Laboratoire de Physique Théorique
Toulouse - UMR 5152

FIG.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 non-overlapping plaquettes on the original lattice. The charge unit cell is 4a*4a. The SO(5) model is defined on the center of the non-overlapping 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.

Recently, a striking feature is revealed in the STM experiments by Hoffman et al.[1], where the local density of states (DOS) near the vortex core show a two dimensional checkerboard-like modulation with a 4a*4a. charge unit cell. Here a is the dimension of the CuO2 plane. This modulation decays exponentially away from the center of the vortex core, with a decay length of about 10a. More recently, a similar pattern has also been seen in the absence of the applied magnetic field [2], possibly induced by the impurities at the surface.

In this work, we propose a specific microscopic state around the vortex core and above Hc2 for the YBCO and BSCO class of superconductors. The insulating version of this state at doping x=1/8 is depicted in Fig. 1. In this state, the d wave hole pairs form a Wigner crystal, dubbed a d wave hole pair checkerboard. The center-of-mass of the d wave hole pair is located at the center of every four plaquette on the original lattice. This state has the 4a*4a checkerboard symmetry as observed in the experiment, and the doping level for the insulating x=1/8 state is reasonably close to the optimal doping level of the cuprates. Therefore, by forming the Wigner crystal state of the hole pairs rather than the holes themselves, the doping level can be compatible with the observed size of the charge unit cell. Here the d wave pair is drawn on the plaquette for illustrative purposes. In the actual system, the pair size can be more extended. We propose that the hole pair checkerboard state is energetically competitive in the microscopic t-J model with short ranged Coulomb interaction. This state takes advantage of the underlying AF exchange energy in the background, while the holes form singlet d wave bonds to minimize their exchange energy as well. Finally, the kinetic energy of the hole pair competes with the Coulomb interaction. At x=1/8, if the Coulomb interaction is stronger than the kinetic energy of the hole pair, the hole pair checkerboard state could win. We believe that the YBCO and BSCO materials are in the regime where the kinetic energy of the hole pairs favor the uniform SC state. However, when the applied magnetic field impedes the coherent hole pair motion either around the vortex core or in the state above Hc2, we argue that the hole pair checkerboard state can be realized, either in the static or dynamically fluctuating form.

1. J. E. Hoffman et al., Science 295, 466 (2002)
2. C. Howald et al., PNAS 100, 9705 (2003)