Magnetothermal properties of the Heisenberg-Ising orthogonal-dimer chain with triangular $XXZ$ clusters

We study a spin-1/2 model with triangular $XXZ$ clusters on the orthogonal-dimer chain in the presence of an external magnetic field. First, we discuss the case where the triangular clusters are coupled via intermediate “classical” Ising spins. Diagonalization of the triangular $XXZ$ clusters yields the exact ground states; finite-temperature properties are computed exactly by an additional transfer-matrix step. A detailed analysis reveals a large variety of ground states at magnetization $M$ equal to fractions 0, $1/4$, and $1/2$ of the saturation magnetization $M=1$. Some of these ground states break translational symmetry spontaneously and give rise to doubling of the unit cell. In a second part, we present complementary numerical data for the spin-1/2 Heisenberg model on the orthogonal-dimer chain. We analyze several examples of $T=0$ magnetization curves, entropy as a function of temperature $T$ and magnetic field, and the associated magnetic cooling rate. Comparison of the two models shows that in certain situations the simplified exactly solvable model yields a qualitatively or sometimes even quantitatively accurate description of the more challenging quantum model, including a case which may be relevant to experimental observations of an enhanced magnetocaloric effect in the two-dimensional compound SrCu${}_2$(BO${}_3$)${}_2$.

Figure 1

“Orthogonal-dimer” chain constructed from triangles of quantum $s=1/2$ spins (large circles) interacting with each other via Heisenberg $XXZ$ interactions (thick solid lines) and assembled to a chain by “classical” Ising spins (small circles). One coupling constant ($J_1$) connects the quantum spins along a vertical line, while a generally different coupling constant ($J$) connects the other two bonds in the quantum triangle. Further coupling constants are assigned to the Ising bonds connecting left ($K_1$) and right ($K_2$) spins with the corresponding “classical” spins ${\sigma }_i$ and ${\sigma }_{i+1}$.

Figure 2

Ground-state phase diagrams for antiferromagnetic $J_1=J>0$ and $K_1=K_2\equiv K$. The top panel shows the case of isotropic interaction ($\Delta ={\Delta }_1=1$). A characteristic feature is the appearance of the macroscopically degenerate ferrimagnetic phase F4 with $M=1/2$. The lower two panels show the case of Ising-type anisotropies, namely, $\Delta =0.8>{\Delta }_1=0.5$ (middle panel) and $\Delta =0.5<{\Delta }_1=0.8$ (lowest panel). Note that $\Delta$ and ${\Delta }_1$ are the prefactors of the $XY$ term in the spin Hamiltonian.

Figure 3

Ground-state phase diagram for the case of ferromagnetic $J_1=J<0$, $K_1=K_2\equiv K$, and $\Delta =2$, ${\Delta }_1=3$ ($XY$-like anisotropy).

Figure 4

Ground-state phase diagrams for the case $J=K_2\equiv K$, $J_1=K_1\equiv \tilde{J}$, which corresponds most closely to the fully quantum orthogonal-dimer chain. The top panel corresponds to isotropic interactions inside the triangles, $\Delta ={\Delta }_1=1$, and antiferromagnetic $\tilde{J}>0$. The middle (lower) panel shows the case of antiferromagnetic (ferromagnetic) $\tilde{J}$ for the anisotropic case $\Delta =2$, ${\Delta }_1=3$ ($XY$-like anisotropy).

Figure 5

Magnetization curve for $J_1=K_1=1$, $J=K_2=1$, and $\Delta ={\Delta }_1=1$. The corresponding ground states can be seen both in the top panel of Fig. 2 and the top panel of Fig. 4. For very low temperature (solid line), the sequence of steplike transitions between the corresponding ground states is recovered. Dashed and dotted lines correspond to $T/J=0.05$ and $T/J=0.1$, respectively.

Figure 6

Magnetization curve for $J=J_1=1$, $K_1=K_2=-1$, and $\Delta =0.5$, ${\Delta }_1=0.8$ (Ising-type anisotropy). The solid line corresponding to $T/J=0.001$ shows sharp transitions between three plateaux: $M=0$, $1/2$, and 1. Dashed and dotted lines correspond to $T/J=0.05$ and $T/J=0.1$, respectively.

Figure 7

Magnetization curve for $J_1=K_1=1$, $J=K_2=1.19034$, and $\Delta =2$, ${\Delta }_1=3$ ($XY$-like anisotropy). The solid line corresponds to $T/J=0.001$ and shows sharp transitions between three plateaux: $M=1/4$, $1/2$, and 1. Dashed and dotted lines correspond to $T/J=0.05$ and $T/J=0.1$, respectively.

Figure 8

Zero-temperature magnetization curves for two cases where all exchange constants are antiferromagnetic: $J_1=K_1=1$, $J=K_2=0.7$ (a) and $J_1=J=K_1=K_2=1$ (b). Dashed lines show numerical results for a Heisenberg chain with $N=40$ spins 1/2 and thick full lines an extrapolation to the thermodynamic limit. Thin full lines show the exact solution of the Heisenberg-Ising model in the thermodynamic limit $N=\infty$. The inset of panel (b) shows the finite-size behavior of the edge of the $M=0$ plateau (spin gap) and the two edges of the $M=1/2$ plateau at $J_1=J=K_1=K_2=1$.

Figure 9

Same as Fig. 8, but for cases which contain ferromagnetic exchange constants, namely, $J_1=K_1=-1$, $J=K_2=1$ (a), $J_1=J=K_2=1$, $K_1=-1$ (b), and $J_1=J=1$, $K_1=K_2=-1$ (c). The inset of panel (c) shows the finite-size behavior of the spin gap (edge of the $M=0$ plateau) at $J_1=J=1$, $K_1=K_2=-1$.

Figure 10

Correlation functions of the $z$ components for the Heisenberg model with $N=32$ sites at $M=0$. $j$ measures the distance in units of cells; due to the periodic boundary conditions, the maximum accessible distance is $N/8=4$. The left panel is for $J_1=\pm 1$, $J=K_2=1$, $K_1=-1$, whereas the right panel shows results for $J_1=J=1$, $K_1=K_2=-1$.

Figure 11

Entropy per spin $S/N$ as a function of $H$ and $T$ for $J_1=K_1=1$, $J=K_2=0.7$. The top panel shows results for the Heisenberg-Ising model in the thermodynamic limit and the lower panel the Heisenberg model with $N=24$ spins. Lines show values of constant entropy, starting with $S/N=0.02$ at the bottom and increasing in steps of $0.02$.

Figure 12

Entropy per spin $S/N$ as a function of $H$ and $T$ for $J_1=J=K_2=1$, $K_1=-1$. The top panel shows results for the Heisenberg-Ising model in the thermodynamic limit and the lower panel the Heisenberg model with $N=24$ spins. Lines show values of constant entropy, starting with $S/N=0.05$ at the bottom and increasing in steps of $0.05$.

Figure 13

Adiabatic cooling rate as a function of magnetic field $H$ for three different temperatures at $J_1=J=K_2=1$, $K_1=-1$. Results for the Heisenberg-Ising model and hard monomers are for the thermodynamic limit, while those for the Heisenberg model are for $N=24$ spins.