Projective symmetry groups beyond spin-$1/2$: Classifying integer-spin quantum spin liquids
Year: 2026
Type: preprint
Abstract: Wen's projective symmetry group (PSG), introduced a quarter of a century ago, reduced the classification of symmetric quantum spin liquids to an algebraic problem and has guided the study of spin-$1/2$ models and materials ever since. Many current candidate materials and quantum simulators, however, have spin-$1$ or larger moments, and no comparable classification exists for them. We formulate the PSG for fermionic partons of arbitrary spin $S$. Integer and half-odd-integer spins already differ in their gauge structure. Half-odd-integer spins give the familiar $\mathrm{SU}(2)$, $\mathrm{U}(1)$, and $\mathbb{Z}_2$ invariant gauge groups, while integer spins give a different hierarchy that includes $\mathrm{U}(1)\rtimes\mathbb{Z}_2$ and $\mathbb{Z}_2\times\mathbb{Z}_2$; the latter has no spin-$1/2$ counterpart. Three lattice-independent results follow for integer spin. Singlet pairing of the partons is odd under site exchange. A lattice symmetry can exchange the particle and hole representations of the local constraint, which forces symmetry-related sites into opposite constraint sectors. And a time-reversal-symmetric $\mathrm{U}(1)$ state with the same constraint on every site can be gapped only if the number of sites per unit cell is a multiple of $2S+1$. We apply the formalism to the spin-$1$ diamond lattice, realized in frustrated A-site spinels, and find ten fully symmetric Ansätze up to second neighbors. In the $J_1$-$J_2$ Heisenberg model at the mean-field level, the ground state depends on how the constraint is imposed. With the particle and hole representations, the lowest states are fully gapped $\mathrm{U}(1)$ states whose two sublattices sit in opposite constraint sectors, and one of them is gauge equivalent to a paired Ansatz. The classification provides a starting point for integer-spin quantum spin liquids on other lattices and in other materials.
Source: arXiv (Cornell University)
Language: English
Subfield: Condensed Matter Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
Awards: G1352909968, G1750626284, G2152757567, G2158694287, G2595738051, SFB 1170: Topologische und korrelierte Elektronik in Ober und Grenzflächen ("ToCoTronics"), G3398237449, G4437173897, G4689434848, G476975023, G5737963569, EXC 2147: Komplexität und Topologie in Quantenmaterialien (CT.QMAT), Kavli Institute for Theoretical Physics, Programs on Critical Problems in Physics, Astrophysics and Biophysics at the Aspen Center for Physics
OpenAlex ID: W7223242301