Random singlet state in ${\mathrm{Ba}}_5{\mathrm{CuIr}}_3{\mathrm{O}}_{12}$ single crystals

We study the thermodynamic and high-magnetic-field properties of the magnetic insulator ${\mathrm{Ba}}_5{\mathrm{CuIr}}_3{\mathrm{O}}_{12}$, which shows no magnetic order down to 2 K, consistent with a spin-liquid ground state. While the temperature dependence of the magnetic susceptibility and the specific heat shows only weak antiferromagnetic correlations, we find that the magnetization does not saturate up to a field of 59 T, leading to an apparent contradiction. We demonstrate that the paradox can be resolved, and all of the experimental data can be consistently described within the framework of random singlet states. We demonstrate a generic procedure to derive the exchange coupling distribution $P\left(J\right)$ from the magnetization measurements and use it to show that the experimental data are consistent with the power-law form $P\left(J\right)\sim J^{-\alpha }$ with $\alpha \approx 0.6$. Thus, we reveal that high-magnetic-field measurements can be essential to discern quantum spin-liquid candidates from disorder dominated states that do not exhibit long-range order.

Figure 1

The depiction of intrinsic disorder in chains of Cu and Ir in the ${\mathrm{Ba}}_5{\mathrm{CuIr}}_3{\mathrm{O}}_{12}$ lattice structure. (a) Cu-Ir chains composed of ${\mathrm{Ir}}^{4+}$ trimers and ${\mathrm{Cu}}^{2+}$ ions (Ba ions fill the space between the chains [24]). Disorder occurs either due to Cu-Ir site mixing or due to Cu being displaced from the prism center [25, 26]. (b) Spin degrees of freedom in a chain segment; here, Ir trimers form effective $J=1/2$ moments that interact with the ${\mathrm{Cu}}^{2+}$ spins. (c) An example of disorder in the position of Cu and Ir leading to exchange disorder. Interchanging the Cu and Ir sites leads to Ir clusters forming low-spin states. The Cu spins interact with each other through perturbatively generated $J^{\prime }$ and $J^{\prime \prime }$, resulting in disorder in the effective magnetic exchange couplings.

Figure 2

Temperature ($T$) dependencies of the magnetic susceptibility ($\chi$) and the specific heat ($C_P$). (a) The magnetic susceptibility data in red ($H\parallel c$) and blue ($H\perp c$). The black dashed line is a fit for $H\parallel c$ with the random singlet model ${\chi }_{\text{RS}}=\partial M_{\text{RS}}/\partial H$ [see Eq. (1)]. Inset: $({\chi }^c-{\chi }_0^c)$ multiplied by a function $f\left(T\right)$. For the colored points we take $f\left(T\right)=3(T-T_W)$ for several values of $T_W$ between $-3$ and $-5$ K, for $H\parallel c$, demonstrating the nonlinearity of the low-temperature dependence. Black points are the RSS contribution $f\left(T\right)={\chi }_{\text{RS}}\left(T\right)/{\mu }_{\text{eff}}^2$. Lines are guides to the eye. At high temperatures all curves converge to ${\mu }_{\text{eff}}^2$. (b) Specific heat divided by temperature. The black line is a fit to the combination of the random singlet model in Eq. (2) and a simplified model for phonons (see text). Inset: The specific heat divided by $T^{0.54}$; the gray band shows the confidence interval of the fit.

Figure 3

The magnetic field dependence of the magnetization at $T=2$ K for the field along the $c$-axis direction. The weak kink near 50 T results from the noise of the equipment. The black dashed line is a fit with the random singlet model [Eq. (1)] using the parameters given in Table 1. The green line represents the magnetization of an $S=1/2$ paramagnet. The Van Vleck contribution $H{\chi }_0$ has been added to both. Inset: Log-log plot of $dM\left(H\right)/dH-{\chi }_0$ for the field along the $c$ axis, and the black line is a power-law fit $0.18H^{-0.6}$. Data for $H\parallel ab$ are not shown due to calibration issues [31].

Figure 4

(a) The energy levels and the ground state of an isolated singlet. The triplet ($S=1$) of excited states at $H=0$ is split in the field, and a change of the ground state occurs at $H_c\left(J\right)$, from singlet ($S=0$) to fully polarized ($S=1$). (b) The random singlet distribution in a magnetic field. Singlets with $J<\mu H$ are broken by the field and are fully polarized, while the ones with $J>\mu H$ remain in the singlet state, leading to a nonsaturating magnetization.