Theory of rare-earth Kramers magnets on a Shastry-Sutherland lattice: dimer phases in presence of strong spin-orbit coupling

Abstract

Shastry-Sutherland magnet is a typical frustrated spin system hosting rich phases. While the Heisenberg limit has been extensively studied, the role of spin-orbit coupling is not well explored. Motivated by newly discovered rare-earth Shastry-Sutherland magnets, we construct a generic effective-spin model that describes the interactions between Kramers doublet local moments on a Shastry-Sutherland lattice. Due to the strong spin-orbit coupling, the model takes the form of extended XYZ interactions on both intra- and inter-dimer bonds. We show that, in addition to the conventional "singlet" dimer phase, strong spin-orbit coupling can stabilize peculiar "triplet" dimer phases. These "triplet" dimer phases, though fully gapped, respond immediately to magnetic fields and evolve smoothly into the fully polarized phase. We present that the recently discovered Shastry-Sutherland magnet Yb$_2$Be$_2$GeO$_7$ belongs to the "triplet" dimer phase, and discuss the implication of our results to a broad class of quantum magnets in general.

Figures (7)

• Figure 1: Crystal field scheme of Yb$^{3+}$ in the rare-earth SS magnet Yb$_{2}$Be$_{2}$GeO$_{7}$. The red arrow represents one hole in the 4$f$ shell. Under strong SOC and CEF the electronic levels split into a set of Kramers doublets. At low temperatures, the magnetism is dominated by the lowest Kramers doublet which is well separated to other Kramers doublets.
• Figure 2: Crystal structure and dipole axes of SS magnets with Kramers local moments. a Crystal structure of Yb$_{2}$Be$_{2}$GeO$_{7}$. Yb$^{3+}$ and O$^{2-}$ are denoted by yellow and red balls, respectively. Mirror planes of Yb$^{3+}$ are denoted by purple lines. The in-plane directions of $\hat{\sigma}^{1}$ and $\hat{\sigma}^{2}$ dipole axes are denoted by the blue arrows, while that of $\hat{\sigma}^{3}$ are indicated by the red arrows. b Illustration of dipole axes. The $\hat{\sigma}^{1}$ and $\hat{\sigma}^{2}$ components are within the mirror plane, while the $\hat{\sigma}^{3}$ component is perpendicular to the mirror plane. $\hat{\sigma}^{1}$ and $\hat{\sigma}^{2}$ are generally non-orthogonal. c Illustration of the SS lattice. Each unit cell contains two dimer bonds denoted by the thick dark green lines and marked by $A$, $B$, respectively. The NN inter-dimer bonds can also be divided into two groups, characterized by $\eta_{ij}=\pm1$ in Eq. (\ref{['eq:inter-dimer']}), and are marked red and blue, respectively, with the directions $i\rightarrow j$ indicated by arrows.
• Figure 3: Phase diagram of the effective model Eq. \ref{['eq:ham']} in the intra-dimer limit $\mathsf{J}_{ij}=0$.a antiferromagnetic $J^{\prime33}>0$ and b ferromagnetic $J^{\prime33}<0$.
• Figure 4: Magnetization process of dimer phases upon external magnetic fields. a singlet dimer phase $|s\rangle$, with parameters $J'^{11}=1.4$, $J'^{22}=1.7$, $J'^{33}=1$. b triplet dimer phase $|t_{3}\rangle$, with parameters $J'^{11}=-1.4$, $J'^{22}=-1.7$, $J'^{33}=1$. In both cases we set parameters $\mu_{B}g_{J}A_{\parallel}^{(1)}=1.1$, $\mu_{B}g_{J}A_{\parallel}^{(2)}=2.1$, $\mu_{B}g_{J}A_{\parallel}^{(3)}=1.0$, $\mu_{B}g_{J}A_{\perp}^{(1)}=2.2$, $\mu_{B}g_{J}A_{\perp}^{(2)}=2.1$. The temperature is set to $T=0.1J'^{33}$. The calculations are performed using the exact diagonalization of decoupled dimers.
• Figure 5: Evolution of energy spectra under external magnetic field $\parallel[001]$. Energy levels and transition energies within a,b the singlet dimer phase, and c,d the triplet dimer phase. In a,c the the singlet energy level is marked by the bold black line, while the three triplet levels are marked by colored solid lines. In b,d the high-field asymptotic behaviors of three excitations are indicated by dotted lines. The system parameters are identical to that in Fig. \ref{['fig:MC']}, except that the temperature $T$ is set to zero.