Classification of spin liquids on the square lattice with strong spin-orbit coupling

TL;DR

This paper classifies $Z_2$ spin liquids on the square lattice with strong spin-orbit coupling, revealing more phases than in SU(2)-symmetric cases and identifying topologically protected Majorana edge states.

Contribution

It extends the projective symmetry group classification to systems with broken SU(2) symmetry, uncovering new spin liquid phases with topological properties.

Findings

More spin liquid phases emerge without SU(2) symmetry.

Many phases exhibit $p+ip$ pairing of spinons.

Presence of topologically protected Majorana edge states.

Abstract

Spin liquids represent exotic types of quantum matter that evade conventional symmetry-breaking order even at zero temperature. Exhaustive classifications of spin liquids have been carried out in several systems, particularly in the presence of full SU(2) spin-rotation symmetry. Real magnetic compounds, however, generically break SU(2) spin symmetry as a result of spin-orbit coupling - which in many materials provides an `order one' effect. We generalize previous works by using the projective symmetry group method to classify $Z_2$ spin liquids on the square lattice when SU(2) spin symmetry is maximally lifted. We find that, counterintuitively, the lifting of spin symmetry actually results in vastly more spin liquid phases compared to SU(2)-invariant systems. A generic feature of the SU(2)-broken case is that the spinons naturally undergo $p+ip$ pairing; consequently, many of these $Z_2$ spin liquids feature a topologically nontrivial spinon band structure supporting gapless Majorana edge states. We study in detail several spin-liquid phases with varying numbers of gapless edge states and discuss their topological protection. The edge states are often protected by a combination of time reversal and lattice symmetries and hence resemble recently proposed topological crystalline superconductors.