Collective coordinates in Fermi systems

Collective coordinates in Fermi systems

Nuclear Physics B142 (1978) 489-509 © North-Holland Publishing Company COLLECTIVE COORDINATES IN FERMI SYSTEMS V. ALESSANDRINI Laboratoire de Physiqu...

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Nuclear Physics B142 (1978) 489-509 © North-Holland Publishing Company

COLLECTIVE COORDINATES IN FERMI SYSTEMS V. ALESSANDRINI Laboratoire de Physique Thborique et Hautes Energies *, Orsay, France D.R. BI~S ** Comision Nacional de Energia Atornica, Buenos Aires, Argentina and Institut de Physique Nuclbaire, Orsay, France B. MACHET Laboratoire de Physique Thdorique et Hautes Energies *, Orsay, France Received 14 June 1978 The problem of developing a consistent perturbation theory for a Fermi system in the case in which the unperturbed system exhibits dynamical symmetry breaking is discussed, by using collective coordinate methods. By adapting to this problem the methods used in the quantization of gauge theories, it is shown how to deal with composite zero-frequency excitations in such a way that the resulting perturbation theory is free of infrared divergencies. Explicit calculations are carried out in the case of a simple quantum mechanical model representing a superfluid Fermi system. 1. Introduction Collective coordinate methOds have been widely used in quantum field theory during the last few years in connection with the semiclassical quantization of extended objects like kinks [ 1 - 3 ] or instantons [4]. These classical solutions break, in general, some symmetry of the theory (like translational invariance in the case of kinks) and consequently the spectrum of small q u a n t u m fluctuations around them includes zero-frequency modes. These zero modes are an inavoidable consequence of the fact that the symmetry has been spontaneously broken, i.e., one is expanding around a vacuum which is not an eigenstate of the generator of the symmetry transformations but rather a coherent superposition of such eigenstates. Due to the presence of these zero modes naive perturbation theory fails. By now, the standard way to tackle this problem is to use the collective coordinate method [ 1 - 4 ] , which is essentially a refined version of the method of separation of variables. The constants of the motion associated with the symmetry broken ~r

Laboratoire associ6 au Centre National de la Recherche Scientifique. Postal address: Brit. 211, ** Universit6 Paris-Sud, 91405 Orsay, France. Fellow of the Consejo Nacional de Investigaciones Cientificas y Tecnicas of Argentina. 489

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V. Alessandrini et al. / Collective coordinates

by the ground state are treated as generalised (collective) coordinates and quantized in a straightforward way. One is then left with the problem of quantizing the reduced system, which is the original one plus some constraints arising from the fact that the constants of the motion have been frozen. Perturbation theory, developed for the reduced system, is free of infrared divergencies because there are no zero frequency modes in this case. The quantization procedure for the reduced system is identical to the one used in the quantization of gauge theories [5]. Indeed, it is well known that a mechanical system with constraints which are constants of the motion is equivalent to a gauge theory since the presence of the constraint transforms the original global invariance into a local (gauge) invariance in the sense that the constrained Lagrangian is invariant under symmetry transformations with time dependent parameters [ 6 - 7 ] . The purpose of this paper is to discuss the same problem for a fermion system such that in some well-defined limiting case in which the theory can be solved, the solution breaks spontaneously a symmetry of the Hamiltonian. The spontaneous symmetry breaking is therefore of dynamical nature. In order to be more precise (though we are not forced to) we shall always keep in mind a superfluid fermi system where the broken symmetry is the U(1) invariance associated with the conservation of the number of fermions N. This means that the ground state one starts with is not an eigenstate of N but a coherent superposition of states with different number of particles, such as the BCS ground state. Because of the Abelian nature of the symmetry group, these systems are the simplest among those which have been treated in nuclear physics with an approach based on the use of collective coordinates. The existence of zero-frequency excitations follows at once from general considerations, and it is intuitively obvious that the collective coordinate method should provide a way to perform a consistent perturbation theory free o f infrared divergencies. However, the major difficulty here is that the zero frequency mode is not just a linear combination of the original canonical variables (as is the case in boson systems), but it is rather a fermion bound state. The way these zero modes are disposed of in the collective coordinate method is rather more subtle, and this is the problem we have addressed our attention to. This paper is organized as follows. In sect. 2 we briefly discuss the application to a Fermi system of the path integral formulation of the collective coordinate method developed by Gervais and Sakita [ 1]. We use here standard methods of quantum field theory and they are briefly reviewed for the benefit of possible readers not quite familiar with them. After the collective coordinate method is set up, one is left with path integrals on fermi variables with delta functionals of composite fields representing bosonic constraints. These delta functionals can obviously not be integrated over, as it is done for example in the case of kinks. However, one can exploit the fact that (due to the presence of constraints) one is actually quantizing a gauge theory, and derive Ward identities [5]. This is done in sect. 3, where it is shown (always in the case of a superfluid fermi system) that Ward identities fix uniquely the Green functions o f

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the composite operators associated to the collective coordinate, such as the number of particles N and the gauge angle qb conjugate to N. One finds that the propagators of these composite operators, in a suitable chosen gauge, are either zero or contact terms which are pure gauge artifacts. The decoupling of the unwanted zero mode from gauge-invariant Green functions is implied by these results. In sect. 4 we introduce a very simple quantum mechanical model consisting of fermions in a single shell with degeneracy 2[2 interacting via a pairing force. This model, well-known in many-body physics, is exactly soluble [8] and we use it as a theoretical laboratory to test how perturbation theory works. The perturbation expansion is in ~ - 1 (quite analogous to the N -1 expansion in quantum field theory) and to leading order the system is a superfluid one. Some general properties of this simple model are discussed. Finally, in sect. 5 we perform some explicit calculations to see how our way of handling the constraint and the gauge condition eliminate the zero-frequency mode. We check to leading order the Ward identities and show explicitly how the spurious state is automatically projected away when the constraint is enforced. Moreover, we also compute some non-trivial effects in next to leading order and obtain complete agreement with the known exact results.

2. The collective coordinate method Let us consider a system of interacting fermions described by a set of creation and destruction operators b+m, b m ; where the index m refers to all the quantum numbers that label the independent particle states. The Hamiltonian of the system will be assumed to commute with the number of particles N=~

÷bm, bm

(2.1)

m

so that the Lagrangian L -- ! ~

gn

b~n/~m - n

(2.2)

exhibits a global U(1) invariance, namely, invariance under time-independent phase transformations of the fermi variables b m ~ e-i¢/2br n , b + -~ ei¢/2b+m •

(2.3)

We shall be concerned with the case in which the Fermi system becomes superfluid and breaks U(1) invariance in some well-defined limit in which the theory can be solved exactly and which is therefore a good starting point to do in principle perturbation theory. In practice, naive perturbation theory cannot be done due to the presence of the zero-frequency bosonic excitations.

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As explained in sect. 1, the correct way to tackle this problem is to introduce collective coordinates. We adopt the path integral formulation of Gervais and Sakita [1], which we now briefly review. Consider for example the vacuum transition amplitude, given by the path integral

Z= f~m CDbm(t)CDb+m(t)exp[i f

(2.4)

L dt] .

In order to separate the collective coordinate one multiplies by 1 in the following way

1 = f ~ p q ) ~ [p(t) - N(b,

b+)]

6F x fcDe~(t) 8 [F0(t) e-'~U)/2 , b + (t) d~U)/~)] II, i~,--~-~/,

(2.S)

where F(b, b +) is an arbitrary function of the fermion creation and destruction operators, and Ilt(dF/8¢(t)) is the Jacobian needed to transform 5 (F) into ~i(¢(t)). Multiplying by 1 the right-hand side of eq. (2.4), going to the "body-fixed frame" by performing the following change of variables in the fermion path integral,

am (t) =bm (t) e -i¢(t)/2 , a +m (t) = b+m(t) e lea(t)/2 ,

(2.6)

and using the constraint N(b, b ÷) = p(t), one finds

z= f Hm

I:ifz( ,

ifp(t) (,)dtl

x ~ [p(0 - N] 8 IF(a, a+)] Z~e, a+).

(2.7)

Let us now define Zn by imposing on (2.7) the boundary condition that p(t) -~ n as t --> +~. This means that we are restricting ourselves to the vacuum to vacuum transition for a given number of particles n. Then the path integral over p(t) and ¢(t) can be computed by standard methods and the final result is

z , = f~cDa,@a+~ exp(i f L dt)6[N(a,a+)-n]6[F(a,a+)]A(a,a+).

(2.8)

This is the starting point for the quantization of the fermionic system with a fixed number of particles. Green functions are computed by adding source terms to the path integral (2.8) and by performing a functional differentiation with respect to the sources. As it is well-known, the constrained system is equivalent to a gauge theory [6-7] in the sense that the action S = f d t L subject to the constraint N = n is invariant

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493

under local (time-dependent) phase transformations of the fermion fields am (t). This is because, if a time-dependent phase transformation is performed, the action transforms as oo

S-+S+i f

dtN(a,a+)~,

(2.9)

but since N(t) = n, the extra term is a total derivative that cancels if the gauge function qS(t) is restricted to the class of functions such that 4~(-oo) = 4)(+00). The subsidiary condition F = 0 is a gauge condition that fixes the gauge and breaks gauge invariance. The Faddeev-Popov determinant [9] iX(a, a ÷) depends of course on the choice o f F and guarantees that Zn is independent of such a choice. A quick way to compute it is to realise that it is proportional to the commutator [N, F]. In order to compute Green functions with the theory defined by eq. (2.8) we can follow the standard practice of gauge theories and transform the Faddeev-Popov determinant [9], gauge condition and constraints into effective interaction terms to be added to the Hamiltonian. If the gauge condition F(b, b +) contains time derivatives of the basic fermion fields, the exponentiation of the determinant demands the introduction of auxiliary fields (FaddeevPopov ghosts) [9]. I f F is a local function of time the determinant gives directly an extra contribution to the Hamiltonian: 2° A(a, a +) = exp [6(0) j

dt ln(N(t), F(t)} ] .

(2.10)

--oo

As far as the gauge condition is concerned, we shall consistently employ 't Hooft's trick [10]. None of the previous arguments are changed if the gauge condition is chosen as F(a, a +) - c(t) = 0, with c(t) an arbitrary function of time but independent of the fermi fields. Both sides of eq. (2.8) are then integrated over c(t) with an arbitrary Gaussian weight

and, since Zn is independent of c(t) one gets (up to a normalization factor) 1 --oo

The normalization factor that is missing is exactly the value of the path integral (2.11), namely, what is needed to obtain 6(F) in (2.12) if the limit c~~ 0 is taken. However, there is no need to take that limit: c~plays the role of a gauge parameter that shows up because the original path integral has been averaged over a class of gauge conditions. The missing normalization factor is irrelevant because Green

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494

functions are always computed as the ratio of two path integrals (by normalising to 1 the vacuum to vacuum amplitude). In general, Green functions of operators that do not commute with N will be gauge variant and depend on the choice of gauge F and also on the parameter a. However, physical observables (like poles and residues of Green functions) must necessarily be gauge invariant and independent ofo~. Finally, in practical calculations the constraint 6-functional will be written as

fiN-n] = HS[N(t)-n] t

= l i m I-I(2n/3)-l/2exP(2t~~ f [ N - u ] Z d t ) , /3-'+0

(2.13)

t

which adds yet another effective interaction to the Hamiltonian. We shall then have to deal with an effective Hamiltonian depending on the parameters a,/3. The normalization factor IIt(27r/3)-1/2 cancels in the calculation of Green functions, or in the calculation of energy levels with respect to the ground-state energy. However, it is important to realise that for finite/3 the constraint has been loosened and that only in the limit/3 -+ 0 do we recover the original gauge theory. Therefore, physical observables will be gauge invariant (and consequently independent of the gauge parameter a) only after the limit/3 -+ 0 is taken. We shall later on perform an explicit check of all these statements in a particular model.

3. Ward identities The physical effect of the 6-functionals representing the constraint and gauge condition in eq. (2.8) is to reduce the phase space of the system and eliminate from the spectrum the collective mode associated with the original invariance of the theory. In cases in which the spontaneous breaking of the symmetry is not dynamical, the zero-frequency mode is one particular integration variable in the path integral and a change of variables suffices to prove the desired decoupling of this mode. In our case, the zero frequency mode is a composite state, and the problem cannot be tackled in this way. Therefore, we shall prove in general the decoupling of the zero frequency mode by deriving Ward identities for some composite operators associated with the collective degree of freedom. These identities are a consequence of gauge invariance, and below, we shall use the standard methods of quantum field theory to derive them [5]. The derivation of Ward identities is particularly simple in a special gauge, in which the gauge function F(a, a+) is taken to be the operator ~(a, a ÷) representing the gauge angle, conjugate to N(a, a+). In this case the Faddeev-Popov determinant is 1. We shall moreover concentrate on the Green functions of the composite operators N(t) and q~(t). The generating functional of such Green functions is (in the

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495

gauge F = qb)

Zn[JN,Joi=f~am~a+mexp(iS-~f~ 2at} × ( i f [ J N N +So@] dt) 6 [ N - n] .

(3.1)

Because of the presence of the constraint, the dependence o f Z n on the source JAr is trivial, and we have

Z , [JN, JO ] = exp[in f JN(t) dt] Z , [ 0 , J 0] - exp[infJNq) dt] Zn [Jo] •

(3.2)

In order to derive a Ward identity for Zn [Jo], we perform an infinitesimal change of variables in the path integral equivalent to an infinitesimal gauge transformation on the fermi fields with gauge angle 6~b(t). The Jacobian of such a transformation is one, so the measure is invariant. Since the constrained action is gauge invariant, the only non-invariant terms are the gauge fixing term and the source term containing • , which is simply shifted by an amount 6~(t). Since a change of variables does not change the value of the integral, the term proportional to 6~(t) in the right-hand side of (3.1) must vanish.

/

O--flXc-l)am@a+mexp i S ~7

×fiN-n]

/

Jo(t)- ~

2dt+i Jo*dt

)

.

This is the Ward identity, which can be written as:

JO (t) Zn [Jo ] + i 6Zn [Jo ] - 0 . 5Jo(t)

(3.4)

This functional differential equation is immediately solved

Zn [JO] = Zn [0] exP(½iafJ~(t) dt) ,

(3.5)

so the generating functional for connected Green functions Wn [JN, JO] defined by

Zn [JN,JO] = exp(iWn [JN,Jo]) ,

(3.6)

is exactly given in this gauge by

Wn [JAr, SO] = - i In Zn(O) + nf

Ju(t) dt + f sg(t) dt .

(3.7)

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All Green functions for the composite operators N and ~ follow at once from this result. Obviously the vacuum expectation value of N(t) is n, and all correlation functions for N ( t ) are zero, as it is obvious from the existence of the constraint. There are no correlations between N and ~ either. The vacuum expectation value of ~(t) is zero, which is quite obvious since we have fixed ~(t) equal to c(t) and then performed a gaussian average over c(t) centered at c = 0. The only non-vanishing Green function is the two-point function (3.8)

(T(cb(t')cb(t))) = - i a 6 (t - t ' ) ,

which is a sort of "contact" term arising from the propagation of an excitation with infinite energy. This is indeed the residual effect of the zero frequency mode. As we shall see in the explicit calculations carried out in the sect. 4, the mechanism that decouples the zero frequency mode works as follows: the gauge breaking term in the Hamiltonian gives the zero mode a non-zero frequency dependent on a, since the new effective Hamiltonian no longer commutes with N. When the constraint is also included in the gaussian approximation (see eq. (2.13)), the spurious state acquires a frequency Wo(a, ~) which goes to infinity for any a as/3 ~ 0. The constraint therefore pushes the unwanted excitation to infinity, but nevertheless there are residual contributions due to the renormalization of the vacuum: the vacuum one starts from is usually not the vacuum for the zero modes. In any case, the correlation function given by eq. (3.8), being proportional to a, is a pure gauge artifact and can be argued away by going to the gauge a = 0, which we call the "unitary" gauge because in this case the zero frequency mode will completely disappear from the theory. Notice also that the unitary gauge is equivalent to keeping the gauge 5(e/,) in the calculation of Green functions. 4. A simple model We shall concentrate our attention on a simple quantum mechanical model of interacting fermions that exhibits (in a limiting case to be discussed below) the phenomenon of dynamical symmetry breaking. It consists of fermions in a degenerate shell interacting through a pairing force. This model was first solved by Racah [8]. It has been discussed in terms of collective coordinates in ref. [11]. Consider 2 ~ degenerate single particle levels, labelled by a quantum number m = -+1, ..., +~2. The states with m negative are the time-reversed states of the corre+ sponding ones with m positive. Let am, am be the set of fermion creation and destruction operators. We shall keep always m > 0 in the summations and write + = a+_m . The model is defined by the pairing Hamiltonian affl (4.1)

H = _½G(A+A + A A + ) ,

where ~2

A = ~ m=l

~2

amain,

A += ~ m=l

amain + + •

(4.2)

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497

The Hamiltonian (2.1) exhibits a global U(1) invariance associated to the conservation of the number of particles, given by ~2

N = ~

m=l

(4.3)

(a+mam + a ~ a m ) .

This symmetry operation corresponds to a phase transformation in the creation and destruction operators +

am -~ e - i ~ / 2 a m ,

"

am -* e i ~ / 2 a + •

(4.4)

The three operators A +, A and 7r = I ( N - f2) satisfy the SU(2) algebra. They are in fact isomorphic to J+, J _ and J z , respectively. Consequently, the eigenvalues of H can be computed exactly, since (4.5)

H = -G(J 2 - J~).

The ground state clearly belongs to the irreducible representation of SU(2) corresponding to the largest eigenvalue of the Casimir operator j2 (which is ~[2 (~[2 + 1)). It is then convenient to define a new quantum number u (seniority) by J = ½([2 - u). The energy levels are then given by: E(u, zr) = -,~G(~2 - u)([2 - u + 2) + GTr2

(4.6)

.

These energy eigenvalues are plotted in fig. 1, as a function of v and 7r, which is the number of pairs of particles with respect to a half-filled shell. For each seniority t,, the dependence oh lr is the one of a typical rotational band, and the ground state corresponds to a configuration with 7:, = 0, namely, N = [2 particles. We are particularly interested in studying the limit [2 ~ 0% keeping G ~ = g fixed. In this case, eq. (4.6) becomes E(u) = -~g([2 + 2) + ½gp + O(1/[2),

(4.7)

meaning that the rotational band flattens and that consequently the energy levels E

8

,V

=

4

,'0=2 •

G~ •

o

~)

=

0

o

N-~2 -52

f2

Fig. I. Exact energy spectrum of the model discussed in the text.

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498

become degenerate (independent of 7r). In particular, one has a degenerate ground state corresponding to states with different numbers of particles. The phenomenon of spontaneous symmetry breaking is therefore clearly exhibited. It only occurs in the limit X2 ~ oo because, as it is well-known, an infinite number of degrees of freedom is needed for a phase transition to occur. The limit ~2 ~ ~ is interesting because an exact solution can be obtained by going to a quasiparticle representation by means of a Bogoliubov transformation. In this limit only bubble diagrams survive (the RPA approximation). One proceeds as usual by defining quasiparticle creation operators as follows

+

--

-I-

3 ~ - Ua~ + Vain ,

(4.8)

with U 2 + V 2 = 1. Destruction operators are obtained by hermitian conjugation of the previous equation. The BCS ground state is defined by

3mtO) = 3~ i0) = 0 ,

(4.9)

and the parameters U, V are determined by minimizing the expectation value of H in the BCS ground state. One may also fLx, by means of a Lagrange multiplier, the mean value of N in this ground state. We shall always choose (N) = ~ ; and in this case one finds U-- V = v ~ . Once the Bogoliubov transformation is done, the Hamiltonian reads:

H= - ~ g I 2 ( 1 - ~ ) 2

+ ¼g(P- P+) 2 ,

(4.10)

where the number of quasiparticles c~ and the operators P, P+ are defined as: ~2

m=l I2

p:_~_l ~ 3,~3m , x/~ m=l 1

(4.11)

~2

These operators satisfy again SU(2) commutation relations of the following form [e, P+] = 1 - ~ / a , [c~,p] = - 2 P ,

(4.12) [c~,p+] = 2p+ .

(4.13)

Moreover, it is very important to keep in mind that the operator 7r representing the number of particles (with respect to a half-fiUed shell) is given in terms of quasi-

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499

particle operators by: 7r = ½x/~ (P + P + ) .

(4.14)

The Hamiltonian (4.10) contains a c-number term ( - ~ g ~ ) which gives the leading contribution to the ground state energy and a term XgC~, which exactly matches the term proportional to u in eq. (4.7), if u is identified with the number of quasiparticles. One recognizes a semiclassical structure, with ~ - 1 playing the role of semiclassical expansion parameter. This is of course not an accident, since the fZ-1 expansion is identical to the"N -1 expansion in quantum field theory. The exact result for the energy levels given by eq. (4.6) shows that, apart from the term gTr2/f2, the ground state energy Eo does not have corrections beyond O(1) in ~ - 1 and that excited states get only a correction of order ~ - ~ . To zeroth order in ~ - 1 , the Hamiltonian reduces to (we leave aside the groundstate energy Eo) HR1,A = l g Q ~ _ g p + p + l g ( p + p+)2 ,

(4.15)

which, together with the neglect of the term of O(1/Q) in the commutator of P and P+ constitutes the RPA approximation, equivalent to the sum of bubble diagrams. Vacuum to vacuum diagrams included in the RPA approximation are shown in fig. 2. Corrections to the RPA approximation arise in this model from the terms ct~/~2 and c~2/~22 in (4.14) (which can be trivially dealt with) and from the corrections to the P, P+ commutator which represent the effect of Fermi statistics since they correspond to exchangediagrams in perturbation theory. The spectrum OfHRp A consists of free quasiparticles with energy _lg, 2 since the only way a free quasiparticle can interact is through an exchange with a vacuum fluctuation (as shown in fig. 3) which is an effect of order f2 -1 . At the two quasiparticle level, the situation is different. All two quasiparticle states orthogonal to P+10)keep their unperturbed energy g. The remaining state is shifted to zero energy and gives the expected zero frequency mode. No consistent perturbation theory in f2 -1 has yet been developed even in this simple model, due to the zero-mode problem. One can try for example to start by considering the free quasiparticle part of H as an unperturbed Hamiltonian, and by multiplying the residual part by an expansion parameter x. It is clear that H does not commute with N if x q: 1 ; so there is no zero frequency mode in this case. How-

I

I

Fig. 2. Diagrams contributing to the vacuum renormalization in the RPA approximation.

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500

i

I

O000C Fig. 3. Exchange of a quasiparticle with a vacuum fluctuation.

ever, it has been shown that the perturbation expansion around x = 0 has a radius of convergence which is always smaller than 1 [12]. Thus the limit x ~ 1 cannot be taken in perturbation theory. We shall explicitly show in sect. 5 how a perturbation theory in [2-1 developed on the basis of the collective coordinate method correctly deals with the zero-mode problem and reproduces the exact result. As a side remark, we would like to discuss a very specific feature of this model, the calculation of the ground-state energy which can be easily understood in the quasiparticle representation as follows. Using the commutation relations (4.12) we rewrite the (P - P+)2 term in (4.10) in terms of (P + P+)2. We then obtain: H = -lg(a+

2) + ½gQ'~ 1+

_~g --~ +lg(p+p+)2_gp+p.

(4.16)

It now follows that the projection over a fixed number of particles of the BCS vacuum, namely 2rr

]0, 7ro) = f o

d~b ei("-Tr°)~j0)

(4.17)

is an exact eigenstate of H. This is because the Hamiltonian commutes with 7r, so the part of H not proportional to 7r2 also commutes with 7r and can be passed through the operator in eq. (4.17) and annihilated against 10). One then finds: HI0, 7ro) =

E-1 g(~2 + 2) +

10, 7to)

(4.18)

which is the exact ground-state energy. This means that all the vacuum renormalization effects will be only due to the zero-frequency collective excitation, since there is no renormalization of the vacuum wave function when i0) is projected on eigenstates of n.

5. Perturbative calculations We consider now the problem of the quantization of this model with a fixed number of particles which we take for simplicity to be n = ~2 or, in terms of n, no = O.

v. Alessandrini et aL / Collective coordinates

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We shall consistently use as a gauge function F the gauge angle cb. Our first task is then to compute the expression of the gauge angle dp in terms of quasiparticle operators. This is quite straightforward since we have two operators A, A ÷ (see eq. (4.2)) satisfying the commutation relations [A, n] = A ,

[A +, 7r] = - A + .

(5.1)

Then, it follows that [i In A, 7r] = i,

[ - i In A ÷ , 7r] = i .

(5.2)

Since dp has to be hermitian, we obtain q5 = ½i (In A - In A + ) .

(5.3)

Writing A and A ÷ in terms of quasiparticle operators, we get

i

~/¢2

[

1

1

L ( P - P + ) + -~ (c~p _ p+c)~) + ~_~ (p3 _ 3p+p2 + 3 p + 2 p _ p + 3 )

+ O(f2-2)1 .

(5.4)

_J

The factor f2 -1/2 occurs because zr is equal to ½x/-~(P +P+). We shall from now on rescale the operators ~r, ~ in such a way that no half-integer powers of f2 occurs. According to our discussion o f sect. 2, we exponentiate the gauge condition and the constraint, and consider the effective Hamiltonian nef f = n - ~

1 (p + p+)2

_ ~

1

[(/19 _ ? + ) + 0 ( ~ , - ~ - 1 ) + ...] 2 .

(5.5)

Notice, however, that we could as well have chosen as gauge function F the leading term of the expansion of ap in powers of ¢2-1 , namely, i ( P - P + ) . In this case there is no need to compute q5 but the F a d d e e v - P o p o v determinant is n o t trivial and one must add a F a d d e e v - P o p o v term to the effective Hamiltonian, which would now be of the form: c~

, nef f =

1 H - ~-fi ( e +

p+)2

1 - ~

p+)2 (e -

- i5(0)

n=l~--rl1 (c/~n \~2]

'

(5.6)

where the last term is the expansion of ln(1 - N / f 2 ) . We shall comment on the 5(0) factor later on. Both ways o f proceeding coincide to leading order and, although the results of higher order corrections may be quite different at first sight, they must agree after the limit/3 ~ 0 is taken, because in this limit we recover the original gauge theory. In any case, the complete Hamiltonian contains now interactions of all orders in f2 -1 . This, in a way, is most welcome for the following reason: we know that the perturbation series for the energy eigenvalues stops at the

502

V. Alessandrini et al. / Collective coordinates

next to leading term. However, exchange effects will produce corrections to any order in ~2-1 in perturbation theory, and they are expected to be compensated by the new terms in the effective Hamiltonian in whichever version (5.5) or (5.6) we choose. Indeed, higher order terms in the expansion of q~ or the F a d d e e v - P o p o v term have the same origin as exchange effects, that is to say, the fact that the commutator [P, P+] is not 1. Let us now examine the effective Hamiltonian (5.5) in the RPA approximation. We have now, instead of (4.17) HRPA = l gQ~ _ g p + p + 1 A ( P + p + ) 2 _ l B ( p _ p + ) 2 ,

(5.7)

where we have defined A = ! g _ __1 2

B = _1

4/3

'

a

(5.8) '

and the collective excitation is calculated in the usual way, by searching for a ladder operator F of the form P = XP - pa°+ ,

X2 - bt2 = 1,

(5.9)

satisfying (5.10)

[HRPA, F] = - - W 0 F •

One easily obtains 600 = 2 V r ~

,

+~

2\~

u =

'

-~

2 ~-

,

=

.

(5.11)

Notice that if we r e m o v e the constraint by letting/3 ~ oo the eigenfrequency o f the spurious state stays finite. This is because the gauge term in the Hamiltonian breaks the conservation of the number of particles and therefore there is no reason to expect a zero mode. If the gauge term is also removed by letting c~-~ o% then w 0 goes to zero and we recover the original difficulty. We remember that the correct limit t o take is/3 ~ 0 for fixed a, and that physical observables should be independent of a. In this limit coo goes to infinity, so in some sense the spurious state is decoupled from the physical spectrum. However, we must still verify that properly normalized Green functions stay finite as/3 ~ 0. The first thing we check is the Ward identity (3.8) which, since we are working at leading order, reduces to the calculation of the propagator of i ( P - P + ) . By inverting eq. (5.9) and its hermitian conjugate P and P+ can be written in terms o f r and P+. We obtain for qb = ix/lc~Oo ( p -- p + ) .

(5.12)

V. Alessandrini et al. / Collective coordinates

503

The propagator for ~p is then computed in terms of the propagator for F (15IT (F(t') F(t)} t0) = e - i t ° ° ( t ' - t)o (t' - t)

(5.13)

where 10) is the state annihilated by F, normalized to 1 because of the fact that vacuum to vacuum diagrams have been divided out. We get as a result of this simple calculation iaCo02

F(w) =

dr eit°r
,

(5.14)

--oo

which exactly agrees with eq. (4.8) as 600 ~ oo. We obtain a finite result because of vacuum renormalization, since the numerator is just the parameter ~2 that controls this effect. Indeed, if ~ = 1 then X = 1 and/~ = 0. It is also amusing to see that if one goes back (wrongly) to the unconstrained theory by the limits/3 ~ ~ , a ~ oo the numerator of (5.14) stays finite and one recovers the pole at w = 0. Next we compute the two-quasiparticle propagator defined by

Gnm@O)

=

?dt

eit°t
n, m = 1 ..... ~2, (5.15)

--oo

by summing all Feynman diagrams representing the free propagator plus the sum of bubbles, exhibited in fig. 4. The free propagator is a diagonal matrix in (n, m) space while the remaining diagrams are independent of n, m due to the specific form of the interactions. We can then write i

anrn(OO) : 8n,m + B(oo) . 6o - g

(5.16)

Let G(6o) be the Fourier transform of . According to (4.1 1), it is ob-

m

n _ -,s-(~--,--_

m

..~-~

n

-_

m ~

-

CY I

<>, I

Fig. 4. The RPA approximation to the two-quasiparticle propagator.

V. Alessandrini et al. / Collective coordinates

504

tained from Gnm(W) by summing over n, m. With our normalizations we obtain G(w) -

i ~-g

+ D,B(~o),

(5.17)

and, by eliminating B(¢o) from the two equations,

Gnm(O) = ~ - gi [ 8

nrn - 1 ] + 1 G(6o)

(5.18)

which shows that the pole in the Green function occurs only in the part of the two quasiparticle Hilbert space orthogonal to P+i0), since the residues are just the matrix elements of the projection operator

9 =I-e+lo)(ole.

(5.19)

The remaining part containing G(co) acts only in the spurious subspace, and is the piece that would contain the zero-frequency pole in a naive calculation. Here instead is just a constant proportional to a, which can be obtained directly from the Ward identities, because the correlation functions containing 7r and ¢ reduce in the RPA limit to correlation functions for (P + P÷) and (P - P+). Then we can directly obtain any Green function containing P and P÷. We find C(,~) = - l i ~ .

(5.20)

The two quasiparticle Green function is gauge dependent because the quasiparticle operators do not commute with zr. However, poles and residues are gauge invariant as expected. In the gauge a = 0 there is no residual effect of the spurious state, and only physical states propagate. The gauge parameter a is playing here the same role as the equivalent gauge parameter of the Landau gauge in QED. Indeed, here as in QED the Ward identities are uniquely fixing the spurious part of the propagators. And the limit a = 0 is analogous to the Lorentz gauge. We now turn to a calculation of an effect at the next to leading order. We have computed several such effects. We shall present here as a sample calculation the simplest one, which is the calculation of the energy shift of a two quasiparticle state Im, m') with m' 4: ~ ; which is a physical state which has no spurious component. A quick glance at the exact result (4.6) for the energy eigenvalues shows that the exact energy difference (E(u = 2) - Eo) is given by the unperturbed value g. We must therefore check that there are no ~2- I corrections to the unperturbed energy. This is indeed what happens in a highly non-trivial way, as the result of cancellations between dit'ferent diagrams which we shall explicitly exhibit because they are the firmest support to the correctness of our procedure. In order to perform the ~ - l expansion it is useful to adopt the language of the "nuclear field theory", used in many body physics to deal with collective coordinates which are fermion bound states [13]. The basic idea is to do perturbation theory by using as independent dynamical variables the fermions and the collective bosons provided by the RPA approximation, in our case the states created and destroyed by I '+ and P. The

V. Alessandriniet al. / Collective coordinates

505

effects of overcompleteness and Fermi statistics are dealt with perturbatively by a set of heuristic Feynman rules whose ultimate justification is the fact that they reproduce the [2 -1 expansion. The vacuum one starts from is the direct product of the quasiparticle vacuum and theyacuum of F since, as we have explained before, quasiparticles will learn about vacuum renormalization only through higher-order processes. We then apply to our Hamiltonian (5.5), the rules of nuclear field theory and enforce the constraint at the end of the calculation by letting/3 --> 0. We consider the Hamiltonian (5.5) and keep only terms of order [2-1 in the expansion of cb2 . After ordering this Hamiltonian in P and P÷, using, of course, the next form of the commutator, it can be written as: ( 1 neff

= lgQ~

+

1 ) i c~(cr~-l) 2 a ~ c ~ _ z~g

8~[2

+ !2 G 4 (p+2+pZ)

--

G2P+P+6H ,

(5.21)

where the last term 6H is the [2 -1 term in the expansion of q~2, and the constants G2 and G4 are given by 1

1

G2 = g - ( A +B) = ½g _ _ -a1 - _ _ 4/3' 1

G4 = A - B

=~g

1

c~ 4/3

(5.22)

It will be useful for later applications to rephrase the results of the RPA calculation for X and/a in terms of these coefficients: G4 ~'= [2Coo(g- G2 - Wo)] 1/2 ,

6o0 - g + G2 •= [2wo(g_ G 2 __O90)]1/2

(5.23)

Next we determine the coupling between a two-quasiparticle state/3~n/3~n-I0) and the spurious state created by F ÷, by using eq. (5.9) and identifying in eq. (5.21) an effective interaction of the form +

+

+

+

+

+

Ab(/3m/3mP +/3m~mP) + Ag~m/3r~P +/3m/3mP ).

(5.24)

The vertices Agand Ab, exhibited in fig. 5, are given by Ag =

Ab

(,uG4- X G 2 ) = ~ 1

k

-

aa) =

(Coo-g)~., 1

( Oo +g)u,

(5.25) (5.26)

where use has been made of eqs. (5.23) to obtain the final result. We can now discuss the calculation of the energy shift AE, to order [2-1, of the two quasiparticle state Im, m') with m' :/: rn. The diagrams contributing to AE are exhibited in figs.

506

V. Alessandrini et al. / Collective coordinates ra

A

+:% m

g

m

m

i

b

m

Fig. 5. Diagrams representing the particle-vibration vertices Ag and Ab. The wavy line represents the RPA collective phonon. 6(a)-(e). Fig. 6a represents the contribution of Hartree-Fock terms in (5.21) 1 4/3~

AEa

1 a~2 '

(5.27)

which are by themselves gauge dependent and potentially dangerous due to the singularity in/3. Fig. 6b represents the contribution of the two-body force 9 [ ( 9 [ - 1)

.__~

m mI (a)

(d)

I~--I m

m ' (b)

m'

m

< (p_p+)>2

(c)

m

m

(e)

Fig. 6. Diagrams contributing to the O(1/KZ)energy shift of a two-quasiparticle state Ira, m'> with m' ~ ff~. The wavy line represents the RPA collective phonon. The dotted line corresponds to the two-body vertex c~(c~ _ 1).

507

V. Alessandrini et al. / Collective coordinates

in n e f f , which gives an energy shift (5.28)

zkE b = - g/2~2 .

Fig. 6c represents the energy shift provided by 6 H . The exact form of this interaction is _6H = 2~

( p _ p + ) 2 + 2 - ~1 ( p _ p + ) 2 c ~ - ~ 1 [4p+p + 2p2 + 2p +2]

1 + 3a--~ [p4 _ 4 p + p 3 + 6p+ 2p2 _ 4p+ 3p + p+ 4]

+

O(~--2)

,

(5.29)

but only the first term contributes to the energy shift we are computing, since the remaining terms will couple to the initial two-quasiparticle state only through effects representing an exchange with a vacuum fluctuation, which give yet another power of ~2-1 . The first term of the right-hand side of (5.29) gives an effective HartreeFock (tadpole) contribution to AE of the form

ace-

2 ((e _ p+)2) =

2

co ao

(5.30)

Finally, fig. 6d represents an exchange effect with a vacuum fluctuation, while fig. 6e represents an interaction with a vacuum fluctuation through the two-body force cK(c~ - 1). They give a contribution to the energy shift z2xEd+e=2A~w 1

o +g

g

(6°0 + g)

I =2A~

Wo

(6°0 + g)2

(5.31)

By making use of eqs. (5.26) and (5.23) one finally obtains 1

Z~XEd+e=

~ (g -- CO0 --

g

G2) - 2~2

wo

1

f2 + a~2

1

4/392 '

(5.32)

which exactly cancels the previous energy shifts given by eqs. (5.27, 28 and 30). We then obtain the expected agreement with the exact result. It is also interesting to see how the same result comes about when the gauge function F is chosen to be i ( P - P + ) instead of q5 and the Hamiltonian is given by eq. (5.6). In this case the calculation of the energy shift is exactly the same, except that the tadpole diagram (5c) (which now arises from the Faddeev-Popov part of the Hamiltonian) has a strength i6(0) instead of ( ( P - P + ) 2 ) / 2 a . However, a careful analysis shows that when the i 6 ( 0 ) factor is properly regularized, one finds that it is naturally replaced by los o. A quick way to see this result is to compare the value o f (~(t) ~(t')) at equal times as given by the Ward identity (3.8) with (q~2) as computed from eq. (5.12). Cancellations between different diagrams of the kind we have just exhibited systematically occur in more involved calculations whose results will be published elsewhere. We will only mention here another interesting example, which is the direct

508

V. Alessandrini et al. / Collective coordinates m

_

~

n

m

0

n

?n

+

m

5-

n

Fig. 7. The two-quasiparticle propagator in terms of the propagator of the RPA collective phonon.

computation of the two-quasiparticle propagator (5.16). This propagator is given, up to order ~2-1 , by the sum of the four Feynman diagrams of fig. 7, which reads Gnm((.o) =

,

t~nm + co - g

i--

~2

+

-

co + m o

6o -

1.

6OoA

When the limit/3 -+ 0 is taken, this expression reproduces (after some tedious algebra) the results (5.18)-(5.20). Notice that in this case not only the singular terms in 13 cancel, but also the double pole at ~o = g disappears as 13~ 0.

6. Conclusions We have extensively discussed in this paper the problem of the quantization of a Fermi system with bosonic constraints associated with a collective coordinate which is a constant of the motion. The methods discussed here are expected to be useful in quantum field theory, whenever one is forced to introduce a collective coordinate in a system of tbrmions that exhibits dynamical symmetry breaking. In many-body physics the way is open for more realistic applications like the problem of real rotations of deformed nuclei, where the only additional complication is a technical one, namely, the fact that one has to deal with a non-Abelian group. However, we must keep in mind that our way of handling the constraint works only in the context of the ~2-1 expansion and implies that the Fermi system has an infinite number of degrees of freedom to start with. For a finite system, the error thus introduced is corrected in successive orders in the ~2-1 expansion. If one starts with a finite number of degrees of freedom, the gaussian approximation to the constraint and the limit/3 -~ 0 do not make sense since some power of the composite fields that occur in the constraint will vanish. In fact, in this case it is not even clear what a delta function of sums of pairs of Fermi variables means. This is of course related to another problem, which is how to perform a canonical quantization of a Fermi system with bosonic constraints, thereby avoiding the use of the path integral representation. The application of Dirac's formulation [6] to a Fermi system is not obvious, since the prescription of replacing Poisson brackets by

V. Alessandrini et al. / Collective coordinates

509

Dirac brackets works for c o m m u t a t i o n relations but cannot be used to m o d i f y the basic a n t i c o m m u t a t i o n relations o f the fermi variables. This is yet a n o t h e r manifestation o f the fact that there are no r e d u n d a n t fermion degrees o f f r e e d o m . In the case o f the simple m o d e l we discussed before, it looks as if the m o d i f i e d c o m m u t a t i o n relations and the basic a n t i c o m m u t a t i o n relations are consistent in the c o n t e x t o f the ~2-1 expansion. This p r o b l e m is currently under investigation. One o f us (D.R.B.) wants to express his gratitude to Prof. R. Vinh-Mau for the hospitality e x t e n d e d to him at the Institut de Physique Nucl6aire d'Orsay.

References [11 J.L. Gervais and B. Sakita, Phys. Rev. D2 (1975) 2943; J.L. Gervais and A. Neveu, Phys. Reports 23 (1976) 237. [2] N. Christ and T.D. Lee, Phys. Rev. D12 (1975) 1606. [3] S. Coleman, Classical lumps and their quantum descendants, lectures delivered at the 1975 Int. School of Subnuclear Physics, Ettore Majorana, ed. A. Zichichi. [4] A.M. Polyakov, Phys. Lett. 59B (1975) 83; A.A. Belavin, A.M. Polyakov, A.S. Schwartz and Yu.S. Tyupkin, Phys. Lett. 59B (1975) 85; G. 't Hooft, Phys. Rev. Lett. 37 (1976) 8; G. 't Hooft, Phys. Rev. D14 (1976) 3422; A.M. Polyakov, Nucl. Phys. B120 (1977) 429. [5] E.S. Abers and B.W. Lee, Phys. Reports 9 (1973) 1. [6] P.A.M. Dirac, Lectures on quantum mechanics, Belfer graduate School of Sciences, Monographs series no. 2 (Yeshiva University, New York, 1964). [7] L.D. Faddeev, Teor. Mat. Fis. 1 (1969) 3. [8] G. Racah, Phys. Rev. 63 (1943) 367. [9] L.D. Faddeev and V.N. Popov, Phys. Lett. 25B (1967) 29. [10] G. 't Hooft, Nucl. Phys. B33 (1971) 173. [ 11 ] D.R. B~s and R.A. Broglia, Proc. Int. School of Physics, Enrico Fermi, Varenna 1976, eds. A. Bohr and R.A. Broglia. [12] D.R. B~s, R.A. Broglia, G.G. Dussel and H.M. Sofia, to be published. [13] P.F. Bortignon, R.A. Broglia, D.R. B~s and R. Liota, Phys. Reports 30 (1977) 305.