Anticommuting <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi mathvariant="double-struck">Z</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> quantum spin liquids

We discuss a class of lattice <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:mi>S</a:mi> <a:mo>=</a:mo> <a:mfrac> <a:mn>1</a:mn> <a:mn>2</a:mn> </a:mfrac> </a:mrow> </a:math> quantum Hamiltonians with bond-dependent Ising couplings and a mutually “anticommuting” algebra of extensively many local <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:msub> <b:mi mathvariant="double-struck">Z</b:mi> <b:mn>2</b:mn> </b:msub> </b:math> conserved charges that was explicated by Pujari []. This mutual algebra is reminiscent of the spin- <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mfrac> <d:mn>1</d:mn> <d:mn>2</d:mn> </d:mfrac> </d:math> Pauli matrix algebra but encoded in the structure of . These models have finite residual entropy density in the ground state with a simple but nontrivial degeneracy counting and concomitant quantum spin liquidity as proved by Pujari []. The spin liquidity relies on a geometrically site-interlinked character of the local conserved <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:msub> <e:mi mathvariant="double-struck">Z</e:mi> <e:mn>2</e:mn> </e:msub> </e:math> charges that is rather natural in the presence of an anticommuting structure. One may contrast this with, for example, the bond-interlinked character of the local conserved <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:msub> <g:mi mathvariant="double-struck">Z</g:mi> <g:mn>2</g:mn> </g:msub> </g:math> charges on the hexagonal plaquettes of the Kitaev honeycomb spin- <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"> <i:mfrac> <i:mn>1</i:mn> <i:mn>2</i:mn> </i:mfrac> </i:math> model, which leads to a mutually commuting local algebra. In this work, we make several exact statements on the kinds of many-body orders that can be present within the class of anticommuting quantum spin liquids coexisting with extensive residual ground state entropy. We elucidate the differences between the many-body order in these models and that found in some gapped quantum spin liquids whose canonical example is the Kitaev toric code. The toric code belongs to the more general class of Levin-Wen or string net constructions that possess mutually commuting algebras for the local conserved charges. We also point out a mutually commuting algebra with local support that is naturally expressed as multilinear Majorana forms in the Kitaev representation of these quantum spin liquids. They capture quantum resonances spread across the lattice in anticommuting <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"> <j:msub> <j:mi mathvariant="double-struck">Z</j:mi> <j:mn>2</j:mn> </j:msub> </j:math> quantum spin liquids.