Projectively implemented altermagnetism in an exactly solvable quantum spin liquid

Abstract

Altermagnets are a new class of symmetry-compensated magnets with large spin splittings. Here, we show that the notion of altermagnetism extends beyond the realm of Landau-type order: we study exactly solvable $\mathbb{Z}_2$ quantum spin(-orbital) liquids (QSL), which simultaneously support magnetic long-range order as well as fractionalization and $\mathbb{Z}_2$ topological order. Our symmetry analysis reveals that in this model three distinct types of ``fractionalized altermagnets (AM$^*$)'' may emerge, which can be distinguished by their residual symmetries. Importantly, the fractionalized excitations of these states carry an emergent $\mathbb{Z}_2$ gauge charge, which implies that they transform \emph{projectively} under symmetry operations. Consequently, we show that ``altermagnetic spin splittings'' are now encoded in a momentum-dependent particle-hole asymmetry of the fermionic parton bands. We discuss consequences for experimental observables such as dynamical spin structure factors and (nonlinear) thermal and spin transport.

Figures (2)

• Figure 1: (a) Square-lattice spin-orbital model with bond-dependent interactions. The staggered out-of-plane magnetization $\langle \mathsf{s}^z_i \rangle \propto n (-1)^i$ is indicated by "$+$" and "$-$" symbols. The model is mapped onto an exactly solvable square-lattice $\mathbb{Z}_2$-gauge theory coupled to spinless fermions, with antiferromagnetism corresponding to charge-density wave order. The purple lines, grey arrows and dotted lines indicate the chosen gauge for the $\pi$-flux ground state of the $\mathbb{Z}_2$ gauge field, the next-nearest neighbor hopping in $H_\lambda$ and third-nearest neighbor hopping described by $H_\kappa$. (b) Low-energy parton band structures near the Dirac points ${\bm{\Sigma}}_\pm$ for antiferromagnetic and altermagnetic states: in the latter, the bands are symmetric under a combination of spin-flip (particle-hole transformation of the parton bands) and lattice rotations. (c) Illustration of parton bands (in red (blue): constant-energy cuts of particle (hole) band) in the three distinct fractionalized altermagnetic phases for small momenta ${\bm{q}}$ near the Dirac points ${\bm{\Sigma}}_\pm$. White lines denote nodal lines with residual particle-hole degeneracy.
• Figure 2: Constant-frequency cut ($\omega = 2.5 J$) of the intravalley contribution to the structure factor $\langle \mathsf{s}^z \mathsf{s}^z \rangle(\omega,{\bm{k}})$ for (a) $\kappa= \lambda =0, n=0.3J$ and (b) $\lambda = 0, n = 0.3J, \kappa n=0.6J$ for which we also show the structure factor in the entire Brillouin zone in panel (c). (d) Berry curvature quadrupole at the ${\bm{\Sigma}}_+$-valley. Blue (red) ellipses denote the parton Fermi surfaces for a positive (negative) applied magnetic field $h$ which are distorted by $\lambda$.