Numerical study of magnetization plateaus in the spin-$\frac{1}{2}$ Heisenberg antiferromagnet on the checkerboard lattice

We present numerical evidence that the spin-1/2 Heisenberg model on the two-dimensional checkerboard lattice exhibits several magnetization plateaus for $m=0, 1/4, 1/2$, and $3/4$, where $m$ is the magnetization normalized by its saturation value. These incompressible states correspond to somewhat similar valence-bond crystal phases that break lattice symmetries, though they are different from the already established plaquette phase for $m=0$. Our results are based on exact diagonalization as well as density-matrix renormalization-group large-scale simulations and interpreted in terms of simple parameter-free trial wave functions.

Figure 1

Sketch of the VBC phase, where $a$ and $b$ denote eigenstates of a plaquette with total spin $a$ and $b$, respectively. For instance, the product state obtained with $(a,b)=(1,2)$ is an exact ground state [26] for $m=3/4$. We argue that similar phases with $(0,1)$ and $(0,2)$ are also realized for $m=1/4$ and $1/2$, respectively. In the $m=0$ case, the ground state can be understood using the $(0,0)$ product state [8, 9, 11].

Figure 2

Magnetization curves of the $S=1/2$ Heisenberg model on the checkerboard lattice for clusters that can accommodate the VBC shown in Fig. 1 for all interesting $m$. Solid lines correspond to ED with periodic boundary conditions, and dashed lines to DMRG with CBC; see text. Inset: Widths of the $m=0, 1/4, 1/2$, and $3/4$ plateaus obtained from ED (open symbols) and DMRG (filled symbols) on various clusters, plotted as a function of the inverse diameter. Data are consistent with finite values for all plateaus in the thermodynamic limit.

Figure 3

Energy gaps vs $m$ for $N=40$ lattice labeled with their quantum numbers. Relevant momenta are labeled as $\mathrm{\Gamma }=(0,0), M=(\pi ,0)$ and 2-fold degenerate $(\pi /2,\pi /2)$. For $m=1/4, 1/2$ and $3/4$, the lowest states correspond to the expected four ones in the Brillouin zone. The magnetization sectors with a gray background are not visited in the magnetization curve, i.e., they are obscured by a magnetization jump.

Figure 4

Dimer-dimer correlations [cf. Eq. (5)] computed on an $N=40$ cluster: positive and negative values are shown by blue and red lines, respectively, and the width is proportional to the data; the reference bond is shown in black. From left to right, data correspond to the ground state for $m=0, 1/4, 1/2$, and $3/4$. The top row corresponds to simple VBC states where computation is done analytically without any free parameter (see text); the bottom row corresponds to ED results for the Heisenberg model.

Figure 5

Bond strengths for $m=0, m=1/4, m=1/2$, and $m=3/4$ (from top to bottom and left to right) on a $16\times 8$ cylinder using DMRG simulations and keeping up to 6000 states. In each plot, the top and bottom lines are identical due to periodic boundary conditions along this direction (CBC). Data larger than 0.01 (in absolute value) are written in the plot.

Figure 6

Local magnetization for $m=1/4, m=1/2$, and $m=3/4$ (from top to bottom) on a $16\times 8$ cylinder using DMRG simulations and keeping up to 6000 states. The radius is proportional to the absolute value. Due to edge effects, for the $m=3/4$ plot, we have used a slightly higher total magnetization ($m=3/4+2/128$) to get a clearer picture of the bulk behavior. Top and bottom lines are identical (CBC).

Figure 7

Connected $S^z$ correlations between the reference site (black circle) and the neighboring ones computed using DMRG at $m=1/2$ on a $16\times 8$ cylinder using CBC. Top panel, $\mathrm{\Delta }=1$; bottom panel, $\mathrm{\Delta }=5$. Only data within the bulk are shown. Positive and negative values are shown in blue and red, respectively.