Canonical states

 

Canonical states are defined as the states that diagonalize the HFB one-body density matrix $\rho({\mbox{{\boldmath {$r$ }}}_1},{\mbox{{\boldmath {$r$ }}}_2})$ of Eq. (30), i.e.,

$$\int \rho({\mbox{{\boldmath {$r$ }}}_1},{\mbox{{\boldmath {$r... ...2 } =v_{i}^{2}\breve\psi_{i}({\mbox{{\boldmath {$r$ }}}}_1)\;,$$ (48)

where, due to the Pauli principle, the canonical occupation numbers v_i² obey the condition $0\leq v_{i}^{2}\leq 1$.

For self-consistent solutions, the canonical occupation numbers v_i² are determined by the diagonal matrix elements h_ii and $\Delta _{i\bar{i }}$ of the particle-hole (p-h) and particle-particle (p-p) Hamiltonians in the canonical basis via the following BCS-like equation [20]:

$$v_{i}^{2}=\frac{1}{2}-\frac{h_{ii}-\lambda}{2E_{i}},$$ (49)

where

$$E_{i}=\sqrt{\left(h_{ii}-\lambda \right)^{2}+\Delta_{i\bar{i}}^{2}} ~.$$ (50)

The chemical potential $\lambda$ is determined from the particle number condition

$$N=\sum\limits_{i}\;v_{i}^{2}=\sum_n N_n,$$ (51)

where N_n denote the norms of the lower HFB wave functions of Eq. (46), i.e.,

$$N_n = \sum_\alpha V^2_{\alpha n} ~.$$ (52)

In the canonical representation, the average (proton or neutron) pairing gap $\widetilde{\Delta}$ [6] is given by the average value of $\Delta _{i\bar{i }}$ in the corresponding (proton or neutron) canonical states,

$$\widetilde{\Delta}=\frac{1}{N}\sum\limits_{i}\; \Delta_{i\bar{i}}v_{i}^{2}~,$$ (53)

where N is the number of nucleons of that type (see Eq.  (51)).

Whenever infinite complete single-particle bases are used in configurational calculations, one may freely expand the upper and lower HFB wave functions of Eq. (46) (the quasiparticle wave functions), as well as the standard eigenstates of the p-h Hamiltonian h, in the canonical basis. These expansions are often extremely slowly converging, however, and any truncation of the basis typically induces large errors. Therefore, in practice, when working with finite bases, one should not expand quasiparticle wave functions and the single-particle eigenstates of h in the canonical basis. The reason is very simple, and it stems from different asymptotic properties of these objects. As discussed in Ref. [6], the quasiparticle spectrum and wave functions are partly discrete and localized and partly continuous and asymptotically oscillating, respectively. These properties are completely analogous to properties of the eigenstates of h, which are also discrete and localized (for negative eigenenergies) or continuous and oscillating (for positive eigenenergies). On the other hand, the properties of eigenvalues and eigenstates of the density matrix (48) are very different, namely the entire spectrum is discrete and all the wave functions are localized. Therefore, even if formally the set of canonical states is complete, it is extremely difficult to expand any oscillating wave function in this basis.

These considerations make it clear that the optimum way of solving the HFB equations is by using the coordinate representation, in which the various asymptotic properties are in a natural way correctly fulfilled. This technique is widely used when spherical symmetry is imposed; then one only has to solve systems of one-dimensional differential equations, which is an easy task. On the other hand, the case of axial symmetry requires solving two-dimensional equations, and that of triaxial shapes requires working with a three-dimensional problem. None of these two latter cases has up to now been effectively solved in coordinate space, although work on the axial solutions is in progress [21].

Therefore, without having access to coordinate-representation solutions, we are obliged to use methods based on a configurational expansion. In this respect, one may clearly distinguish two classes of finite single-particle bases, each of which aims at a reasonable solution of the HFB equations (46). One uses a truncated basis composed of eigenstates of h [22,8,9]. This basis is partly composed of discrete localized states and partly of discretized continuum and oscillating states. Technically it is very difficult to include many continuum states in the basis, especially when triaxial deformations are allowed. In practice, Refs.[22,8,9] included states up to several MeV into the continuum. Such a small phase space is certainly insufficient to describe spatial properties of nuclear densities at large distances, although some ground-state properties, like total binding energies, will be at most weakly affected.

The second uses a truncated infinite discrete basis. The most common of course is the HO basis, which has been used in numerous HFB calculations, especially those employing the Gogny effective interaction (see, e.g., Refs. [23,24,25,26]), and in Hartree-Bogoliubov calculations based on a relativistic Lagrangian (see, e.g., Refs. [27,28]). Because it uses a basis with a similar structure to the canonical basis (infinite and discrete), this approach can be viewed as aiming at the best possible approximation to the canonical states and not the quasiparticle states. In this sense, the amplitudes U_n and V_n that appear in Eq. (46) should be considered more as expansion coefficients of quasiparticle states in a basis similar to the canonical basis than as quasiparticle wave functions themselves.

Our approach, which we discuss in greater detail below, belongs to the second class. The THO basis defined and described in Sec. 2 is a model that aims at an optimal description of the canonical states. Therefore, in the following we adapt properties of the THO basis, and in particular the value of the decay constant $\kappa$ (12), to the asymptotic properties of canonical states. In fact, the unique decay constant of all THO basis states is exactly the desired property of canonical states. As discussed in Ref. [7], the asymptotic properties of the most important canonical states (those having average energies close to the Fermi energy) are governed by a common unique decay constant,

$$\kappa=\sqrt{\frac{2m(E_{\min}-\lambda)}{\hbar^2}} ~,$$ (54)

where $E_{\min}$ is the lowest quasiparticle energy E_n. This should be contrasted with decay constants associated with the eigenstates of h, which are all different and depend on the single-particle eigenenergies.