Triangular lattice models of the Kalmeyer-Laughlin spin liquid from coupled wires

Chiral spin liquids (CSLs) are exotic phases of interacting spins in two dimensions, characterized by long-range entanglement and fractional excitations. We construct a local Hamiltonian on the triangular lattice that stabilizes the Kalmeyer-Laughlin CSL without requiring fine-tuning. Our approach employs coupled-wire constructions and introduces a lattice duality to construct a solvable chiral sliding Luttinger liquid, which is driven toward the CSL phase by generic perturbations. By combining symmetry analysis and bosonization, we make sharp predictions for the ground states on quasi-one-dimensional cylinders and tori, which exhibit a fourfold periodicity in the circumference. Extensive tensor network simulations demonstrating ground-state degeneracies, fractional quasiparticles, nonvanishing long-range order parameters, and entanglement signatures confirm the emergence of the CSL in the lattice Hamiltonian.

💡 Research Summary

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The paper presents a concrete lattice Hamiltonian on the triangular lattice that stabilizes the Kalmeyer‑Laughlin chiral spin liquid (CSL) without fine‑tuning, by combining the coupled‑wire construction with a novel lattice duality. The authors start from an array of one‑dimensional spin‑½ chains, each described by a Luttinger‑liquid Hamiltonian with bosonic fields (φ, θ). In the decoupled limit each wire enjoys a U(1) symmetry, but generic inter‑wire couplings typically drive the system into conventional symmetry‑broken phases such as XY order, valence‑bond solid (VBS), or Ising antiferromagnet.

To overcome this competition, the authors engineer a “chiral sliding Luttinger liquid” (SLL) fixed point. The key ingredient is a non‑integer spin‑rotation operator Rα = exp(−iπ α Sᶻ) that rotates spins about the z‑axis by a fractional angle. By inserting Rα into a four‑spin exchange term they construct a lattice interaction H◇(α) that preserves Sᶻ on each wire while generating a highly non‑local coupling in the bosonized language. Together with an Ising‑type Sᶻ‑Sᶻ coupling HZZ(Jz∥, Jz⊥), the full Hamiltonian HCSLL = H◇ + HZZ respects a separate U(1) for every wire, thereby realizing the SLL.

A renormalization‑group analysis of the bosonized theory shows that the chiral cosine operator O_chiral = cos(φ_y − φ_{y+1}) (which drives the ν = 1/2 Laughlin‑type CSL) becomes the most relevant perturbation at the SLL fixed point, while competing operators such as O_XY, O_VBS, and O_Ising acquire larger scaling dimensions and are rendered irrelevant. The triangular geometry is crucial: the usual XY cosine term cancels due to a combined parity‑time‑reversal symmetry, which is absent on square lattices.

The authors then explore the quasi‑one‑dimensional limit by wrapping the wire array into cylinders of N wires. They predict a four‑fold periodicity in N: when N is not a multiple of four, a weak symmetry‑breaking order parameter appears, decaying exponentially with N; when N = 4 k, the ground state is unique and symmetric, corresponding to two distinct symmetry‑protected topological (SPT) phases that become degenerate as N → ∞, reproducing the twofold topological degeneracy of the 2D CSL.

Extensive tensor‑network simulations validate these predictions. Density‑matrix renormalization group (DMRG) and variational uniform matrix product state (VUMPS) calculations on finite and infinite cylinders confirm the expected ground‑state degeneracies, the four‑fold periodicity, and the presence of a non‑vanishing string order parameter that vanishes only exponentially with N. Spin‑pumping experiments on the cylinders reveal fractional charge ½, and the entanglement spectrum exhibits the characteristic chiral conformal‑field‑theory counting (central charge c = 1). Infinite projected entangled‑pair states (iPEPS) on the full 2D lattice, optimized variationally, display a topological entanglement entropy γ ≈ ln √2 and a chiral edge mode, both hallmarks of the Kalmeyer‑Laughlin CSL.

In summary, the work delivers (i) a realistic, local (up to four‑spin) Hamiltonian that hosts the CSL without fine‑tuning, (ii) a clear analytical pathway from coupled wires to a stable chiral phase via the SLL fixed point, (iii) a quantitative link between 1D SPT sectors and 2D topological degeneracy, and (iv) robust numerical evidence using state‑of‑the‑art tensor‑network methods. The approach is versatile and suggests straightforward extensions to other frustrated lattices (e.g., kagome) and to more exotic non‑Abelian chiral spin liquids.