An arXiv preprint reports a sharp split in the model’s phase diagrams. Under traditional regularization, or TRS, the three prescriptions give substantially different domains for a finite-momentum instability; under the medium separation scheme, or MSS, they give nearly overlapping domains over the physically relevant range of M0.
The comparison comes from a local stability study of the homogeneous solution in the chiral-limit two-flavor Nambu-Jona-Lasinio model. It examines how the model’s finite-momentum instability and moat patterns compare across the regularization treatments.
How the calculation was set up
Researchers expanded the effective action to quadratic order and examined eigenvalues of the static two-point function. In practical terms, that tests the homogeneous solution’s response to disturbances at different momenta and identifies a local loss of stability. It is not a calculation of every possible nonuniform state.
The comparison uses two treatments of the momentum integral. In TRS, the ultraviolet regulator is applied to the complete integral. In MSS, the divergent vacuum terms are first isolated, while the ultraviolet-finite medium contribution is left unregularized. Both treatments are paired with a three-dimensional cutoff, Pauli-Villars regularization and proper-time regularization.
The calculation distinguishes a local finite-momentum instability from the moat regime. A local finite-momentum instability is marked by a negative minimum of the static two-point function at nonzero momentum. The moat regime is treated as a finite-momentum precursor that can arise while the homogeneous phase remains locally stable. Moat onset therefore marks a local change in the calculation, not the fully developed modulated state.
The regulator split shows up in the phase diagrams
Under TRS, the three prescriptions draw instability domains that differ substantially, with sizable shifts and different turning structures. The difference concerns both the domains’ locations and the shapes of their boundaries.
Under MSS, the three vacuum prescriptions give nearly overlapping instability domains over the physically relevant M0 range. The authors attribute the apparent scheme dependence mainly to regulation of ultraviolet-finite medium contributions near the Fermi surface.
One phase-diagram slice fixes M0 at 400 MeV and covers the (T, μ) plane. In that slice, TRS shows incompatible high-density behavior across the prescriptions. Under MSS, the instability boundaries are nearly coincident across the three prescriptions, and the moat onset is quantitatively stable.
In the MSS phase diagram, the moat boundary bends toward lower chemical potential as temperature increases.
A result inside a model
The authors interpret the near agreement under MSS as evidence that moat behavior and the subsequent finite-momentum instability are genuine, quantitatively robust features of the dense-medium response rather than artifacts of the ultraviolet prescription. That interpretation remains within the tested Nambu-Jona-Lasinio model and its local stability calculation.
The analysis establishes local onset of spatial modulation but does not determine the fully developed inhomogeneous ground state or fluctuation corrections. The phase diagrams therefore do not by themselves identify which nonuniform state would ultimately be realized.
The supplied document is an arXiv v1 preprint dated 26 August 2026. Its acknowledgments report partial support from CNPq, FAPERGS and CAPES, along with support from INCT-FNA and the Serrapilheira Institute.
Paper data and sources
Original title: Robust Finite-Momentum Instabilities in Dense Matter
Authors: André G. da Silva, Ricardo L. S. Farias, Theo Motta, William R. Tavares
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text