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 $\mathrm{SU}(2)$, $\mathrm{U}(1)$, 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.