Figure 3
Understanding doping dependence of the spin dynamical structure factor S (q, ω). (a) A cartoon illustrating the energy cost of spin excitations with doping within a ‘locally static hole’ model: (top row) ground state with AF correlations; (bottom row) a single spin-flip excitation. Green dashed lines represent the paths of hole delocalization by three-site terms, blue saw-tooth lines represent broken AF bonds and green saw-tooth lines represent broken three-site bonds. Undoped, a single spin-flip costs an energy of 2J from the four broken AF bonds. With doping, this is reduced by the dilution of the AF background. With the three-site terms the overall energy cost increases compared with the undoped system due to the reduction in hole-delocalization energy. The side panel shows the process of hole-motion represented by the three-site terms, similar to the superexchange process. For electron doping, a particle-hole transformation can be applied so that a site with an open circle represents double occupancy. (b) Nearest neighbour spin-spin correlations 〈S0S1› as a function of electron concentration n from DQMC. The solid lines represent a fit of the doping dependence in the ‘locally static hole’ model (see Methods). (c) A comparison between ED results for S(q, ω) in HHubbard, Ht−J, and Ht−J+H3s (see Methods) for different values of n at (2π/3, 0). The three-site terms lead to hardening of spin excitations in qualitative agreement with the results from the Hubbard model. We calculate S(q, ω) on the three model Hamiltonians with the parameters J=0.4t, t=0.25t, U=10t (corresponding to J=0.4t by the relation J=4t2/U) and a Lorentzian broadening HWHM=0.05t.
