{\guillemotleft}Anticommuting{\guillemotright} $\mathbb{Z}_2$ quantum spin liquids

We discuss a class of lattice $S=\frac{1}{2}$ quantum Hamiltonians with bond-dependent Ising couplings and a mutually {\guillemotleft}anticommuting{\guillemotright} algebra of extensively many local $\mathbb{Z}_2$ conserved charges that was explicated in [arXiv:2407.06236]. This mutual algebra is reminiscent of the spin-$\frac{1}{2}$ Pauli matrix algebra but encoded in the structure of \emph{local conserved charges}. These models have finite residual entropy density in the ground state with a simple but non-trivial degeneracy counting and concomitant quantum spin liquidity as proved in [arXiv:2407.06236]. The spin liquidity relies on a geometrically site-interlinked character of the local conserved $\mathbb{Z}_2$ charges that is rather natural in presence of an {\guillemotleft}anticommuting{\guillemotright} structure, as opposed to for example the bond-interlinked character of the local conserved $\mathbb{Z}_2$ hexagonal plaquette charges of the Kitaev honeycomb spin-$\frac{1}{2}$ model which leads to a mutually commuting local algebra. In this work, we make several exact statements on the many-body order that can be present within the class of {\guillemotleft}anticommuting{\guillemotright} quantum spin liquids. We elucidate the differences between the many-body order in these models and that found in some gapped quantum spin liquids with mutually commuting local algebras, e.g. the Kitaev toric code or Levin-Wen models. We also point out a mutually commuting algebra with local support that are naturally expressed as multi-linear Majorana forms in the Kitaev representation of these quantum spin liquids. They capture non-trivial quantum resonances throughout the lattice in these {\guillemotleft}anticommuting{\guillemotright} $\mathbb{Z}_2$ quantum spin liquid Hamiltonians.