Magnetically Hidden State on the Ground Floor of the Magnetic Devil’s Staircase

We investigated the low-temperature and high-field thermodynamic and ultrasonic properties of ${\mathrm{SrCu}}_2({\mathrm{BO}}_3{)}_2$, which exhibits various plateaux in its magnetization curve above 27 T, called a magnetic Devil’s staircase. The results of the present study confirm that magnetic crystallization, the first step of the staircase, occurs above 27 T as a first-order transition accompanied by a sharp singularity in heat capacity $C_p$ and a kink in the elastic constant. In addition, we observe a thermodynamic anomaly at lower fields around 26 T, which has not been previously detected by any magnetic probes. At low temperatures, this magnetically hidden state has a large entropy and does not exhibit Schottky-type gapped behavior, which suggests the existence of low-energy collective excitations. Based on our observations and theoretical predictions, we propose that magnetic quadrupoles form a spin-nematic state around 26 T as a hidden state on the ground floor of the magnetic Devil’s staircase.

Figure 1

Schematic illustration of a one-particle energy diagram for $S^z=0$, 1, and 2 levels in magnetic fields based on the ESR study [16]. At around 20 T, the $S^z=2$ level (red band) falls below the $S^z=1$ level (blue line). Far below (above) $H^{*}$, the state is regarded as a gas (solid) phase of the bound triplets. A possible intermediate phase between the gas and solid phases appears within the proximity of $H^{*}$. Importantly, many-body effects among multiple particles in real materials complicate this energy diagram around $H^{*}$, which should make its details unpredictable. The inset illustrates the crystal structure of ${\mathrm{SrCu}}_2({\mathrm{BO}}_3{)}_2$ viewed along the $c$ axis. The dashed rectangles represent dimers of the spin-$1/2$ ${\mathrm{Cu}}^{2+}$ ions. The thick black and gray lines indicate the antiferromagnetic exchange interactions $J$ and $J^{\prime }$, respectively.

Figure 2

(a) Low-temperature magnetization data taken from Ref. [13]. Each plateau is illustrated by the colored bars. According to the theory [17], regions other than the plateaux are attributed to the domain-wall (DW) phase and supersolid (SS), respectively. (b) Magnetic field dependence of the ultrasonic properties, the relative change in elastic constant $\mathrm{\Delta }{C}_{66}/C_{66}$ (left axis) and its field derivative $d(\mathrm{\Delta }{C}_{66}/C_{66})/dH$ (right axis) at 0.6 K. The red (blue) curve exhibits the data measured in the up (down) field sweep. The colored areas, determined by the dips of $d(\mathrm{\Delta }{C}_{66}/C_{66})/dH$, correspond to the colors used in (a).

Figure 3

(a) High-field $C_p$ as a function of temperature. The data were measured by the dual-slope (DS) method and the quasiadiabatic (QA) method. The 22.0 T data reported in Ref. [28] are also shown. The dotted curves obtained for 22.0 T and 24.0 T exhibit the Schottky behavior. Note that the data at 28.0 T are plotted on a semilogarithmic scale for clarity. (b) Magnetic field dependence of $C_p$ measured by the ac (AC) method at various temperatures. For comparison, the dataset measured by the DS method at 0.56 K is also shown. The arrows indicate the anomalies in $C_p(H)$ curves caused by the crystallization and the transformation of the spin structure.

Figure 4

(a) Temperature-field phase diagram of ${\mathrm{SrCu}}_2({\mathrm{BO}}_3{)}_2$. The dashed curve is a visual guide to emphasize the crystal phase. The colors of the shaded areas correspond to those of the phases shown in Figs. 2 and 2. (b) Quasiadiabatic magnetocaloric effect in a pulsed magnetic field. The red (blue) curve represents the field-dependent temperature in the up-sweep (down-sweep). The dashed curve, the colored areas, and the symbols are the same as those shown in (a). The black square indicates the field where the MCE curve shows a kink. (c) Temperature dependence of entropy at each field. The black dotted lines represent the calculated curves of the Schottky-type entropy with the energy gap $\mathrm{\Delta }/k_B$. The dashed curves overlapping with the 24.0 T and 26.0 T data were derived from the data at 24.1 T and 26.2 T for the QA measurements. The thin black line indicates 0.72 J/Kmol, $(R\ln 2)/8$, which corresponds to the entropy of the $1/8$ plateau at $T_{\text{crystal}}$.