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

TL;DR

This work provides a concrete lattice realization of the Kalmeyer-Laughlin CSL on a triangular lattice by embedding a coupled-wire construction within a lattice duality that yields a chiral sliding Luttinger liquid fixed point. By tuning inter-wire couplings through a nonlocal duality and a controlled perturbative scheme, the authors promote the CSL instability while suppressing competing orders, achieving a CSL ground state that retains topological features such as degeneracy and fractionalization. The authors support their analytical framework with extensive tensor-network simulations, including DMRG, VUMPS, and iPEPS, demonstrating ground-state degeneracies, fourfold circumference periodicity on cylinders, nonlocal string order, and chiral entanglement spectra that match Laughlin-like counting. Overall, the paper establishes a robust route from coupled-wire theory to microscopic lattice models realizing CSLs and outlines paths to generalize the approach to other topological phases and higher SU(2) levels.

Abstract

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.

Figures (27)

• Figure 1: Schematic figure for the coupled-wire construction. Gapping out counterpropagating modes on the nearest chains gives an inert bulk. The unpaired modes remain gapless at the boundary.
• Figure 2: Triangular lattice is defined by the unit vectors $\bm{b}_0$, $\bm{b}_+$, and $\bm{b}_-$ connecting neighboring sites. On a cylinder (shown here for $N=4$), different transverse boundary conditions are possible. The XC-$N$ boundary condition is equivalent to choosing the red unit cell. The XC-$N(0)$ boundary condition uses the green zig-zag unit cell, which is only possible when $N$ is even.
• Figure 3: Scaling dimensions at the decoupled Luttinger liquid fixed point. The operator $\mathcal{O}_{\text{chiral}}$ leading to the CSL has the largest scaling dimension. In particular, this value is greater or equal to $2$ for any $K$, meaning it is at most marginal. In contrast, the scaling dimensions of the other operators can be smaller than $2$, in which case they are relevant.
• Figure 4: Comparison between weakly coupled $XXZ$ chains and Hamiltonian $H_\diamondsuit$ in Eq. \ref{['eqn.HD']}. For each red bond, the in-plane spin exchange interaction $S^+S^-$ is replaced by the product of the four operators at the corners of the red diamond in the right panel. The spin rotation operator $R^\alpha$ is a linear combination of $\mathbbm{1}$ and $S^z$ [cf. Eq. \ref{['eqn.SzRotation']}], i.e., $H_\diamondsuit$ contains two-, three-, and four-spin interactions.
• Figure 5: Scaling dimensions as functions of $\alpha$ in the two-dimensional limit ($N = \infty$) with $J^z_{\perp} = \lambda = 0$. In the two-parameter space of $K$ and $\alpha$, the operator $\mathcal{O}_{\text{chiral}}$ can become a relevant perturbation ($\Delta_{\text{chiral}} < 2$). Moreover, the regime in which $\mathcal{O}_{\text{chiral}}$ has the least scaling dimension among its competitors is finite.