Uniform and staggered magnetizations induced by Dzyaloshinskii-Moriya interactions in isolated and coupled spin-$1∕2$ dimers in a magnetic field

We investigate the interplay of Dzyaloshinskii-Moriya interactions and an external field in spin-$1∕2$ dimers. For isolated dimers and at low field, we derive simple expressions for the staggered and uniform magnetizations which show that the orientation of the uniform magnetization can deviate significantly from that of the external field. In fact, in the limit where the $\mathbf{D}$ vector of the Dzyaloshinskii-Moriya interaction is parallel to the external field, the uniform magnetization actually becomes perpendicular to the field. For larger fields, we show that the staggered magnetization of an isolated dimer has a maximum close to one-half the polarization, with a large maximal value of $0.35g{\mu }_B$ in the limit of very small Dzyaloshinskii-Moriya interaction. We investigate the effect of interdimer coupling in the context of ladders with density-matrix renormalization-group (DMRG) calculations and show that, as long as the values of the Dzyaloshinskii-Moriya interaction and of the exchange interaction are compatible with respect to the development of a staggered magnetization, the simple picture that emerges for isolated dimers is also valid for weakly coupled dimers with minor modifications. The results are compared with torque measurements on ${\mathrm{Cu}}_2{\left({\mathrm{C}}_5{\mathrm{H}}_{12}{\mathrm{N}}_2\right)}_2{\mathrm{Cl}}_4$.

Figure 1

(Color online) Pictorial representation of the model of Eq. (1) of a dimer with DM interaction in a magnetic field.

Figure 2

(Color online) Field dependence of the uniform and staggered magnetizations per site $m_u^y$, $m_u^z$, and $m_s^x$ of the isolated dimer model for $D∕J=0.04$ and $\theta =\pi ∕4$. The dashed lines are the analytical results derived in the limit $D∕J⪡1$. Note the difference of scale for positive and negative magnetizations.

Figure 3

(Color online) Ladder with staggered DM interaction.

Figure 4

(Color online) Structure of a buckled ladder. In such a ladder, a staggered DM interaction in the $y$ direction is allowed by symmetry.

Figure 5

(Color online) Examples of the variation of the uniform magnetization along the $z$ axis with the field. Inset: Plot of $m_u^z$ as a function of ${(D\sin \theta ∕J)}^{2∕5}$ slightly below $H_{c1}$, which confirms the scaling predicted in Ref. 8.

Figure 6

(Color online) Lower panel: $y$ component of the uniform magnetization as a function of the field for $\theta =\pi ∕6$ and various values of $D∕J$. Significant values appear between $H_{c1}$ and $H_{c2}$, and far outside this interval as soon as $D∕J$ is not too small. Upper panel: Angle ${\varphi }_u$ between the magnetic field and the uniform magnetization ${\mathbf{m}}_u$ as a function of the field for $\theta =\pi ∕6$ and several values of $D∕J$. Note that $\theta -{\varphi }_u$ goes to $\pi ∕2$ in the low-field limit, in agreement with the prediction for an isolated dimer [Eq. (6)].

Figure 7

(Color online) Staggered magnetization as a function of the magnetic field for several values of $D∕J$ and $\theta$. Large values are achieved between $H_{c1}$ and $H_{c2}$, and far outside this interval as soon as $D∕J$ is not too small. The value between $H_{c1}$ and $H_{c2}$ depends relatively weakly on $D∕J$ and $\theta$, and is of the same order as the maximal value in the case of an isolated dimer $\left(0.35g{\mu }_B\right)$. In contrast, the value outside this interval depends very strongly on the magnitude of $D\sin \theta$.

Figure 8

(Color online) Upper panel: Field dependence of the excitation gap ▵ for $J_{\Vert }∕J=0.2$ and several values of $D∕J$: $D∕J=0.01$ and $\theta =\pi ∕2$ (black circle), $D∕J=0.02$ and $\theta =\pi ∕2$ (red square), $D∕J=0.04$ and $\theta =\pi ∕2$ (green diamond), and $D∕J=0.08$ and $\theta =\pi ∕2$ (blue triangle) for $N=80$ (DMRG). Lower left panel: Scaling of the excitation gap as a function of $1∕N$ for $D∕J=0.02$ and $\theta =\pi ∕2$. Lower right panel: Scaling of the gap as a function of $D∕J$ for $N=80$ (see text).

Figure 9

(Color online) Transverse uniform magnetization $m_u^y$ for $D_y∕J=0.05$, $D_z∕J=0.086$, and $J_{\Vert }∕J=0.2$ (blue circles) and torque divided by field [experimental curve obtained on Cu(Hp)Cl] as a function of the field (red line). Inset: torque measurement raw data (dashed-dotted black line) and torque divided by field (red line).