Incommensurate charge ordered states in the t-t′-J model

Abstract

We study the incommensurate charge ordered states in the t t ′J model using theGutzwillermean field theory on large systems. In particular, we explore the properties of incommensurate charge modulated states referred to as nodal pair density waves (nPDW) in the literature.nPDWstates intertwine site and bond charge order with modulated d-wave pair order, and are characterized by a nonzero amplitude of uniform pairing; they also manifest a dominant intra-unit cell d-density wave form factor. To compare with a recent scanning tunneling microscopy (STM) study (Hamidian et al 2015 Nat. Phys. 12 150) of the cuprate superconductor BSCCO-2212, we compute the continuum local density of states (LDOS) at a typicalSTM tip height using the Wannier function based approach. By Fourier transforming Cu andOsub-lattice LDOS we also obtain bias-dependent intra-unit cell form factors and spatial phase difference.Wefind that in thenPDWstate the behavior of form factors and spatial phase difference as a function of energy agrees remarkably well with the experiment.This is in contrast to commensurate charge modulated states, which we show do not agree with experiment. Wepropose that thenPDWstates are good candidates for the charge density wave phase observed in the superconducting state of underdoped cuprates.