Fig. 2

Strength of biquadratic exchange K within Emery model and enhanced nematic spin fluctuations. a Renormalized quadrupolar oxygen density response function \({\tilde{\mathrm {\Pi}}}_{{\boldsymbol{k}} = 0}^\eta = \frac{1}{2}\left[ {\left( {{\mathrm{{\Pi}}}_{{\boldsymbol{k}} = 0}^\eta } \right)^{ - 1} - U_{{\boldsymbol{k}} = 0}} \right]^{ - 1}\) as a function of p-orbital holes n_p (per planar d-orbital) obtained within the three-band Emery model at low temperature T = 10⁻²t_pp and fixed n_d = 1. The interaction of the mobile holes with the antiferromagnetic Néel background of d-orbital spins is fully taken into account. Other parameters are set to t_pd = 1, Δ = 2.5, U_dd = 11, and V_pd = V_pp such that \(U_{{\boldsymbol{k}} = 0} = V_{pp} - \frac{{U_{pp}}}{8}\). We use U_pp = 5.5 for V_pp = 5.5 and U_pp = 4.5 for all other values of V_pp. Amplitude of oxygen quadrupolar fluctuations increases with V_pp and smaller oxygen bandwidth, e.g., smaller t_pp. The inset shows the resulting value of K/J ∝ (J′²/J)(n_p/t_pp) (at small n_p) from which we conclude that an enhancement of the fluctuations by V_pp is crucial for a significant biquadratic exchange coupling. Note that we have approximated \({\mathrm{{\Pi}}}_{\boldsymbol{k}}^\eta \approx {\mathrm{{\Pi}}}_{{\boldsymbol{k}} = 0}^\eta\) for simplicity, which does not affect our conclusion. b, c show the static nematic susceptibility χ_nem in Eq. (7) for the H_t−J−K model of Eq. (2) at half-filling as a function of temperature T obtained by Monte-Carlo simulations of classical spins. A non-zero K enhances the response in the nematic B_1g channel only. Inset phase diagram shows that we are investigating χ_nem above the Néel ordered state. For consistency with the known spin-wave spectrum, we consider a small ferromagnetic next-nearest-neighbor exchange J₂ = −0.1J