Spin-orbital liquid and quantum critical point in ${\mathrm{Y}}_{1-x}{\mathrm{La}}_x{\mathrm{TiO}}_3$

The specific heat, the susceptibility under pressure, and the dielectric constant were measured for single crystals ${\mathrm{Y}}_{1-x}{\mathrm{La}}_x{\mathrm{TiO}}_3$. The observed $T^2$-dependent specific heat at low temperatures for $0.17\le x\le 0.3$ samples shows a spin-orbital liquid state between the ferromagnetic/orbital ordering $(x<0.17)$ and antiferromagnetic/possible orbital liquid phase $(x>0.3)$. The nonmonotonous pressure dependence of $T_{\text{C}}$ and the glassy behavior of the dielectric loss for the $x=0.23$ sample suggest that it is approaching a possible quantum critical point. All these properties result from the coupling between the strong spin and orbital fluctuations while approaching the phase boundary.

Figure 1

(a) Temperature dependencies of the specific heat ($C_{\text{P}}$) for ${\mathrm{Y}}_{1-x}{\mathrm{La}}_x{\mathrm{TiO}}_3$ samples; here the lattice contribution is simulated by using the method described in Ref. [19]. (b) The temperature dependencies of the magnetic specific heat ($C_{\text{mag}}$) for ${\mathrm{Y}}_{1-x}{\mathrm{La}}_x{\mathrm{TiO}}_3$. (c) The variation in magnetic entropy $\Delta S_{\text{mag}}$ below 40 K. Inset: $\Delta S_{\text{mag}}/R\ln 2$ vs the La content ($x$). (d) $C_{\text{mag}}$ for $x=0$ and 0.3 samples (open symbols); the solid and dashed lines are the fits as described in the main text. The La-content ($x$) dependencies of the fitting parameters (e) $\alpha$ and (f) $\gamma$ for $C_{\text{mag}}$.

Figure 2

The magnetic susceptibility $\chi$ at different pressures, the $d\chi /dT$ $vs$ $T$ curves, and the pressure dependencies of $T_{\text{C}}$ for $x=0$ (a)–(c), 0.17 (d)–(f), 0.23 (g)–(i), and 0.3 (j)–(l), respectively. The solid lines in (c), (f) and (l) are linear fittings. Insert of (i): the pressure dependence of $\chi$ for $x$ = 0.23 at 6 K.

Figure 3

(a)-(b) Temperature dependencies of dielectric constant at different fields with constant frequency $f=20$ kHz. (c) and (d) Temperature dependencies of dielectric constant under different frequencies at $H$ = 9 T. Here, data in (a) and (c) is the real part ${\varepsilon }^{\prime }$ and data in (b) and (d) is the imaginary part ${\varepsilon }^{\prime \prime }$. To clearly show the behavior of dielectric constant, an offset is made for (b)–(d) respectively. Insert: $\ln f$ $vs$ the inverse dielectric loss peak temperature ($T_0$).

Figure 4

The schematic phase diagram for ${\mathrm{Y}}_{1-x}{\mathrm{La}}_x{\mathrm{TiO}}_3$. The magnetic order temperatures (solid symbols) were obtained from the studies here and Ref. [24]. The red regime represents the orbital order phase (OO), the yellow regime represents the spin-orbital liquid phase (SOL), and the blue regime represents the orbital liquid phase(OL). The critical point (QCP) is around $x=0.23$. With $0\le x\le 0.23$, the system is FM below $T_{\text{C}}$. With $0.23<x\le 1.0$, the system is AFM below $T_{\text{N}}$. The question mark for OL means that the OL state is still under debate.