Antiferroquadrupolar Order and Rotational Symmetry Breaking in a Generalized Bilinear-Biquadratic Model on a Square Lattice

The magnetic and nematic properties of the iron chalcogenides have recently been the subject of intense interest. Motivated by the proposed antiferroquadrupolar and Ising-nematic orders for the bulk FeSe, we study the phase diagram of an $S=1$ generalized bilinear-biquadratic model with multineighbor interactions. We find a large parameter regime for a ($\pi$, 0) antiferroquadrupolar phase, showing how quantum fluctuations stabilize it by lifting an infinite degeneracy of certain semiclassical states. Evidence for this $C_4$-symmetry-breaking quadrupolar phase is also provided by an unbiased density matrix renormalization group analysis. We discuss the implications of our results for FeSe and related iron-based superconductors.

Figure 1

The phase diagram derived from the site-factorized wave function studies, as a function of $K_2$ and $K_3$. We have fixed $J_1=J_2=1/4$, and $K_1=-1$. Collinear AFM represents an antiferromagnet with wave vectors $(\pi ,0)$ or $(0,\pi )$, and Néel AFM with $(\pi ,\pi )$. $(\pi ,0)$ AFQ corresponds to the antiferroquadrupolar phase, which has an infinite degeneracy that is to be lifted by quantum fluctuations. FQ corresponds to the ferroquadrupolar state. The red open circle, green open diamond, and blue open square are three parameter points for which the DMRG results will be presented.

Figure 2

(a) Illustration of the $(\pi ,0)$ AFQ state found within the site-factorized wave function studies. The red bars are the directors $\mathbf{d}$ labeling the quadrupolar direction. Generically, there are four sublattices per unit cell, and the two independent directors are specified by one independent angle $\theta$. (b) Illustration of the square network consisting of the lattice for performing flavor-wave theory calculations. The unit cells, which contain four sublattices, are connected by the vectors ${\mathbf{e}}_1\equiv \widehat{x}$ and ${\mathbf{e}}_2\equiv \widehat{y}$, where we set the lattice constant $a\equiv 1$.

Figure 3

Energy per site of the $(\pi ,0)$ AFQ state versus $\theta$ obtained in the flavor-wave theory calculations. The system we use in the numerics consists of $100\times 100$ unit cells, with four sites per unit cell. The parameters for this calculation are $(K_2,K_3)=(1,-1)$.

Figure 4

The spin dipolar [(a)–(c)] and quadrupolar [(d)–(f)] structure factors $m_s^2(\mathbf{q})$ and $m_Q^2(\mathbf{q})$ with $(J_1,J_2,J_3,K_1)=(1/4,1/4,0,-1)$ on an $L_y=8$ lattice for Néel order [(a),(d)] with $(K_2,K_3)=(1,1)$, FQ order [(b),(e)] with $(K_2,K_3)=(-1,-1)$, and $(\pi ,0)$ AFQ order [(c),(f)] with $(K_2,K_3)=(1,-1)$. The color represents the peak height of the corresponding structure factor.