Magnetization plateau of the classical Ising model on the Shastry-Sutherland lattice: A tensor renormalization-group approach

We study the magnetization for the classical antiferromagnetic Ising model on the Shastry-Sutherland lattice using the tensor renormalization-group approach. With this method, one can probe large spin systems with little finite-size effect. For a range of temperature and coupling constant, a single magnetization plateau at one third of the saturation value is found. We investigate the dependence of the plateau width on temperature and on the strength of magnetic frustration. Furthermore, the spin configuration of the plateau state at zero temperature is determined.

Figure 1

The Shastry-Sutherland lattice. $J$ bonds (dashed lines) are the exchange couplings along the edges of the squares and $J^{\prime }$ bonds (solid lines) are the diagonal intradimer couplings.

Figure 2

(Color online) Checkerboard decomposition of the Shastry-Sutherland lattice and the corresponding tensor network.

Figure 3

(Color online) (a) Rewiring: the original rank-four tensors are decomposed to two rank-three tensors. (b) Decimation: the tensor $T^{\prime }$ is obtained by summing over the indices around the square.

Figure 4

Under the TRG procedure, a tensor network is transformed into a coarse-grained tensor network.

Figure 5

Magnetization curves for three different temperatures. The size of the system is $2^{10}\times 2^{10}$. The parameters are $J^{\prime }=1$ and $D_{\text{cut}}=18$.

Figure 6

(a) Magnetization versus temperature $T$ and magnetic field $h$. The parameters are $J^{\prime }=1$ and $D_{\text{cut}}=18$. (b) Phase diagram of the magnetization plateau. The theoretical values of the critical fields at zero temperature are denoted by filled circles.

Figure 7

(a) Magnetization versus frustration $J^{\prime }$ and magnetic field $h$. The parameters are $T=0.2$ and $D_{\text{cut}}=18$. (b) Phase diagram of the magnetization plateau. Dashed lines are the theoretical phase boundaries at zero temperature.

Figure 8

(Color online) (a) Spin configuration for the Néel state. Solid and empty circles represent spin-up and spin-down states, respectively. (b) Spin configuration for the collinear state. (c) Spin configuration for the magnetization plateau at $m/m_s=1/3$. The large squares with dashed-dotted lines indicate possible choices of unit cells.