Space Group Symmetry Fractionalization in a Chiral Kagome Heisenberg Antiferromagnet

The anyonic excitations of a spin liquid can feature fractional quantum numbers under space group symmetries. Detecting these fractional quantum numbers, which are analogs of the fractional charge of Laughlin quasiparticles, may prove easier than the direct observation of anyonic braiding and statistics. Motivated by the recent numerical discovery of spin-liquid phases in the kagome Heisenberg antiferromagnet, we theoretically predict the pattern of space group symmetry fractionalization in the kagome lattice SO(3)-symmetric chiral spin liquid. We provide a method to detect these fractional quantum numbers in finite-size numerics which is simple to implement in the density matrix renormalization group. Applying these developments to the chiral spin liquid phase of a kagome Heisenberg model, we find perfect agreement between our theoretical prediction and numerical observations.

Figure 1

(a) Symmetry operations and two typical edges in numerical studies of the kagome chiral spin liquid. (b) Entanglement spectrum $\{E_a\}=-\log ({\rho }_L)$ of the YC8 $s$ sector, plotted against the momentum $k_a$ around the cylinder. The lowest pair is a “Kramers doublet” under the antiunitary $R_x$, indicating $(R_x^2{)}_s=-1$. Since the elements of the pair occur at the same $T_1$ momentum, $(R_x{T}_1{)}^2=-1$ as well, as expected from $(R_x{T}_1{)}_s^2=-1$.

Figure 2

Finite DMRG geometries used for XC8. Geometries (a) and (b) have an odd number of edge spins, trapping a spinon, from which we compute $Q_s(I_h),Q_s(I_s)$. From (c), we compute both $Q_{\mathbb{1}}(I_h)$ and $Q_{\mathbb{1}}(I_s)$. The results are tabulated in Fig. 3. A similar set of geometries is used for YC8, whose edge differs by 90°.

Figure 3

The measured reflection quantum numbers $Q_{\mathbb{1}/s}(I_{h/s})$ inferred from 16 CPSs overlaps $a_{\sigma }/a_{I_{h/s}\sigma }$. Results were obtained using both finite and infinite DMRGs on YC8 and XC8 cylinders.