Density matrix renormalization group study of nematicity in two dimensions: Application to a spin-1 bilinear-biquadratic model on the square lattice

Nematic order is an exotic property observed in several strongly correlated systems, such as iron-based superconductors. Using large-scale density matrix renormalization group (DMRG) techniques, we study at zero temperature the nematic spin liquid that competes with spin dipolar and quadrupolar orders. We use these nematic orders to characterize different quantum phases and quantum phase transitions. More specifically, we study a spin-1 bilinear-biquadratic Heisenberg model on the square lattice with couplings beyond nearest neighbors. We focus on parameter regions around the highly symmetric $SU\left(3\right)$ point where the bilinear and biquadratic interactions are equal. With growing further-neighbor biquadratic interactions, we identify different spin dipolar and quadrupolar orders. We find that the DMRG results for cylindrical geometries correctly detect nematicity in different quantum states and accurately characterize the quantum phase transitions among them. Therefore, spin-driven nematicity, here defined as the spontaneous breaking of the lattice invariance under a ${90}^{\circ }$ rotation, is an order parameter which can be studied directly in DMRG calculations in two dimensions in different quantum states.

Figure 1

Sketch illustrating bonds in the middle of the RC6 cylinder used for calculating the nematic order parameters in both the $B_{1g}$ and $B_{2g}$ channels. The dashed lines represent mirror symmetries along $e_x, e_y, e_{x+y}$, and $e_{x-y}$. The blue, red, purple, and orange lines represent bonds along the $e_x, e_y, e_{x+y}$, and $e_{x-y}$ directions, respectively.

Figure 2

Quantum phase diagram of the $SU\left(3\right)$ model with additional $K_2$ interaction at $K_3=0$. (a) and (b) Sketches illustrating the AFQ2 and AFM23 orders. (c) Quantum phase diagram of the spin-1 $SU\left(3\right)$ Heisenberg model on the square lattice with $J_1=K_1=1.0,K_3=0$, and varying $K_2$. The regime of the nematic spin liquid around the $SU\left(3\right)$ point is indicated by red shading. (d)–(f) Spin ($m_S^2$) and quadrupolar ($m_Q^2$) structure factors for different values of $K_2$, which are obtained from the middle $6\times 12$ site region of a long RC6 cylinder. The top and bottom panels are for $m_S^2$ and $m_Q^2$, respectively. (g) The real-space quadrupolar correlation functions for the AFQ2 state with $K_2=-0.4$ in the middle of the RC6 cylinder. (h) The real-space spin correlation functions for the AFM23 state with $K_2=1.0$ in the middle of the RC6 cylinder. The green site is the reference site; blue and red denote positive and negative correlations of the sites with respect to the reference site, respectively. The areas of circles are proportional to the magnitude of the correlations.

Figure 3

Interaction dependence of order parameters and correlation functions in the $SU\left(3\right)$ model with additional $K_2$ coupling. (a) and (b) Spin ($m_S^2$) and quadrupolar ($m_Q^2$) order parameters for $J_1=K_1=1.0,K_3=0.0$ at different momenta as a function of $K_2$ on the RC6 and RC9 cylinders. Both order parameters are obtained from the middle $6\times 12$ sites on the RC6 cylinder or the middle $9\times 18$ sites on the RC9 cylinder. (c) and (d) The log-linear plots of spin $\langle {\mathbf{S}}_i·{\mathbf{S}}_j\rangle$ and quadrupolar $\langle {\mathbf{Q}}_i·{\mathbf{Q}}_j\rangle$ correlations as a function of the distance on the RC6 cylinder.

Figure 4

Quantum phase diagram of the $SU\left(3\right)$ model with additional $K_3$ interaction. (a)–(c) Sketches illustrating the AFQ4, AFM3, and AFQ3 phases, respectively. (d) Quantum phase diagram of the spin-1 $SU\left(3\right)$ Heisenberg model on the square lattice with $J_1=K_1=1.0,K_2=0.0$, and varying $K_3$. The regime of the nematic spin liquid around the $SU\left(3\right)$ point is indicated by red shading. (e)–(h) Spin ($m_S^2$) and quadrupolar ($m_Q^2$) structure factors for different values of $K_3$, which are obtained from the middle $6\times 12$ sites of the RC6 cylinder. The top and bottom panels are for $m_S^2$ and $m_Q^2$, respectively. (i) The real-space quadrupolar correlation functions for the AFQ4 state with $K_3=-0.4$ in the middle of the RC6 cylinder. (j) The real-space quadrupolar correlation functions for the AFQ3 state with $K_3=0.4$ in the middle of the RC6 cylinder. The green site is the reference site; blue and red denote positive and negative correlations of the sites with respect to the reference site, respectively. The areas of circles are proportional to the magnitude of correlations.

Figure 5

Interaction dependence of the order parameters and correlation functions in the $SU\left(3\right)$ model with additional $K_3$ coupling. (a) and (b) Spin ($m_S^2$) and quadrupolar ($m_Q^2$) order parameters for $J_1=K_1=1.0,K_2=0.0$ at different momenta vs $K_3$ on the RC6 and RC9 cylinders. Both order parameters are obtained from the middle $6\times 12$ sites on the RC6 cylinder or the middle $9\times 18$ sites on the RC9 cylinder. (c) and (d) Log-linear plots of the spin $\langle {\mathbf{S}}_i·{\mathbf{S}}_j\rangle$ and quadrupolar $\langle {\mathbf{Q}}_i·{\mathbf{Q}}_j\rangle$ correlations as a function of the distance on the RC6 cylinder for different values of $K_3$.

Figure 6

Nematic order parameters in the $B_{1g}$ (${\sigma }_{B1g}^S$ and ${\sigma }_{B1g}^Q$) and $B_{2g}$ (${\sigma }_{B2g}^S$ and ${\sigma }_{B2g}^Q$) channels (a) vs $K_2$ at $K_3=0$ and (b) vs $K_3$ at $K_2=0$ on the RC6 and RC9 cylinders. The insets show the corresponding quantum phase diagrams.