Vortices and charge order in high-$T_c$ superconductors

We theoretically investigate the vortex state of the cuprate high-temperature superconductors in the presence of magnetic fields. Assuming the recently derived nonlinear $\sigma$-model for fluctuations in the pseudogap phase, we find that the vortex cores consist of two crossed regions of elliptic shape, in which a static charge order emerges. Charge density wave order manifests itself as satellites to the ordinary Bragg peaks directed along the axes of the reciprocal copper lattice. Quadrupole density wave (bond order) satellites, if seen, are predicted to be along the diagonals. The intensity of the satellites should grow linearly with the magnetic field, in agreement with the result of recent experiments.

Figure 1

(a) Brillouin zone and hotspots connected with the antiferromagnetic wave vector $\mathbf{Q}$. Hotspots can be organized in terms of two quartets $L=1$ and 2. (b) The hotspots forming the $L=1$ quartet are indicated, and the velocities are defined via the angle $\eta$. (c) Wave vectors of the different orderings are presented. QDW order is modulated with ${\mathbf{Q}}_{1,2}$ while the CDW is modulated with ${\mathbf{Q}}_{x,y}$.

Figure 2

(a) Result of numerical studies of Eqs. (14) and (15) for a single vortex: superconducting order parameter ${\Delta }_{\mathrm{SC}}$ (red, solid), charge order ${\Delta }_{\mathrm{QDW}}$ (blue, dashed), and magnetic field $\overline{B}$ (green, dotted) in arbitrary units as a function of the dimensionless radius $\overline{r}$. The nonlinear constraint on the total order parameter (1) leads to the rise of charge order triggered by simultaneous decay of the superconducting order. The magnetic field penetrates deep into the plane. (b) Checkerboard density wave order inside the vortex core of radius $\sim \xi$.

Figure 3

(a) Vortex contribution to the intensity of one $\left(200\right)$ satellite peak. The linear behavior is clearly visible. (b),(c) Peak structure of the QDW and CDW order, respectively. The satellites to the (000) and (100) Bragg peaks are shown.