Anomalous Lattice Softening Near a Quantum Critical Point in a Transverse Ising Magnet

We have investigated the elastic response of a transverse Ising magnet ${\mathrm{CoNb}}_2{\mathrm{O}}_6$ by means of ultrasound velocity measurement. A huge elastic anomaly in the $C_{66}$ mode is observed near a quantum critical point when sweeping a magnetic field perpendicular to the Ising axis. This anomaly appears to become critical only for the Faraday configuration (field parallel to the sound propagation direction) but is much less pronounced for the Voigt geometry (field perpendicular to the sound propagation direction). We propose that the relativistic spin-orbit interaction plays a crucial role in the quantum critical regime resulting in the elastic anomaly, which is enhanced by quantum fluctuations.

Figure 1

(a) Crystal structure of ${\mathrm{CoNb}}_2{\mathrm{O}}_6$. Black arrows are the spin directions in the magnetically ordered phase. $J_0$ and $H$ are the intrachain exchange interaction and transverse magnetic field, respectively. (b) Schematic illustration of the strain for the modes $C_{33}$ and $C_{66}$. (c) Temperature dependence of the relative change of the elastic constant $\mathrm{\Delta }{C}_{33}/C_{33}$ at 0 T, obtained by measuring the velocity of longitudinal ($\mathit{k}\parallel \mathit{u}$) ultrasound propagating along the $c$ axis ($\mathit{k}\parallel c$). (d) Temperature dependence of the relative change of the elastic constants $C_{22}$ ($\mathit{k}\parallel \mathit{u}\parallel b$), $C_{44}$ ($\mathit{k}\parallel b,\mathit{u}\parallel c$), and $C_{66}$ ($\mathit{k}\parallel b,\mathit{u}\parallel a$). (e) Temperature dependence of the magnetization with a field of 0.01 T applied along the $a$, $b$, and $c$ direction ($M_a$, $M_b$, and $M_c$) measured in field-cooled (FC) mode.

Figure 2

(a) Magnetic-field dependence of the elastic constant $C_{66}$ at various temperatures. For this transverse ultrasound wave propagates along $b$ and the atoms oscillate along the $a$ direction. (b) Magnetic-field dependence of the elastic constants $C_{22}$, $C_{33}$, and $C_{44}$ at 1.3 K.

Figure 3

(a) Magnetic-field dependence of the relative change of the elastic constant $C_{66}$ obtained from the measurements of two $C_{66}$ modes in the Faraday ($\mathit{k}\parallel b\parallel \mathit{H}$, $\mathit{u}\parallel a$, blue line) and Voigt ($\mathit{k}\parallel a$, $\mathit{u}\parallel \mathit{H}\parallel b$, red line) configurations at 1.9 K. (b) Schematic illustration of the rotational invariance effect.

Figure 4

(a) Temperature dependence of the relative change of the elastic constant $C_{66}$ ($\mathit{k}\parallel \mathit{H}$) in various applied magnetic fields. The black dashed lines are Curie-Weiss fits (see main text). We shifted each curve for better visibility. (b) Magnetic-field dependence of the parameters ($\mathrm{\Theta }$ and $E_{\mathrm{JT}}$) from the Curie-Weiss fits.