Collinear $p$-wave magnetism and hidden orbital ferrimagnetism

Abstract

In the absence of spin-orbit coupling, collinear magnets are classified as even-wave magnets, i.e., either ferro-, antiferro-, or altermagnets. It is based on the belief that collinear magnets always feature an inversion-symmetric band structure, which forbids odd-wave magnetism. Here, we show that collinear magnets, which break time reversal symmetry in the non-magnetic sector, can have an inversion symmetry broken band structure and lead to unconventional types of collinear magnets. Hence, collinear odd-wave magnets do exist, and we explain that a magnetic field-induced Edelstein effect is their unique signature. We propose minimal models based on the coexistence of AFM order with compensated loop current orders for all types of collinear magnets. Our work provides a new perspective on collinear magnets and the spin-space group classification.

Figures (5)

• Figure 1: Collinear $p$-wave magnets. (a) Minimal model of an 1D collinear $p$-wave magnet. Inset: An AFM state on a quasi-1D lattice with an orbital magnetic field. Each plaquette carries a flux $\Phi$ which is realized by the complex hoppings, where the current direction is indicated by the arrow. The sign of the magnetization at each site is indicated by the color. The inversion even current pattern reduces the symmetry of the state from $[\mathcal{T}||\mathcal{T}\mathcal{P}]$ to $[\mathcal{T}||\mathcal{P}] = [C_2||\mathcal{P}]$. Panel (a) shows the band structure (color coded by spin) for $\Phi=0.5$ and $m=2t$. (b-d) Stacking the 1D minimal model leads to a minimal model for a 2D collinear AFM. Inset (b): A compensated LCO (arrows indicate current directions, green (orange) plaquettes indicate that the orbital moment is directed out of (into the) plane) coexists with a $(\pi,\pi)$-AFM. (b) The spin-split band structure for $m=z=t$ along the high symmetry path indicated in black in panel (d). (c) The Edelstein susceptibility $\chi^{z[1,1]} = \chi^{zx} + \chi^{zy}$ along the spin-splitting direction is only non-zero in the presence of an in-plane magnetic field. (d) The typical FS of a collinear $p$-wave magnet has a mirror and spin-flip mirror symmetry.
• Figure 2: Unconventional LCO-induced magnetic phases in the square lattice. (a,d,g) shows the band structure along a high symmetry line through the Brillouin zone and the inset depicts the LCO and the magnetic state. (c,f,i) shows a representative Fermi surface of this magnetic state. The reduced Brillouin zone (magnetic Brillouin zone) is indicated in grey (dashed). (a-c) An inversion-broken AFM. (b) shows the non-reciprocal, second order longitudinal Drude conductivity $\sigma^{(2)}_{xxx}$ (derived in the SM supplement) as characteristic observable, for inversion symmetry breaking. (d-f) A hidden orbital AFM. (e) shows the magnetization as a characteristic observable, quantifying the spin splitting for various Fermi levels. (g-i) A LCO-induced AM. (h) shows the spin splitter angle $\alpha$, where $\tan \alpha/2 = (\sigma^{\uparrow}_{xy}-\sigma^{{\downarrow}}_{xy})/(\sigma^{\uparrow}_{xx}+\sigma^{{\downarrow}}_{xx})$ as a characteristic observable which quantifies the spin splitting for various Fermi levels.
• Figure 3: Origin of the magnetic Edelstein effect. The Fermi surface colored by spin (a,c) and the momentum-resolved Edelstein susceptibility (b,d) of a collinear $p$-wave magnet for zero magnetic field (a,b) and finite magnetic field (b,d). A magnetic field lifts the band crossings in the band structure. The sidepanels are histrogram plots of the momentum-resolved Edelstein susceptibility, i.e., $\sum_{k_x=\pm k_y} \chi^{z,[1,1]}(\bm{k})$. The top panel of (b) shows that the positive contributions (green) and negative contributions (purple) of the momentum-resolved Edelstein susceptibility cancel for each momentum exactly. At finite magnetic field (d) the avoided crossing does not lead to a contribution to the momentum-resolved Edelstein susceptibility. Hence, the contribution of the outer bands (green) is uncompensated at this momentum which leads to Gaussian peaks in the top panel of (d).
• Figure 4: Additional examples of LCOs coexisting with Neél AFM states. (a,c) shows the band structure along a high symmetry line where the inset depicts the state and (b,d) a representative Fermi surface of the state. (a,b) is based on the historic LCO introduced as $d$-density wave by Ref. chakravarty_hidden_2001. It is a conventional AFM even though a LCO is present such that $[C_2||\bm{t}]$ is broken, because $[\mathcal{T}||\mathcal{T} \mathcal{P}]$ is preserved. The state shown in (c,d) constitutes a second example for a hidden orbital ferrimagnet.
• Figure 5: Unconventional magnetic phases on the Lieb lattice. (a,b) is an inversion-broken AFM. The LCO is identical with the intra unit cell time reversal violating state $\Theta_{II}$, introduced by Varma in 2006 varma_theory_2006. The staggered version of this LCO, i.e., with an ordering vector $\bm{Q}=(\pi,\pi)$, leads to a $p$-wave magnet, shown in (c,d). A superposition of the orderings of the former two LCOs, i.e., the LCO forms exclusively around the spin-up sublattice, leads the a hidden orbital ferrimagnet (e,f). On the Lieb lattice already a $\bm{Q}=(\pi,\pi)$ LCO can induce an AM, see (g,h).