Choice of occupied quasiparticle states

In practical applications, one often assumes that the quasiparticle energies in matrix $E$ are all positive. However, this is not at all required by the variational principle. Indeed, while eigenvalues of $\mathcal{H}$ must come in pairs of opposite quasiparticle energies $(E_\mu,-E_\mu)$, the theory says nothing on whether a positive or a negative one enters matrix $E$.

This question is evidently related to the mundane problem of the order in which we arrange eigenvectors of $\mathcal{H}$ in Eq. (7). Any such order is allowed, provided one eigenvector of the pair $(E_\mu,-E_\mu)$ is put into the first half of the spectrum (empty quasiparticle states), and the other one is put in the second half of the spectrum (occupied quasiparticle states). However, which one goes where is in principle arbitrary. The signs of quasiparticle energies in $E$ are arbitrary too, and thus variational equations have not one, but many solutions. Naturally, in practical applications, the choice of occupations is motivated by physics as the energy of the system may be different depending on how the arbitrariness of choosing signs of quasiparticle energies is resolved. It is then plausible, and often assumed, that the lowest total energy is obtained by occupying states corresponding to negative quasiparticle energies, and leaving empty those corresponding to positive quasiparticle energies, i.e., by assuming that positive quasiparticle energies are collected in matrix $E$. One should stress at this point, however, that there is no a priori reason why such a choice must guarantee obtaining the lowest total energy, and examples to the contrary are available in numerous applications. Thus, which quasiparticle states should be occupied is a matter of a specific physical situation, and not a rule cast in stone.

The problem of choosing occupied quasiparticle states is particularly conspicuous when iterative (self-consistent) methods are used for solving the non-linear eigenproblem (7). The self-consistent procedure can be described as the sequence of the following steps:

1. Find eigenstates of the quasiparticle Hamiltonian (6).
2. Choose the occupied quasiparticle states.
3. Calculate the generalized density matrix (5), i.e., the density matrix ($\rho$) and pairing tensor ($\kappa$).
4. Calculate the particle-hole ($\Gamma$) and particle-particle ($\Delta$) mean fields to determine the Hamiltonian (6) in the next iteration.
5. Return to step 1.

Again, this self-consistent sequence of steps is described in all textbooks; however, the crucial step 2 is seldom ever mentioned. However, in many applications, this step is absolutely essential for obtaining a convergent algorithm. Moreover, choosing negative occupied quasiparticle energies at each iteration may sometimes lead to divergent iterations.

The problem here is in finding ways of identifying quasiparticle states (tags), which are independent of their energies, and which would allow for pinning down the structure of each individual state. Ideally, tags could be provided by quantum numbers of conserved symmetries. For example, spherical symmetry allows each state to be characterized by the standard quantum numbers $nljm$. Then, decisions of occupying quasiparticle states can be made based on lists of quantum numbers.

A particular version of using conserved quantum numbers as tags of quasiparticle states was pioneered in nuclear physics under the name of non-collective rotation, or non-collective cranking approximation [21,17,18], and this method is a central theme of our discussion. It consists of using in Eq. (6) the single-particle Routhian $h'$ instead of the single-particle Hamiltonian $h$, where

$$h' := h - \lambda_S S,$$ (8)

and $S$ is the operator of a conserved symmetry. This may look like an attempt of introducing a Lagrange multiplier to constraining the average value of $S$, but in fact it is not. Indeed, since quasiparticle states are eigenstates of $S$, the quasiparticle energies are simply reordered as a function of $\lambda_S$ without any modification of the structure of quasiparticle states or total energy. Using Routhian $h'$ may thus facilitate selecting proper quasiparticle states into the set of occupied ones, depending on the physical situation corresponding to the chosen symmetry $S$. In short, non-collective cranking can be viewed as a configuration selector.

The method fails when quasiparticle states are not eigenstates of symmetry operators, which can happen in unrestricted calculations. The practical solution consists of calculating overlaps of quasiparticle states with a fixed set of wave-functions and establishing in this way tags that are related to the closest similarities of structure between both sets.

Only when a set of properly tagged quasiparticle states is occupied at step 2 of each iteration can one have a good chance of obtaining a converged solution. At the end of the day, one can check whether such a solution corresponds to the negative quasiparticle states being occupied or not.