Quantum spin models of commensurate $p$-wave magnets

Abstract

The $p$-wave magnet has emerged as a new type of magnetism exhibiting odd-parity, time-reversal-symmetric spin splitting in momentum space, and has attracted considerable interest as a promising platform for spintronic applications. However, the theoretical understanding of the fundamental mechanism responsible for stabilizing this phase remains limited. In this work, we identify a microscopic interacting model that realizes the $p$-wave magnet as its ground state. We first introduce a Hubbard model and derive the corresponding low-energy spin Hamiltonian. At the classical level, we find that the $p$-wave magnet is stabilized but remains energetically degenerate with competing noncoplanar states. Quantum fluctuations lift this degeneracy, selecting the $p$-wave magnet as the unique ground state. The resulting electronic structure exhibits finite spin accumulation via the Edelstein effect, highlighting the potential of $p$-wave magnetism for spintronic applications. We further discuss the relevance of our theory to quasi-two-dimensional honeycomb magnets such as Ni$_2$Mo$_3$O$_8$. Our findings establish the possibility of spontaneous $p$-wave magnetism.

Figures (4)

• Figure 1: (a) Classical phase diagram of the model given in Eq. (\ref{['eq:spin']}) for $\hat{\boldsymbol d}_\gamma = \hat{\gamma}$. For $0.85 \le \theta < \pi/2$, the model stabilizes the $p$-wave magnetic and noncoplanar states, which are energetically degenerate. (b) Spin configuration of the $p$-wave magnet, which is noncollinear. The canting angle $\phi$ between the local spin-polarization axes $S^{\tilde{z}}_A$ and $S^{\tilde{z}}_B$ quantifies the noncollinearity between the two sublattices. The magnetic unit cell, indicated by the dashed green outline, is twice as large as the crystallographic unit cell. (c,d) Eigenvalue spectra of the LT interaction matrix for $\theta = \pi/5$ and $\pi/3$. The minimum of the lowest-energy band is located at an incommensurate wave vector for the former and at the M point for the latter. (e,f) Nearest-neighbor spin correlators $\langle S^{\tilde{z}}_{i} S^{\tilde{z}}_{j} \rangle$ and $\langle S^{\tilde{z}}_{i} S^{\tilde{x}}_{j} \rangle$ for the $p$-wave magnet.
• Figure 2: (a) Quantum phase diagram of the spin-1/2 model in Eq. (\ref{['eq:spin']}) for $\hat{\boldsymbol d}_\gamma = \hat{\gamma}$. The simulations are performed on an infinite cylinder with $L_x \times L_y \times 2 = 6 \times 4 \times 2$ sites per unit cell, where $L_x$ is repeated along the infinite direction and $L_y$ is the periodic circumference. Quantum fluctuations select the $p$-wave magnet over the noncoplanar states that are energetically degenerate at the classical level. (b,c) Nearest-neighbor spin correlators $\langle S^{\tilde{z}}_{i} S^{\tilde{z}}_{j} \rangle$ and $\langle S^{\tilde{z}}_{i} S^{\tilde{x}}_{j} \rangle$ for the $p$-wave magnet at $\theta = \pi/3$. The magnetic unit cell is indicated by the dashed green outline in (b).
• Figure 3: (color online) (a) Band spectrum of the model in Eq. (\ref{['eq:kondo']}) with parameters $t = 1$ eV and $J_d = 0.6$ eV. The bands exhibit $p$-wave type spin polarization in momentum space, consistent with Eq. (\ref{['eq:momentum_spin']}). Spin degeneracy is restored along the high-symmetry $\Gamma$–$M$ line at the Brillouin-zone boundary. (b) At $J_d = 0$, four spin-degenerate Dirac cones appear at the Fermi level within the first Brillouin zone of the crystallographic unit cell. (c) For $0 < J_d < J_d^c$, eight spin-polarized Dirac cones appear at the Fermi level in the first Brillouin zone of the crystallographic unit cell. At the critical coupling $J_d = J_d^c$, Dirac cones annihilate in pairs and a gap opens. Pairs of Dirac cones are connected by guiding lines for visualization. (d) Calculated $\chi^{\text{even}}_{\perp x}$ for $t = 1$ eV and $\Gamma = 0.01$ eV with $J_d = 0.2$, $0.4$, and $0.6$ eV. The magnitude of $\chi^{\text{even}}_{\perp x}$ increases with $J_d$.
• Figure 4: (a) Classical phase diagram of the model given in Eq. (\ref{['eq:spin']}) for the $\hat{\boldsymbol d}_{\gamma}$ specified in Eq. (\ref{['eq:d_2']}). For $0.67 \le \theta < 1.25$, the model stabilizes the $p$-wave magnetic state and a competing noncoplanar state, which are energetically degenerate. The canting angle $\phi$ is defined for the $p$-wave magnet and quantifies the noncollinearity between the two sublattices. (b,c) Eigenvalue spectra of the LT interaction matrix for $\theta = \pi/4$ and $5\pi/12$. The minimum of the lowest-energy band is located at the M point for the former and at the K point for the latter. (d) Quantum phase diagram of the corresponding spin-1/2 model.