Phase diagram of dilute nuclear matter: Unconventional pairing and the BCS-BEC crossover

We report on a comprehensive study of the phase structure of cold, dilute nuclear matter featuring a SD condensate at non-zero isospin asymmetry, within wide ranges of temperatures and densities. We find a rich phase diagram comprising three superfluid phases, namely a Larkin-Ovchinnikov-Fulde-Ferrell phase, the ordinary BCS phase, and a heterogeneous, phase-separated BCS phase, with associated crossovers from the latter two phases to a homogeneous or phase-separated Bose-Einstein condensate of deuterons. The phase diagram contains two tricritical points (one a Lifshitz point), which may degenerate into a single tetracritical point for some degree of isospin asymmetry.

💡 Research Summary

The paper presents a comprehensive theoretical investigation of the phase structure of dilute nuclear matter under conditions of finite isospin asymmetry, spanning a wide range of temperatures and densities. The authors focus on neutron–proton pairing in the spin‑triplet, isospin‑singlet ³S₁‑³D₁ channel, which dominates the attractive interaction at low densities. Using the Nambu‑Gor’kov formalism, they derive an 8×8 Green’s‑function matrix and solve the Dyson equation with explicit treatment of both relative and center‑of‑mass (CM) coordinates. This allows them to treat homogeneous phases (BCS with zero CM momentum Q, the normal unpaired phase) as well as inhomogeneous phases where the Cooper pairs acquire a finite CM momentum (the Larkin‑Ovchinnikov‑Fulde‑Ferrell, LOFF, phase) and spatially separated mixtures of superfluid and normal components (phase‑separated, PS, states).

The key control parameters are the chemical‑potential mismatch δµ = (µₙ – µₚ)/2, the pairing gap Δ₀ at zero mismatch, the temperature T, the total nucleon density ρ, and the asymmetry α = (ρₙ – ρₚ)/(ρₙ + ρₚ). By minimizing the free‑energy functional with respect to the CM momentum Q and the filling fraction x of the normal component, the authors map out the equilibrium phases. They employ the realistic Paris nucleon‑nucleon potential in the ³S₁‑³D₁ channel, and verify that the qualitative features are robust against the choice of interaction by also testing Skyrme energy‑density functionals (SkI2, SLy4).

The results reveal a rich phase diagram containing four distinct regions: (i) the normal (unpaired) phase at high temperature; (ii) the conventional BCS superfluid at moderate asymmetry and temperature; (iii) the LOFF phase, which appears in a narrow band of low temperature and relatively high density where the Fermi‑surface mismatch can be partially compensated by a finite pair momentum Q; and (iv) a phase‑separated region where a symmetric BCS (or BEC) component coexists with a normal neutron‑rich component. As the density is lowered, the BCS superfluid continuously evolves into a Bose‑Einstein condensate (BEC) of deuterons, characterized by a sign change of the average chemical potential (\bar\mu) and a crossover of the coherence length ξ from ξ≫d (d is the inter‑particle spacing) to ξ≪d. The PS‑BCS region similarly transforms into a PS‑BEC state, i.e., a mixture of a deuteron BEC and a dilute neutron gas.

Two tricritical points are found for each value of the asymmetry α; one of them is a Lifshitz point where the LOFF phase meets the BCS and normal phases. For a particular asymmetry (α≈0.255) the two tricritical points merge into a tetracritical point, indicating simultaneous coexistence of four phases. Most phase boundaries are second‑order (continuous), except the PS‑BCS to LOFF transition, which is first‑order. The LOFF region expands with increasing asymmetry but shrinks as temperature rises, reflecting the reduced phase‑space overlap of neutron and proton Fermi surfaces.

The authors discuss astrophysical implications, noting that the densities and temperatures explored correspond to conditions in supernova envelopes, proto‑neutron stars, and the outer layers of neutron stars, where deuteron formation and isospin‑asymmetric pairing are expected. The presence of LOFF and PS phases could affect neutrino transport, cooling rates, and the equation of state in these environments.

In summary, the paper extends the conventional BCS‑BEC crossover framework to imbalanced nuclear matter by incorporating both finite‑momentum pairing (LOFF) and spatial phase separation. It provides the first global phase diagram for dilute, isospin‑asymmetric nuclear matter, identifies the locations and nature of critical points, and highlights the relevance of these exotic superfluid phases to nuclear astrophysics and to other imbalanced fermionic systems such as ultracold atomic gases and color‑superconducting quark matter.