Magnetic phase diagram of a frustrated ferrimagnetic ladder: Relation to the one-dimensional boson Hubbard model

We study the magnetic phase diagram of two coupled mixed-spin $(1,\frac{1}{2})$ Heisenberg chains as a function of the frustration parameter related to diagonal exchange couplings. The analysis is performed by using spin-wave series and exact numerical diagonalization techniques. The obtained phase diagram—containing the Luttinger liquid phase, the plateau phase with a magnetization per rung $M=1∕2$, and the fully polarized phase—is closely related to the generic $(J∕U,\mu ∕U)$ phase diagram of the one-dimensional boson Hubbard model.

Figure 1

Phase diagram of the model (1) in the $(J_2,h)$ plane. The phase boundary $h_s=h_s\left(J_2\right)$ of the fully polarized phase $(M=3∕2)$ is exact. The dashed curve $abcd$ marks the area occupied by the $M=1∕2$ plateau phase, as obtained in a linear spin-wave approximation. The boundary position of this phase, as obtained from exact numerical diagonalization of periodic clusters, is traced by squares $(L=10)$ and crosses $(L=12)$. The dotted curve shows the phase boundary where ${\Delta }_0^{(-)}\left(h\right)=0$ [see Eq. (4)], as obtained from the first-order spin-wave theory. The dashed curve starting from the point $b$ and going upwards left as well as the piece $bc$ represents in a linear spin-wave approximation the boundary where ${\Delta }_{\pi }^{(-)}\left(h\right)=0$. The canted spin phase exists for $h=0$ in the interval $0.342<J_2<J_c$, where $J_c=0.399$ is the phase transition point from the canted phase to the Luttinger liquid phase. The rest of the area is occupied by the Luttinger liquid phase.

Figure 2

Magnetization curves of the model (1) for different values of the frustration parameter $J_2$ ($=0.38$, 0.42, 0.5, and 1). $m=M∕M_s$ is the reduced magnetization, $M$ being the magnetic moment per rung, and $M_s=3∕2$. The symbols mark the midpoints of the magnetization steps, as obtained from the ED data for $L=12$. For clarity, we do not show the data points for $J_2=0.38$ and 0.42 in the interval $h_c<h<h_s$ since they closely follow the presented data for $J_2=0.5$.