«Anticommuting» ${\mathbb{Z}}_2$ quantum spin liquids

Abstract

We discuss a class of lattice $S=\frac{1}{2}$ quantum Hamiltonians with bond-dependent Ising couplings and a mutually «anticommuting» 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 «anticommuting» 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 «anticommuting» 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 «anticommuting» ${\mathbb{Z}}_2$ quantum spin liquid Hamiltonians.

arXiv comment24 pages, 15 figures. Additional elaboration in Sec. IV-C on comparison with Ising gauge theories. An additional point in Sec. II connecting the local multilinear Majorana form symmetries to a physical global Z2 symmetry. Results from previous version unaffected. A few references added following recent communication. Comments welcome

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