Despite significant progress in resonant inelastic x-ray scattering (RIXS) experiments on cuprates at the Cu L-edge, a theoretical understanding of the cross section remains incomplete in terms of elementary excitations and the connection to both charge and spin structure factors. Here, we use state-of-the-art, unbiased numerical calculations to study the low-energy excitations probed by RIXS in the Hubbard model, relevant to the cuprates. The results highlight the importance of scattering geometry, in particular, both the incident and scattered x-ray photon polarization, and they demonstrate that on a qualitative level the RIXS spectral shape in the cross-polarized channel approximates that of the spin dynamical structure factor. However, in the parallel-polarized channel, the complexity of the RIXS process beyond a simple two-particle response complicates the analysis and demonstrates that approximations and expansions that attempt to relate RIXS to less complex correlation functions cannot reproduce the full diversity of RIXS spectral features.

The coupling between lattice and charge degrees of freedom in condensed matter materials is ubiquitous and can often result in interesting properties and ordered phases, including conventional superconductivity, charge-density wave order, and metal-insulator transitions. Angle-resolved photoemission spectroscopy and both neutron and nonresonant x-ray scattering serve as effective probes for determining the behavior of appropriate, individual degrees of freedom-the electronic structure and lattice excitation, or phonon dispersion, respectively. However, each provides less direct information about the mutual coupling between the degrees of freedom, usually through self-energy effects, which tend to renormalize and broaden spectral features precisely where the coupling is strong, impacting one's ability to quantitatively characterize the coupling. Here, we demonstrate that resonant inelastic x-ray scattering, or RIXS, can be an effective tool to directly determine the relative strength and momentum dependence of the electron-phonon coupling in condensed matter systems. Using a diagrammatic approach for an eight-band model of copper oxides, we study the contributions from the lowest-order diagrams to the full RIXS intensity for a realistic scattering geometry, accounting for matrix element effects in the scattering cross section, as well as the momentum dependence of the electron-phonon coupling vertex. A detailed examination of these maps offers a unique perspective into the characteristics of electron-phonon coupling, which complements both neutron and nonresonant x-ray scattering, as well as Raman and infrared conductivity.

Exploring cuprate chains Superconductivity in cuprates takes place in their two-dimensional (2D) layers but solving even the simplest model of interacting fermions in 2D is a challenge. The theory problem simplifies in 1D, with experiment becoming the tricky part. Chen et al . synthesized a cuprate that consists of parallel chains and behaves like a 1D system. Crucially, the material could be doped over a wide range of hole concentrations. The researchers showed that including a near-neighbor attractive interaction in a 1D model of interacting fermions was necessary to explain their photoemission measurements. —JS

A Wigner crystal, a regular electron lattice arising from strong correlation effects 1-6 , is one of the earliest predicted collective electronic states. This many-body state exhibits quantum and classical phase transitions 7 and has been proposed as a basis for quantum information processing applications 8, 9 . In semiconductor platforms, two-dimensional Wigner crystals have been observed under magnetic field 10-17 or moiré-based lattice potential 18-21 where the electron kinetic energy is strongly suppressed. Here, we report bilayer Wigner crystal formation without a magnetic or confinement field in atomically thin MoSe2 bilayers separated by hexagonal boron nitride. We observe optical signatures of robust correlated insulating states formed at symmetric (1:1) and asymmetric (4:1 and 7:1) electron doping of the two MoSe2 layers at cryogenic temperatures. We attribute these features to the bilayer Wigner crystals formed from two commensurate triangular electron lattices in each layer, stabilized via inter-layer interaction 22, 23 . These bilayer Wigner crystal phases are remarkably stable and undergo quantum and thermal melting transitions above a critical electron density of up to 6 ´ 10 12 cm -2 and at temperatures of ~40 K. Our results demonstrate that atomically thin semiconductors provide a promising new platform for realizing strongly correlated electronic states, probing their electronic and magnetic phase transitions, and developing novel applications in quantum electronics and optoelectronics 24-28 .Atomically thin heterostructures made of graphene and transition metal dichalcogenide (TMD) monolayers can host a variety of correlated electronic states [29][30][31][32][33] . Recent advances in materials growth and heterostructure fabrication have enabled the preparation of high-quality heterostructures with minimal disorder [34][35][36][37][38] . The large effective masses of charge carriers 39, 40 and the weak Coulomb screening in TMDs suppress the Fermi energy and enhance electron