Momentum-dependent light scattering in insulating cuprates
Résumé
We investigate the problem of inelastic x-ray scattering in the spin-1 / 2 Heisenberg model on the square lattice. We first derive a momentum-dependent scattering operator for the A 1g and B 1g polarization geometries. On the basis of a spin-wave analysis, including magnon-magnon interactions and exact diagonalizations, we determine the qualitative shape of the spectra. We argue that our results may be relevant to help interpret inelastic x-ray scattering experiments in the antiferromagnetic phase of the cuprates. The advances made in third-generation light sources have recently provided new insights into the study of electron dynamics in strongly correlated systems via resonant inelastic x-ray scattering ͑RIXS͒. Detailed information has already been obtained on Mott gap excitations and orbital transitions as a function of doping in the cuprate families. 1-6 However, due to limitations in resolution, ϳ300 meV near the elastic line, excitations at low energies remain hidden. On the other hand, low-energy excitations at small photon momentum transfers, e.g., phonons, magnons, and electron-hole excita-tions in metals and superconductors, have been well characterized via Raman spectroscopy. 7 Ultimately, achieving a detailed understanding of the momentum and polarization dependence of low-energy excitations in strongly correlated matter would greatly clarify the interplay between various degrees of freedom, such as antiferromagnetism, charge-density-wave order, and superconductivity, often present in system with many competing interactions. 8 In the energy range below a few hundred meV lies one of the most prominent features observed via Raman scattering in Heisenberg antiferromagnets-the two-magnon feature at energy ϳ2.7J, with J the nearest-neighbor magnetic exchange. While the peak frequency of the broad two-magnon peak in Raman scattering is well understood, the asymmetry of the line shape as well as the polarization dependence remain unexplained, even though it has been lavished with attention. 9 Preliminary RIXS studies on the Cu K edge in LaCuO 4 ͑Ref. 10͒ and M edge in CaCuO 2 ͑Ref. 11͒ have shown evidence for low-energy two-magnon scattering. Since in the near future this low-energy window will open for inelastic x-ray studies, it will afford an opportunity to study the dynamics of magnon excitations via the charge degrees of freedom that will complement neutron and Raman scattering studies. Lorenzana and Sawatzky 12 investigated the properties of phonon-assisted absorption in two-dimensional ͑2D͒ Heisen-berg antiferromagnets, and investigated the bimagnon spectrum probed by infrared conductivity. In this paper we extend these calculations to present a theory of inelastic x-ray scattering of the two-magnon response in a Heisenberg antifer-romagnet. In particular we show that the momentum and polarization dependence can provide detailed information on the nature of magnon-magnon interactions in the parent insulating cuprate compounds. We remark at the outset that we neglect specific resonant matrix elements relevant to the RIXS process including transitions resulting from the creation of the core hole, and work in the restricted model of the half-filled single-band Hubbard model on the square lattice to capture scattering via magnon creation. Thus we neglect specific pathways of charge exci-tation, such as the Cu K edge 1s-4p transition, and how double spin flips may occur near the site where the core hole is created. Such kinematic details are indeed important to determine accurately the RIXS intensity as well as the proper symmetry and polarization of spin excitations that may be probed by specific x-ray transitions. However, in order to obtain a preliminary understanding of how the two-magnon response can be probed and how polarization may enter, we first simply focus on the evolution of the two-magnon response , far from any specific resonance, for nonzero photon momentum transfers q. Transitions via light scattering can be created via dipole or multipole matrix elements involving states within the conduction band or out of the valence band. These transitions may be selected by orienting incident and scattering polarization light vectors, ê i and ê f , respectively. The photons entering in the scattering process are represented by ͑ i,f , k i,f , ê i,f ͒ where the indices i and f represent the incoming and outgoing photon, respectively. Since we are interested in the insulating phase, the scattering of light is induced by the interband part of the operator. Following Refs. 13 and 14, we derive a finite momentum transfer q, q ϵ k i − k f , scattering operator for different polarization ge-ometries. The interband part of the scattering matrix element is given by ͗f͉M r ͉i͘ = ͚ ͩ ͗f͉J k f · ê f ͉͉͗͘J −k i · ê i ͉i͘ ⑀ − ⑀ i − i + ͗f͉J −k i · ê i ͉͉͗͘J k f · ê f ͉i͘ ⑀ − ⑀ i + f ͪ ͑1͒ where represents states out of the lower Hubbard band, and J k is the current operator J k = ͚ p, ⑀ץ͑ p / ץp͒c p+k/2, † c p−k/2, , with ⑀ p =−2t͓cos͑p x a͒ + cos͑p y a͔͒ for a square lattice with lattice constant a, and nearest-neighbor hopping t. The current may be expressed as PHYSICAL REVIEW B 75, 020403͑R͒ ͑2007͒
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