Chiral Spin Liquid in a Frustrated Anisotropic Kagome Heisenberg Model

Kalmeyer-Laughlin (KL) chiral spin liquid (CSL) is a type of quantum spin liquid without time-reversal symmetry, and it is considered as the parent state of an exotic type of superconductor—anyon superconductor. Such an exotic state has been sought for more than twenty years; however, it remains unclear whether it can exist in a realistic system where time-reversal symmetry is breaking ($T$ breaking) spontaneously. By using the density matrix renormalization group, we show that KL CSL exists in a frustrated anisotropic kagome Heisenberg model, which has spontaneous $T$ breaking. We find that our model has two topological degenerate ground states, which exhibit nonvanishing scalar chirality order and are protected by finite excitation gap. Furthermore, we identify this state as KL CSL by the characteristic edge conformal field theory from the entanglement spectrum and the quasiparticles braiding statistics extracted from the modular matrix. We also study how this CSL phase evolves as the system approaches the nearest-neighbor kagome Heisenberg model.

Figure 1

(a) Illustration of YC8 ($L_y=4$) kagome lattice. Here, we plot the bond spin correlation distribution of two ground states $⟨{\mathit{S}}_i·{\mathit{S}}_j⟩-e_{\alpha }$, where $e_{\alpha }$ is the average of the bond spin correlations, $-0.206450$ for ${\psi }_1$ and $-0.206486$ for ${\psi }_s$. (b) Two topological degenerate ground states ${\psi }_1$ and ${\psi }_s$ in CSL. Response of YC8 cylinder under flux insertion: (c) The energy evolution. (d) The one column overlap between states at different fluxes $F_1(\theta )=|⟨\psi (\theta )|\psi (\theta +\pi /3)⟩|$ and $F_2(\theta )=|⟨\psi (0)|\psi (\theta )⟩|$. Here, $\psi (0)={\psi }_1$ and $\psi (2\pi )={\psi }_s$.

Figure 2

(a) Spin correlation $⟨{\mathit{S}}_i·{\mathit{S}}_{i+r}⟩$ versus distance $r$. The chiral correlation $⟨{\chi }_i{\chi }_{i+r}⟩$ versus the distance $r$: (b) Results from the iDMRG. (c) Results from the finite DMRG. The system sizes we calculated are $3\times 4\times 24$ and $3\times 6\times 16$ YC cylinders. Here, $J^{\prime }=1$.

Figure 3

Entanglement spectrum of YC12 cylinder ($L_y=6$) (a) ${\psi }_1$ and (b) ${\psi }_s$. The horizontal axis is the momentum along the $y$ direction, $p_y=0,2\pi /6,\dots ,5\times 2\pi /6$ (up to a global shift). The leading spectra are marked in red, above which there is a spectrum gap marked with a dashed line. Other geometry and size samples have similar results.

Figure 4

$J^{\prime }$-dependent behavior of the two topological degenerate sectors: (a) normal sector ${\psi }_1$ and (b) spinon sector ${\psi }_s$. Here, we show results of calculations for the YC8 cylinder: (I) the chirality order, $\chi$ [Eq. (2)], where for the real code, we define $\chi =\sqrt{\underset{r\to \infty }{\lim }⟨{\chi }_i{\chi }_{i+r}⟩}$; (II) the correlation length $\xi$ of the ground state; (III) the singlet gap ${\mathrm{\Delta }}_s$ of the ground state.