Stabilize or reinterpret the holographic BEC–BCS crossover construction

Develop an improved holographic construction of a phase describing the crossover region between pion condensation and quark pairing at nonzero isospin chemical potential, or explain why the metastable pion-condensate–quark configuration cannot become thermodynamically preferred.

Background

The pion-condensate–quark (πQ) configuration combines a pion condensate with isospin-carrying string sources and continuously connects the pion-condensed and quark phases. Its density composition approaches the expected BEC–BCS pattern at large isospin chemical potential, with the quark contribution dominating asymptotically.

However, the configuration is never energetically favored in the parameter studies reported in the review. Determining whether a modified construction can realize a stable crossover, or understanding the obstruction to stability, remains unresolved.

References

We have not found any region in the $T$-$\mu_I$ plane where this main conclusion is different: The $\pi$Q phase is never energetically preferred. This reinforces in hindsight the phase diagram of Fig.\ \ref{fig:PionT}, no new phase has to be added. We have, however, not checked the entire space of model parameters systematically. For instance, unphysically large values of the pion mass may lead to a qualitatively different phase diagram . We can therefore not exclude that the $\pi$Q phase becomes stable in some regime of the phase diagram if the model parameters are chosen differently.

— Advances in the holographic description of hot and dense QCD matter  (2608.27023 - Kovensky et al., 27 Aug 2026) in Section 6, subsection “Towards a holographic BEC-BCS crossover”