A Data-Driven Approach to Understanding Hole Localization in Spin Ladder Cuprates

Determining the nature of the electronic phases that compete with superconductivity in hole-doped cuprates becomes one of the deepest problems in condensed matter physics. Debate exists over whether charge carriers form a ‘stripe’ phase, because of modulation by a distortion in the crystal structure through the electron-lattice interaction, or a hole crystal driven by many-body interactions. Distinguishing between the two requires determining whether the charge density is modulated in real space or holes are localized. The advancements in both modern neutron spectrometers and diffractometers motivated us to revisit this long-lasting problem. These developments allow for low-noise and high-resolution multi-dimensional data acquisition in an achievable experimental time frame and dramatically changes interpretation of scattering. Although neutrons are not able to directly measure the charge order in cuprates, neutron scattering combined with data-driven machine learning analysis can reveal great details in lattice structure and vibrations (phonons), which are sensitive to various electronic phases through mediating an effective electron-electron interaction. Using a two-leg ladder cuprate compound, Sr_14-xCa_xCu₂₄O₄₁ (x = 11.5), as an example, our recent inelastic neutron scattering measurements have unveiled a local optical phonon mode induced by a variation in coupling strength with oxygen sites on the two neighboring rungs in the ladder layer. High-resolution diffraction measurements further reveal a modulated out-of-plane distortion of oxygen atoms along the chain layers. A modulated variation in electric potential along the ladder results from this chain distortion, leading to a variation in coupling strength with oxygen sites on the rung. The minima of the potential also serve as hole pinning sites, from which the hole crystalline structure can be determined. This determination of hole distribution and hole crystalline structure represents a major step in understanding the charge order in cuprates.