Spontaneous spin polarization due to tensor self-energies in quark matter

We study the magnetic properties of quark matter in the Nambu–Jona-Lasinio model with the tensor interaction. The spin-polarized phase given by the tensor interaction survives even when the quark mass is zero, while that given by the axial-vector interaction disappears. There are two kinds of spin-polarized phases: one appears in the chiral-broken phase, and the other in the chiral-restored phase where the quark mass is zero. The latter phase can appear independently of the strength of the tensor interaction.

Figure 1

The energy constant surfaces for $e-U_0=3M_q$ and $s=-1$, when (a) $U_T=3M_q$ and when (b) $U_A=3M_q$.

Figure 2

Coupling constant versus the critical Fermi momentum for the spontaneous spin polarization.

Figure 3

$F_T(0)$ with (a) PM1, (b) PM2, and (c) the dynamical quark mass as functions of ${\rho }_B/{\rho }_0$. In panel (a) the dashed, solid and dotted-dashed lines represent the results results when $G_T/G_s=-0.9$, $-1.2$ and $-1.5$, respectively. In panel (b) the dotted line represents the result when $G_T/G_s=-0.6$, and the dashed and solid lines are the same as those in panel (a). The thin lines in panels (a) and (b) indicate $F_T(0)$ when $M_q=0$. In panel (c), the solid and long dashed lines indicate $M_q/M_N$ with PM1 and PM2, respectively.

Figure 4

Spin-polarization properties with PM1. (a) and (c): The tensor densities normalized by the normal nuclear density. The solid and dotted lines represent the results in the chiral-broken and chiral-restored phases, respectively. (b) and (d): The dynamical quark mass normalized by nucleon mass in the spin-polarized (solid lines) and spin-saturated phases (dashed lines). The left and right panels show the results when [(a) and (b)] $G_T=-1.2$ and [(c) and (d)] $G_T=-1.5$, respectively.

Figure 5

$F_T(U_T)$ versus $U_T/M_N$ with PM1 and $G_T=-1.2G_s$. The dotted, dashed, long-dashed, dotted-dashed, and solid lines represent results when ${\rho }_B/{\rho }_0=0.5$, 1, 2, 3, 3.4, respectively.

Figure 6

Spin-polarization properties with PM2. (a), (c), and (e): The T densities normalized by the normal nuclear matter density. The solid and dotted lines represent the results in the chiral-broken and chiral-restored phases, respectively. (b), (d), (f): The dynamical quark mass normalized by nucleon mass in the spin-polarized (solid lines) and spin-saturated phase (dashed lines). The left, middle, and right panes show the results when [(a) and (b)] $G_T/G_s=-0.9$, [(c) and (d)] $G_T/G_s=-1.2$, and [(e) and (f)] $G_T/G_s=-1.5$, respectively.

Figure 7

Critical density between the spin-saturated and spin-polarized phases as functions of $G_T{M}_N^2$ when (a) $M_q\ne 0$ and (b) $M_q=0$. In panel (a) the solid and dashed lines represent ${\rho }_{SC}^{(\mathrm{I})}/{\rho }_0$ with PM1 and PM2, respectively. In panel (b) the solid line shows the critical density of SP-II, ${\rho }_{SC}^{(\mathrm{II})}/{\rho }_0$, and the dotted and dashed lines indicate the critical densities of $CPT$, ${\rho }_c/{\rho }_0$, with PM1 and PM2, respectively.

Figure 8

The tensor density normalized by (a) the normal nuclear matter density and (b) the dynamical quark mass with PM1 and $G_T=-G_s$. The dotted-dashed and solid lines represent the results with and without the vacuum contribution (VC), respectively. In panel (a), the dotted and dashed lines indicate the tensor density with and without VC in the chiral-restored phases, respectively.