Ising Phases of Heisenberg Ladders in a Magnetic Field

We show that Dzyaloshinskii-Moriya (DM) interactions can substantially modify the phase diagram of spin-$1/2$ Heisenberg ladders in a magnetic field provided they compete with exchange. For nonfrustrated ladders, they induce a local magnetization along the DM vector that turns the gapless intermediate phase into an Ising phase with broken translational symmetry, while for frustrated ladders, they extend the Ising order of the half-integer plateau to the surrounding gapless phases of the purely Heisenberg case. Implications for experimental ladder and dimer systems are discussed.

Figure 1

Sketch of the frustrated ladder of Eqs. (1, 2). The thick solid lines stand for $J$, the thin ones for $J_{\parallel }$, the dashed ones for $J_X$, and the arrows for $\mathbf{D}$. We have also shown the 3 generators of the space group: the ${\sigma }_{xz}$ and ${\sigma }_{yz}$ mirror planes and a twofold rotation axis $C_2(z)$.

Figure 2

Variational phase diagram for (a) $D=0$ and (b) $D/(J_X+J_{\parallel })=0.1$. On the left and bottom axes, $J_X$ and $H-H_0$, where $H_0=J+(J_X+J_{\parallel })/2$, are given in the units of $J_X+J_{\parallel }$. In (a), the spins forming singlets and triplets in the gapped phase are represented by open and solid circles. In the gapless $O(2)$ phase the arrows show a typical mean-field ground state. In (b), the $O(2)$ symmetry is broken by finite $D$: a unique state is selected in the nonfrustrated case, a twofold degeneracy is found in the frustrated case. The Ising phase of the plateau in (a) develops continuously by turning on $D$. Along the red dashed line the site-factorized wave function is an exact wave function in the large-$J$ limit. $R$ and $R^{\prime }$ denote the tricritical points.

Figure 3

(a) Uniform magnetizations of neighboring rungs ($m_{0+}^z$ and $m_{1+}^z$) and average ($m$) in the upper Ising phase and (b) staggered magnetizations $m_{0+}^x$ and $m_{1+}^x$. Solid lines: variational approach; symbols and dotted lines: DMRG for 58 sites; shaded area: plateau phase when $D=0$.

Figure 4

Magnetizations in the lower Ising phase. Solid lines: variational approach; symbols and dotted lines: DMRG for 80 sites.

Figure 5

Schematic representation of the phase transitions. Away from the $\Delta =1$ Heisenberg line, the ${\mathcal{H}}_{\mathrm{eff}}$ loses the axial symmetry around the total field: the spins develop staggered order along the long dashed blue arrows if $\Delta <1$ and along the short dashed red arrows if $\Delta >1$.