The odd-parity altermagnetism: A spin group study

Abstract

Following recent intensive studies on altermagnetism(ALM) characterized by non-relativistic even-parity spin splitting, realizing unconventional odd-parity magnetism has also attracted increasing interest. Here, using symmetry arguments based on spin-group analyses, we elucidate the sufficient conditions for the presence of odd-parity spin splitting in collinear antiferromagnetic systems, which is further established as the standard odd-parity ALM. Then, we utilize the well-known Haldane-Hubbard model to identify the odd-parity spin splitting in the collinear ALM ground state where the nonmagnetic TRS is broken by sublattice currents coming from the Haldane hopping. A comprehensive phase diagram involving the Haldane hopping strength and the onsite electron Coulomb repulsion is established to elucidate correlation-driven electronic phases.

Figures (3)

• Figure 1: (Color online) The schematic cartoon of the spin splitting in the (a)even- and (b)odd-parity altermagnets with $d$- and $f$-wave spin splitting in momentum space.
• Figure 2: (Color online) The schematic illustration of the compensated collinear magnets with opposite sublattice currents at two sublattices denoted by blue solid arrows for the (a) square and (b) hexagon lattice, where the red and blue solid circles represent the up- and down-spin polarized sublattice, respectively. Here the black dashed line denotes the mirror reflection $M_{yz}$ perpendicular to $x$ axis.
• Figure 3: (Color online) The electron spectrum function $A_{\sigma}(\bm{k},\omega)$ as a function of momentum at $\lambda=0.3$ and $U=5.5$ in the reduced first Brillouin zone with $\omega=$ 1(a) and 2(b), respectively, where the CCE for up- and down-spin electrons are plotted in red and blue color, respectively; (c)The phase diagram of the half-filled HH model as a function of Haldane hopping $\lambda$ and onsite electron Coulomb repulsion $U$, which is composed of three regimes: CI with the TKNN number $C=2$ at small $U's$, the odd-parity ALM CI with $C=2$ and $|M|>0$ for intermediate interaction strengths, and odd-parity ALMI in the strong interaction regime. (d)The momentum dependence of the energy gap function $\Delta_{\bm{k}\sigma}$[See Eq.\ref{['Gap-Function']}] along the zigzag direction in the first Brillouin zone[See Fig.1 in the supplemental materialZeng25-supp] with $\lambda=0.3$ as well as $U=$ 4.8(black), 5.1(red), 5.5(blue), and 6.1(magenta), where $\Delta_{\bm{k}\sigma}$ for up- and down-spin electron quasiparticles are plotted by solid and dashed lines, respectively.