Gauge invariance of the pseudopotential

Besides the Galilean invariance mentioned above, the standard Skyrme force has been also proved to be invariant with respect to a more general local gauge invariance, and to give rise to the energy density that is invariant under the same symmetry when specific relations between the coupling constants are set [26,27].

The gauge transformation acts on a many-body wave function by multiplying it with a position-dependent phase factor, that is,

$$\vert\Psi'\rangle = \exp \left( i \sum_{j=1}^{A}\phi(r_j) \right)\vert\Psi\rangle ,$$ (19)

and its action transferred onto the pseudopotential is,

$$\hat{V'}= e^{-i\phi(r'_2)} e^{-i\phi(r'_1)} \hat{V} e^{i\phi(r_1)} e^{i\phi(r_2)} .$$ (20)

Apart from zero order, the terms of the pseudopotential are not trivially invariant with respect to the transformation of the Eq. (20) and, in general, the transformed pseudopotential $\hat{V}'$ is different than the original pseudopotential $\hat{V}$. To impose the gauge invariance on the pseudopotential, one has to derive a list of constraints among the parameters, which can be done using the condition

$$[\phi(r_1),\hat{V}]+ [\phi(r_2),\hat{V}] = 0 .$$ (21)

As expected, at second order, all the 7 terms of the pseudopotential listed in Table 3 fulfill condition (21). Then they all are the stand-alone gauge invariant terms of the pseudopotential, which in the last column of the Table is marked by the letter Y. On the other hand, at fourth order, only two of the terms of the pseudopotential listed in Table 4, those that correspond to parameters $C_{11,00}^{31}$ and $C_{11,20}^{31}$, fulfill condition (21). At sixth order, none of the terms are stand-alone gauge invariant.

At fourth order, the gauge invariance forces seven parameters of the pseudopotential to be specific linear combinations of four independent ones. In Table 4, they are marked by letters D and I, respectively. In Appendix B, we list such relations between the dependent and independent parameters. One should note that other choices of the four independent parameters are also possible, that is, at fourth order, there are simply four different gauge-invariant linear combinations of terms of the pseudopotential (1). Moreover, at this order, there are also two terms that alone are gauge non-invariant - those that correspond to parameters $C_{11,11}^{31}$ and $C_{22,11}^{22}$; in Table 4, they are marked by letters N. Similarly, at sixth order, there are six gauge-invariant linear combinations of terms of the pseudopotential, that is, sixteen dependent parameters are related to six independent ones, see Appendix B, and there are also four alone gauge non-invariant terms corresponding to parameters $C_{11,11}^{51}$, $C_{22,11}^{42}$, $C_{31,11}^{31}$, and $C_{33,11}^{33}$.

A comparison between the numbers of terms of the Galilean-invariant pseudopotential and the gauge-invariant pseudopotential is plotted in Fig. 1. Again we note that at each order, the numbers of gauge-invariant parameters (2 for the zero order, 7 for the second order, 6 for the fourth order, and 6 for the sixth order) are exactly the same as the numbers of independent coupling constants of the EDF in each isospin channel with the gauge invariance imposed, cf. Table VI of Ref. [2]. Again, this observation will be crucial when we proceed to derive relations between the isoscalar and the isovector parts of the EDF, stemming from the gauge-invariant pseudopotential. We also remark that whereas the second-order spin-orbit term, corresponding to parameter $C_{11,11}^{11}$, is gauge invariant, all higher-order spin-orbit terms, corresponding to parameters $C_{\tilde{n} \tilde{L},11}^{\tilde{n}' \tilde{L}'}$ with $\tilde{n}+\tilde{n}'>2$ do violate the gauge symmetry.

Figure 1: (Color online) Number of terms of the pseudopotential (2), plotted as a function of the order in derivatives.