BCS-BEC crossovers and unconventional phases in dilute nuclear matter

We study the phase diagram of isospin-asymmetrical nuclear matter in the density-temperature plane, allowing for four competing phases of nuclear matter: (i) the unpaired phase; (ii) the translationally and rotationally symmetric, but isospin-asymmetrical BCS condensate; (iii) the current-carrying Larkin-Ovchinnikov-Fulde-Ferrell phase; and (iv) the heterogeneous phase-separated phase. The phase diagram of nuclear matter composed of these phases features two tricritical points, in general, as well as crossovers from the asymmetrical BCS phase to a Bose-Einstein condensate (BEC) of deuterons plus a neutron gas, for both the homogeneous superfluid phase (at high temperatures) and the heterogeneous phase (at low temperatures). The BCS-BEC-type crossover in the condensate occurs as the density is reduced. We analyze in detail some intrinsic properties of these phases, including the Cooper-pair wave function, the coherence length, the occupation numbers of majority and minority nucleonic components, and the dispersion relations of quasiparticle excitations about the ground state. We show by explicit examples that the physics of the individual phases and the transition from weak to strong coupling can be well understood by tracing the behavior of these quantities.

Figure 1

Phase diagram of dilute nuclear matter in the temperature-density plane for several isospin asymmetries $\alpha$ (see also Ref. [36]). Here ${\rho }_0=0.16{\mathrm{fm}}^{-3}$ is the nuclear saturation density for $\alpha =0$. Included are four phases: unpaired phase, BCS (BEC) phase, LOFF phase, and PS-BCS (PS-BEC) phase. For each asymmetry there are two tricritical points, one of which is always a Lifshitz point [37]. For special values of asymmetry, these two points degenerate into a single tetracritical point at ${log}_{10}(\rho /{\rho }_0)=-0.22$ and $T=2.85$ MeV when ${\alpha }_4=0.255$ (shown by a square). The LOFF phase disappears at the point ${log}_{10}(\rho /{\rho }_0)=-0.65$ and $T=0$ (shown by a triangle) for $\alpha =0.62.$ The boundaries between BCS and BEC phases are identified by the change of sign of the average chemical potential $\overline{\mu }$. The diamonds (red online) mark the density and temperature values in the diagram that are used in the representative study (see Table 1) of weak-coupling, intermediate-coupling, and strong-coupling regimes (from right to left).

Figure 2

Gap as a function of temperature for asymmetry values $\alpha =0.0$ (black), $\alpha =0.1$ (blue), $\alpha =0.15$ (red), and $\alpha =0.2$ (magenta). Results allowing for the LOFF phase are traced by solid lines; those restricted to the BCS phase are traced by dashed lines.

Figure 3

Gap as a function of asymmetry at constant density $\rho =0.1{\mathrm{fm}}^{-3}$ for $T=0.5$ (black), $T=1.0\mathrm{MeV}$ (blue), $T=1.5\mathrm{MeV}$ (red), $T=2.0\mathrm{MeV}$ (magenta). Results allowing for the LOFF phase are traced by solid lines; those restricted to the BCS phase are traced by by dashed lines.

Figure 4

Dependence of the kernel $K\left(k\right)$ on momentum in units of Fermi momentum for fixed $T=0.2\mathrm{MeV}$, $\alpha =0.3$, and various densities indicated in the plot.

Figure 5

Dependence of the kernel $K\left(k\right)$ on momentum in units of Fermi momentum for fixed $\rho =0.04{\mathrm{fm}}^{-3}$, $\alpha =0.3$, and various temperature indicated in the plot.

Figure 6

Dependence of the kernel $K\left(k\right)$ on momentum in units of Fermi momentum for fixed $\rho =0.04{\mathrm{fm}}^{-3}$, $T=0.2\mathrm{MeV}$, and various values of asymmetry indicated in the plot.

Figure 7

Dependence of the kernel $K\left(k\right)$ on momentum in units of Fermi momentum at fixed $\rho =0.04{\mathrm{fm}}^{-3}$, $T=0.2\mathrm{MeV}$, and $\alpha =0.3$ for the LOFF phase, where $\theta$ is the angle formed by the c.m. and relative momenta in Eq. (27). In the case $\theta =0^{\circ }$ the full result (solid line with circles) is decomposed into components with $r=-a$ (long dashed line) and $r=a$ (dash-dotted line).

Figure 8

Dependence of $\Psi \left(r\right)$ on $r$ for the three coupling regimes and various values of asymmetry (see Table 1 for values of density and temperature).

Figure 9

Dependence of $r^2{\left|\Psi \left(r\right)\right|}^2$ on $r$ for the three coupling regimes. Conventions are the same as in Fig. 8.

Figure 10

Dependence of $\Psi \left(r\right)$ on $r$ in the regime WCR for two different angles $\theta$ for asymmetry $\alpha =0.49$ at which the LOFF phase is the ground state.

Figure 11

Dependence of $r^2{\left|\Psi \left(r\right)\right|}^2$ on $r$ in the regime WCR for two different angles $\theta$ for asymmetry $\alpha =0.49$ at which the LOFF phase is the ground state.

Figure 12

Dependence of the neutron and proton occupation numbers on momentum $k$ (in units of Fermi momentum) for the three coupling regimes and various asymmetries indicated in the legend.

Figure 13

Dependence of the neutron and proton occupation numbers on momentum $k$ (in units of Fermi momentum) in the WCR for two asymmetries where the LOFF phase is the ground state. The three angles indicated refer to the neutron occupation numbers. The proton occupation numbers are plotted for angles ${180}^{\circ }-\theta$.

Figure 14

Dispersion relations for quasiparticle spectra in the case of the BCS condensate as functions of momentum in units of Fermi momentum. For each asymmetry, the upper branch corresponds to $E_{+}^{+}$ and the lower to the $E_{-}^{+}$ solution.

Figure 15

Dispersion relations for quasiparticle spectra in the LOFF phase for three angles and $\alpha =0.49$.

Figure 16

Illustration of Fermi surfaces in the asymmetrical BCS state (solid lines) and LOFF phase (dashed lines). The LOFF phase is characterized by the following values of parameters: $\alpha =0.49$, $\delta \mu =6.45\mathrm{MeV}$, $\Delta =1.27\mathrm{MeV}$, and $Q=0.4{\mathrm{fm}}^{-1}$.