Magnetization process in the classical Heisenberg model on the Shastry-Sutherland lattice

We investigate classical Heisenberg spins on the Shastry-Sutherland lattice and under an external magnetic field. A detailed study is carried out both analytically and numerically by means of classical Monte Carlo simulations. Magnetization pseudoplateaus are observed around 1/3 of the saturation magnetization for a range of values of the magnetic couplings. We show that the existence of the pseudoplateaus is due to an entropic selection of a particular collinear state. A phase diagram that shows the domains of existence of those pseudoplateaus in the $\left(h,T\right)$ plane is obtained.

Figure 1

(Color online) The Shastry-Sutherland lattice (left panel) and the topologically identical structure realized in the (001) plane of ${\text{SrCu}}_2{\left({\text{BO}}_3\right)}_2$ and rare-earth tetraborides (right panel). $J^{\prime }$ bonds are the magnetic couplings along the edges of the squares (clear dashed lines),and $J$ bonds (black bold lines) are the diagonal dimer couplings.

Figure 2

(Color online) Lowest-energy configuration of the SSL in the (001) plane for the particular ratio $J^{\prime }/J=1/2$. The 12-spin unit cell is represented in bold clear lines. Left panel: the $120°$ structure with three kinds of spin orientations. Right panel: the “up-up-down” (UUD) state at $M/M_{\text{sat}}=1/3$. Each triangle contains two up-spins and one down-spin. Up-spins are represented in red (dark gray) and down-spins in yellow (light gray).

Figure 3

(Color online) From left to right: the umbrella and the up-up-down states are possible spin configurations at $M/M_{\text{sat}}=1/3$. The “$Y$ state” and the “canted state” are the quasi-long-range-ordered configurations discussed in Sec. 3B.

Figure 4

(Color online) Lowest branch of the fluctuation matrix $\mathcal{M}$ above the UUD state in Fourier space $\left(k_x,k_y\right)$. In the darkest regions, the black lines represent the lines of soft modes whose analytical expressions are given in the text.

Figure 5

(Color online) Susceptibility and magnetization (inset) of the SSL at $J^{\prime }/J=1/2$ and $T=0.02$ for various system sizes.

Figure 6

(Color online) Susceptibility and magnetization (inset) of the SSL at $J^{\prime }/J=1/2$ and for a system size $L=18$. The temperature $T$ varies.

Figure 7

(Color online) Specific heat of the SSL with $J^{\prime }/J=1/2$ at $T=0.02$ and for various system sizes.

Figure 8

(Color online) Phase diagram of the classical SSL for the ratio of the magnetic couplings $J^{\prime }/J=1/2$.

Figure 9

(Color online) Snapshots of the MC simulations of the spin texture in an $L=12$ system for $J^{\prime }/J=1/2$ and $T\sim 0.003$. The circles represent the projection on the $z$ axis, red (dark gray) if $S^z>0$, and yellow (light gray) if otherwise. The radius is proportional to $\left|S^z\right|$. The arrow denotes the projection in the lattice plane. From upper to lower panel: $Y$ configuration at $h=1.2$, UUD configuration at $h_{1/3}=1.5=h_{\text{sat}}/3$, and canted state at $h=1.8$.

Figure 10

(Color online) MC results for the susceptibility and magnetization (inset) of the SSL at $J^{\prime }/J=0.4$ and $T=0.02$ for various system sizes.

Figure 11

(Color online) Phase diagram for the SSL for the ratio $J^{\prime }/J=0.4$.