Antiferromagnetism and Hole Pair Checkerboard in the Vortex State of High $T_c$ Superconductors

We propose a microscopic state for the vortex phase of BSCO superconductors. Around the vortex core or above $H_{c2}$, the $d$ wave hole pairs form a checkerboard localized in the antiferromagnetic background. We discuss this theory in connection with recent STM experiments.

Figure 1

Illustration of the $d$ wave hole-pair checkerboard state at $x=1/8$. In this state, the $d$ wave hole pairs occupy every four nonoverlapping plaquettes on the original lattice. The charge unit cell is $4a\times 4a$. The SO(5) model is defined on the center of the nonoverlapping plaquettes. Such a state could be realized around the vortex core, whose center is depicted by the cross. In the actual realization of this state, the hole pair can be much more extended, and the AF ordering could be much reduced from the classical value.

Figure 2

Charge distribution around the vortex core. Parameters of the Hamiltonian are $V_1/J=3$, $V_2/J=2.2$, ${\Delta }_S/J=1.2$, and ${\tilde{\Delta }}_c/J=-0.12$. The center of the figure is the center of the vortex. The grid size is $2a$.

Figure 3

From top to bottom: Distribution of the SC, AF, and $\pi$ order parameters around the vortex core. The relative size of the AF and SC regions can be varied by varying $J_s/J_c$. Notice that the $\pi$ order parameter is nonzero only in the region where the AF and SC order parameters overlap.