Signature symmetry

In this section we apply methods of occupying quasiparticle states, outlined in Sec. 2.1, to a physical situation where the system has a conserved signature symmetry, which can be used within the cranking approximation [21,17,18]. The signature operation [22] is a rotation by $\pi$ around one direction in space, which is conventionally called the $x$-axis:

$$\hat{R}_x = \exp({-i\pi \hat{J}_x}),$$ (13)

where $\hat{J}_x$ denotes the total angular-momentum operator along the $x$ axis. The signature operator is manifestly unitary, $\hat{R}^+_x\hat{R}_x=1$. Since a rotation by $2 \pi$ reverses the phase of fermion wave functions, the square of the signature operator gives the particle-number parity, $\hat{R}^2_x=\pi_N$. Therefore, $\hat{R}_x$ is hermitian and antihermitian in even and odd spaces, respectively. In particular, in the single-particle space, the signature is a unitary antihermitian operator.

Since the signature $\hat{R}_x$ and time-reversal $\hat{T}$ operators commute,

$$\hat{T}^+\hat{R}_x\hat{T}=\hat{R}_x,$$ (14)

signature is a time-even operator. Therefore, for non-rotating systems (i.e., without time-odd fields), signature is equivalent to the time-reversal symmetry $\hat{T}$, but it is more convenient to use, because $\hat{R}_x$ is a linear and not an antilinear operator [23]. For states with nonzero angular momentum, where $\hat{T}$ is internally broken, $\hat{R}_x$ is often still preserved [23,24,25]. The single-particle (and one-quasiparticle) states may then be classified according to the signature exponent quantum number $\alpha$ [26]:

$$\hat{R}_x \vert\alpha k\rangle = e^{-i\pi \alpha} \vert\alpha k\rangle,$$ (15)

where $\alpha$ takes the values of $\pm ^1/_2$. For conserved signature, the HFB mean fields $h$ and $\Delta$ commute and anticommute with $\hat{R}_x$, respectively, and the HFB equations (7) can be written in a good signature basis:

$$\mathcal{H}_\alpha \vert\alpha \mu\rangle = E_{\alpha\mu} \vert\alpha \mu\rangle,$$ (16)

where the HFB Hamiltonian matrix (6) in one signature reads

$$\mathcal{H}_\alpha = \begin{array}{cc} (h-\lambda)_{\alpha ... ...,\alpha k'} & (-h+\lambda)_{-\alpha k,-\alpha k'} \end{array}$$ (17)

and the two-component quasiparticle wave function is

$$\vert\alpha \mu\rangle = \left ( \begin{array}{c} U^{\alpha\mu} \\ V^{\alpha\mu} \end{array} \right ).$$ (18)

A quasiparticle state with good signature is a linear combination of states in time-reversed orbits [23]:

$$\vert 1/2, k\rangle = {\textstyle{\frac{1}{\sqrt{2}}}}\left[-\vert k\Omega_k\rangle +\pi (-1)^{\Omega_k-1/2} \hat{T}\vert k\Omega_k\rangle\right],$$ (19)

$$\vert-1/2, k\rangle = {\textstyle{\frac{1}{\sqrt{2}}}}\left[\hat{T}\vert k\Omega_k\rangle +\pi (-1)^{\Omega_k-1/2} \vert k\Omega_k\rangle\right],$$ (20)

where $\Omega$ is the eigenvalue of the $z$ component of the single-particle angular momentum, $\hat{J}_z$, and we have adopted the phase convention according to which $\hat{T}\vert\pi j\Omega\rangle = \pi (-1)^{j+\Omega} \vert\pi j,-\Omega\rangle$ [27,28], where $\pi=\pm1$ is the parity quantum number. If the Kramers degeneracy is present, the description in terms of Kramers doublets and signature doublets is equivalent. It is for polarized systems having time-odd mean fields that the use of the signature symmetry is superior.

The HFB equation (18) has the (quasiparticle-quasihole) symmetry (7): for each state $\vert\alpha \mu\rangle$ of a given signature, there exists a conjugate state of opposite signature $\vert-\alpha \tilde{\mu}\rangle$, opposite energy:

$$E_{-\alpha\tilde{\mu}} = - E_{\alpha \mu},$$ (21)

and the quasiparticle wave function given by

$$\vert-\alpha \tilde{\mu}\rangle = \left ( \begin{array}{c} V^{\alpha\mu *} \\ U^{\alpha\mu *} \end{array} \right ).$$ (22)

By this symmetry, one needs to solve the HFB equation (18) only for one signature, obtaining positive and negative quasiparticle energies $E_{\alpha \mu}$. Therefore, the entire set of negative quasiparticle energies is composed of two groups: (i) the negative ones $E_{\alpha \mu}$ obtained directly from the HFB equation solved for signature $\alpha$, and (ii) the inverted positive ones (23), which correspond to states of signature $-\alpha$.

The zero-quasiparticle HFB reference state (10), representing the lowest configuration for a system with even number of fermions, corresponds to a filled sea of Bogoliubov quasiparticles with negative energies (Fig. 1(a)). In a one-quasiparticle state, representing a state in an odd nucleus, a positive-energy state is occupied and its conjugated partner is empty (Fig. 1(b)).

Figure 1: (Color online) Quasiparticle content of three configurations: (a) vacuum; (b) the lowest one-quasi-particle state with $\alpha$=1/2, accessible via 2FLA; (c) the lowest two-quasi-particle state with $\alpha$=0, not accessible via 2FLA.

The exchange of the eigenvectors $(U^\mu,V^\mu)$ and $(V^{\mu *},U^{\mu *})$, that have opposite signatures, corresponds to the exchange of columns in the $\varphi$ and $\chi$ matrices discussed in Sec. 2.1 and reverses the particle-number parity $\pi_N$. The density matrix and a pairing tensor of a one-quasiparticle state (12) can be obtained from Eqs. (11) [29,30]: where $\rho^{0}$ and $\kappa^{0}$ correspond to the reference state (10).