Details of the calculations

The present study is limited to the simplest chiral s.p. configuration, $\pi{}h_{11/2}^1~\nu{}h_{11/2}^{-1}$, which has been assigned in the literature to the candidate chiral bands observed in the $N=75$ isotones. In order to check whether the existence of the HF chiral solutions is not a particular feature of one Skyrme parameter set, all calculations were repeated with two parametrizations, SLy4 [27] and SKM* [28]. The parity was kept as a conserved symmetry, and the pairing correlations were not included.

In forming the chiral geometry, orientations of several angular-momentum vectors play a crucial role. Their behavior depends, among others, on the nucleonic densities which are odd under time reversal, like the current and spin densities. Therefore, taking those densities and corresponding terms in the energy density functional into account seems important for the microscopic description of the chiral rotation. Self-consistent methods are best suited for such a task, and investigating the role of the time-odd densities was one of our priorities in the present study.

The Skyrme energy density functional depends on time-even and time-odd nucleonic densities with coupling constants $C_t^\rho$, $C_t^{\Delta\rho}$, $C_t^\tau$, $C_t^J$, $C_t^{\nabla J}$ (10 time-even terms) and with coupling constants $C_t^s$, $C_t^{\Delta s}$, $C_t^T$, $C_t^j$, $C_t^{\nabla j}$ (10 time-odd terms) [42]. The index $t=0,1$ denotes the isoscalar and isovector parts. In standard parametrizations, which are used in the present work, $C_t^\rho$ and $C_t^s$ additionally depend on the isoscalar particle density. If the assumption of the local gauge invariance is made [42], there are several following relations between the time-even and time-odd coupling constants,

$$C_t^j=-C_t^\tau~, \qquad C_t^J=-C_t^T~, \qquad C_t^{\nabla j}=+C_t^{\nabla J}~.$$ (4)

In the present calculations, the coupling constants $C_t^J$ of the time-even terms for $t=0,1$ were always set to zero, like in the original fits of the forces SLy4 and SkM*. In order to conform to the local gauge invariance, whose consequence is Eq. (4), the coupling constants $C_t^T$ of the time-odd terms were set to zero, too. Apart from $C_t^J$, all other time-even coupling constants were taken as they come from the parameters of the Skyrme force.

To examine the role of the time-odd densities, we have performed three variants of calculations, throughout the text denoted as $N$, $G$, and $T$, and defined in the following way:

• Variant $N$: All time-odd coupling constants are set to zero and the local gauge invariance conditions in Eq. (4) are violated.

• Variant $G$: Coupling constants $C_t^j$ and $C_t^{\nabla j}$are taken as they come from the parameters of the Skyrme force and as required by the local gauge invariance (4); all other time-odd coupling constants are set to zero.

• Variant $T$: Apart from $C_t^T$, all other time-odd coupling constants are taken as they come from the parameters of the Skyrme force and as required by the local gauge invariance (4).

The density-dependent and independent components of $C_t^s$ were suppressed or not simultaneously. Setting or not some time-odd coupling constants to zero implies excluding or including in the calculations the corresponding time-odd terms of the mean field. In variant $N$, the mean field contained only the time-even contributions (apart from the cranking term) as in the case of the non-selfconsistent s.p. potentials.

The calculations were carried out by using the code HFODD (v2.05c) [29,30,31]. The code expands the s.p. wave-functions onto the HO basis, and uses the iterative method to solve the HF equations. Twelve entire spherical HO shells were included in the basis. It has been verified that increasing this number up to 16, changes the quantities important for the present considerations (deformation, moments of inertia, alignments etc.) by less than 1%.