Projective symmetry groups beyond spin-1/2: Classifying integer-spin quantum spin liquids
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 SU(2), U(1), and ℤ2 invariant gauge groups, while integer spins give a different hierarchy that includes U(1)rtimesℤ2 and ℤ2×ℤ2; 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 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 J1-J2 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 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.
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BibTeX
@misc{maity2026projective,
title = {{Projective symmetry groups beyond spin-$1/2$: Classifying integer-spin quantum spin liquids}},
author = {Atanu Maity and Chunxiao Liu and Rhine Samajdar and Yasir Iqbal},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30048},
eprint = {2610.30048},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30048},
}
RIS
TY - UNPB
TI - Projective symmetry groups beyond spin-1/2: Classifying integer-spin quantum spin liquids
AU - Atanu Maity
AU - Chunxiao Liu
AU - Rhine Samajdar
AU - Yasir Iqbal
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.30048
UR - https://arxiv.org/abs/2610.30048
AB - 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 SU(2), U(1), and ℤ_2 invariant gauge groups, while integer spins give a different hierarchy that includes U(1)rtimesℤ_2 and ℤ_2×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 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 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.
ER -