Current jumps in flat-band ladders with Dzyaloshinskii-Moriya interactions

Localized magnons states, due to flat bands in the spectrum, is an intensely studied phenomenon and can be found in many frustrated magnets of different spatial dimensionality. The presence of Dzyaloshinskii-Moriya (DM) interactions may change radically the behavior in such systems. In this context, we study a paradigmatic example of a one-dimensional frustrated antiferromagnet, the sawtooth chain in the presence of DM interactions. Using both path integrals methods and numerical density matrix renormalization group, we revisit the physics of localized magnons and determine the consequences of the DM interaction on the ground state. We have studied the spin current behavior, finding three different regimes. First, a Luttinger-liquid regime where the spin current shows a step behavior as a function of parameter $D$, at a low magnetic field. Increasing the magnetic field, the system is in the Meissner phase at the $m=1/2$ plateau, where the spin current is proportional to the DM parameter. Finally, further increasing the magnetic field and for finite $D$ there is a small stiffness regime where the spin current shows, at fixed magnetization, a jump to large values at $D=0$, a phenomenon also due to the flat band.

Figure 1

Sketch of the Sawtooth chain. (a) A four sites unit cell is used along the paper. (b) Representation of the Magnon crystal ground state at the critical coupling ${\alpha }_c$.

Figure 2

Local normalized magnetizations in the unit cell as functions of total magnetization for $D=0, \alpha ={\alpha }_c$, and $N=60$ sites. The spin imbalance between the spins ${\mathit{S}}_1$ and ${\mathit{S}}_3$ occurs at the magnetization plateau, where $m=1/2$. There, the spin ${\mathit{S}}_1$ is completely polarized, i.e., $\langle S_1^z\rangle =1$ (see Fig. 1). Lower magnetization sectors are not shown for clarity. (Inset) We change the frustration coupling, setting $\alpha =\tilde{\alpha }={\alpha }_c+\frac{{\alpha }_c}{10}$ and the spin imbalance is still present. In both main figure and inset, $\langle S_2^z\rangle =\langle S_4^z\rangle$ and the respective curves are overlapped.

Figure 3

Magnetization as a function of the applied magnetic field for different values of the Dzyaloshinskii-Moriya interaction $D$, at the critical coupling ${\alpha }_c$, for a $N=60$-site system. At $D=0$ a macroscopic jump in the magnetization profile is observed as a manifestation of the magnon condensation. Lower magnetization sectors are not shown for clarity.

Figure 4

Spin current as a function of the Dzyaloshinskii-Moriya interaction $D$, corresponding to the critical coupling ${\alpha }_c$, and $N=60$ sites. The sawtooth profile for the current is a consequence of the Luttinger liquid behavior at low magnetic field. Black symbols correspond to $m=1/2$ plateau where a Meissner phase is clearly observed. (Inset) Spin current as a function of $D$ corresponding to $m=1/2$. Notice that at high values of $D$ the system departs from the linear behavior associated to the Meissner phase.

Figure 5

Spin current as a function of DM interactions, at fixed magnetization, for a 20-spin system. In dashed lines (solid lines), the results using open (periodic) boundary conditions. The discontinuities in the spin current for $D$ in range $(0,1/10)$, described by the Luttinger-liquid theory, are a consequence of the periodic boundary conditions taken in the numerical calculations.

Figure 6

Spin current as a function of the Dzyaloshinskii-Moriya interaction corresponding to the critical coupling $\alpha ={\alpha }_c, N=60$ sites, for different fixed magnetization sectors. At $D=0$, the current is discontinuous for $1/2<m<1$, due to the flat band. Inset: Spin current as a function of $D$, setting $\alpha =\tilde{\alpha }={\alpha }_c+\frac{{\alpha }_c}{10}$. The system no longer has a flat band and the spin current converges continuously to zero as $D$ tends to zero.

Figure 7

Ground state spin current as a function of the applied magnetic field for diferent $D$ values, for the critical coupling $\alpha ={\alpha }_c$ and $N=60$. As $h$ varies, the system accesses different magnetization sectors as we show in Fig. 3. Here, for the ground state spin current, there are three regimes. First, the low-field regime ($m<1/2$), where the system is in the Luttinger-liquid phase and the ground state spin current is a discontinuous function of $D$. Increasing the applied magnetic field, the system enters in the Meissner phase, it is gapped and has $m=1/2$. Finally, at high field, the system is in the low-stiffness phase, with higher continuous spin currents and $1/2<m<1$.

Figure 8

Spin ladders that for $\alpha ={\alpha }_c$ and $J^{\prime }=J(2\mathrm{\Delta }+1)/(\mathrm{\Delta }+1)$ exhibit a flat band, leading to a magnetization plateau of localized magnons and a magnetization jump to saturation. A sawtooth chain is shown here for comparison with the kagome-like ladders (b) and (c). With dotted gray line we denote the unit cell that breaks the lattice translation symmetry. In red thick line, we denote the sites in which there is a finite probability to find a magnon when each ladder has a flat band.