Spin parity effects in a monoaxial chiral ferromagnetic chain

While spin parity effects, physics crucially depending on whether the spin quantum number $S$ is half-odd integral or integral, have for decades been a source of new developments for the quantum physics of antiferromagnetic spin chains, the investigation into their possible ferromagnetic counterparts has remained largely unchartered, especially in the fully quantum (as opposed to the semiclassical) regime. Here we present such studies for monoaxial chiral ferromagnetic spin chains. We start by examining magnetization curves for finite-sized systems, where a magnetic field is applied perpendicular to the helical axis. For half-odd integer $S$, the curves feature discontinuous jumps identified as a series of level crossings, each accompanied by a shift of the crystal momentum $k$ by an amount of $\pi$. The corresponding curves for integer valued $S$ are continuous and exhibit crossover processes. For the latter case, $k=0$ throughout. These characteristics are observed numerically when the strength of the Dzyaloshinskii-Moriya interaction (DMI) $D$ is comparable to or larger than that of the ferromagnetic exchange interaction $J$. Solitons are known to be responsible for stepwise changes seen in magnetization curves in the classical limit. These findings therefore prompt us to revise the notion of a soliton, for arbitrary $S$, into a quantum mechanical entity. To unravel this phenomenon at the fully quantum level as is appropriate to spin chains with small $S$, we examine in detail special limiting Hamiltonians amenable to rigorous analysis, consisting of only the DMI and the Zeeman energy. Dubbed the $DH$ model (for $S=\frac{1}{2}$) and the projected $DH$ ($\mathrm{p}DH$) model (for general $S$), they have a set of $2S$ conserved quantities, each of which is the number of solitons of a specific integer-valued height (as measured in the $S_z$ basis), which ranges from 1 to $2S$. We discuss how to determine the exact crystal momentum of the lowest-energy state belonging to a sector with a given set of the $2S$ soliton numbers. Combined with energetic considerations, this information enables us to reproduce the spin parity effect in the magnetization curves. Finally, we show that the ground states of the special models have substantial numerical overlap with those for generic systems with a finite exchange interaction, suggesting the same physics to be valid there as well.

Figure 1

Magnetization curves of finite-sized spin chains for $S=\frac{1}{2},1,\frac{3}{2},$ and 2, where we set $D/J=1$.

Figure 2

Magnetization curves of finite-sized spin chains for $S=\frac{1}{2},1,\frac{3}{2},$ and 2. Here we have set $D/J=50$.

Figure 3

(Left) The magnetization curve of a finite-sized spin chain for the $S=\frac{1}{2}DH$ model with $L=16$. In addition to the crystal momentum $k$, the eigenvalue $N$ of the ground state is shown. The dotted line shows the position on the magnetic field axis at which the $N=1$ state (indicated by the red arrow) is the ground state. (Right) The energy spectrum, i.e., the set of eigenenergy and the crystal momentum of the system with $L=40$. The magnetic field is fixed at the value $H=0.831\left|D\right|$, where the $N=1$ state is the ground state. The ground state is indicated by the red arrow. The red and blue points are the eigenstates for which $N=1$ and 2, respectively. The open green circle is an $N=0$ eigenstate.

Figure 4

Schematic illustration of the sign ${(-1)}^{\delta \left(\mathit{n}\right)}$ for $S=\frac{1}{2}$. The symbol $\delta \left(\mathit{n}\right)$ was defined by Eq. (13).

Figure 5

Schematic illustration of a one-site translation of a single soliton. The underlined three sites indicate those allowing the outcome of the action of the local Hamiltonian ${\widehat{h}}_i$ on the state to be nonzero.

Figure 6

Left panel: Magnetization curves for the two $S=1$ models ${\widehat{\mathcal{H}}}_{DH}$ and ${\widehat{\mathcal{H}}}_{\mathrm{p}}$. The system size is set at $L=8$. The crystal momentum in the ground state is always zero for both models. The number of solitons $N_f$ of heights $f=1,2$ present in the ground state $|{\mathrm{g}}_{\mathrm{p}}\rangle$ for ${\widehat{\mathcal{H}}}_{\mathrm{p}}$ is also shown. Right panel: Overlap $|\langle {\mathrm{g}}_{\mathrm{p}}|{\mathrm{g}}_{DH}\rangle |$ between the ground states $|{\mathrm{g}}_{\mathrm{p}}\rangle ,|{\mathrm{g}}_{DH}\rangle$ for the two models for the system sizes $L=8$, 10, 12. The magnetic field lies in the range $H/D=1.32–1.44$, where single-soliton states of maximal height ($=2$) dominate the ground state of ${\widehat{\mathcal{H}}}_{DH}$. The inset of the right panel is a blowup of the vertical axis.

Figure 7

Left panel: Magnetization curves for $S=\frac{3}{2}{\widehat{\mathcal{H}}}_{DH}$ and ${\widehat{\mathcal{H}}}_{\mathrm{p}}$ for finite-sized systems with $L=8$. The crystal momentum in the ground state changes by $\pm \pi$ accompanying the discontinuous changes in the magnetization. The number of solitons $N_f$ of heights $f=1,2$ present in the ground state $|{\mathrm{g}}_{\mathrm{p}}\rangle$ for ${\widehat{\mathcal{H}}}_{\mathrm{p}}$ is also shown. Right panel: Overlap $|\langle {\mathrm{g}}_{\mathrm{p}}|{\mathrm{g}}_{DH}\rangle |$ between the ground states $|{\mathrm{g}}_{\mathrm{p}}\rangle ,|{\mathrm{g}}_{DH}\rangle$ for the two models for $S=1$ with $L=8$, 10, 12 for the magnetic field in the range of $H/D=1.86–2.00$, where the single-soliton states with maximum height ($=3$ for $S=\frac{3}{2}$) dominate the ground state of ${\widehat{\mathcal{H}}}_{DH}$. The inset of the right panel is a blowup of the vertical axis.

Figure 8

Schematic illustration of the sign ${(-1)}^{\delta \left(\mathit{n}\right)}$ for $S=1$ (upper panel) and $S=\frac{3}{2}$ (lower panel). The symbol $\delta \left(\mathit{n}\right)$ was defined by Eq. (13).

Figure 9

Examples of signed basis for negative $D$. The upper two panels represent basis states for $S=\frac{1}{2}$ corresponding to those shown in Fig. 4. The sign ${(-1)}^{{\delta }^{(-)}\left(\mathit{n}\right)}$ uses ${\delta }^{(-)}\left(\mathit{n}\right)$ defined in Eq. (63). The lower two panels represent basis states for $S=1$ and $\frac{3}{2}$, which correspond to the figures shown in Fig. 8. The numbers underneath the figures represent $L-j$ where $j$ is the site index.

Figure 10

Size dependence of the weight of the one-soliton states in the ground state of ${\widehat{\mathcal{H}}}_{\mathrm{ch}}$ for $J=D$ and $S=\frac{1}{2}$ (left panel) and $S=1$ (right panel). The magnetic field is set at $H/D=0.29$ ($H/D=0.5$) for $S=\frac{1}{2}$ ($S=1$). The insets of both panels magnify the vertical axis.

Figure 11

Spherical coordinate and geometry used in the text.