Quantum melting of the hole crystal in the spin ladder of Sr$_{14-x}$Ca$_x$Cu$_{24}$O$_{41}$

The ``spin ladder'' is a reduced-dimensional analogue of the t-J model that has been shown theortically to exhibit close competition between d-wave superconductivity and a ``hole crystal" (HC) phase in which the carriers form a static lattice. An example of a real doped spin ladder is the cuprate Sr$_{14-x}$Ca$_{x}$Cu$_{24}$O$_{41}$, which exhibits superconductivity at $x=13.6$ (under pressure) and a commensurate HC at $x=0$. In this talk I will present a resonant soft x-ray scattering (RSXS) study of the effects of discommensuration on this HC, i.e. how it evolves with the hole density. As $x$ is varied the HC forms only with the commensurate wave vectors $L=1/5$ and $L=1/3$; for incommensurate values it ``melts.'' A simple scaling between $L$ and temperature is observed, $\tau_{1/3} / \tau_{1/5} = 5/3$, indicating an inverse relationship between the interaction strength and the HC period. Our results suggest that the HC consists of hole pairs crystallized through an interplay between lattice commensuration and a poorly screened Coulomb interaction. I will discuss the relationship between the HC and the static ``stripe" phase that has been observed in the closely related system La$_{2-x}$Ba$_x$CuO$_4$.