Double-stage nematic bond ordering above double stripe magnetism: Application to ${\mathrm{BaTi}}_2{\mathrm{Sb}}_2\mathrm{O}$

Spin-driven nematicity, or the breaking of the point-group symmetry of the lattice without long-range magnetic order, is clearly quite important in iron-based superconductors. From a symmetry point of view, nematic order can be described as a coherent locking of spin fluctuations in two interpenetrating Néel sublattices with ensuing nearest-neighbor bond order and an absence of static magnetism. Here, we argue that the low-temperature state of the recently discovered superconductor ${\mathrm{BaTi}}_2{\mathrm{Sb}}_2\mathrm{O}$ is a strong candidate for a more exotic form of spin-driven nematic order, in which fluctuations occurring in four Néel sublattices promote both nearest- and next-nearest-neighbor bond order. We develop a low-energy field theory of this state and show that it can have, as a function of temperature, up to two separate bond-order phase transitions, namely, one that breaks rotation symmetry and one that breaks reflection and translation symmetries of the lattice. The resulting state has an orthorhombic lattice distortion, an intra-unit-cell charge density wave, and no long-range magnetic order, all consistent with reported measurements of the low-temperature phase of ${\mathrm{BaTi}}_2{\mathrm{Sb}}_2\mathrm{O}$. We then use density functional theory calculations to extract exchange parameters to confirm that the model is applicable to ${\mathrm{BaTi}}_2{\mathrm{Sb}}_2\mathrm{O}$.

Figure 1

Multistage melting of the magnetic order in a square lattice as occurs in (a) single stripe (SS) magnetism and (b) double stripe (DS) magnetism. The nearest- (next-nearest-) neighbor ferromagnetic bonds are indicated with blue (yellow) ovals. While in (a) there is one nematic bond-order degree of freedom associated with rotational symmetry breaking, in (b) two bond-order degrees of freedom are associated with rotation, translation, and reflection symmetry breaking. The new two-site unit cell associated with the translation symmetry is indicated by the red dashed line.

Figure 2

The four degenerate ground states characterized by different configurations of ${\mathbf{M}}_a$'s and signs of corresponding order parameters $\phi , {\psi }_{x,y}$, and ${\psi }_{+}$. Again, the ferromagnetic bonds are indicated with blue/yellow ovals. The dashed black line shows the mirror plane symmetry broken by ${\psi }_{\pm }$.

Figure 3

Signs of ${\phi }_{\mathrm{even}/\mathrm{odd}}$ in a double stripe order (left) and plaquette order (right).

Figure 4

Two examples of how $\phi$ and ${\psi }_{+}$ orders develop, with the proxy for the transition temperature ${\overline{r}}_0$ plotted versus $u/g_1$ for two values of the relative strength of the biquadratic terms $g_3/g_1$. The upper, red line indicates the development of rotational symmetry breaking ($\phi$), while the lower, blue line indicates the dimerization (${\psi }_{+}$), which breaks the diagonal mirror reflection symmetry. Solid lines indicate second-order transitions, while dashed lines indicate first-order transitions, with the double-dashed line indicating simultaneous first-order transitions. The regions of different classes of behavior are indicated in Fig. 5.

Figure 5

Classes of phase transition behavior as the relative strength of the biquadratic terms $g_3/g_1$ and $u/g_1$ are varied. I: simultaneous first-order transitions of $\phi$ and ${\psi }_{+}$; II: second-order transition to $\phi$ followed by a first-order transition to ${\psi }_{+}$; III: distinct second-order phase transitions of $\phi$ and ${\psi }_{+}$; IV: distinct first-order transitions of $\phi$ and ${\psi }_{+}$; V: first-order transition to $\phi$ followed by a second-order transition to ${\psi }_{+}$.

Figure 6

Schematic illustrating the different magnetic patterns considered in our collinear calculations. The inequivalent Ti sites are labeled as $M_1$ and $M_2$ in panel (d), and indicate NNN FM bonds bridging oxygen or vacancy sites, respectively. The relative sizes of the circles representing the Ti sites show the variation in local moment amplitudes (based on $\mathrm{LSDA}+U$ calculations with $U=3.5\text{eV}$) across the magnetic patterns. (a) Ferromagnetic (FM), (b) checkerboard (CB), (c) parallel stripes, (d) double stripes (DS), (e) oxygen-centered plaquettes, (f) vacancy-centered plaquettes.

Figure 7

In ${\mathrm{BaTi}}_2{\mathrm{Sb}}_2\mathrm{O}$, there are two Ti sites [Ti(1), hollow circles and Ti(2), solid black circles] per unit cell, that are differentiated by the orientation of their oxygen coordination (red circles). The 2 Ti unit cell is shown by the blue, dashed lines. At high temperatures, these are equivalent and must carry the same charge, due to the rotation symmetry. However, below $T_{\phi }$, the ferromagnetic bonds break rotation symmetry, and one Ti sublattice will have ferromagnetic bonds that cross oxygen sites, while the other will not, allowing a charge disproportionation to develop, such that $T_{\phi }=T_{\text{CDW}}$.

Figure 8

Schematics illustrating the four interpenetrating Néel sublattices (shown as different color vectors) that form the double stripe magnetic configuration and the two different $0^{\circ }\le \theta \le {180}^{\circ }$ noncollinear rotations used in our DFT calculations to obtain the biquadratic terms $K_1$ and $K_2$. Red lines indicate the dimensions of the lateral supercell in each calculation. (a) Rotation analogous to a fluctuation of the ${\psi }_{\pm }$ order parameter (see Fig. 2). (b) Rotation analogous to a fluctuation of the $\phi$ order parameter (see Fig. 2), which rotates between the $\mathbf{Q}=(\pi /2,\pi /2)$ and $(-\pi /2,\pi /2)$ configurations.

Figure 9

Noncollinear energies as a function of the rotation angle $\theta$. The dashed lines are the fits to the model. The insets in each panel show the inequivalent Ti local moments as a function of $\theta$. The red circles refer to NNN FM O-bridging Ti sites and the blue triangles to NNN FM vacancy-bridging Ti sites when $\theta =0^{\circ }$. (a) Noncollinear energies for the rotations analogous to fluctuations of the ${\psi }_{\pm }$ order parameter [see Fig. 8]. (b) Noncollinear energies for the rotations analogous to fluctuations of the $\phi$ order parameter [see Fig. 8].

Figure 10

Noncollinear energies for rotations connecting the two double stripe patterns where one of the local Ti moments has collapsed. $\theta =0^{\circ }$ corresponds to moments with oxygen-bridging FM NNN interactions and $\theta ={180}^{\circ }$ to moments with vacancy-bridging FM NNN interactions. The dashed line is the fit to the model. The inset shows the Ti local moment as a function of $\theta$.