Spin Chain Network Construction of Chiral Spin Liquids

We show that a honeycomb lattice of Heisenberg spin-$1/2$ chains with three-spin junction interactions allows for controlled analytical studies of chiral spin liquids (CSLs). Tuning these interactions to a chiral fixed point, we find a Kalmeyer-Laughlin CSL phase which here is connected to the critical point of a boundary conformal field theory. Our construction directly yields a quantized spin Hall conductance and localized spinons with semionic statistics as elementary excitations. We also outline the phase diagram away from the chiral point where spinons may condense. Generalizations of our approach can provide microscopic realizations for many other CSLs.

Figure 1

$Y$ junction and 2D honeycomb network of spin-$1/2$ chains. (a) Single $Y$ junction. Three Heisenberg chains of equal length $\ell \gg 1$ (with lattice spacing $a=1$) are labeled by $\alpha =1$, 2, 3. They are coupled at the junction ($x=0$) by the three-spin boundary coupling $J_{\chi }$ in Eq. (1). At the chiral fixed point, $J_{\chi }=J_{\chi }^c$, incoming (left-moving, $L$) spin currents are perfectly rerouted in a counterclockwise (for $J_{\chi }>0$) sense [35, 36]. (b) Adding $Y$ junctions at each chain boundary $x=\ell$, and iterating the process, one obtains the shown 2D spin chain network. Down-pointing triangles (yellow) mark the positions $\mathbf{V}$. The neighboring junctions correspond to up-pointing triangles (green). The three vectors ${\mathit{\delta }}_{\alpha }$ in Eq. (4) are also indicated [39]. For $J_{\chi }=J_{\chi }^c$, the field theory reduces to decoupled chiral bosons describing circulating loop spin currents (blue), parametrized by the hexagon centers $\mathbf{R}$ (open circles) and the 1D coordinate $\xi \in [0,6\ell ]$. For a strip geometry, the theory includes gapless edge modes (red). The dotted line highlights the cluster $C(\ell ,2)$ used in DMRG simulations.

Figure 2

Spinons in the CSL phase. (a) A local operator acting on a single site of a given chain $c$ (gray circle) excites the zero modes of two neighboring hexagons. These excitations with $\mathrm{\Delta }N(\mathbf{R})=\pm 1/2$ (red crosses) correspond to spinons. The dashed line connecting two spinons defines an open string $w$. (b) The operator transporting a spinon around a loop $\mathcal{L}$ (purple line) produces a $\pi$ phase shift if a spinon is present inside $\mathcal{L}$.

Figure 3

Schematic phase diagram for the 2D network Eq. (1), where the shaded region indicates a stable CSL phase. In addition to quantum paramagnetic (PM) and valence bond crystal (VBC) phases, we also expect antiferromagnetically ordered phases (AFM1 and AFM2).

Figure 4

DMRG results for the energy gaps ${\mathrm{\Delta }}_{s=1,2}$ to the first excited state with total spin $S_{\mathrm{tot}}^z=s$ versus the coupling $g$ for $C(\ell =8,n)$ clusters. The singlet-triplet gap ${\mathrm{\Delta }}_1$ drops to zero for $g\gtrsim 3.7$. The gap ${\mathrm{\Delta }}_2$ has a minimum with ${\mathrm{\Delta }}_2^{(\min )}\propto 1/(n\ell )$ at $g=g_c(\ell )$. Inset: Magnified view of $n\ell {\mathrm{\Delta }}_2$ versus $g$.