Separation of the monopole contribution to the nuclear Hamiltonian
Abstract
It is shown that the nuclear Hamiltonian can be separated rigorously as For sufficiently smooth forces the monopole part is entirely responsible for HartreeFock selfconsistency and hence saturation properties. The multipole part contains the “residual” terms  pairing, quadrupole, etc.  that will be analyzed in a companion paper. We include a review of the basic results often needed when working with multipole decompositions and average monopole properties.
pacs:
21.60.Cs, 21.60.Ev, 21.30.xyIt has not been possible, yet, to construct interactions that could satisfy simultaneously three basic conditions:

to be realistic, i.e. consistent with the nucleonnucleon () phase shifts,

to ensure good saturation properties, i.e. correct binding energies at the observed radii,

to provide good spectroscopy.
As a consequence many forces have been designed for specific contexts or problems : pairing plus quadrupole [1, 2], density dependent potentials for meanfield approaches [3, 4], LandauMigdal parametrizations for studies of the giant resonances [5], direct fits to twobody matrix elements for shellmodel calculations [6] and many others. A way out of this unsatisfactory state of affairs would be to exhibit an interaction consistent with conditions A,B,C above. If we only assume that it exists and could be reduced to an effective form smooth enough to do HartreeFock (HF) variation, many of its properties can be discovered. The basic tool we shall need is the following separation property:
Given a sufficiently smooth Hamiltonian ,, it can be separated as Only the monopole field is affected by spherical HartreeFock variation. Therefore it is entirely responsible for global saturation properties and single particle behaviour.
As long as we do not find ways of reconciling conditions A and B  a major problem  must be treated phenomenologically. The phenomenological enforcement of good saturation properties is quite feasible in a shellmodel context and leads to the pleasing result that the multipole part  which can be extracted rather uniquely from the realistic interactions  has an excellent behaviour [7, 8]. Therefore conditions A and C, as well as B and C, are mutually compatible.
An elementary argument explains the situation. The observed nuclear radii, fm, imply average interparticle distances of some fm, and therefore the nucleons “see” predominantly the medium range of the potential. This is a region that is well understood theoretically [9] and well described by the realistic forces.
The picture that emerges is quite simple: the short range part of the interaction is not well understood, but it is certainly responsible for the repulsive terms necessary for saturation. Therefore, with our present knowledge of the nuclear forces, the phenomenological treatment of is a necessity, but it may perhaps lead to some fresh ideas. The argument is that a Fock representation that can accomodate all possible nonrelativistic interactions. Although in principle enormously many matrix elements have to be specified, is described by a small fraction of the total and by some formal manipulations we shall be able to isolate the very few that count (essentially the ones describing the bulk properties of nuclear matter). Therefore it may possible to exhibit some simple that contains few parameters and describes nuclear data satisfactorily. Since something simple in Fock space may be complicated in coordinate space, it may point to ways  so far overlooked  of reconciling conditions A and B.
There are three steps in the task of giving a complete characterization of .
1. The first is technical: we must prove the separation property, and prepare the tools that make it possible to take the greatest advantage of the underlying symmetries: angular momentum,isospin and the unitary symmetries of the monopole world.
2. As mentioned, we know much about , but this knowledge comes in huge arrays of matrix elements. It would be far more useful if we could extract from this mass of numbers the truly important ones.
3. It remains to specify the monopole field.
We shall deal with the first point here, and with the second in a companion (next) paper [10] where it is shown that is indeed simple and a hint emerges about . The hint has been taken up in [11], a study of nuclear masses that provides a first approximation to point 3.
Our main purpose now is to prove the separation property of the monopole field . The proof is far more compact if multipole (i.e. angular momentum) techniques are introduced. Furthermore, they are essential for the next paper. French lectures [12] are the fundamental reference on the subject and  supplemented by a good book on angular momentum [13]  they contain most of the results we may need. In assembling them in the first section and the appendix, we have tried to give a selfcontained account, and compress in a few pages many of the things of use in daily shell model practice, putting some emphasis on the connection between coupled and uncoupled representations, and on questions of phase.
The second section deals with separation and HF properties and it aims at making the reader familiar with basic technical and conceptual aspects of the monopole field.
Notations are regrouped at the end of the appendix.
I Basic Couplings and Representations
This section deals with angular momentum coupling and recoupling of operators following the techniques of French. There is a little here that is not adapted directly from [12]: some changes in notation, the use of scalar product whenever possible, a lesser reluctance to go over the scheme for simple operations and the introduction of the two possible analogues of hermitean conjugation in the coupled schemes.
The reader is reminded that in the Appendix, eqs.(A1621) contain the elementary formulae that may be needed in handling and symbols.
i.1 Technical preliminaries
The uncoupled representation or scheme
We shall work in spaces containing orbits labeled each associated with operators that act on a vacuum
(1) 
For a given number of particles , we can construct different states, each associated with a state operator :
(2) 
Operators of rank take the form (we use * for Hermitean  or complex  conjugation so as to reserve the + superscript to creation operators)
(3) 
We shall be mainly interested in the case, in the Hamiltonian of rank (kinetic and potential energies and respectively):
(4) 
and in the special body operator
(5) 
whose explicit form is dictated by its rank and the fact that it has eigenvalue 1 when acting on any state of particles.
The particle hole (ph) transformations
(6) 
consist in interchanging creation and annihilation operators for some orbits, by interchanging vacua
(7) 
In its simplest form, and all operators have to be brought to normal order. In particular
(8) 
and
(9) 
Coupled representations
To take advantage of the basic symmetries, operators have to be coupled to good angular momentum and isospin . The scheme labels will be replaced by pairs (, where specifies the subshell to which the orbit belongs, and its projection quantum numbers. Following French [12], we shall introduce a product notation in which expressions are independent of the coupling scheme. In formalism, for example, a single tensorial index will represent pairs in spinisospin space
(10) 
(Note that as tensor index has not exactly the same meaning as label of shell (i.e. , where is the principal quantum number). No confusion can possibly arise from this convention.)
Expressions involving these indices will stand for products as in
(11) 
more generally:
(12) 
where may be some or ClebshGordan coefficient or similar functions as in
(13) 
In formalism, also called neutronproton (np), we do not couple explicitly to good , the tensorial indeces refer to a single space , and the identifications are , , etc. (Note that when used as label, must specify whether the shell is a neutron or a proton one). In and formalisms, we have and product spaces respectively, and the necessary identifications are straightforward as long as we do not recouple and explicitly to good .
Let us introduce the coupled and normalized two body state operators written in terms of ordered pairs (:
(14) 
means that the sum is restricted to if
means that must be such that if
We can always relax the ordering through
.
Let us make the following identifications
(15) 
and take advantage of the ordering in eq.(14) to write directly
(16a)  
(16b)  
(16c)  
(16d) 
Rotational and isospin invariance are made manifest through the scalar products, which in angular momentum manipulations must be rewritten as zero coupled pairs. There are two natural ways of doing so, depending on the order of the couplings:
(17a)  
(17b) 
where has been replaced by the most convenient tensor, which is either , the conjugate of , or , the adjoint of . The existence of these operators follows from the commutation rules of and with any irreducible tensor :
(18) 
To stress the unavoidable ambiguity in sign, we have taken the unusual step of introducing two operators, the adjoint and conjugate , that are the inverse of one another:
(19) 
If the two operators are identical. If , it becomes impossible to ensure  with a single definition  the same sign for scalar product and zero coupling of operators. Which version we choose depends on the problem at hand. For normal ordering as in eq.(17a), and in particular
(20) 
the natural choice of basic tensors is and , for which we adopt the notation of French:
(21) 
With antinormal ordering as in eq.(17b), needed in ph transforms we find
(22) 
which simply amounts to a notation
as the bar over the operators means ph transform. But it also means adjoint, and indeed:
by definition and by definition of and by the last equality in (19). Therefore the ph transform of a state operator is its adjoint. The antitransform of a state operator is its conjugate by exactly the same argument. Note that for the uncoupled operators ph transform is Hermitean conjugation, better represented by its dual analogues in (18) than by the arbitrary choice of one of them. From (22) we have the basic transforms:
(23) 
The coupled operators quadratic in and are
(24) 
Obviously . From the easily proved identity
(25) 
(equally valid for conjugation), we obtain
(26) 
For reduced matrix elements we use Racah’s definition
(27) 
For any operator it is true that
(28) 
and by applying (27) to both sides it follows that
(29) 
where we have omitted complex conjugation on the rhs because our reduced matrix elements will be real.
The coupled form of a rank 1 operator is deduced from the uncoupled one in (3) by using (27) and the definition of in (24):
(30) 
We can always rewrite an arbitrary in terms of the symmetric and antisymmetric operators
(31) 
Then calling , the expansion becomes
(32) 
The phase ensures that the spherical harmonics are symmetric if (Condon and Shortley’s choice) , which leads to and vanishing of the term. The phase convention should be changed to if . Then , and . Note that now instead of the usual but and always, since makes no sense. Still, positive definite zero coupling for tensors of integer rank could be obtained with the change to . As we have seen, for half integer rank, no convention will ensure that zero coupling is always definite positive.
Recoupling.
Let us consider a ph transformation (6) and using the identification (15) assume that orbits and are transformed while and (i.e. and ) are left untouched. Then we have
(33) 
Now use (16b) , relax the ordering for projections in the contraction in (33) , and introduce the tensors, permuting the middle ones
(34) 
For clarity the internal couplings are indicated by different notations: and . Recoupling through a normalized symbol yields
obtained by reorganizing the couplings with the help of 3 symbols. All these operations can be summed up by
(37) 
and its inverse
(38) 
where the first term can be written by inspection but the contraction needs some care. The hybrid procedure we have adopted  of mixing and  points to a paradox: French’s notation and techniques simplify many complex coupling problems but complicate a few simple ones. (Hint: try to obtain (37) and (38) using the  impressive  artillery in Hsu’s appendix to [12]; a good exercise).
i.2 The V and representation of
Equations (37) and (38) were derived to prepare for a ph transform of some orbits. However, beyond the possibility of changing vacua  which may be quite useful occasionally  we are interested in the different representation(s) of the Hamiltonian that these operations entail. Accordingly, we shall always keep the untransformed notation and for the orbits.
First we consider the case in which all orbits are transformed. Introducing , the coupled version of eq.(8) becomes,
(39)  
All contractions are calculated using (36) .
Next we examine the transformations associated to (37) and (38) in which only the middle operators are interchanged. It is convenient to allow for complete flexibility in the summations and we shall relax the restrictions by replacing the factors by the convention
(40) 
so that the sums could be interpreted as restricted or not restricted. We write therefore
(41) 
and from now on we set etc.
According to (37), can be transformed into
(42) 
where
(43a)  
(43b) 
(Remember that means that we sum over Pauli allowed ). Eq.(38) suggests an alternative to (42)
(44) 
where each multipolarity is associated with a two body operator. The obvious check that (44) is indeed (42) comes from
(45) 
The proof is left as an exercise (use (43a) and Racah sum rule (A.19)).
In a representation, by introducing explicitly the isospin in values etc and the 6
(46) 
we find
(47a)  
(47b) 
and reciprocally
(48a)  
(48b) 
Ii the Monopole Field and the Separation
Before going into the technical problems, it is worth explaining how the monopole field appears. Several of the statements that follow need a formal proof. It will be supplied in the body of the section.
The separation of the Hamiltonian into an “unperturbed” and a “residual” part , , is at the heart of manybody physics, and the idea that , could be represented by some central  i.e. single particle  field is of great heuristic and qualitative value. However, to decide whether it makes sense quantitatively, we must understand how , of rank2, can be approximated by a rank1 operator.
What can be done cleanly is to define as plus a two body part so that
(49) 
has two equivalent properties:
Complete Extraction. When written in a multipole representation ’ contains no number operators. It means that all the terms that are diagonal in the basis we are using have migrated to .
Trace Equivalence. The expectation value of for any basic state is the average energy (i.e. the trace) of the configuration to which it belongs. (A configuration is the set of states with fixed number of particles in each orbit).
To recover and examine the notion of central field we choose as vacuum some determinantal state, i.e. we set below the Fermi orbit and above. The single  particle and single  hole energies are then calculated from eq.(49), by adding or removing a particle from the vacuum. In a HF calculation these energies play an important role and invite the interpretation that the determinantal state generates a mean field in which the particles move. This interpretation is very good when there is a dominant agent in the Hamiltonian that is responsible for most of the field, as in atoms and planets.
In nuclei it is simply wrong. What the HF calculation produces is a basis of orbitals. When is written in that basis, can be extracted, and it will yield indeed the HF value for the vacuum and single fermion energies. However, there is no reason to linearize (i.e. to approximate it by a central field) when estimating energies of other configurations: the quadratic effects grow fast. To fix ideas: typically keV in the pf shell, not a large number, but it multiplies the operator that may become or even , and drastically modify the effects of the central field.
There is much empirical evidence that at fixed total number of particles it is a good approximation to keep the same basis for all orbits in the vicinity of the Fermi level, and therefore is fixed. However we also know that, when we change the number of particles, the orbitals and therefore must evolve.
Since the evolution of the orbitals is associated to unitary transformations of the basis, the operators become linear combinations of ones. Therefore we must generalize the definition of , by extending the notion of “complete extraction” to all operators, not only the diagonal ones. The resulting object is what we call . As the separation is invariant (representationindependent), provides the  mathematically and physically  natural definition of unperturbed Hamiltonian.
ii.1 Separation of
Although there is only one , it will have different aspects in and formalisms. We differenciate them by a tag:
is constucted by extracting all the possible and contributions to eqs.(42) or (44), while contains all the possible terms. Obviously the kinetic energy is part of , whose rank 2 component must have the form of the contributions to (44) which we call . To give them a transparent aspect we replace and by
(50) 
which, for , reduce to . The two body operators
(51a)  
(51b) 
in turn become for ,
(52a)  
(52b) 
and the contribution to can be written as
(53) 