  To capture various experimental results in the pseudogap regime of the
underdoped cuprate superconductors for temperature $T<T^{*}$, we propose a
four-component pair density wave (PDW) state, in which all components compete
with each other. Without random field disorders (RFD), only one of the PDW
components survives. If the RFD is included, this state could become phase
separated and consist of short range PDW stripes, in which two PDW components
coexist but differ in magnitudes, resulting in charge density waves (CDW) and a
time-reversal symmetry breaking order, in the form of loop current, as
secondary composite orders. We call this phase-separated pair nematic (PSPN)
state, which could be responsible for the pseudogap. Using a phenomenological
Ginzburg-Landau approach and Monte Carlo simulations, we found that in this
state, RFD induces short range static CDW with phase-separated patterns in the
directional components and the static CDW is destroyed by thermal phase
fluctuations at a crossover temperature $T_{CO}<T^{*}$, above which the CDW
becomes dynamically fluctuating. The experimentally found CDW with
predominantly d-wave form factor constrains the PDW components to have
$s^{\prime}\pm id$ pairing symmetries. We also construct a lattice model and
compute the spectral functions for the PSPN state and find good agreement with
ARPES results.
