Canonical-basis HFB method
Naoki Tajima
Department of Applied Physics, Fukui University, Bunkyo 3-9-1, Fukui, 910-8507, Japan
Advantages of a new method to obtain solutions of the Hartree-FockBogoliubov (HFB) equation directory in the HFB canonical form are discussed
in comparison with the ordinary method in which the solutions are expressed as
the vacuum of Bogoliubov quasiparticles. A byproduct and necessary special
cares are also described.
In order to treat the neutron-rich side of the nuclear chart, one has to take into
account the pairing correlation in the continuum part (= positive-energy part) of thesingle-particle spectrum. One has to do so not only for nuclei very close to the neutron
drip line because, for a reliable treatment of the pairing correlation in the presence of shellfluctuations, one has to consider explicitly at least (empirically speaking) half of a major
shell above the Fermi level in BCS calculations. Namely, one has to treat the positiveenergy states explicitly if ²F > − 12 h̄ωosc . This condition applies to roughly half of the4nuclei in the nuclear chart and the number of such nuclei is about 102 .
One also has to consider the long tail for large
r for neutron-richnuclei. The large-r
³
´2
√
−κr
asymptotic form of the density looks like ρ(r) ∼ e /r , κ = −2m²F /h̄, where ²F isthe energy of the last occupied Hartree-Fock (HF) orbital. In the presence of the pairing
(min)
(min)
correlation, ²F is replaced with ²F − Eqp, where Eqp
is the smallest quasiparticle
excitation energy and is roughly equal to the pairing gap. This means that the tail isshortened by the pairing correlation. Therefore calculations of the density tail require areliable treatment of the pairing.
In order to treat pairing correlations in the continuum and long density tails simultaneously, one has to employ the coordinate-space HFB method. Here, not a radial gridbut a three-dimensional mesh representation is preferable because all the nuclei except
near spherical magics are deformed. The three-dimensional mesh is, however, feasible
only for Skyrme forces, for which the mean field becomes local. The canonical-basis
HFB method [1, 2, 3, 4, 5] is a new method to obtain HFB solutions for Skyrme forcesin three-dimensional Cartesian mesh representation with positive-energy states explicitlyconsidered.
In the ordinary method to solve the HFB[6, 7, 8], the solutions are expressed as thevacuum of the Bogoliubov quasiparticles.
|ψi =#basis
Yi=1

bi |0i,bi =XZ
snod3 r φ∗i (~r, s) a(~r, s) + ψi (~r, s) a† (~r, s) ,(1)where b†i are the creation operators of quasiparticles. The two kinds of amplitudes can beobtained as the eigenvectors of the HFB super matrix,
Ã−h h̃h̃ h!Ã

φi
ψi!

Ã= ²i

φi
ψi!

.

(2)

This method is not very efficient because the quasiparticle wavefunctions contain by far
more information than is necessary to describe the ground state. Most of their information
concerns particle or hole excitations.
HFB solutions can be expressed in the BCS form as|Ψi =iY
max ³

ui + vi a†i a†ı̄´

|0i,i=1a†i =XZ

d3 r ψi (~r, s) a† (~r, s),(3)swhere a†i is the creation operator of a nucleon in the ith HFB canonical basis state.
This expression is exact when imax = 12 #basis according to the Bloch Messiah theorem.
The essential advantage of this expression is that one may neglect v 2 ¿ 1 states as far
as the ground state is concerned. One can truncate the number of canonical basis byimax = O(A) ¿ #basis to a good approximation. The canonical-basis HFB method isa much more economical way to express HFB ground states of neutron rich nuclei thanexpressing them as quasiparticle vacua.
The canonical-basis method determines the canonical basis and the uv factors without
the full knowledge of the quasiparticle states. This can be pursued in the gradient method
under constraint of orthonormality among the canonical basis.
X
δE=H
ψ
=
λij ψj ,iiδψi∗j

Hi = vi2 h + ui vi h̃,(4)where h is the HF Hamiltonian and h̃ is the pairing Hamiltonian. This expression ofHi gives a new insight: The positive energy canonical basis are the bound states of thepairing Hamiltonian. Consequently, they are guaranteed to be spatially localized (by h̃,not by h) and form a discrete spectrum. See Ref.[4] for explanations.
The following table is a summary of the comparison between the two methods.
method#basisquasi particle
canonical basis∝ box volume∝ nuclear volume

orthogonality
condition
redundant
essential

pairing force
δ-func, + dens. dep.
+ mom. dep.

In the canonical basis method, the Hamiltonian becomes state-dependent and thus
the orthogonality among the canonical basis states should be explicitly taken care of. SeeRef.[4] for how it can be made possible.
The canonical-basis method needs momentum-dependent (or finite-range) pairing interaction in order to circumvent the point collapse problem, which is peculiar to thismethod. It is depicted in Fig. 1. I have tried several ways to avoid the collapse to findunwanted side effects for many of them. See Ref. [4] for a detailed discussion.

Fig. 1: Point collapse of a high-lying
canonical orbital. The canonical orbital
whose occupation probability is smaller
than some critical value shrinks into apoint unless the pairing Hamiltonian hasa kinetic term.

Fig. 2: The pairing gap ∆ versus theFermi level ²F for the global-minimum
HFB solution (thick bold line) and possible local-minimum solutions (thick dashline).

In my talk I have shown the result of a simulation of the approach to the neutron drip
line as a demonstration of the method and suggested a possibility to obtain approximate
localized solutions for nearly bound systems (Fig. 2). Details can be found in Ref. [5].
I have also discussed on the extension of the method to the full Skyrme interaction
with time-reversal symmetry imposed, which will be published in future.

References[1] K. Mühlhans, K.Neergård, and U. Mosel, Nucl. Phys. A420, 204 (1984).
[2] P.-G. Reinhard, M. Bender, K. Rutz, and J.A. Maruhn: Z. Phys. A358, 277 (1997).
[3] N. Tajima, in proc. Innovative Computational Methods in Nuclear Many-Body Problems, Osaka, 1997, p. 343; nucl-th/0002061.
[4] N. Tajima, Phys. Rev. C, 69, 034305 (2004).
[5] N. Tajima, in proc. Exotic Nuclei and Atomic Masses, Pine Mountain, 2004; DOI:
10.1140/epjad/i2005-06-076-2; nucl-th/0503036.
[6] J. Dobaczewski, H. Flocard, and J. Treiner, Nucl. Phys. A422, 103 (1984).
[7] M.V. Stoitsov, W. Nazarewicz, and S. Pittel: Phys. Rev. C58, 2092 (1998).
[8] E. Terán, V. E. Oberacker, and A. S. Umar, Phys. Rev. C 67, 064314 (2003).