Spin Ferroquadrupolar Order in the Nematic Phase of FeSe

We provide evidence that spin ferroquadrupolar (FQ) order is the likely ground state in the nonmagnetic nematic phase of stoichiometric FeSe. By studying the variational mean-field phase diagram of a bilinear-biquadratic Heisenberg model up to the 2nd nearest neighbor, we find the FQ phase in close proximity to the columnar antiferromagnet commonly realized in iron-based superconductors; the stability of the FQ phase is further verified by the density matrix renormalization group. The dynamical spin structure factor in the FQ state is calculated with flavor-wave theory, which yields a qualitatively consistent result with inelastic neutron scattering experiments on FeSe at both low and high energies. We verify that FQ can coexist with $C_4$ breaking environments in the mean-field calculation, and further discuss the possibility that quantum fluctuations in FQ act as a source of nematicity.

Figure 1

Variational mean-field phase diagram of the Hamiltonian Eq. (1) with $J_1=1$, $J_2=0.8$, and periodic boundary condition ($2\times 2$ and $4\times 4$ unit cells yield exactly the same results) [37]. The dashed lines denote shifted phase boundaries when breaking $C_4$ symmetry in Eq. (1) by hand, using $J_1^{x,y}=(1\pm 0.2)J_1$.

Figure 2

Static spin and quadrupolar structure factors obtained from DMRG on $\mathrm{RC}L-2L$ cylinders with $J_1=1$, $J_2=0.8$, $K_2=-1$. (a),(b) $m_S^2(\mathit{q})$ for $L=8$. (c),(d) $m_Q^2(\mathit{q})$ for $L=8$. (e),(f) Finite-size scaling of $m_S^2[\mathit{q}=(0,\pi )]$ and $m_Q^2[\mathit{q}=(0,0)]$ as a function of the inverse cylinder width, where the lines are guides to the eye.

Figure 3

Dispersion and dynamical spin structure factor in the FQ phase obtained from flavor-wave calculation with $J_1=1$, $J_2=0.8$, $K_1=-1.65$, $K_2=-0.8$. (a) Dispersion plotted in the 1st BZ. (b) Energy-momentum dependence of $S(\mathbf{q},\omega )$. (c)–(f) Constant-energy cuts of $S(\mathbf{q},\omega )$ in $\mathit{q}$ space. (c) $\omega /J_1=2$. (d) $\omega /J_1=4$. (e) $\omega /J_1=6$. (f) $\omega /J_1=8$. A Lorentzian broadening factor $\lambda =0.8J_1$ is used for approximating the delta functions.

Figure 4

The density-density interactions between the ${\mathit{Q}}_{1,2}$ modes when approaching the FQ-CAFM phase boundary $K_1^c=-1.6$. The parameters used in this plot are $J_1=1$, $J_2=0.8$, and $K_2=-0.8$.