Hartree-Fock-Bogoliubov theory

The simplest route to the HFB theory is to employ the variational principle to a two-body Hamiltonian using Thouless states (6) as trial wave functions. The variation of the average energy with respect to the antisymmetric matrix $Z$ results in the HFB equation in the matrix representation, ${\cal{}H}{\cal{}U}={\cal{}U}{\cal{}E}$, or explicitly,

$$\left(\begin{array}{cc} T+\Gamma & \Delta \\ -\Delta^*& -{... ...\left(\begin{array}{cc} E & 0 \\ 0 & -E \end{array}\right) ,$$ (7)

where $T_{\mu\nu}$ is the matrix of the one-body kinetic energy, $\Gamma_{\mu\nu}=\sum_{\mu'\nu'}V_{\mu\mu';\nu\nu'}\rho_{\nu'\mu'}$ and $\Delta_{\mu\mu'}=\frac{1}{2}\sum_{\nu\nu'}V_{\mu\mu';\nu\nu'}\kappa_{\nu\nu'}$ are the so-called particle-hole and particle-particle mean fields, respectively, obtained by averaging two-body matrix elements $V_{\mu\mu';\nu\nu'}$ with respect to the density matrix $\rho_{\nu'\mu'}=\langle\Phi\vert a^+_{\mu'}a_{\nu'}\vert\Phi\rangle$ and pairing tensor $\kappa_{\nu\nu'}=\langle\Phi\vert a_{\nu'}a_{\nu}\vert\Phi\rangle$, and $E$ is the diagonal matrix of quasiparticle energies[11].

The matrices ${\cal{}H}$ and ${\cal{}U}$ are referred to as the HFB Hamiltonian and Bogoliubov transformation, respectively, and columns of ${\cal{}U}$ (eigenstates of ${\cal{}H}$) are vectors of quasiparticle states. The HFB equation (7) possesses the quasiparticle-quasihole symmetry. Namely, for each quasiparticle state $\chi_\alpha$ (the $\alpha$-th column of ${\cal{}U}$) and energy $E_\alpha$ there exists a quasihole state $\phi_\alpha$ of opposite energy $-E_\alpha$,

$$\chi_\alpha=\left(\begin{array}{c}U_{\mu\alpha}\\ V_{\mu\alp... ...array}{c}V^*_{\mu\alpha}\\ U^*_{\mu\alpha}\end{array}\right) .$$ (8)

That is, the spectrum of ${\cal{}H}$ is composed of pairs of states with opposite energies. In most cases, the lowest total energy is obtained by using the eigenstates with $E_\alpha>0$ as quasiparticles $\chi_\alpha$ and those with $E_\alpha<0$ as quasiholes $\phi_\alpha$, that is, by occupying the negative-energy eigenstates. States $\chi_\alpha$ and $\phi_\alpha$ can usually be related through a self-consistent discrete symmetry, such as time reversal, signature, or simplex.[12,13,14].

The HFB equation (7) is also valid in a more general case, when the total energy is not equal to the average of any many-body Hamiltonian. Within the DFT, it stems from the minimization of the binding energy given by an EDF ${\cal{}E}(\rho,\kappa,\kappa^*)$, subject to the condition of the generalized density matrix being projective, that is, ${\cal{}R}^2={\cal{}R}$ for

$${\cal{}R} = \left(\begin{array}{cc} \rho & \kappa \\ -\ka... ... \end{array}\right)$} = \sum_\alpha \phi_\alpha\phi_\alpha^+ .$$ (9)

In this case, the mean fields are obtained as functional derivatives of EDF: $\Gamma_{\mu\nu}=\partial{\cal{}E}/\partial\rho_{\nu\mu}$ and $\Delta_{\mu\mu'}=\partial{\cal{}E}/\partial\kappa^*_{\mu\mu'}$. As is the case in DFT, densities (here the density matrix and pairing tensor) become the fundamental degrees of freedom, whereas the state $\vert\Phi\rangle$ acquires the meaning of an auxiliary entity (the Kohn-Sham state[15]). Indeed, for any arbitrary generalized density matrix ${\cal{}R}$ (9), one can always find the corresponding state $\vert\Phi\rangle$. For that, one determines the Bogoliubov transformation ${\cal{}U}$ as the matrix of its eigenvectors, ${\cal{}R}{\cal{}U}={\cal{}U}\left(\begin{array}{cc}0&0\\ 0&1\end{array}\right)$; the Thouless state $\vert\Phi\rangle$ (6) corresponds to $Z=VU^{-1}$. Consequently, the paired state $\vert\Phi\rangle$ of DFT is not interpreted as a wave function of the system - it only serves as a model for determining one-body densities. Nonetheless, these densities are interpreted as those associated with the (unknown) exact eigenstate of the system.

Unrestricted variations of the EDF are not meaningful. Indeed, since Thouless states (3) are mixtures of components with different particle numbers, absolute minima will usually correspond to average particle numbers that are unrelated to those one would like to describe. In particular, for self-bound systems governed by attractive two-body forces (nuclei), by adding more and more particles one could infinitely decrease the total energy of the system. Therefore, only constrained variations make sense, that is, one has to minimize not the total energy ${\cal{}E}(\rho,\kappa,\kappa^*)$, but the so-called Routhian, ${\cal{}E}'(\rho,\kappa,\kappa^*)={\cal{}E}(\rho,\kappa,\kappa^*)+{\cal{}C}(\rho)$, where ${\cal{}C}$ is a suitably chosen penalty functional, ensuring that the minimum appears at prescribed average values of one-body operators. In particular, the average total number of particles can be constrained by ${\cal{}C}(\rho)=-\lambda\langle\Phi\vert\hat{N}\vert\Phi\rangle=-\lambda\mbox{Tr}(\rho)$ (linear constraint) or ${\cal{}C}(\rho)=C_N[\mbox{Tr}(\rho)-N_0]^2$ (quadratic constraint),[16,17] where $\lambda$ becomes the Fermi energy corresponding to $N_0$ fermions.

For different systems and for different applications, various constraints ${\cal{}C}(\rho)$ can be implemented; for example, in nuclei one can simultaneously constrain numbers of protons and neutrons, as well as multipole moments of matter or charge distributions. When the total energy is a concave function of relevant one-body average values, quadratic constraints are mandatory[16,17]. The minimization of ${\cal{}E}'(\rho,\kappa,\kappa^*)$ requires solving the HFB equation for the quasiparticle Routhian ${\cal{}H}'$, which, for the simplest case of the constraint on the total particle number, reads ${\cal{}H}'={\cal{}H}-\lambda\left(\begin{array}{cc}1&0\\ 0&-1\end{array}\right)$.

Finally, let us mention that in the coordinate space-spin(-isospin) representation, the HFB equation (7) acquires particularly interesting form, which in condensed matter and atomic literature is called Bogoliubov-de Gennes equation[4]. In the coordinate representation, quasiparticle vectors become two-component wave functions, which - in finite systems - acquire specific asymptotic properties[18,19,20,21] determining the asymptotic behavior of local densities. The quasiparticle energy spectrum of HFB contains discrete bound states, resonances, and non-resonant continuum states. As illustrated in Fig. 1, the bound HFB solutions exist only in the energy region $\vert E_i\vert\leq-\lambda$. The quasiparticle continuum with $\vert E_i\vert > -\lambda$ consists of non-resonant continuum and quasiparticle resonances, see Sec. 6.

Figure 1: Quasiparticle spectrum of the HFB Hamiltonian. The bound states exist in the energy region $\vert E_i\vert\leq-\lambda$, where $\lambda$ is the chemical potential (always negative for a particle-bound system).