Symmetry constrained field theories for chiral spin liquid to spin crystal transitions

TL;DR

This work analyzes transitions from the Kalmeyer-Laughlin chiral spin liquid (CSL) to regular noncoplanar magnetic orders under symmetry $G = SO(3) imes p6$, highlighting a necessary compatibility between the CSL's anomaly and the ordered-state topological invariants. It develops two coherent field-theory frameworks: (i) coupled $O(3)$ nonlinear sigma models with a chiral interaction, and (ii) a multi-flavor Chern-Simons-matter theory, later extended to a matrix-parton description with a $U(2)$ field. A central result is that the CSL anomaly enforces site-dependent spin representations (e.g., half-integer on kagomé sites, integer on triangular sites) in any continuous CSL-to-order transition, constraining which RMOs can be neighboring states. The theory yields explicit examples, notably octahedral order on the kagomé lattice and tetrahedral order on triangular/honeycomb lattices, and proposes numerical and tensor-network tests to validate the predictions. Overall, the paper provides a topological, symmetry-aware framework for understanding and predicting continuous CSL-to-order transitions in frustrated magnets.

Abstract

We consider the spin rotationally invariant Kalmeyer-Laughlin chiral spin liquid (CSL) in systems with broken time-reversal symmetry and explore symmetry constraints on possible conventional spin crystal states accessible via a direct transition. These constraints provide a framework to identify topological invariants of the magnetically ordered state. We show that the existence of a direct transition from a CSL requires a precise compatibility condition between the topological invariants of the ordered state and the anomaly of the CSL. The lattice symmetries also constrain the functional form of the low-energy theory to describe these transitions. This allows us to construct explicit Chern-Simons-matter field theories for the transition into a class of noncoplanar orders identified as candidates directly accessible from the CSL, including the octahedral spin crystal on the kagomé lattice, and the tetrahedral order on the triangular and honeycomb lattice. These transitions can either be described using coupled fractionalized $ \mathbb{CP}^1 $ theories or fractionalized matrix principal chiral models. We also discuss extensions to more general magnetic ordering transitions out of the CSL.

Figures (3)

• Figure 1: The three high-symmetry points $a, b,$ and $c$ on a lattice with $p6$ spatial symmetry consisting of two translations $T_{\mathbf{a}_1}$ and $T_{\mathbf{a}_2}$ (marked in blue). The unit cell consists of one $a$ site, two $b$ sites, and three $c$ sites. The $a$ sites are $C_6$ centers, $b$ sites are $C_3$ centers, and $c$ sites are $C_2$ centers. On a triangular lattice, $a$ sites are the vertices, $b$ sites are the plaquette centers, and $c$ sites are the edge centers. On a honeycomb lattice, $a$ sites are the plaquette centers, $b$ sites are the vertices, and $c$ sites are the edge centers. On a kagomé lattice, $a$ sites are the hexagonal plaquette centers, $b$ sites are triangular plaquette centers, and $c$ sites are the vertices.
• Figure 2: The octahedral order on the kagomé lattice with the spins pointing along the six vertices of an octahedron and spanning a $2\times 2$ unit cell with $12$ sites. Each of the three sublattices exhibits a Néel order at the three $\mathbf{M}$ points in the Brillouin zone. The triangular plaquettes host a uniform scalar spin chirality.
• Figure 3: The tetrahedral order on the triangular (Left) and honeycomb (Right) lattices with the spins pointing along the four vertices of a tetrahedron spanning a $2\times 2$ unit cell with $4$ and $8$ sites respectively. The tetrahedral order can be obtained by overlapping three Néel orders at the three $\mathbf{M}$ points in the Brillouin zone (see Table \ref{['table: Noncoplanar RMOs expectations']}). It exhibits uniform scalar spin chirality just like the octahedral order.