Hidden Zeeman Field in Odd-Parity Magnets: An Ideal Platform for Topological Superconductivity

Abstract

Odd-parity magnets (OPMs) have emerged as a fundamental class of unconventional magnetisms, characterized by time-reversal-preserving non-relativistic spin splitting (NSS). Despite growing interest, the fundamental understanding of OPMs remains critically incomplete, as previous studies have focused exclusively on NSS while overlooking the intrinsically broken time-reversal symmetry ($\mathcal{T}$) inherent to magnetic order. In this work, we reveal that OPMs universally host a hidden Zeeman field rooted in this $\mathcal{T}$-breaking, which fundamentally reshapes their band structure. Through an analytical $f$-wave magnet model, we show that NSS microscopically originates from an emergent gauge field, manifesting as a real-space spin loop current order. Crucially, the large NSS (eV scale) enables conventional superconductivity to coexist robustly with the hidden Zeeman field, with Zeeman splitting reaches hundreds of meV. This unique band structure establishes OPMs as an ideal platform for topological superconductors (TSCs), supporting large topological regions. Based on OPMs, we engineer a series of TSCs hosting distinct Majorana boundary modes, including unidirectional Majorana edge states. Our work corrects a fundamental misconception about OPMs and establishes them as a versatile platform for field-free and robust TSCs.

Figures (8)

• Figure 1: Band structures of OPMs. (a) and (b) Schematic illustration of models $H_1$ and $H_2$, respectively. The blue dashed lines denote the unit cells. (c) Schematic illustration of mapping $H_1$ to a spin loop-current order by $U(\bm r)$. (d) The plot of Brillouin zones for $H_1$ (blue) and original triangle lattice (gray). (e) The bulk bands of model $H_1$ and inset plots its spin textures. (f) The bulk energy bands of model $H_2$. In (c), $\phi=-\pi/3$. In (c), anti-PBC denotes the abbreviation of anti-periodical boundary conditions. In (e) and (f), $t=1$, $J=0.5$, and $\mu=-2.5$ for the inset plot in (e).
• Figure 2: Self-consistently calculated transition temperature $T_c$ versus $J/t$ for model $H_{\text{BdG}}^{(1,2)}$. The common model parameters are $t=1$, $J=0.5$, $U=1.5$, and $\mu=-1$.
• Figure 3: (a) Energy spectrum of model $H_{\text{BdG}}^{(2)}$ in a nanowire geometry along the $y$ direction. (b) Color map of the minimum absolute energy $|E|_{\text{min}}$ of $h_0(\bm k)$ with $s$-wave paring. (c) Schematic illustration of model $H_{\text{BdG}}^{(2)}$, mapping it to stacked superconducting 1D helimagnets with compensated Zeeman fields. Energy bands of $H_{\text{BdG}}^{(2)}$ in a nanowire geometry along the $y$ direction without and with relatvistic SOC for (d) and (e), respectively. The model parameters are taken as $t=1$, $J=0.5$, $\Delta=0.2$. $\mu=1$ for (a) and (b), $\mu=3.5$ for (d) and (e).
• Figure S1: (a) Schematic illustration of model $H_{\text{BdG}}^{(2)}$, mapping it to stacked superconducting 1D helimagnets with additional compensated Zeeman fields. (b) and (c) Energy bands of $H_{\text{BdG}}^{(3)}$ and $H_{\text{BdG}}^{(2)}$ in a nanowire geometry along the $y$ direction. (d) Schematic illustration of considering relativistic spin-orbital coupling term $i\lambda\sigma_y$ in $H_2$. The model parameters are taken as $t=1$, $J=0.5$, $\Delta=0.2$, $\lambda=0.1$, and $\mu=3.5$.
• Figure S2: (a), (b), and (c) Schematic illustration of the theoretical models $\mathcal{H}_1$, $\mathcal{H}_2$, and $\mathcal{H}_3$, realizing type-I, type-II, and type-III OPMs, respectively. The numbers $(xyz)$ beneath the magnetic atoms denote the directional vectors of the local magnetic moments. Solid lines indicate that different magnetic sublattices are related by specific symmetries. (d), (e), and (f) Bulk energy bands for models $\mathcal{H}_1$, $\mathcal{H}_2$, and $\mathcal{H}_3$, respectively. (g), (h), and (i) Eigenvalues near zero energy for models $\mathcal{H}_{\text{BdG}}^{(i)}$ under the open boundary conditions. The adopted chemical potential is marked by black dashed lines in (d)–(f)), lying within the magnetic gap. The model parameters are taken as $t=1$, $J=0.5$, and $\Delta=0.1$.