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Interplay between pair density waves and random field disorders in the pseudogap regime of cuprate superconductors

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.