  In a previous paper [M. E. Gusakov, A. I. Chugunov, and E. M. Kantor, Phys.
Rev. Lett. 112, 151101 (2014)], we introduced a new scenario that explains the
existence of rapidly rotating warm neutron stars (NSs) observed in low-mass
X-ray binaries (LMXBs). Here it is described in more detail. The scenario takes
into account the interaction between superfluid inertial modes and the normal
(quadrupole) $m=2$ $r$-mode, which can be driven unstable by
Chandrasekhar-Friedman-Schutz (CFS) mechanism. This interaction can only occur
at some fixed "resonance" stellar temperatures; it leads to formation of the
"stability peaks" which stabilize a star in the vicinity of these temperatures.
We demonstrate that a NS in LMXB spends a substantial fraction of time on the
stability peak, that is, in the region of stellar temperatures and spin
frequencies, that has been previously thought to be CFS unstable with respect
to excitation of $r$-modes. We also find that the spin frequencies of NSs are
limited by the CFS instability of normal (octupole) $m=3$ $r$-mode rather than
by $m=2$ $r$-mode. This result agrees with the predicted value of the cutoff
spin frequency $\sim 730$ Hz in the spin distribution of accreting millisecond
X-ray pulsars. In addition, we analyze evolution of a NS after the end of the
accretion phase and demonstrate that millisecond pulsars can be born in LMXBs
within our scenario. Besides millisecond pulsars, our scenario also predicts a
new class of LMXB descendants - hot and rapidly rotating nonaccreting NSs ("hot
widows"/HOFNARs). Further comparison of the proposed theory with observations
of rotating NSs can impose new important constraints on the properties of
superdense matter.
