically trivial and can be removed by other definitions of the currents. In particular the more usual choices of generators made by physicists do exactly that. The change of sign of the anomalous commutator between the fermionic generators and the total gauge generators is fairly subtle and, although it follows directly enough from the general formalism and our calculations, it may depend on the assumption of rapid fall-off of the group matrix elements. This is a natural consequence of only having low momentum transfer by the group in two dimensions, but the large density of states in higher dimensions makes the manipulations more formal. This work was supported by grants NSF-DMR-84-15063, NSF-DMR-83-16981. We would like thank Qian Nui, Yigao Liang, and Eduardo Fradkin for useful conversations.
Appendix The Berry phase originally appeared in the context of the adiabatic approximation (or its equivalent, the Born-Oppenheimer approximation). If we have a time-dependent hamiltonian H(t) then a solution ~(t) to the time-dependent
M. Stone, W.E. Goff / Berry's phase
259
Schr~Sdinger equation
iOta(t) =H(t)+(t)
(A.1)
is given, for slowly varying H(t), in terms of solutions q~, to the "snapshot", time-independent, eigenvalue problem
H(t)+t=E(t)+t
(A.2)
as
q,(t) =q,,exp(-if'E(t)dt+
i~,(t)),
(A.3)
provided
iOt~+
(~10t~) = 0.
(A.4)
The Berry phase, ~, is needed because we have an arbitrary phase choice to make for each t in (A.2) but the dynamics in (A.1) determine a definite phase at each t and thus we need a phase factor to make up the slack. In general there is a different phase % for each evolving eigenstate ~p,(t) idcp, + (+nldhb,) = O,
(A.5)
so the total phase for transport round a curve F is
i~.(r) =
- ~r=aa(q~.ld+.)
= - fA(dq,,hd+,).
(A.6)
Now first-order perturbation theory gives (+,,[d~b,)
(+mldHl~b,)
,
m =~ n,
(A.7)
E n -- E m
therefore (d~b, ld~b,) =
-
(q,.IJ21@.)
=
- ½(~b,] [J,, Ja.]l~b,) dk; A d M ,
(A.8)
or
icp,(F) = ~(q~,IJ21~b,)
(A.9)
Thus the curvature two-form can be expressed as the expectation value of the commutator of the operators J. This commutator is not necessarily diagonal in the
260
M. Stone, W.E. Goff / Berry'sphase
~ , basis however. It is worth showing that this formula for the curvature is the same as (2.16), especially as they both contain the commutator of two J ' s but with opposite sign. We begin by showing that d J - j 2 really is diagonal as claimed in the text. To do this it is convenient to introduce the projection operators Pn = IG)(GI and to notice that the exterior derivative of P, is given by d P , = Id~k,)(q, nl +
bk.)(dq,.I
= [ J , Pn]"
(A.IO)
Since the d 2 of anything is 0 it then follows that
O=d[J, P n ] = [ d J - J 2 , Pn]
(A.11)
and thus d J - j 2 commutes with all operators Pn and must be diagonal. N o w look at the identity 0 = Tr(P~J),
(A.12)
which follows from the fact that J is defined to have no diagonal elements. Take the d of this to get 0 = T r ( d P . J + P.dJ)
= Tr([J,P~]J + P.J) = T r ( J P j - p.j2 + P.dJ) = T r ( - 2P~J 2 + P~dJ),
(A.13)
where we have used the cyclic property of the trace and the anticommuting nature of the one-form J at the last step. We have proved that Tr(P~(dJ-
j2)) = Tr(Png2),
(A.14)
and so the two forms of the curvature are equivalent.
References
[1] J. Schwinger, Phys. Rev. Lett. 3 (1959) 296 [2] T. Goto and T. Imamura, Prog. Theor. Phys. 14 (1955) 396 [3] R. Jackiw in Current algebra and anomalies, ed. S.B. Treiman et al., (Princeton Univ. Press, Princeton, 1985) [4] P. Jordan, Z. Phys. 93 (1935) 464 [5] S. Tomonaga, Prog. Theor. Phys. 5 (1950) 544 [6] S. Coleman, Phys. Rev. Dll (1975) 2088; S. Mandelstam, Phys. Rev. Dll (1975) 3026;
M. Stone, W.E. Goff / Berry's phase
[7]
[8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
261
M.B. Halpern, Phys. Rev. D12 (1975) 1684; T. Banks, D. Horn and H. Neuberger, Nucl. Phys. B108 (1976) 119; E. Witten, Commun. Math. Phys. 92 (1984) 455 V.G. Knizhnik and A.B. Zamolodchikov, Nucl. Phys. B247 (1984) 83; I.B. Frenkel and V.G. Kac, Inv. Math. 62 (1980) 83; G. Segal, Commun. Math. Phys. 80 (1981) 301 S. Jo, Nucl. Phys. B259 (1985) 616; Phys. Lett. B 163 (1985) 353 L.D. Faddeev, Phys. Lett. 145B (1984) 81 L. Bonora and P. Cotta-Ramusino, Commun. Math. Phys. 87 (1983) 589 P. Nelson and L. Alvarez-Gaum~, Commun. Math. Phys. 99 (1985) 103 A.J. Niemi and G.W. Semenoff, Phys. Rev. Lett. 55 (1985) 927, 2627; 56 (1986) 1019 N.S. Manton, Ann. Phys. 159 (1985) 220 B. Zumino, Nucl. Phys. B253 (1985) 477 W.A. Bardeen and A.R. White, eds., Anomalies, geometry and topology, (World Scientific, Singapore (1985) A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Nucl. Phys. B241 (1984) 333; D. Friedan, Z. Qiu and S. Shenker, Phys. Rev. Lett. 52 (1984) 1575 M. Kobayashi, K. Seo, and A. Sugamoto, Nucl. Phys. B273 (1986) 607 M.V. Berry, Proc. R. Soc. Lond. A 392 (1984) 45 B. Simon, Phys. Rev. Lett. 51 (1983) 2167 H. Sonoda, Nucl. Phys. B266 (1986) 410, Phys. Lett. 156B (1986) 220 S. Hosono, Nagoya (Japan) preprint DPNU-86-44-Rev (1987) S.-G. Jo, MIT preprint CTP#1419 (1987) L.D. Faddeev and S.L. Shatashvili, Phys. Lett. B167 (1986) 225 G.W. Semenoff, in Topological and geometric methods in field theory, ed. J. Hietarinta and J. Westerholm (World Scientific, Singapore) pp. 433-443 A.J. Niemi, Lecture at Workshop on Skyrmions and anomalies, Krakow, Poland, preprint DOE/ER/01545-392 (1987) S. Hosono and K. Seo, Gifu (Japan) preprints GWJC-3, GWJC-4 (1987) M. Stone, Phys. Rev. D33 (1986) 1191 For example, L.N. Chang and Y. Liang, Comm. Math. Phys. 108 (1987) 139
B E R R Y ' S PHASE, C O M M U T A T O R S , AND T H E DIRAC SEA Michael STONE and William E. GOFF University of Illinois at Urbana-Champaign, Department of Physics, Loomis Laboratory, 1110 W Green St., Urbana, IL 61801, USA
Received 22 May 1987 (Revised 2 November 1987)
We discuss the connection between the Berry phase and the Schwinger term in generators of symmetry groups, and stress the relative minus sign between the anomalous commutator of the currents and the anomalous commutator of the full group generators.
1. Introduction Careful computation of the commutators of fermion currents in a quantum field theory reveals that they do not always have the form anticipated from naive manipulations using the canonical commutation relations. Additional terms, usually called Schwinger terms [1-3], were first noticed by Jordan [4] in connection with his neutrino theory of light, and were in fact the key ingredient in this first example of a Fermi-Bose equivalence. Some years later, Tomonaga, in one of his seminal works on q u a n t u m field theory, also used the additional terms to show that a one-dimensional electron gas can be equivalently regarded as a boson system [5]. It is in this p a p e r of T o m o n a g a ' s that one sees clearly stated, for the first time, the equations of what are now called a Kac-Moody algebra. Later Schwinger [1] gave general arguments based on current conservation, the equations of motion and the existence of a ground state to show that additional terms were to be expected in all current algebras. A substantial amount of physics has grown out of these early ideas: the Bose-Fermi equivalences in two dimensions [6], the theory of Kac-Moody algebras and their connection with conformally invariant field theories and strings [7], some current algebra in four dimensions and an enhanced understanding of the origin of anomalies in gauge theories [8-15]. One can obtain these Schwinger terms in many ways, ranging from being careful with the normal ordering of the operators and then using the properties of the set of states to which the operators are to be a p p l i e d - as was done by the early authors - to observing that, in the 1 + 1 dimensional field theories which we will be concerned with here, the short distance current-current correlation function of two 0169-6823/88/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
244
M. Stone, W.E. Goff / Berry'sphase
fight-handed fermions goes as z-2 and thus, using the complex variable technology recently developed for conformally invariant theories [7, 16], the commutator must contain the derivative of a delta function. Other popular procedures involve separating the points at which the two Fermi operators act or taking a BJL limit [8, 3,17] to extract the commutator from Feynman diagram computations. In the last few years, starting with the work of Faddeev [9], we have begun to appreciate that these extra terms in the algebra of currents, and the implied projective representations of the symmetry groups generated by the currents, have an origin in the complex topology of the configuration spaces of the system on which the groups act [10]. There have been several works connecting the Schwinger terms with the nonabelian anomaly and the Wess-Zumino action via the descent equations [14,15], as well as directly with the Berry phase [18-21] which, in some sense, is the origin of all these effects. Despite these insights actual computations of the anomalous commutators for the full operators generating the symmetries - that is the electric field operators as well as the purely fermionic charges - seem to have caused some difficulty in that different authors report apparently incompatible results. For example there has been extensive work on the problem of anomalous commutators by Jo [8,22] who argues that the electric field operators do not commute in general, namely in 3 + 1 dimensions. In 1 + 1 dimensions, he finds that they do commute but also calculates a zero contribution from the cross-term piece. In ref. [22], Jo shows that the BJL method agrees with that of Faddeev and Shatashvili [23] in 1 + 1 dim, but not in 3 + 1 dim. Kobayashi, Seo, and Sugamoto [17] also use the BJL limit and concur with Jo's result, but no statement is made about the contribution from the various pieces. Niemi and Semenoff [12, 24, 25] take commuting electric fields and find the cross-terms between the fermionic currents and the electric fields to be zero, so the anomalous commutator of the full Gauss-law operator comes entirely from the anomalous commutator of the charges. In this paper we will argue that the cross-terms are non-zero and that the commutator of the full generators differs in sign from that of the fermionic charges. We also find ourselves at variance with the viewpoint expressed by Faddeev and Shatashvili [23] and Hosono [21], that the anomalous commutator arises because of a c-number subtraction of an A-dependent expectation value from the current. This gives the result, which we believe to be erroneous, that, in our notation (see sect. 3), the anomaly arises from 7/([A, B] s) rather than ~/([As, BS]). While we were revising this paper, two papers by Hosono and Seo [26] appeared. They use different methods to arrive a relation similar to ours. Their method is such that 8/8A acts on regulators rather than on the fermionic annihilation and creation operators as we have done. They account for differences between their covariant method and the BJL method by the difference between the consistent and covariant currents (those that give the consistent and covariant anomalies). These currents differ only by a local gauge dependent c-number redefinition. They also hypothesize
M. Stone, W.E. Goff / Berry'sphase
245
that the commutativity or non-commutativity of the electric fields depends on the regularisation procedure used. In their covariant formalism, and assuming that the electric field operators commute, they obtain our 2d relationship. They also obtain the same relationship in 4d, though with some reservations of validity. (A factor of i in one of their definitions may obscure this agreement.) Our intent in this paper is to provide a simple demonstration, in the context of quantum mechanics and in the case of two-dimensional systems, of the commutation of the full gauge generators including a careful treatment of the cross-terms between the fermionic charges and those parts of the symmetry generators which act on the Bose variables. Sect. 2 will review some of the essential geometry, in particular the way that the Berry phase provides a connexion on the bosonic part of the configuration space and, following the general theory introduced by Sonoda [20], show how this connexion influences the commutation relations of symmetry generators. In sect. 3 we will apply this formalism to show how the Berry phase of the vacuum, or Dirac sea, is responsible for the Schwinger term. In the appendix we prove a few useful relations between different formulae for the Berry phase.
2. Berry phase, connexions and commutators In this section we will briefly review the well known ideas of M.V. Berry, as interpreted by Simon in ref. [19], and discuss the significance of these ideas for the action of symmetry groups. Suppose we have a family of hamiltonians H ( h ) labeled by points h ~ ; / / where ~ ' is some manifold. In the systems of interest the parameters h will be some bosonic gauge fields or condensates, such as Higgs fields or charge density waves, and the hamiltonians H(X) will be those of a system of fermions interacting with these bosons. Consider the eigenvalue problem for each H ( h ) H ( h ) q J , = En(X)~b . .
(2.1)
If it does not become degenerate with another state at any point on the manifold, a particular state can be regarded as part of a family of states ~kn(h). Associated with this family of states there is a one-form A = A i d h i on the manifold iAn= -
(2.2)
This is the Berry phase which would occur in the state q~n if we adiabatically changed the hamiltonian from H ( h ) to H ( h + d h ) and it can be regarded as defining a parallel transport which takes eigenstates of one hamiltonian to the corresponding eigenstate of the slightly changed hamiltonian. In more technical language, A n is a connexion on the bundle of one-dimensional vector spaces (i.e. the multiples of ~kn(X)) over the manifold ;/g.
M. Stone, W.E. Goff / Berry's phase
246
As a toy, but essentially generic, example consider the family of hamiltonians corresponding to a spin-½ particle in a magnetic field [27,18] //. = #o -n,
(2.3)
where n is a point in S 2 (the direction of the magnetic field). For each n there are two eigenstates with eigenvalues _+#. If we restrict ourselves to the family of states ~p+ with eigenvalue +/~ we find that we have to be careful about the choice of phases for these states. We notice that
1--( cOS½0) ~P+-
(2.4)
sin ½0 ei~
is a smooth choice of phases for the states in any region not including the south pole (where it is ambiguous since all values of q~ correspond to the same point), while cos ½0 e - i~ ~2+= t
(2.5)
sin½O
is a smooth choice everywhere except at the north pole. Of course,
#1 = ei~+]
(2.6)
everywhere that both phase choices are defined. Eq. (2.5) is the transition function connecting the two patches. This is an example of a non-trivial fibre bundle in that there is no simple way of regarding the combination of base manifold S 2 and the one-dimensional Hilbert spaces at each point as a simple product, except in patches of the base manifold. The transition function sews the patches together in a topologically non-trivial way. There are superficially different A's for each choice:
iAl=-
(~plld~bl+)=- li(-cos0
+ 1) dq~,
(2.7)
iA2+= _ (~2+ [d~pZ+) = _ ½ i ( - c o s 0 - 1) dq~.
(2.8)
while
AI+ and A 2+ differ by a gauge transformation A I _- A +2-
dq~,
(2.9)
which is well-defined except at the poles where dq, is singular. Since these connexions or gauge-fields differ only by a gauge transformation they have the same
M. Stone, W.E. G o l f / Berry's phase
247
curvature two-form F + = d A + = - ½sin0d0 A dq)= - ½d[area].
(2 .lO)
This means that were we to adiabatically rotate the direction of the magnetic field so that it swept out a region on the two-sphere, we would find that the state would return to its original value times a phase equal to minus one-half the area of the region. The integral, divided by 2~r, of the curvature two-form over the base manifold S 2 must be an integer called the first Chern class of the bundle. In this case, since the area of the two-sphere is 4rr we have fs2F = -2~r.
(2.11)
There are possible interesting consequences when the states form a non-trivial bundle over the family of hamiltonians. If the parameters in the family of hamiltonians represent dynamical variables which are being treated in the Born-Oppenheimer approximation, it is well known that the connexion A has an effect on the d y n a m i c s - it gives rise to the Wess-Zumino term in the effective action for the Bose fields. It is also possible that there is an effect on the implementation of symmetry operations on the dynamical system consisting of the internal fermionic (in this case the spin- ~) degrees of freedom and the external Bose (orientation of the magnetic field) variables. A nice account of this has been given by Sonoda [20] and we will follow his argument in the following. Define an operator valued one-form J = J~ dXi by its matrix elements between the states ~k, ( q,,,IJI 4',,,)
-
( q,,,Id i l l +,,,)
n_-/=m
E m - E,
=0
n = m.
(2.12)
Then first-order perturbation theory and the absence of diagonal elements in J tells us that d~b, = & k ,
(2.13)
is the change in the state +n as it is Berry-transported from H to H + dH. Consequently the operator ~ r = Peers
(2.14)
must be diagonal in the ~kn basis and give the total phase change in each state after it has been Berry transported around a closed curve F. For an infinitesimal
248
M. Stone, W.E. Goff / Berry'sphase
parallelogram of sides d~ 1 and d~ 2 this reduces to <~p, l P e x p ~ / l a p , ) = 1 + <~n]OIJ2 - 02J 1 -t- [J2, J1]llPn) dXl d2,2 •
(2.15)
In other words, if we define the operator valued two-form F by i F . ~ " 7tF~j 1 • d X ~A
dXj
= 1( O i j j _ O j J i _
=dJ-J
[j/, jj]) dXi A dM
2,
(2.16)
= i~[~b >(~P.lF("),
(2.17)
n it is the curvature two-form. Another expression for the expectation, in energy eigenstates, of the same curvature form is derived in the appendix. It is
(~.lJ21~.) = i F (").
(2.18)
When the variables ~ become dynamical, the wavefunctions of the combined system are q'.(X), the amplitude that the parameters have values ~, and the fermionic "internal" eigenstate is ~b.. If a Lie group f¢ acts on the manifold Jg, if: ,./g ~ ..g,
(2.19)
then, in the absence of "internal" degrees of freedom, the group would act on the wave functions by (gq')(X) = qS(g- aX)
(2.20)
and the generators of infinitesimal transformations (being a trifle unconventional in the omission of factors of i) would be the differential operators on wave functions G , , = - X a, where X a = X ~ 3 i is the right-invariant vector field of infinitesimal displacements (corresponding to multiplying on the left by elements of the group infinitesimally close to the identity). The X a will obey the defining relations of a Lie algebra: [ X o, Xb] = --fab'Xc.
(2.21)
(The minus sign in the structure constants is the one that usually occurs for right-invariant vector fields. It is compensated by the minus sign in the definition of G which itself originates in the g-1 in the action of the group on the wave
249
M. Stone, W.E. Goff / Berry'sphase
function.) If we now include the action of the group on the "internal" states ~bn we can do so in a number of ways. The one chosen by Sonoda is to Berry transport the internal state using the operator J. The total group generator is now a sum of two parts G~ = - X a + J ( X a )
=
- -
(2.22)
XtOi-~ JiX~.
The commutator of these Ga generators can be evaluated by taking note that the X ' s also act on the J ' s . We find
[-- Xa + J( Xo), --
+ JC xO] = L;C-- xO +
[J(Xo), J(X )I
-- J([ Xa,
- ( X o J ( X O - X j ( X o ) --J([Xo, =f~aC(-X~+J(X~)) - (dJ-JZ)(x~,
Xb), (2.23)
where we have used d J ( X a , Xb) = X~J( Xb) -- X b J ( X ~ ) - J([ X~, X6] ) .
(2.24)
[Ga, Gb] =fabCGc -- i~aO(~),
(2.25)
O~b ( X ) = 3~ ( X a, Xb; X) = X~ X[F,j( X )
(2.26)
Succinctly,
where the anomalous piece
is made out of the Berry curvature (2.16) and the vector fields. This procedure of Sonoda's does not produce the conventional generators of the symmetry because of its deliberate suppression of the diagonal elements of J in the energy-eigenstate basis. Before we argue that it is a reasonable, if unconventional, procedure let us see how it works in the toy example of the spin-½ particle. Under a rotation, 8o~, H changes by 8 H = 8to A n . o.
(2.27)
We find that (2.28) is the operator " J " corresponding to the rotation. It differs from the more usual operator generating rotations, - ½io, by diagonal terms corresponding to our phase choices. The commutator of the J ' s is in this case diagonal (it is not in general, if there are more than two states) and equal to [Z8,ol, Zs~,~ ] = ½in. (8~ 1 A 8 ~ 2 ) ( n - o ) .
(2.29)
250
M. Stone, W.E. Goff / Berry'sphase
We recognize that the diagonal terms are in fact the Berry phases for the states concerned as a consequence of (2.18). The commutator of the total generators Jso, = L • 8~o + Xso, (L being the usual angular momentum operator) is (2.30) and again we see the Berry phase operator occuring as an additional term although with opposite sign compared to the commutator of the spin generators. The additional terms in the commutation relations mean that we have a projective, or ray, representation of the action of the group on the states. However, in the case of our toy example, we know that we can replace our generators by the conventional ones and recover the usual commutation relations. The difference between the conventional and "Berry" generators is just (n. 8~0)n. o which is nowhere singular on the two-sphere. We cannot always do this if the bundle of internal states is non-trivial [11]. To see this, suppose the group acts on the manifold without leaving any point fixed, (this is not so in our toy example) then the orbit of any point on the manifold is a copy of the group manifold. Suppose further, that (as is the case of our simple example) the bundle over this orbit is non-trivial and there is therefore no section of normalized states (that is there is no way to choose representative states smoothly over the orbit), then there can be no conventional group action; if there were, we could just apply a unique group element to a state over some chosen point in the manifold and obtain a definite state at the image of our chosen point. This would be a smooth section in contradiction to our assumption. While there may be no conventional, vector (as opposed to ray) representation of the group, the "Berry" action always provides a valid implementation of the symmetry which is compatible with the topology. Any other implementation of the symmetry must be obtainable from the Berry action via a (globally defined) redefinition of the generators. A good analogy comes from parallel transport on a two-sphere: There are many possible connexions but they must all have curvature that integrates up to 4~r (2~r times the Euler character) because of the Gauss-Bonnet theorem. Having found one all the others can be obtained by the addition of a globally defined vector field to the connexion form. We cannot give a simple criterion foretelling when the Berry prescription can be reduced to a vector representation this way - the toy example shows that it is not just a question of whether the bundle is trivial or not - but there is a well-known formal framework [10,15] for stating the question. Objects like : - ( X , Y; 2~) are antisymmetric functions of elements of the Lie algebra and are functions on the manifold ~¢~'. They are therefore function valued forms on the Lie algebra (which need not always be obtained as pull-backs of forms on J-/). We can define an operation 8 which acts on them and behaves like the ordinary exterior derivative d in that it increases the number of antisymmetric slots by one and gives zero if
M. Stone, W.E.Goff / Berry'sphase
251
applied twice. On e-number one-forms s¢(X; X) we have
&s~C(X,Y,X)=Xssc(Y;X)- Yd(X;?~)-.s¢([X,Y];X)
(2.31)
and on two-forms
&~( X, Y, Z; X) = X.,~(Y, Z; X) + (cyclic permutations) - . ~ ( [ X, Y], Z; X) + (cyclic permutations).
(2.32)
We can define 8 for general objects of this type which we will call members of An(g, ~[). 8 increases the number of antisymmetric slots for elements of the Lie algebra n
n+l
8: A ( N , ~gg) --* A (~q, ~ ' )
(2.33)
and satisfies 62 = 0. If we replace G~ by Ga + ~'~ then o~ changes to o~+ 8~¢. The Jacobi identity for Ga requires 8o~= 0 so ~" is an element of the second cohomology group H2U¢, Jg). Thus the question of whether the additional o~ term can be removed is restated by asking if there exists a globally defined d e AI(~, ~ ' ) such that o~= 8 d . That is, whether o~ is cohomologically trivial*. As the toy case of the spin-½ particle shows the additions to G, need not necessarily be c-numbers. Most of the definitions of generators will differ from Sonoda's by having non-zero, but globally defined, diagonal elements (the off-diagonal ones must coincide with his). This can be taken into account by writing
f = J + ~f(")l~b.)@.l.
(2.34)
n
Here f(") is a globally defined 1-form of diagonal elements on s/g, and J is the Sonoda generator with zero diagonal elements in the energy-eigenstate basis. Then an easy computation shows that
df-f
2= d J - J 2 +
y'df(")p, = ion,
(2.35)
and therefore
[ - X a + f ( X a ) , - - X b + f ( X b ) ] =fabC(-Xc+f(Xc)) -io~(Xa, Xb), (2.36) where, in general, f ' ( X , , X,) is no longer the Berry phase curvature. * Actuallyin a field theorywe require that ~¢ be a localfunctionalof the field so somekind of "local cohomology"is needed. See ref. [10] for details.
252
M. Stone, W.E. Goff / Berry'sphase
The Berry phase curvature for any state can still be found from the formula
i ~ = ( n If2ln )
(2.37)
as the diagonal terms do not contribute because of the antisymmetry.
3. 1 + 1 dimensional Fermi systems
In this section we will apply the general formalism to the specific example of 1 + 1 dimensional Fermi systems. These have the advantage that the geometric properties of the anomalous commutators are not obscured by the more complicated renormalizations needed in higher dimensions while retaining all the essential ideas [28]. Consider the fermi field ~b(x) and expand it in terms of the eigenstates of a first quantized hamiltonian Hf(A) which depends on a gauge field A.
~ ( x ) = Ea,.u,.(x),
(3.1)
A )u. = E.u. .
(3.2)
H for a fight-moving particle is, for example,
Hf(A) = - i ( O x + A x ( x ) ) .
(3.3)
There is an ordinary action of a time-independent gauge group ~(~(1) o n Hi(A ) and Un~
u.(x) L (gu.)(x) = EUm(X)(mlgln) = EUm(X)(m, glgln, g),
(3.4)
Hf(A ) L gHfg -1 = Hf( gAg -1 - Oxgg -t) = Hf( Ag).
(3.5)
Here [m) and Ira, g) represent, respectively, the single particle states u,(x) and gun(x) before and after the group action. For mathematical convenience we choose to work on a periodic interval x ~ [0, 2¢r] and g : [0, 2~r] ~ ~, g(0) = g(2~r) = identity ~ N is an element of the "loop group" N(1) of the ordinary Lie group ~¢. The restriction of g to the identity at the endpoints of the interval ensures that the group acts freely (i.e. without fixed points). We want to use this action on the first quantized theory to induce an action of the group on the second quantized theory. That is, we seek to construct operators U
M. Stone, W.E. Goff / Berry'sphase
253
(corresponding at the moment just to the " J " ' s ) which act on ~ as follows:
U(g)+U-l(g) = g - ~ ,
(3.6)
U(g)amU-l(g) = E (mlg-tln)a,,= a,,(g),
(3.7)
n
U(g)atmU-l(g) = E a ~ ( n [ g l m ) - a~(g).
(3.8)
l1
Notice that qJ = EanU n = Ean(g)(gu),. The second-quantized Fermi hamiltonian is constructed from the first-quantized one in the usual way, HF(A ) = +tH,(A) +,
(3.9)
U ( g ) H F ( A ) U - I ( g ) = Hv(gAg -1 - Oxgg -1) = HF(Ag ) .
(3.10)
and we see that
The total hamiltonian includes a hamiltonian HB(il ) for the gauge fields Htot(A ) = H F ( A ) + HB(.4 ) .
(3.11)
There is also a bosonic operator UB which acts on the operator ~f by
u.(g)
fu
1 = g-l
g + g-laxg
(3.12)
so that the total hamiltonian is invariant under the combined effect of both operators
UB(g)U(g)Htot(UB(g)U(g)) -1 = Utot(g)HtotUtotl(g) = Htot .
(3.13)
To study the algebra of the symmetry generators of the group, we construct infinitesimal generators for which an initial guess would be
o(dgg -1) & Y',at,(g)a,,(g)(n, gldgg-llm, g).
(3.14)
Under "naive" manipulations this behaves as expected since, e.g., da¢,(g) = ~,a~(mJdg{n )
= ~_,a*.,(g)(mlg-~dgln) = E a ~ ( g ) ( m , gldgg-lln, g) & [o(dgg-'),a~(g)].
(3.15)
M. Stone, W.E. Goff / Berry'sphase
254
The question marks are inserted because p is ill-defined as it stands since the sum is infinite. It is very easy to make mistakes in these manipulations so for maximal security we will set up an algebraic operation which makes well-defined secondquantized operators from first-quantized ones. We begin with a matrix An,,, which we will always assume to be such that An, . ---, 0 rapidly for n - m ~ oo, and then define a mollified matrix A~,~ by
A~,,,, = ( n l A
Im)L(E.)L(E,,,),
(3.16)
where f~(E,) is a cutoff function with the properties that f~(E,) =f~(lEn[ ) ~ 0 as n ~ oo Vs and f~(En)~ 1 as s ~ 0 Vn. Then, using :A: to denote normal ordering of a and a t with respect to the vacuum of their hamiltonian, we define P(A ~) =
1 "f ET[an, am](nlA[m)fs(E,)f~(Em)
(3.17)
and
. p(AS)='~,.~[an, .1 t am]: (nlAlm)L(En)£(Em),
(3.18)
SO
0(a
= : 0 ( A ' ) - ln(A'),
(3.19)
where ~( A')
=
(3.20)
Esgn( E,)(nlA [n)fsZ( En). n
has a well-defined limit as s ~ 0. With these definitions we have well-defined operators and can compute
[P(A'),P(B~)] =P([A',B'])+!n([A',B'])+i°~(
,
(3.21)
where
ion'(AS,
B s) = - ½ Y'~ (sgn(E.) -
sgn(Em))(nlA [m) (mlB[n)fs2(En)fs2(Em),
n~m
(3.22) by observing that the operators [p(AS), p ( B ' ) ] and p([A s, B q ) are equal up to a (finite) c-number and i.~" is the vacuum expectation of [O(W), p(BS)] and - ~1 / t h e vacuum expectation of p([A ~, Bq). There is so far little profit in these manipulations however, since it is easy to see that ½~([A s, BS]) +
i ~ ( A s, B ~) - O.
(3.23)
M. Stone, W.E. Goff / Berry's phase
255
The problem is that we want o([A, B]S), not p([W, Bq), if we are to make contact with the well-behaved algebra of the first-quantized operators. The cutoff on the sum over the intermediate states in o([A s, B~]) prevents us from preceeding directly. Now
([.4, BI ~- [A ', B']).., =L(e.)L(e.,) E (1 - L2(E,.)) m
×((nlAlm)(mlBln' ) - (nlBlm)(mlAtn'))
(3.24)
has entries only at large values of m, and thus also only at large values of n, n' because of our assumption that matrix elements rapidly vanish if m, n are far apart. (Essentially this means no high Fourier components in dgg -1 - not a great restriction.) If we apply our operators only to "Tomonaga" states [5], for which there exists an Emax and E~n such that all one-particle states are empty if E n > Emax and all full if En < Emin, and normal order the operators, then
:O([W,B'I):-:o([A,B]~):=:O([W,B~I-[A,
BI'): ~O
as s-*O. (3.25)
So,
o([A ~, B']) + -~,~rA', 8'1) = :0([A', 8"1)" 2 \t
--.:o([A,B]S): = o([A, B] s) + ~ ( [ A , B]~),
(3.26)
we see that on Tomonaga states and for small s,
[p(A'),p(B~)I=P([A,B]S)+(i~(AS,
BS)+~n([A,B]')),
(3.27)
and now the two c-numbers do not cancel. For example, if we take A,,, = [shift(+p)] nm ~-- ~n-prn,
(3.28)
B,,, = [shift( +p')] ,,, - 6,_p,,,,
(3.29)
the commutator [A, B] is well-defined (only finite sums in the products) and zero, but i~=pSp_p, and p(A)= lims~0P(A' ) obeys [0 (shift(p)), 0 (shift( p'))] = -pSp_p, as found long ago by Jordan.
(3.30)
M. Stone, W.E. Goff /
256
Berry's phase
Specializing now to the case in hand, we construct our operators which will correspond to the J ' s out of the annihilation and creation operators and matrix elements of the group transformation at the current value of Ag on the orbit, p ( d g g -1) = lira E ~1 [ a ,t ( g ) ,
am(g)](n,gldgg-llm, g)L(E,)L(Em).
s-'~0 t/~m
(3.31)
Notice that the cutoffs on the m and n sums mean that we cannot just replace the g's in a(g) and the In, g) by g = (identity) in this expression although we could perform sums inside the matrix element to rewrite ( n, g [dgg- 1[m, g) = ( n Ig- ld g[ n ) since they converge without a cutoff by our assumption about the matrix elements. Writing the commutator as the product of forms we can express it compactly as
[p(dgg-1)] 2= p((dgg-1) 2) -[-i~-[-
1T/((dgg-1)2).
(3.32)
From this, using the formula for the Berry curvature of an individual state as the expectation value of the commutator,
i~(n) = (nlJ21n),
(3.33)
/
we can read off the curvature of any state
a~l(g)a~2(g)... [0; g) = [nxn2...; g)
(3.34)
as
L~"('''2") = E(ni, gl(dgg
1 2
) [ni, g) + i.~.
(3.35)
i
In words, the Berry curvature of the many-particle state is the sum of each particle's first-quantized curvature, (n i, g](dgg-1)2]ni, ~), plus a contribution equal to the vacuum curvature - and it is latter which appears as the anomalous commutator. To compute the commutator of the total generator, i.e. including the piece that acts on the gauge field, we need dp(dgg -1) - (p(dgg-1)2). 1 t P ( ( d g g - 1 ) s ) = E ~[a.(g),am(g)](n,
gldgg-llm, g)f~(E.)fs(Em)
n,m ---- ~] ~1 [ a .t ( g ) , n,m Therefore, if the cutoff on dp((dgg-~)~) = Z
a,,,(g)](n,elg-ldgtm,e)fs(E,,)fs(Em).
(3.36)
p((dgg-ly ') is much milder than on p ( ( d g g - l y ) ,
([P((dgg-1)¢),½[a:(g),am(g)l](n,elg-'dgl re,e)
n~m 1 ? +-~[a,(g),am(g)](m,el
--
(g-ldg)2ln,e))f~(E.lf,(E.,)
.
(3.37)
257
M. Stone, W.E. Goff / Berry'sphase
The s ~ 0 limit is insensitive to the ordering of the cutoff and gives
dp((dgg-~)s) : [p((dgg-a)"), p((dgg-1)s)] - p((dgg-1) 2) ""~°2(p(dgg-1))2- p((dgg-1)2),
(3.38)
or
do (dgg -1)
-(p(dgg-1))2= (p (dgg-')) 2 -
p((dgg- 1)2)
= ion+ ½~((dgg-1)2). So, writing
(3.39)
p(dgg-l(Xa)) as P(Xa) and similarly for 4, we have
[-Xa + p( Xa), --Xb + p(Xb)] =f~bc(-xc + p( Xc) ) - (i~-( X~, Xb) + ½f~bcn(xc)) (3.40) A simple interpretation is obtained by rewriting this in terms of normal ordered currents. We use the following: d~ ((dgg-t) s) = - ~(((dgg-~)2) s)
(3.41)
and p(dgg-1) =
:p(dgg-1) "- ~im½~/((dgg 1)s),
(3.42)
to find dp (dgg -1) = d: p (dgg -1): - ½d~/(dgg -1) = d:p(dgg-1): + ½*/((dgg-1)2),
(3.43)
and finally
d:p(dgg-1):-(:o(dgg
1);) 2--" i ~ .
(3.44)
This expression is a c-number, and equal to the vacuum phase, because :p(dgg-1): has diagonal elements, by which it differs from the Sonoda expression: :p(dgg -1) -- :p(dgg-1)Sonoda + ~
:af~(g)a,(g):(n, gldgg-lln, g). (3.45)
The last term is just a sum, of the form used in (2.35), of projection operators onto the states. Using (2.35), we see that this is just what is needed to remove the operator part of the anomalous commutator occuring in the Sonoda approach due
M. Stone, W.E. Goff / Berry's phase
258
tO the Berry phases of the one-particle states. The operator term is cohomologically trivial since there exists a normal ~(1) representation on the one particle states. The normal-ordered currents therefore have c-number Schwinger terms. Thus [ - X a + :p(Xa):,--Xb+ :p(Xb): ] =f~bc(--Xc+ :p(Xc): )
-i~'(Xa, Xb)
(3.46)
while
[:p(Xo):,
=Lbc:o( xc): +
xo,
(3.47)
This relative minus sign between the anomalous commutator of the total generators and the fermionic ones is an inevitable consequence of the Sonoda formalism but does not seem to have been widely appreciated.
4. Conclusion We have seen that, despite the extra analytic care necessitated by the infinite Dirac sea, the computations of the anomalous commutators are entirely consistent with the general geometric formulation of the problem by Sonoda. In particular we can see that the Berry phases of the states other than the vacuum are not in general zero - although as long as there is a way to implement the group action on the first-quantized states they are cohomologically trivial and can be removed by other definitions of the currents. In particular the more usual choices of generators made by physicists do exactly that. The change of sign of the anomalous commutator between the fermionic generators and the total gauge generators is fairly subtle and, although it follows directly enough from the general formalism and our calculations, it may depend on the assumption of rapid fall-off of the group matrix elements. This is a natural consequence of only having low momentum transfer by the group in two dimensions, but the large density of states in higher dimensions makes the manipulations more formal. This work was supported by grants NSF-DMR-84-15063, NSF-DMR-83-16981. We would like thank Qian Nui, Yigao Liang, and Eduardo Fradkin for useful conversations.
Appendix The Berry phase originally appeared in the context of the adiabatic approximation (or its equivalent, the Born-Oppenheimer approximation). If we have a time-dependent hamiltonian H(t) then a solution ~(t) to the time-dependent
M. Stone, W.E. Goff / Berry's phase
259
Schr~Sdinger equation
iOta(t) =H(t)+(t)
(A.1)
is given, for slowly varying H(t), in terms of solutions q~, to the "snapshot", time-independent, eigenvalue problem
H(t)+t=E(t)+t
(A.2)
as
q,(t) =q,,exp(-if'E(t)dt+
i~,(t)),
(A.3)
provided
iOt~+
(~10t~) = 0.
(A.4)
The Berry phase, ~, is needed because we have an arbitrary phase choice to make for each t in (A.2) but the dynamics in (A.1) determine a definite phase at each t and thus we need a phase factor to make up the slack. In general there is a different phase % for each evolving eigenstate ~p,(t) idcp, + (+nldhb,) = O,
(A.5)
so the total phase for transport round a curve F is
i~.(r) =
- ~r=aa(q~.ld+.)
= - fA(dq,,hd+,).
(A.6)
Now first-order perturbation theory gives (+,,[d~b,)
(+mldHl~b,)
,
m =~ n,
(A.7)
E n -- E m
therefore (d~b, ld~b,) =
-
(q,.IJ21@.)
=
- ½(~b,] [J,, Ja.]l~b,) dk; A d M ,
(A.8)
or
icp,(F) = ~(q~,IJ21~b,)
(A.9)
Thus the curvature two-form can be expressed as the expectation value of the commutator of the operators J. This commutator is not necessarily diagonal in the
260
M. Stone, W.E. Goff / Berry'sphase
~ , basis however. It is worth showing that this formula for the curvature is the same as (2.16), especially as they both contain the commutator of two J ' s but with opposite sign. We begin by showing that d J - j 2 really is diagonal as claimed in the text. To do this it is convenient to introduce the projection operators Pn = IG)(GI and to notice that the exterior derivative of P, is given by d P , = Id~k,)(q, nl +
bk.)(dq,.I
= [ J , Pn]"
(A.IO)
Since the d 2 of anything is 0 it then follows that
O=d[J, P n ] = [ d J - J 2 , Pn]
(A.11)
and thus d J - j 2 commutes with all operators Pn and must be diagonal. N o w look at the identity 0 = Tr(P~J),
(A.12)
which follows from the fact that J is defined to have no diagonal elements. Take the d of this to get 0 = T r ( d P . J + P.dJ)
= Tr([J,P~]J + P.J) = T r ( J P j - p.j2 + P.dJ) = T r ( - 2P~J 2 + P~dJ),
(A.13)
where we have used the cyclic property of the trace and the anticommuting nature of the one-form J at the last step. We have proved that Tr(P~(dJ-
j2)) = Tr(Png2),
(A.14)
and so the two forms of the curvature are equivalent.
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