Nuclear Matter Spin-Polarization

• Nuclear matter spin-polarization is characterized by differing densities of spin-up and spin-down nucleons, crucial for modeling the EOS and phase transitions.
• The spin symmetry energy quantifies the cost of aligning spins, with its steep density dependence stiffening the EOS at supranuclear densities.
• Experimental probes in heavy-ion collisions and astrophysical observations provide constraints on spin-polarized models, advancing our understanding of neutron star interiors.

Nuclear matter spin-polarization refers to the alignment of the intrinsic spins of nucleons (neutrons and protons) within dense many-body nuclear systems under external or internal mechanisms favoring unequal spin-up and spin-down populations. Spin-polarization is a key degree of freedom in the thermodynamics, phase structure, and observable manifestations of nuclear matter, from terrestrial heavy-ion collisions to the magnetized interiors of neutron stars. Its influence is encoded quantitatively in effective many-body Hamiltonians, and enters the equation of state (EOS), symmetry energies, susceptibilities, and instability thresholds of nuclear matter. Modern theoretical treatments embrace Hartree-Fock, Brueckner-Hartree-Fock (BHF), Dirac-Brueckner, relativistic Hartree-Fock, variational (LOCV), and coalescence models, each with a distinctive approach to incorporating spin and spin-isospin channels.

1. Definitions and Formalism

Spin-polarization in nuclear matter is characterized by distinct densities of spin-up ($\rho_\uparrow$) and spin-down ($\rho_\downarrow$) nucleons. The spin-polarization parameter is generally defined as

$$\delta_s = \frac{\rho_\uparrow - \rho_\downarrow}{\rho_\uparrow + \rho_\downarrow}$$

For multicomponent systems (neutrons and protons), separate polarization parameters $\delta_n$ and $\delta_p$ may be introduced. In astrophysical contexts (e.g., neutron stars), polarization is often parametrized with $\Delta \equiv \Delta_n \approx -\Delta_p$ for the relevant configuration. Fully polarized matter corresponds to $|\delta_{s}| = 1$, while the unpolarized case has $\delta_{s} = 0$.

The total energy density $\mathcal{E}(\rho, \Delta)$ is decomposed into kinetic and interaction components, with the interaction part carrying explicit dependence on the spin and spin-isospin channels via density-dependent parameters (e.g., $F_{st}(\rho)$, with $\rho_\downarrow$0 for spin and $\rho_\downarrow$1 for isospin channels) (Khoa et al., 2022).

2. Spin Symmetry Energy and Equation of State

The spin symmetry energy $\rho_\downarrow$2 quantifies the quadratic energy cost to polarize nuclear matter, analogous to the nuclear symmetry energy $\rho_\downarrow$3 in isospin-asymmetric systems. The energy per nucleon expands as

$\rho_\downarrow$4

and

$\rho_\downarrow$5

There exists a strong density-dependent correlation between $\rho_\downarrow$6 and $\rho_\downarrow$7. At saturation density, $\rho_\downarrow$8 MeV, $\rho_\downarrow$9 MeV, and the slope parameters $$\delta_s = \frac{\rho_\uparrow - \rho_\downarrow}{\rho_\uparrow + \rho_\downarrow}$$0 MeV, $$\delta_s = \frac{\rho_\uparrow - \rho_\downarrow}{\rho_\uparrow + \rho_\downarrow}$$1 MeV in typical mean-field models (Khoa et al., 2022). $$\delta_s = \frac{\rho_\uparrow - \rho_\downarrow}{\rho_\uparrow + \rho_\downarrow}$$2 rises more rapidly than $$\delta_s = \frac{\rho_\uparrow - \rho_\downarrow}{\rho_\uparrow + \rho_\downarrow}$$3 at supranuclear densities ($$\delta_s = \frac{\rho_\uparrow - \rho_\downarrow}{\rho_\uparrow + \rho_\downarrow}$$4), producing a stiffer EOS in spin-polarized matter.

The pressure increases ("stiffens") significantly with greater spin-polarization, with implications for neutron star mass and radius and for phase stability (Tan et al., 2020, Bigdeli et al., 2010).

3. Spin-Polarized Phases and Magnetic Instabilities

The competition between kinetic and interaction energies under finite spin-polarization has direct consequences for the stability of nuclear matter:

• Most microscopic many-body calculations (BHF, Dirac-Brueckner) with realistic interactions show that the energy per nucleon increases monotonically with $$\delta_s = \frac{\rho_\uparrow - \rho_\downarrow}{\rho_\uparrow + \rho_\downarrow}$$5, i.e., the ground state is unpolarized up to at least $$\delta_s = \frac{\rho_\uparrow - \rho_\downarrow}{\rho_\uparrow + \rho_\downarrow}$$6 (Sammarruca, 2011). Consequently, neither a ferromagnetic nor an antiferromagnetic transition is realized in these models within the range of densities probed in neutron stars.
• Parametric effective-interaction models (e.g., finite-range SEI) can, for particular choices of parameters controlling the repulsion in spin-triplet-odd channels, produce an antiferromagnetic phase at high densities ($$\delta_s = \frac{\rho_\uparrow - \rho_\downarrow}{\rho_\uparrow + \rho_\downarrow}$$7) but exclude a ferromagnetic phase unless a pathological attraction is present (Behera et al., 2015).
• The inverse magnetic susceptibility $$\delta_s = \frac{\rho_\uparrow - \rho_\downarrow}{\rho_\uparrow + \rho_\downarrow}$$8 remains positive for all densities and polarizations considered in realistic models, precluding spontaneous spin ordering at zero and finite temperature (Bigdeli et al., 2010, Tachibana et al., 18 Jul 2025).

4. Implications for Neutron Stars and Astrophysical Observables

Strong internal magnetic fields in neutron stars (magnetars) may generate significant, albeit partial, spin-polarization of baryons, especially in the outer core. In the presence of large polarization $$\delta_s = \frac{\rho_\uparrow - \rho_\downarrow}{\rho_\uparrow + \rho_\downarrow}$$9, the following features emerge (Tan et al., 2020, Khoa et al., 2022):

• The symmetry energy $\delta_n$0 and spin symmetry energy $\delta_n$1 are stiffened at high densities, substantially increasing the proton and electron fractions.
• The proton fraction $\delta_n$2 is enhanced with increasing $\delta_n$3, crossing critical values for the onset of rapid (direct) Urca processes in magnetar cooling at lower densities.
• The EOS becomes stiffer, supporting more massive neutron stars and altering their mass-radius relation. Masses above $\delta_n$4 and radii $\delta_n$5–$\delta_n$6 km are predicted for $\delta_n$7, in agreement with GW170817 and pulsar limits.
• Excessive polarization ($\delta_n$8) produces radii and pressures that violate multimessenger observational constraints, limiting the physically admissible degree and spatial profile of spin alignment.

5. Spin-Polarization Effects in Heavy-Ion Collisions and Laboratory Probes

In noncentral ultra-relativistic heavy-ion collisions, the global orbital angular momentum induces vorticity, polarizing produced quarks and, through hadronization, baryons. The spin polarization of $\delta_n$9 hyperons has been widely measured via their self-analyzing weak decays, providing a probe of net vorticity (Liu et al., 17 Aug 2025). Proton spin polarization, more sensitive to $\delta_p$0, $\delta_p$1 quark dynamics, is less directly accessible, but can be reconstructed from hypertriton weak-decay observables using a linear relation: $\delta_p$2 where $\delta_p$3 and $\delta_p$4 are the global polarizations extracted from angular distributions of the hypertriton and $\delta_p$5 decay products, respectively.

These experimental strategies, in synergy with theoretical EOS and symmetry energy constraints, provide avenues for mapping spin dynamics and constraints on spin-polarized nuclear EOS in the laboratory, offering connections to neutron-star physics.

6. Impact on Liquid-Gas Phase Transitions and Spinodal Instabilities

Spin-polarization modifies the phase structure of nuclear matter at sub-saturation densities:

• The critical temperature $\delta_p$6 for the liquid-gas transition decreases with increasing polarization parameter $\delta_p$7; for $\delta_p$8, $\delta_p$9 is reduced by 20–30% (Rezaei et al., 2015).
• The extent of the two-phase coexistence (spinodal) region shrinks for both parallel ($\Delta \equiv \Delta_n \approx -\Delta_p$0) and especially antiparallel ($\Delta \equiv \Delta_n \approx -\Delta_p$1) alignment (Polls et al., 2020). In the limit of full antiparallel polarization, the spinodal region is virtually quenched.
• Spinodal instabilities remain dominantly isoscalar (total-density fluctuation), but isospin distillation—the tendency for the dense phase to become more symmetric—is suppressed by spin-polarization.
• Magnetic fields in dense astrophysical environments thus act to stabilize homogeneous matter and inhibit phase separation and fragmentation in neutron star crusts.

7. Correlations among Symmetry Energy Parameters and Theoretical Constraints

The density dependence of the spin symmetry energy, commonly characterized by its slope parameter $\Delta \equiv \Delta_n \approx -\Delta_p$2, shows systematic correlations with the slope $\Delta \equiv \Delta_n \approx -\Delta_p$3 of the nuclear symmetry energy. In relativistic Hartree-Fock calculations:

• For isoscalar polarization ($\Delta \equiv \Delta_n \approx -\Delta_p$4), $\Delta \equiv \Delta_n \approx -\Delta_p$5 and $\Delta \equiv \Delta_n \approx -\Delta_p$6 exhibit a negative correlation: $\Delta \equiv \Delta_n \approx -\Delta_p$7, so constraints on $\Delta \equiv \Delta_n \approx -\Delta_p$8 from nuclear experiment translate to bounds on spin-polarization effects (Tachibana et al., 18 Jul 2025).
• For isovector polarization ($\Delta \equiv \Delta_n \approx -\Delta_p$9), $|\delta_{s}| = 1$0 is nearly independent of $|\delta_{s}| = 1$1.

This interdependence underpins consistent modeling of nuclear matter structure across varying isospin and spin channels, and further constrains neutron star properties and their multimessenger signatures.

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In summary, the spin-polarization of nuclear matter exerts a profound quantitative influence on the EOS, symmetry energies, susceptibilities, and phase boundaries in both astrophysical and laboratory contexts. State-of-the-art many-body calculations and experimental observables jointly indicate that substantial—but bounded—spin alignment is both physically realizable and essential for reconciling nuclear microphysics with macroscopic neutron-star observations, providing a unified framework linking laboratory, theoretical, and astrophysical nuclear science (Tan et al., 2020, Khoa et al., 2022, Tachibana et al., 18 Jul 2025, Liu et al., 17 Aug 2025, Polls et al., 2020, Rezaei et al., 2015, Bigdeli et al., 2010, Behera et al., 2015, Sammarruca, 2011).