Magnetization plateaus in the spin-$\frac{1}{2}$ antiferromagnetic Heisenberg model on a kagome-strip chain

The spin-$\frac{1}{2}$ Heisenberg model on a kagome lattice is a typical frustrated quantum spin system. The basic structure of a kagome lattice is also present in the kagome-strip lattice in one dimension, where a similar type of frustration is expected. We thus study the magnetization plateaus of the spin-$\frac{1}{2}$ Heisenberg model on a kagome-strip chain with three-independent antiferromagnetic exchange interactions using the density-matrix renormalization-group method. In a certain range of exchange parameters, we find twelve kinds of magnetization plateaus, nine of which have magnetic structures breaking translational and/or reflection symmetry spontaneously. The structures are classified by an array of five-site unit cells with specific bond-spin correlations. In a case with a nontrivial plateau, namely a 3/10 plateau, we find long-period magnetic structure with a period of four unit cells.

Figure 1

Structure of a kagome-strip chain. The black solid, red dashed, and blue broken lines denote the exchange interactions $J_X,J_1$, and $J_2$, respectively. We set $J_X=1$.

Figure 2

(a) Magnetization curve of the kagome-strip chain with $N=200(=5\times 40)$ for $J_X=J_1=J_2=1$ at zero temperature. The symbol $A\left(B\right)$ represents the 3/5 (1/5) plateau. (b) The nearest-neighbor spin correlation $\langle {\mathbf{S}}_i·{\mathbf{S}}_j\rangle \text{−}\langle S_i^z\rangle \langle S_j^z\rangle$ and the local magnetization $\langle S_i^z\rangle$ in the 3/5 plateau for $5\times 3$ sites at the center of the chain. Black solid (purple dashed) lines connecting two nearest-neighbor sites denote negative (positive) values of the spin correlation, and their thicknesses represent the magnitudes of correlation. Blue (red) circles on each site denote a positive (negative) value of $\langle S_i^z\rangle$, and their diameters represents its magnitude. (c) Same as (b) but for the 1/5 plateau.

Figure 3

Magnetic structures of a five-site $J_X\text{−}{J}_1$ unit. Black solid (purple dashed) lines connecting two nearest-neighbor sites denote negative (positive) values of spin correlation defined as $\langle {\mathbf{S}}_i·{\mathbf{S}}_j\rangle$ -$\langle S_i^z\rangle \langle S_j^z\rangle$, and their thicknesses represents the magnitudes of correlation. Blue (red) circles on each site denote a positive (negative) value of local magnetization $\langle S_i^z\rangle$, and their diameters represent its magnitude. We use symbols “$X$,” “$\mathrm{\Xi }$,” “$\alpha$,” “$\beta$,” and “$Z$” to represent each magnetic structure. The total spin $S$ and its $z$ component $S_z$ in each state are equal, i.e., $S=S_z$, with 3/2, 1/2, 1/2, 1/2, and 1/2 for “$X$,” “$\mathrm{\Xi }$,” “$\alpha$,” “$\beta$,” and “$Z$,” respectively. The state “$Z$” is given by a linear combination of “$\mathrm{\Xi }$,” “$\alpha$,” and “$\beta$” (see the Appendix).

Figure 4

Magnetization curve of the kagome-strip chain with OBC for $J_X=1$ at zero temperature. (a) $J_1=1.0$ and $J_2=1.5$ for $N=200(=5\times 40)$. (b) $J_1=1.3$ and $J_2=0.6$ for $N=195(=5\times 39)$. (c) $J_1=0.9$ and $J_2=0.1$ for $N=200$. (d) $J_1=1.0$ and $J_2=0.3$ for $N=200$. (e) $J_1=1.1$ and $J_2=0.4$ for $N=200$. (f) $J_1=1.1$ and $J_2=0.7$ for $N=200$. (g) $J_1=1.3$ and $J_2=0.9$ for $N=235(=5\times 47)$. Magnetic structures of the plateaus $C,D,E,F,G,H,I,J,K$, and $L$ shown in each panel are displayed in Fig. 5. The magnetic structures of the plateaus, $A^{\prime }$ and $D^{\prime }$, are almost the same as that of the plateau $A$ in Fig. 2 and $D$, respectively. (h) The position of the plateaus in the $J_2$ vs $J_1$ plane.

Figure 5

The nearest-neighbor spin correlation $\langle {\mathbf{S}}_i·{\mathbf{S}}_j\rangle$ -$\langle S_i^z\rangle \langle S_j^z\rangle$ and the local magnetization $\langle S_i^z\rangle$ in each plateau for $5\times 7$ sites at the center of the chain. (a) The plateau $C$, (b) $D$, (c) $G$, (d) $H$, (e) $I$, (f) $J$, (g) $K$, (h) $L$, (i) $E$, and (j) $F$ in Fig. 4. Black solid (purple dashed) lines connecting two nearest-neighbor sites denote negative (positive) values of the spin correlation, and their thicknesses represent the magnitudes of correlation. Blue (red) circles on each site denote a positive (negative) value of $\langle S_i^z\rangle$, and their diameters represents its magnitude.

Figure 6

The difference of the upper and lower magnetic fields, $h_{\mathrm{u}}-h_{\mathrm{l}}$, for $M/M_{\mathrm{sat}}=4/5$ as a function of $J_1$ at $J_2=1.5$ for $N=200$. The 4/5 plateau exists at $0.86\lesssim J_1\lesssim 1.03$. Inset: the size scaling of $h_{\mathrm{u}}$ and $h_{\mathrm{l}}$ for several value of $J_1$ as a function of $1/N$. Both $h_{\mathrm{u}}$ and $h_{\mathrm{l}}$, denoted by blue and red circles, respectively, are extrapolated by a linear line.

Figure 7

Finite-size scaling of the lower and upper magnetic fields of magnetization plateaus. (a) The 7/15 plateau $\left(L\right)$ in Fig. 4. (b) The 1/3 plateau $\left(E\right)$ and (c) the 3/10 plateau $\left(F\right)$ in Fig. 4. The solid lines show fitted results with scaling functions of $h_{\mathrm{u}\left(\mathrm{l}\right)}=a_{\mathrm{u}\left(\mathrm{l}\right)}{(1/N)}^2+b_{\mathrm{u}\left(\mathrm{l}\right)}(1/N)+c_{\mathrm{u}\left(\mathrm{l}\right)}$ for the upper (lower) field.