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 high temperature superconductors that was predicted to exhibit both superconductivity and an electronic charge density wave or ``hole crystal'' (HC). Both phenomena have been observed in the doped spin ladder system Sr$_{14- x}$Ca$_{x}$Cu$_{24}$O$_{41}$ (SCCO), which at x = 0 exhibits a HC which is commensurate at all temperatures. To investigate the effects of discommensuration we used resonant soft x-ray scattering (RSXS) to study SCCO as a function of doped hole density, $\delta $. The HC forms only with the commensurate wave vectors $L_{L}$ = 1/5 and $L_{L}$ = 1/3 (Not at 1/4!) and exhibits a simple temperature scaling $\tau _{1/3}$/ $\tau _{1/5}$ = 5/3. For incommensurate values the HC ``melts''. During this study, the distribution of holes in ladder ($n_{L})$ and chain ($n_{c})$ of SCCO are redetermined as a function of $x$ using polarization-dependence x-ray absorption spectroscopy (XAS). An interpretation of polarization dependent XAS is proposed. Based on our interpretation, for $x$ = 0, the estimation of $n_{L}$ and $n_{c}$ is 2.8 and 3.2, respectively. The number of holes in the ladder is linearly increasing with $x$. For $x$ = 11, the estimation of $n_{L}$ and $n_{c}$ is 4.4 and 1.6, respectively. This number of holes is matching well with the model of paired of holes needed to explain the RSXS result.