  According to the recycling scenario, millisecond pulsars (MSPs) have evolved
from low-mass X-ray binaries (LMXBs). Their orbits are expected to be circular
due to tidal interactions during the binary evolution, as observed in most of
the binary MSPs. There are some peculiar systems that do not fit this picture.
Three recent examples are PSRs J2234$+$06, J1946$+$3417 and J1950$+$2414, all
of which are MSPs in eccentric orbits but with mass functions compatible with
expected He white dwarf companions. It has been suggested these MSPs may have
formed from delayed accretion-induced collapse of massive white dwarfs, or the
eccentricity may be induced by dynamical interaction between the binary and a
circumbinary disk. Assuming that the core density of accreting neutron stars in
LMXBs may reach the density of quark deconfinement, which can lead to phase
transition from neutron stars to strange quark stars, we show that the
resultant MSPs are likely to have an eccentric orbit, due to the sudden loss of
the gravitational mass of the neutron star during the transition. The
eccentricities can be reproduced with a reasonable estimate of the mass loss.
This scenario might also account for the formation of the youngest known X-ray
binary Cir X$-$1, which also possesses a low-field compact star in an eccentric
orbit.
