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## Local Density Approximation

Approximation of the many-body energy (58) by a functional of the one-body density matrix (70) can be further simplified in the coordinate representation. Namely, it appears that the HF density matrix (75) influences the energy mostly through the local density Neg70,Neg72,Neg75. This observation defines the local density approximation (LDA).

Neglecting for simplicity the spin-isospin degrees of freedom, we can write the interaction energy [the second term in Eq. (70)] in the form

 (94)

For local effective interaction, the non-antisymmetrized matrix element is given by the potential ,
 (95)

and the interaction energy reads
 (96)

The first term (direct) depends only on the local density matrix (equal arguments), while the second term (exchange) involves the full one-body density matrix. Therefore, the local density plays a special role due to locality of the effective interaction.

It is therefore convenient to represent the one-body density matrix (59) in total and relative coordinates, i.e.,

 (97)

where
 (98)

Denoting the local density by single argument, , and noting that by translational invariance the potential depends only on the relative coordinate, we have
 (99)

We see that direct and exchange terms, , have markedly different dependence on the density matrix, and thus have to be treated separately.

In the direct term, we can use the fact that the range of the effective force is smaller than the typical distance at which the density changes. Indeed, the nuclear density is almost constant inside the nucleus, and then falls down to zero within the region called the nuclear surface, which has a typical width of about 3fm. Hence, within the range of interaction, and for the purpose of evaluating the direct interaction energy, the density can be approximated by the quadratic expansion,

 (100)

and
 (101)

where = . When inserted into Eq. (99), this expansion gives [for scalar interactions ] the direct interaction energy:
 (102)

where coupling constants and are given by the moments of the interaction:
 (103)

In the exchange term, the situation is entirely different, because here the range of interaction matters in the non-local, relative direction . In order to get a feeling what are the properties of the one-body density matrix in this direction, we can calculate it for infinite matter,

 (104)

where the s.p. wave functions (plane waves) are integrated within the Fermi sphere of momenta . Obviously, depends only on the relative coordinate, i.e.,
 (105)

Function in square parentheses equals 1 at =0, and has the first zero at fm, i.e., in the non-local direction the density varies on the same scale as it does in the local direction.

Therefore, the quadratic expansion of the density matrix in the relative variable

 (106)

where derivatives = are always calculated at =0, is, in principle, sufficient for the evaluation of the exchange interaction energy. However, we can improve it by introducing three universal functions of , , , and , which vanish at large , i.e., we define the LDA by:
 (107)

Since for small , Eq. (107) must be compatible with the Taylor expansion (106), the auxiliary functions must fulfill conditions at =0,
 (108)

In order to conserve the local-gauge-invariance properties of the interaction energy Dob95e, we also require that
 (109)

The auxiliary functions and can be calculated a posteriori, to give the best possible approximation of a given density matrix . However, they can also be estimated a priori by making momentum expansion around the Fermi momentum . This gives the density-matrix expansion (DME) of Ref. Neg72, in which
 (110)

where are the spherical Bessel functions.

The term depending on the non-local density in the exchange integral (99) now reads

 (111)

and gives the exchange interaction energy:
 (112)

where coupling constants and are given by the following integrals of the interaction:
 (113)

The exchange interaction energy also depends on densities (119) and (120) that we define below. It is obvious that when the pure Taylor expansion is used to approximate the density in the non-local direction, Eq. (106), i.e., for , the direct and exchange coupling constants are equal, and .

Altogether, quadratic approximations to the one-body density matrix allow expressing the direct and exchange interaction energies as integrals of local energy density. Such energy density depends on the local density, on derivatives of the local density, and on several other densities that represent properties of the one-body density matrix in the non-local direction.

We should stress that the validity of the LDA depends on different scales involved in properties of nuclei. Namely, the scale of distances characterizing the ground-state one-body density matrix is significantly larger than the range of effective forces. Therefore, the LDA may apply only to selected, low-energy phenomena where the spatial structure of the density matrix is not very much affected.

Moreover, we see that the low-energy nuclear properties may depend on an extremely restricted set of properties of effective interactions. Within the LDA, only a few numbers [the coupling constants of Eqs. (103) and (113)] determine the energy density. This is entirely in the spirit of the effective field theory; separation of scales results in a transmission of a very limited information from one scale to another. Once this information (in our case - the coupling constants) is either evaluated, or fit to data, properties of the system can be properly calculated at the larger scale.

We also see that the coupling constants can be evaluated by assuming any effective interaction that has a smaller range than the physical range. In doing so, we can even go down to zero range, and nothing will change, provided we fix the parameters of the zero-range force so as to properly describe the moments of the force, Eqs. (103) and (113), and thus properly reproduce the coupling constants.

We can now proceed to the real world by putting back into our description the spin and isospin degrees of freedom. Based on the results above, we can first construct the most general set of local densities, with derivatives up to the second order taken into account, and then build the local energy density. The complete such construction has been performed only very recently Per03; it involves the full proton-neutron mixing and treats both the particle-hole and particle-particle channels of interaction.

We begin by writing the one-body density matrix (59) with all variables shown explicitly,

 (114)

and we define the densities in total and relative coordinates (97) as
 (115)

The spin-isospin components can now be separated,
 (116)

where and are the spin (19) and isospin (27) Pauli matrices. The scalar-isoscalar , scalar-isovector , vector-isoscalar , and vector-isovector densities can be obtained in a standard way by taking appropriate traces with the Pauli matrices. All necessary local densities can now be obtained by calculating at =0 the derivatives in the total and relative coordinates, up to the second order.

Without the proton-neutron mixing, which we neglect from now on in order to simplify the presentation, only the third components of isovectors are non-zero, and we can use the notation

 (117)

The list of all required local densities then reads Eng75:
 (118) (119) (120) (121) (122) (123)

where
 (124)

are momentum operators in the relative and total coordinate, and index =0, 1 distinguishes between the isoscalar and isovector components. The kinetic densities are usually defined in terms of the derivatives acting on the and coordinates (98), i.e., . There is also one density depending on (tensor-kinetic density) Flo75,Per03, which we do not discuss here because it appears only for tensor interactions. Since the Pauli matrices and momenta are time-odd operators, wee see that densities , , and are time-even, and densities , , and are time-odd.

For an arbitrary central finite-range local potential with the full spin-isospin dependence [cf. the Gogny interaction in Eq. (63)],

 (125)

we can now repeat the derivation of the LDA functional, by using expansions (100) and (107) in each spin-isospin channel. As a result, we obtain the interaction energy (direct and exchange terms combined) in the form
 (126)

where . The energy density depends on six isoscalar and six isovector coupling constants that are simple moments of potentials, i.e.,
 (127)

and
 (128)

where (for , , , or )
 (129)

 (130)

Again we see, that for , the direct and exchange coupling constants are equal, and , and hence only six coupling constants in energy density (126) are independent. This requires that the so-called time-odd coupling constants are linear combinations of the so-called time-even coupling constants Dob95e:

 (131)

and
 (132)

It is well known that the local energy density (126) is also obtained for the Skyrme zero-range momentum-dependent interaction Sky56,Sky59,Vau72. Without density-dependent and spin-orbit terms, this interaction reads

 (133)

where acts to the left, and acts to the right. For this interaction, the interaction energy has exactly the form given in Eq. (126), with coupling constants Eng75,Dob95e that depend on parameters , , , , , and ,
 (134)

and
 (135)

For , the Skyrme interaction (133) exactly reproduces the LDA of the finite-range interaction (125), provided the Skyrme parameters are given by
 (136) (137) (138)

Coupling constants of the Skyrme functional fulfill constraints (131) and (132). When the better approximation of the non-local density matrix is used, i.e., for or in Eq. (107), the Skyrme interaction cannot reproduce the LDA energy density. However, it is enough to release constraints (131) and (132), and treat all the twelve coupling constants as independent parameters, to recover the full freedom of the LDA local energy density.

Again we explicitly see that (exactly in the spirit of the effective field theory), the zero-range interaction can reproduce the same properties of nuclear systems as does the real effective interaction, provided the coupling constants in the energy density are either adjusted to data, or calculated from the real finite-range interaction. It is also clear that the zero-range interaction cannot be treated literally - it is significant only as a generator'' of the proper energy density, while all physical results depend only on this energy density, and not on the interaction itself. In particular, it is incorrect to look for exact eigenstates of the system interacting with the zero-range interaction; we know that for such an interaction the ground state does not exist because of the collapse. However, even for the finite-range effective interaction (for which the ground state does, in principle, exist) the exact ground state is irrelevant, because the interaction has been built to act only in the space of Slater determinants, see Sec. 4.2.

Of course, there is nothing magic or fundamental in the LDA to the energy density. It just reflects the fact that the nuclear one-body density matrix varies on a larger scale of distances than does the nuclear effective interaction. Validity of this approximation depends on the fundamental assumption that the total energy can be described as a functional of the one-body density matrix. The fact that we assumed a local effective interaction is not crucial - for non-local interactions the direct term becomes more complicated, but the LDA still holds Neg72. However, effective interactions must, in fact, also depend on energy (Secs. 4.2 and 4.3), so the presented derivation of the LDA is not complete. One usually goes beyond the local energy density derived from approximations to one-body density, and one includes also terms that depend on local densities in a more complicated way, cf. the density-dependent term of the Gogny interaction (63).

Some people say: the LDA is just fitting of parameters - it is enough to have many parameters to fit anything one wants. This point of view simply disregards the success of the LDA in nuclear phenomenology. The effective field theory point of view is, in my opinion, more interesting, and potentially more fruitful. It regards the success of phenomenological LDA as indication that scales between quark-gluon QCD interactions and low-energy nuclear phenomena are indeed very well separated, and hence few numbers only are enough to define latter in terms of the former. The challenge of course remains: to look for derivations of these few numbers by decent fundamental theory, and to adjust these numbers to data and look for phenomena where the adjustments fail.

We finish this section by recalling the form of the HF equation (73), and that of the HF mean-field Hamiltonian (72), corresponding to the local-energy-density functional (126). Upon variation of the energy with respect to local densities, one obtains the HF equation (74) in spatial coordinates,

 (139)

where numbers the neutron (=) and proton (=) orbitals, and
 (140) (141)

The isoscalar (=0) and isovector (=1) time-even and time-odd mean fields read
 (142) (143)

where we defined the following mean-field potentials as functions of densities
 (144) (145) (146) (147) (148) (149)

Since neither in the effective interactions, (125) and (133), nor in the energy density (126), we showed the spin-orbit, tensor, or density-dependent terms, such contributions are not shown in the mean fields above. The mean-field Hamiltonian resulting from the LDA is simply given by local one-body potentials, with a complete dependence on spin, and by momentum-dependent terms that have the form of generalized effective-mass and spin-momentum couplings.

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Next: About this document ... Up: MANY-NUCLEON SYSTEMS Previous: Conserved and Broken Symmetries
Jacek Dobaczewski 2003-01-27