Dimensional Crossover of Charge-Density Wave Correlations in the Cuprates

Short-range charge-density wave correlations are ubiquitous in underdoped cuprates. They are largely confined to the copper-oxygen planes and typically oscillate out of phase from one unit cell to the next in the $c$ direction. Recently, it was found that a considerably longer-range charge-density wave order develops in ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+x}$ above a sharply defined crossover magnetic field. This order is more three-dimensional and is in-phase along the $c$ axis. Here, we show that such behavior is a consequence of the conflicting ordering tendencies induced by the disorder potential and the Coulomb interaction, where the magnetic field acts to tip the scales from the former to the latter. We base our conclusion on analytic large-$N$ analysis and Monte Carlo simulations of a nonlinear sigma model of competing superconducting and charge-density wave orders. Our results are in agreement with the observed phenomenology in the cuprates, and we discuss their implications to other members of this family, which have not been measured yet at high magnetic fields.

Figure 1

The CDW structure factor at the $a$ and $b$ incommensurate peaks as function of $c$-axis wave vector, $l$, for $T=0.2{\rho }_s$ and various magnetic fields. The inset depicts the YBCO unit cell. Only copper atoms (brown balls) and oxygen atoms (blue balls) are shown. The ${\mathrm{CuO}}_2$ planes (light blue) host the SC and CDW orders. The doped oxygens go into the (orange) ${\mathrm{CuO}}_x$ chain layers and are the main source of disorder.

Figure 2

A magnetic field strongly enhances the $l=1$ CDW structure factor peak in the $b$ direction but only weakly affects the $l=1/2$ signal. Inset: in-plane, ${\xi }_b$, and out-of-plane, ${\xi }_c$, correlation lengths. Results are for $T=0.2{\rho }_s$.

Figure 3

(a) Low-temperature high-field saturation of the $l=1/2$ CDW $b$-peak vs (b) increase of the $l=1$ signal. (c) In-plane and out-of-plane $l=1$ correlation lengths at $B=0.1B_0$.

Figure 4

Suppression of the $l=1$ CDW $b$ peak by in-plane disorder, $\tilde{V}$, as revealed by its dependence on magnetic field (at $T=0.2{\rho }_s$) and on temperature (at $B=0.1B_0$).