One- and three-dimensional quantum phase transitions and anisotropy in ${\mathrm{Rb}}_2{\mathrm{Cu}}_2{\mathrm{Mo}}_3{\mathrm{O}}_{12}$

Single crystal samples of the frustrated quasi-one-dimensional quantum magnet ${\mathrm{Rb}}_2{\mathrm{Cu}}_2{\mathrm{Mo}}_3{\mathrm{O}}_{12}$ are investigated by magnetic, thermodynamic, and electron spin resonance (ESR) measurements. Quantum phase transitions between the gapped, magnetically ordered, and fully saturated phases are observed. Surprisingly, the former has a distinctive three-dimensional character, while the latter is dominated by one-dimensional quantum spin fluctuations. The entire $H-T$ phase diagram is mapped out and found to be substantially anisotropic. In particular, the lower critical fields differ by over 50% depending on the direction of applied field, while the upper ones are almost isotropic, as is the magnetization above saturation. The ESR spectra are strongly dependent on field orientation and point to a helical structure with a rigidly defined spin rotation plane.

Figure 1

(a) Typical single crystal ${\mathrm{Rb}}_2{\mathrm{Cu}}_2{\mathrm{Mo}}_3{\mathrm{O}}_{12}$ sample used in this work. (b),(c) Schematic view of the crystal structures of ${\mathrm{Rb}}_2{\mathrm{Cu}}_2{\mathrm{Mo}}_3{\mathrm{O}}_{12}$ (monoclinic, space-group $C2/c$).

Figure 2

Typical measured constant-temperature field scans of heat capacity in ${\mathrm{Rb}}_2{\mathrm{Cu}}_2{\mathrm{Mo}}_3{\mathrm{O}}_{12}$ for the transverse (a) and longitudinal (b) field geometries. Additional solid curves in the vicinity of sharp peaks are empirical power-law fits used to pinpoint the transition fields. Arrows indicate positions of additional broad features in the data, as described in the text. For visibility the scans are offset by 0.1 J/mol ${\mathrm{K}}^2$ relative to one another.

Figure 3

Symbols: temperature scans of the heat capacity measured in ${\mathrm{Rb}}_2{\mathrm{Cu}}_2{\mathrm{Mo}}_3{\mathrm{O}}_{12}$ at different fields. (a) Zero applied field. The solid line is a fit of an exponentially activated form, as described in the text. (b) Intermediate fields. Arrows indicate lambda anomalies that represent long range magnetic ordering. (c),(d) Magnetic fields $H_{c1}$ and $H_{c2}$ corresponding to quantum phase transitions at gap closure and full saturation, respectively. The solid lines are power laws with exponents of $3/2$ and $1/2$.

Figure 4

Symbols: magnetic phase diagrams of ${\mathrm{Rb}}_2{\mathrm{Cu}}_2{\mathrm{Mo}}_3{\mathrm{O}}_{12}$ in the transverse (a) and longitudinal (b) geometries. The backgrounds show corresponding false color maps of $C(T,H)/T$. The phase regions are labeled as follows: three-dimensional long-range order (3D LRO), gapped, Tomonaga-Luttinger spin liquid (TLL), and quantum critical regime (QC). Dashed lines are guides for the eye.

Figure 5

Magnetization curves measured in ${\mathrm{Rb}}_2{\mathrm{Cu}}_2{\mathrm{Mo}}_3{\mathrm{O}}_{12}$ at $T=0.1$ K in fields applied along three crystallographic directions (symbols). Solid lines are linear and square-root fits to the data in the vicinity of $H_{c1}$ and $H_{c2}$, respectively. Two data sets are offset by $0.2{\mu }_{\mathrm{B}}$ and $0.4{\mu }_{\mathrm{B}}$ for visibility.

Figure 6

Frequency-field diagram ESR excitations measured at $T=1.4$ K. Solid lines are the fits of the linear function $\mathrm{\Delta }+g^{\prime }{\mu }_{\mathrm{B}}H/h$. The insets show typical raw ESR spectra with characteristic transmission dips marking the resonance frequency.