We discuss a class of lattice S = 12 quantum Hamiltonians with bond-dependent Ising couplings and a mutually "anticommuting" algebra of extensively many local Z2 conserved charges that was explicated by Pujari [arXiv:2407.06236]. This mutual algebra is reminiscent of the spin-1 2 Pauli matrix algebra but encoded in the structure of local conserved charges. 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 [arXiv:2407.06236]. The spin liquidity relies on a geometrically site-interlinked character of the local conserved Z2 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 Z2 charges on the hexagonal plaquettes of the Kitaev honeycomb spin-21 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 Z2 quantum spin liquids.
We discuss a class of lattice S = 12 quantum Hamiltonians with bond-dependent Ising couplings and a mutually "anticommuting" algebra of extensively many local Z2 conserved charges that was explicated by Pujari [arXiv:2407.06236]. This mutual algebra is reminiscent of the spin-1 2 Pauli matrix algebra but encoded in the structure of local conserved charges. 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 [arXiv:2407.06236]. The spin liquidity relies on a geometrically site-interlinked character of the local conserved Z2 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 Z2 charges on the hexagonal plaquettes of the Kitaev honeycomb spin-21 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 Z2 quantum spin liquids.
We discuss a class of lattice S = 1 2 quantum Hamiltonians with bond-dependent Ising couplings and a mutually “anticommuting” algebra of extensively many local Z 2 conserved charges that was explicated by Pujari []. This mutual algebra is reminiscent of the spin- 1 2 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 Z 2 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 Z 2 charges on the hexagonal plaquettes of the Kitaev honeycomb spin- 1 2 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 Z 2 quantum spin liquids.
We introduce a framework of constructing topological subsystem codes based on the class of anticommuting quantum spin liquids described in [Phys. Rev. B 113, 064402 (2026)]. A canonical model from this class can be considered as a spatial modification of the toric code that voids its stabilizer code property. Rather, these models contain an extensive set of anticommuting local conserved operators that lead to an extensive ground state degeneracy. This degeneracy forms the basis of the subsystem degrees of freedom in the associated quantum error correcting code. The code inherits the many-body topological order of the quantum spin liquid, making it a topological subsystem code. We present two concrete and detailed examples for constructing these codes on a square lattice and a kagome lattice geometry, requiring weight-4 and weight-3 local check operator measurements respectively. In contrast to other subsystem codes, a unique property of these codes is the presence of an extensive number of local gauge qubits that are left undisturbed by the check operators apart from the logical qubits. Our construction provides a template for generating this new category of topological subsystem codes on different lattice or graph geometries, suitable for implementation on various quantum hardware platforms.
An exact mechanism is written down to guarantee extensive residual ground state entropy and spin liquidity in spin-1/2 lattice models with bond-dependent couplings. It is based on the presence of extensively large and mutually non-commuting (``«anticommuting»'') sets of local conserved quantities with a gauge-like character. This mutual algebra is similar to those of spin-1/2 degrees of freedom however arising in the structure of local conserved charges whose support is not restricted to a single lattice site. The general theorem is first pedagogically illustrated through a variant of the familiar one-dimensional quantum Ising model featuring such an «anticommuting»structure. This leads to classical spin liquidity co-existing with quantum Ising order. The rest of the paper is then devoted to applications in higher dimensions with more general «anticommuting»structures which voids spin or magnetic ordering altogether. Proofs of the resultant quantum spin liquidity are given through an analysis of static and dynamic n-point spin correlators relying solely on the «anticommuting»algebraic structure of the constructed models. It is not evident if they admit exact solutions using known techniques. The precise nature of these quantum spin liquids is thus an open question including the existence of a quasiparticle description for these models. We compare and contrast them with other well-known quantum spin liquids.
The higher spin Kitaev model prominently features the extensive locally conserved quantities the same as the spin-1/2 Kitaev honeycomb model, although it is not exactly solvable. It remains an open question regarding the physical meaning of these conserved quantities in the higher spin model. In this Letter, by introducing a Majorana parton construction for a general spin-S we uncover that these conserved quantities are exactly the Z_{2} gauge fluxes in the general spin-S model, including the case of spin-1/2. Particularly, we find an even-odd effect that the Z_{2} gauge charges are fermions in the half integer spin model, but are bosons in the integer spin model. We further prove that the fermionic Z_{2} gauge charges are always deconfined; hence, the half integer spin Kitaev model would have nontrivial spin liquid ground states regardless of interaction strengths in the Hamiltonian. The bosonic Z_{2} gauge charges of the integer spin model, on the other hand, could condense, leading to a trivial product state, and this is indeed the case at the anisotropic limit of the model.
We study a class of Hamiltonians with a structure that we call "dynamical splitting": the Hamiltonian terms can be divided into two sets acting on the same degrees of freedom such that every term in one set commutes with every term in the other, although terms within either set do not all commute. This structure yields an algebraic duality to effective degrees of freedom on which the two parts of the Hamiltonian act disjointly, enabling exact diagonalization on lattices with approximately twice as many spins as usually accessible. We exploit it in the "chiral XYZ model", a geometrically frustrated spin-1/2 model on the triangular lattice which was previously introduced as a special limit of a Majorana-Hubbard model. This model also possesses anticommuting noncontractible line symmetries, which enforce an exact, topology-dependent degeneracy between locally indistinguishable states. We first study a ZN clock generalization and find, at large N, a gapless ground state with three subsystem-symmetry-protected nodal lines. Exploiting dynamical splitting and the subsystem symmetries, we carry out exact diagonalization of the N=2 model on lattices up to 9×9 spins. The many-body gap and bipartite entanglement provide strong evidence for a gapless state consistent with the large-N nodal structure: the entanglement scales as LlogL and exhibits 1+1-dimensional CFT-like chord scaling on cylinders. Finally, we study instabilities and proximate phases. In particular, we find evidence that a subsystem-symmetry-preserving deformation drives a finite coupling transition to a gapped phase with Z2×Z2 topological order.
In the field of frustrated magnetism, Kitaev models provide prototypical examples of exactly solvable quantum spin liquids, in which the spin degrees of freedom fractionalize into Majorana fermions coupled to an emergent Z2 gauge field. While the ground states of these models are already well understood as exhibiting different kinds of Majorana (semi)metals, it has still been an open task to achieve a detailed understanding of the thermodynamics, and, namely, the low-temperature ordering of the gauge field into different Z2 flux configurations. In this thesis, we investigate the thermodynamics of Kitaev systems in two and three spatial dimensions with sign-problem-free quantum Monte Carlo simulations. In a first study of elementary 3D Kitaev models, we verify that the ground state Z2 flux sectors of these systems are entirely determined by their elementary plaquette length – a result which shows the validity of Lieb’s theorem for lattice geometries which lack the geometric requirements for its proof. We closely investigate the low-temperature phase transition associated with gauge-ordering, which is a particular realization of an inverted Ising phase transition, as it occurs in general lattice Z2 gauge theories. Our results corroborate the understanding of this transition as separating different regimes in terms of vison-loop excitations, by showing a clear correlation between the critical temperature and the vison gap. We also introduce the concept of “gauge frustration”, which, for a particular 3D Kitaev model, leads to the suppression of the phase transition and a more complex physical behavior at low temperatures. A second quantum Monte Carlo study focuses on a generalized 2D Kitaev model on a five-coordinated lattice system. Here, the gauge ordering is accompanied by the spontaneous breaking of time-reversal symmetry, a scenario which not only allows for a variety of topological ground states, but also for the occurrence of a phase transition in two spatial dimensions. We show that this model exhibits such a transition at a particularly high temperature scale. Moreover, the system possesses a number of phases where the Z2 fluxes are only partially ordered.