Spontaneous spin-nematic ordering in a spin-chain system

The spontaneous onset of the spin-nematic order in spin-chain systems is studied. The ordering persists due to distortions of either magnetic ions, or ligands taking part in the superexchange between magnetic ions. The spin-nematic order yields a nonzero biaxial magnetic anisotropy for the spin chain, violating the U(1) symmetry. The influence of the external magnetic field and nonzero temperature is considered. We predict how the spin-nematic ordering can manifest itself in the temperature and magnetic field behavior of magnetoacoustic characteristics.

Figure 1

The surface $a=(C/d^{2}J)x-(\partial E_{gs}^s/J\partial x)$ as a function of $x$ and $C/d^{2}J$. The crossing of the surface with the zero plane defines possible values of $x$ for the antiferromagnetic spin chain with the strong easy-plane anisotropy $\mu =\pi /2-0.3$.

Figure 2

The ground-state energy $E_0=E_{gs}+C{x}^2/2d^2$ for the antiferromagnetic spin chain with the easy-plane anisotropy $\mu =\pi /2-0.3$ as a function of $x$ and $C/d^{2}J$.

Figure 3

The right-hand side of Eq. (20) for $H=0$ and several values of $T(T={10}^{-7}J$, the solid black line; $T=0.2J$, the dotted blue line; and $T=J$, the dashed red line) as a function of $x$. When the line $C/2J{d}^2$ crosses those lines, the solution for the equation Eq. (20) exists.

Figure 4

The cross section of the surface $f_1$ as the function of $C/J{d}^2$ and $t_c=T_c/J$ with zero plane determines the transition temperature of the spin chain to the spin-nematic ordered phase as a function of $C/J{d}^2$.

Figure 5

The low-temperature $\left(T={10}^{-7}J\right)$ behavior of the right-hand side of Eq. (20) for several values of the field (from bottom to top: $H=0.1J,H=0.3J,H=0.5J,H=0.7J$, and $H=0.9J)$.

Figure 6

The temperature behavior of the renormalization of the sound velocity in the spin-chain system with the spin-nematic ordering at $T_c=0.1J$. The red dashed line describes the transverse sound mode, while the black solid line corresponds to the longitudinal mode.

Figure 7

The temperature behavior of the renormalization of the quadrupolar contribution to the sound velocity $\mathrm{\Delta }{v}_q/v$ in the spin-chain system with the spin-nematic ordering at $T_c=0.1J$. The red dashed line describes the transverse sound mode, while the black solid line corresponds to the longitudinal mode.

Figure 8

The magnetic field behavior of the renormalization of the sound velocity in the spin-chain system with the spin-nematic ordering at $T_c=0.1J$. The red dashed line describes $\mathrm{\Delta }v/v$ for $T=0.15J$, while the red solid solid line describes the case $T=0.05J$. for the transverse mode; the black dashed and solid lines are related to the cases of $T=0.15J$ and $T=0.05J$, respectively, for the longitudinal sound mode.