Ground-state structures in Ising magnets on the Shastry-Sutherland lattice with long-range interactions and fractional magnetization plateaus in ${\mathrm{TmB}}_4$

A method for the study of the ground states of lattice-gas models or equivalent spin models with extended-range interactions is proposed. It is shown that effect of longer-range interactions can be studied in terms of the solution of the ground-state problem for a model with short-range interactions. The method is applied to explain the emergence of fractional magnetization plateaus in $\mathrm{TmB}{}_4$ regarded as a strong Ising magnet on the Shastry-Sutherland lattice with long-range interactions.

Figure 1

(a) Extended Shastry-Sutherland lattice and (b) the lattice formed by magnetic ${\mathrm{Cu}}^{2+}$ ions in $\mathrm{SrCu}{}_2\left(\mathrm{BO}{}_3\right){}_2$ (it is topologically equivalent to the SS lattice). The lattice formed by magnetic ions in $\mathrm{TmB}{}_4$ or in $\mathrm{ErB}{}_4$ is similar, but the triangles are equilateral.

Figure 2

Top: Coordination circles and respective neighbors of the site depicted in black on the lattice that is topologically equivalent to the SS lattice. Bottom: Similar neighbors on the SS lattice. The sites contained in a “windmill” cluster (see Fig. 3) together with the central (black) site are depicted in yellow.

Figure 3

Clusters considered here: (a) “turtle,” (b) “windmill,” (c) “screw,” and (d) “seahorse.” At the top and at the bottom, the clusters are depicted on the lattice shown in Fig. 1 and Fig. 1, respectively.

Figure 4

Ground-state phase diagram of the Ising model on the extended SS lattice for $0<J_3<{\tilde{J}}_2$. Phases 1, 3, 4, and 5 are, respectively, the fully polarized phase, the Néel phase, the 1/3-plateau phase (or the UUD phase), and the collinear phase. Phases 6 and 7 are 1/2-plateau phases. (See Ref. [12] for more detail.) The dashed line parallel to the $h$ axis $(h$ denotes a magnetic field) represents a magnetization process in the case of $0<{\tilde{J}}_1<2({\tilde{J}}_2-J_3)$.

Figure 5

(a) Néel structure (phase 3), (b) 1/3-plateau structure (phase 4), and (c) a general disordered structure at the boundary of the relative phases (the yellow background indicates ferromagnetic chains).

Figure 6

Configurations of the “windmill” cluster for the structures at the boundary between the Néel phase and the 1/3-plateau phase.

Figure 7

The similar structure as in Fig. 5, but with the number of the “windmill” configuration indicated in the center of each “empty” square.

Figure 8

A subcluster of the “windmill” cluster and its two nonequivalent positions in the cluster: central and lateral (enveloped by an ellipse).

Figure 9

Region of variation for the quantities $k_1$ and $k_2$. Vertices $B(0,1),A(2/3,0)$, and $C(2/5,1/5)$ of the triangle $ABC$ correspond to the Néel structure, the 1/3-plateau structure, and the 1/5-plateau structure, respectively.

Figure 10

1/5-plateau structure that emerges from the boundary between phases 3 and 4, if the interaction range reaches the seventh neighbors $\left(k_1=k_3=2/5,k_2=1/5\right)$.

Figure 11

Configurations of the “turtle” cluster which generate all the ground-state structures at the boundary between the Néel phase and the 1/3-plateau phase. Each “empty” square contains the number of the “windmill” configuration with the center in this square (see Fig. 6).

Figure 12

Maximum subcluster (in red) which can occupy two nonequivalent positions in the “turtle” cluster.

Figure 13

Subcluster configurations which generate relations (10).

Figure 14

Region of variation for the quantities $k_1,k_2$, and $l_5$. Each vertex corresponds to a fully dimensional structure.

Figure 15

Structure with magnetization 1/7 that emerges from the boundary between phases 3 and 4 if the range of interaction reaches the 21st neighbors (five square lattice constants). For this structure $k_1=\frac{2}{7},k_2=\frac{3}{7}$, and $l_5=0$.

Figure 16

Pairwise interactions on the SS lattice which can be taken into account by considering the “turtle” cluster (see Fig. 3). The corresponding neighbors of the central site (the black circle) are depicted in yellow.

Figure 17

Structure with magnetization 1/4 that emerges from the boundary between phases 3 and 4 if the range of interaction reaches the 32nd neighbors (six square lattice constants). For this structure $k_1=\frac{1}{2},k_2=\frac{1}{8}$, and $l_5=\frac{1}{8}$.

Figure 18

Coordination circles and respective neighbors of the site depicted in black on the Archimedean lattice $3^2.4.3.4$ that is topologically equivalent to the SS lattice. The sites are enumerated in the same manner as in Fig. 16. The sites which enter a “turtle” cluster (see Fig. 3) together with the central (black) site are depicted in yellow.

Figure 19

Configurations of the “windmill” cluster for the structures at the boundary between phases 4 and 6.

Figure 20

“Seahorse” (in red) is the maximum subcluster that can occupy two nonequivalent positions in the “windmill” cluster.

Figure 21

Some configurations of the “seahorse” subcluster.

Figure 22

A fragment (shown in red) which generates a half-stripe of configurations 1 and/or 4.

Figure 23

Half-stripes generated by configurations 15, 16, 17, and 21.

Figure 24

An example of a structure at the boundary between phases 4 and 6. In each “empty” square, a number of the “windmill” configuration with the center in this square is indicated.

Figure 25

Tetrahedron of variation for the quantities $k_3,k_5$, and $k_6$.

Figure 26

Structures $6a$ and $6b$. They are stabilized by ferro- and antiferromagnetic interactions of the fourth neighbors (antiferro- and ferromagnetic interactions of the fifth neighbors), respectively.

Figure 27

Structure with the maximum value of $k_6=\frac{2}{7}$ (the magnetization $m=\frac{3}{7})$. It emerges from the boundary between phases 4 and 6 when the interaction range reaches the 11th neighbors (three square lattice constants).

Figure 28

Configurations of the “turtle” cluster generating all the structures at the boundary between phases 4 and 6. In each “empty” square, a number of the “windmill” configuration with the center in this square is indicated.

Figure 29

Polyhedron of variation for the quantities $k_1,k_5$, and $l_{14}$. Each vertex corresponds to a fully dimensional structure.

Figure 30

The structure given rise by the antiferromagnetic 15th-neighbor interaction from the boundary between phases 4 and 6. This structure possesses maximum number of “windmill” configurations 1 (or 3) with no “windmill” configurations 5 and “turtle” configurations 14.

Figure 31

Configurations of the “screw” cluster for the boundary between phases 1 and 6.

Figure 32

The maximum subcluster that can occupy two nonequivalent positions in the “screw” cluster.

Figure 33

Polyhedron of variation for the quantities $m,p_3+p_6$, and $p_3+p_7$.

Figure 34

A structure emerging from the boundary between phases 1 and 6. It corresponds to the vertex $D$ for which $p_3=\frac{2}{3},p_5=\frac{1}{3}$, and $m=\frac{2}{3}$. The unit cell is indicated.

Figure 35

Partially disordered stripe structure emerging from the boundary between phases 1 and 6. It corresponds to the vertex $E$ for which $p_3=p_7=\frac{4}{9},p_{10}=\frac{1}{9}$, and $m=\frac{5}{9}$.

Figure 36

Configurations of the “screw” cluster at the boundary between phases 3 and 7.

Figure 37

Configurations of the “windmill” cluster for the boundary between phases 3 and 7.

Figure 38

An example of a structure at the boundary between phases 3 and 7. The number of the “windmill” configuration in an “empty” square with the center in this square is indicated in each such square. $k_6=k_{10}=\frac{1}{5},k_1=k_3=k_4=k_5=k_8=k_9=\frac{1}{10}$.

Figure 39

Structures $7a,7b$, and $4a$.

Figure 40

An example of a structure at the boundary between phases 3 and 7. This structure can be treated as a simple mixture of structures 3 and $4a$. The number of the “windmill” configuration with the center in an “empty” square is indicated in each square.

Figure 41

Polygon of variation for the quantities $m$ and $k_1+2k_5+k_{10}$. Its vertices correspond to the structures shown in Fig. 39 and the Néel structure 3.

Figure 42

The structure at the boundary between phases 3 and 7 which corresponds to the vertex $E$ of the polyhedron $ABCDE$ (Fig. 43). Each “empty” square is labeled by the number of the “windmill” configuration with the center in this square. The structure is generated by the “windmill” configurations 4 and 11 $\left(k_4=\frac{1}{3},k_{11}=\frac{2}{3}\right)$.

Figure 43

Polyhedron of variation for the quantities $m,k_1+2k_5+k_{10}$, and $2k_2+k_3$. Its vertices correspond to the structures shown in Figs. 39 and 42 and the Néel structure 3.

Figure 44

Configurations of the “turtle” cluster which generate all the ground-state structures at the boundary between the Néel phase and the 1/3-plateau phase $4a$. Each “empty” square contains the number of the “windmill” configuration with the center in this square (see Fig. 37).