Formation of twin compact stars in low-mass X-ray binaries: Implications on eccentric and isolated millisecond pulsar populations

Millisecond pulsars (MSPs) are laboratories for stellar evolution, strong gravity, and ultra-dense matter. Although MSPs are thought to originate in low-mass X-ray binaries (LMXBs), approximately 27% lack a binary companion, while others are found in systems with large orbital eccentricities. Understanding how these systems form may provide insight into the internal properties of neutron stars (NSs). We study the formation of a twin compact star through rapid first-order phase transitions in NS cores due to mass accretion in LMXBs. We investigate whether this mechanism, possibly coupled with secondary kick effects such as neutrino or electromagnetic rocket effects, may leave an observable long-lasting imprint on the orbit. We simulate mass accretion in LMXBs consisting of a NS and a low-mass main sequence companion, following the evolution of the NS mass, radius, and spin until a strong phase transition is triggered. For the NS structure, we assume a multipolytrope equation-of-state that allows for a sharp phase transition from hadronic to quark matter and satisfies observational constraints. We find that in compact binaries with relatively short pre-Roche lobe overflow orbital periods, an accretion-induced phase transition may occur during the LMXB phase. In contrast, in systems with wider orbits, this transition may take place during the spin-down phase, forming an eccentric binary MSP. If the transition is accompanied by a secondary kick (w > 20 km/s), the binary is likely to be disrupted, forming an isolated MSP or reconfigured to an ultra-wide orbit. Our results suggest that accretion in LMXBs provides a viable path for forming twin compact stars, potentially leaving an observable imprint on the orbit. The eccentricity distribution of binary MSPs with long (> 50 d) orbital periods could provide constraints on first-order phase transitions in dense nuclear matter.