``Equation of state'' of asymmetric polarized nuclear matter

In INM $\Delta \rho_{t t_3} (\vec{r}) %= \delta \; \Delta \rho_{t t_3} (\rvec) = \Delta... ...{\leftrightarrow}{J}_{t t_3} (\vec{r}) %= \delta \tensor{J}_{t t_3} (\rvec) = 0$. We choose pure neutron and proton states, which leads to $\rho_{1, \pm 1} = 0$, $\rho_1 := \rho_{1,0}$, and similarly for all other densities. We take the z axis as the quantization axis for the spin, i.e., s_{t, x} = s_{t, y} = 0, s_t := s_{t, z}, and for the kinetic spin density $\vec{T}$. As discussed in Refs. [72], this breaks the isotropy of INM, leading to an axially deformed Fermi surface, an effect which we neglect. Adding the kinetic term, the total energy per nucleon (i.e. the ``equation of state'') for the energy functional (7) and (8) is given by

$$\frac{{\cal H}}{\rho_0} {\textstyle\frac{{3}}{{5}}} {\textstyle\frac{{\hbar^2}}{{2m}}} \, \beta \, \rho_0^{2/3} \, F_{5/3}^{(0)}$$ =

$$+ \big( C_0^{\rho} + C_1^{\rho} \, I_\tau^2 + C_0^{s} \, I_\sigma^2 + C_1^{s} \, I_{\sigma \tau}^2 \big) \, \rho_0$$

$$+ {\textstyle\frac{{3}}{{5}}} \big( C_0^{\tau} F^{(0)}_{5/3} + C_1^{\tau} I_\tau F^{(\tau)}_{5/3}$$

$$+ C_0^{T} I_\sigma F^{(\sigma)}_{5/3} + C_1^{T} I_{\sigma \tau} F^{(\sigma\tau)}_{5/3} \big) \, \beta \, \rho_0^{5/3} \, .$$ (45)

For unpolarized INM one has $I_\sigma = I_{\sigma \tau} = 0$ which recovers the expression given in Ref.[23].

An interesting special case is polarized neutron matter, which is discussed in [70] for the Skyrme interactions. A stability criterion derived there from the two-body force point of view as outlined in Appendix 8 was used to constrain the parameters of the SLyx forces [23,24]. In this limiting case, one has $\rho_{{\rm n}\uparrow}= \rho_0$, $\rho_{{\rm n}\downarrow}= \rho_{{\rm p}\uparrow}= \rho_{{\rm p}\downarrow}= 0$, which is equivalent to $I_\tau = I_\sigma = I_{\sigma \tau} = 1$ and leads to

$$\frac{{\cal H}}{\rho_0} 2^{4/3} \beta {\textstyle\frac{{3}}{{5}}} \left[ {\textstyle\frac... ...( C_0^{\tau} + C_1^{\tau} + C_0^{T} + C_1^{T} \big) \rho_0 \right] \rho_0^{2/3}$$ =

$$\quad + \big( C_0^{\rho} + C_1^{\rho} + C_0^{s} + C_1^{s} \big) \, \rho_0 \quad .$$ (46)

Expressions (34) for an antisymmetrized Skyrme force imply that $C_0^{\rho} + C_1^{\rho} + C_0^{s} + C_1^{s} = 0$, and

$$\frac{{\cal H}}{\rho_0} = 2^{4/3} \beta {\textstyle\frac{{3... ...} \, t_2 \, (1 + x_2) \, \rho_0 \right] \rho_0^{2/3} \quad .$$ (47)

The stability of polarized neutron matter for all densities requires $x_2 \approx -1$ [70], so the SLyx interactions take $x_2 \equiv - 1$ [23,24]. However, from the energy-density-functional point of view, the coupling constants are independent, and the second term in Eq. (50) also contributes to the stability condition.