Pair density wave characterized by a hidden string order parameter

A composite pairing structure of the superconducting state is revealed by a density matrix renormalization group study in a two-leg t−J model. The pairing order parameter is composed of a pairing amplitude and a phase factor, in which the latter explicitly depends on the spin background with an analytic form. Such a stringlike phase factor is responsible for a pair density wave (PDW) induced by spin polarization m with a wave vector Q_PDW=2πm. By contrast, the pairing amplitude remains smooth, unchanged by the PDW. Moreover, local spin polarization can give rise to a sign change of the order parameter across the local defect. Unlike in a Fulde-Ferrell-Larkin-Ovchinnikov state, the nonlocal phase factor here plays the role of the new order parameter characterizing the PDW, whose origin can be traced back to the essential sign structure of the doped Mott insulator.