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arXiv:2407.06236

A theorem on extensive ground state entropy, spin liquidity and some related models

Sumiran Pujari

Published Jul 7, 2024• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2407.06236

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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 nnn-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.
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.
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.
ABSTRACT Quantum spin liquids (QSLs) are magnetically frustrated phases characterized by spin fractionalization, emergent gauge fields, and long‐range entanglement. Quantum spin–orbital liquids (QSOLs) form a subset of QSLs with fluctuating orbital degrees of freedom, normally adding a layer of complexity to an already involved research field. This topical review provides guidelines for understanding a specific type of QSOL in which the orbital operators facilitate the analysis of quantum liquids, thereby providing adequate starting points for /exploring these phases. Such models are extensions of the spin‐1/2 Kitaev honeycomb model (KHM), in the sense that their exact solutions depend on an extensive number of conserved quantities, combined with a mapping to a problem of Majorana fermions hopping on a static gauge field. The starting points to understand such models are classical Hamiltonians characterized by bond‐dependent Ising interactions. Such classical models are exactly solvable in spin basis thanks to their extensive symmetries and can be directly connected to classical spin ice (CSI) systems that satisfy a Gauss law. QSOLs are easily stabilized in these models by applying a transverse field or introducing other exchange mechanisms that preserve the conserved local operators. The theory of such Ising QSOLs bridges the KHM to specific types of CSIs, thus providing further insight into paradigmatic forms of spin liquids. Furthermore, bond‐dependent Ising models are related to minimal Hamiltonians describing Rydberg‐atom simulations, which provide experimental grounds for investigating them.
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 introduce a geometrically frustrated classical Ising model, dubbed the “spin-vorticity model,” whose ground-state manifold is a classical spin liquid: a 2-form Coulomb phase. We study the thermodynamics of this model both analytically and numerically, exposing the presence of algebraically decaying correlations and an extensive ground-state entropy, and give a comprehensive account of its ground-state properties and excitations. Each classical ground state may be decomposed into collections of closed two-dimensional membranes, supporting fractionalized string excitations attached to the edges of open membranes. At finite temperature, the model can then be described as a gas of closed strings in a background of fluctuating membranes. The emergent gauge structure of this spin liquid is naturally placed in the language of 2-form electrodynamics, which describes one-dimensional charged strings coupled to a rank-2 antisymmetric gauge field. After establishing the classical spin-vorticity model, we consider perturbing it with quantum exchange interactions, deriving an effective “membrane exchange” model of the quantum dynamics, analogous to ring exchange in quantum spin ice. We demonstrate the existence of a Rokhsar-Kivelson point where the quantum ground state is an equal-weight superposition of all classical ground-state configurations, i.e., a quantum spin liquid. The quantum aspects of this spin liquid are exposed by mapping the membrane exchange model to a strongly coupled frustrated 2-form U(1) lattice gauge theory. We further demonstrate how to quantize the string excitations by coupling a 1-form string field to the 2-form U(1) gauge field, thus mapping a quantum spin model to a 2-form U(1) gauge-Higgs model. We discuss the stability of the gapless deconfined phase of this gauge theory and the possibility of realizing a class of quantum phases of matter: 2-form U(1) quantum spin liquids.