Chiral and critical spin liquids in a spin-$\frac{1}{2}$ kagome antiferromagnet

The kagome spin-$\frac{1}{2}$ systems have attracted intensive attention in recent years as the primary candidate for hosting different gapped spin liquids (SLs). To uncover the nature of the novel quantum phase transition between the SL states, we study a minimum $XY$ model with the nearest-neighbor (NN) ($J_{xy}$), the second-NN, and the third-NN couplings ($J_{2xy}=J_{3xy}=J_{xy}^{\prime }$). We identify the time-reversal-symmetry-broken chiral SL (CSL) with the turn on of a small perturbation $J_{xy}^{\prime }\sim 0.06J_{xy}$, which is fully characterized by the fractionally quantized topological Chern number and the conformal edge spectrum as the $\nu =1/2$ fractional quantum Hall state. Interestingly, the NN $XY$ model ($J_{xy}^{\prime }=0$) is shown to be a critical SL state adjacent to the CSL, characterized by the gapless spin singlet and spin triplet excitations. The quantum phase transition from the CSL to the gapless critical SL is driven by the collapsing of the neutral (spin singlet) excitation gap. The effect of the NN spin-$z$ coupling $J_z$ is also studied, which leads to a quantum phase diagram with an extended regime for the gapless SL.

Figure 1

(a) Schematic phase diagram of the spin-$\frac{1}{2}$ kagome model with Hamiltonian Eq. (1) where both the CSL and a critical SL with gapless singlet excitations are identified. The gapped SL [31] may exist neighboring with the critical SL. (b) Log-linear plot of chiral correlation $\langle {\chi }_i{\chi }_j\rangle$ versus the distance of triangles $R_{ij}$ along the $x$ direction for the ground state in the vacuum sector. (c) and (d) are the ES of the ground states obtained using the infinite DMRG (iDMRG) in the vacuum and spinon sectors, respectively. ${\lambda }_i$ is the eigenvalue of the reduced density matrix.

Figure 2

(a) and (b) are the chiral correlations for the $XY$ model ($J_z=0$) with $J_{xy}^{\prime }=0.1$ and 0.3 on the cylinders with $L_y=4–6$. (c) and (d) are the spin correlations of the $xy$ component $|\langle S_i^x{S}_j^x+S_i^y{S}_j^y\rangle |$ and the $zz$ component $|\langle S_i^z{S}_j^z\rangle |$ for the same systems with $L_y=4$ and 6.

Figure 3

(a) Spin magnetization $\langle S_{x,y}^z\rangle$ at $R_i=(x,y)$ after adiabatically inserting a flux $\theta =\pi /4$. The area of the circle is proportional to the amplitude of $\langle S_{x,y}^z\rangle$. The blue (red) color represents the positive (negative) $\langle S_{x,y}^z\rangle$. (b) Accumulated spin magnetization $\langle S_x^z\rangle ={\sum }_y\langle S_{x,y}^z\rangle$ in each column with inserting flux. (c) Net spin transfer $\mathrm{\Delta }{S}^z{|}_{\mathrm{edge}}$ as a function of $\theta$ on the $L_y=6$ cylinder. The inset shows the results on the $L_y=4$ cylinder. (d) Phase diagram of the $XY$ kagome model ($J_z=0$).

Figure 4

The ES flow with inserting flux $\theta$ for $J_z=0,J_{xy}^{\prime }=0.1$ on a $L_y=6,L_x=16$ cylinder system obtained using iDMRG.

Figure 5

(a) Evolution of the ED low-energy spectrum in the $k=(0,0)$ sector with $J_{xy}^{\prime }$ for the $XY$ model on the $3\times 3\times 4$ torus. The singlet gap between the lowest two ground states and the higher-energy states is denoted as $E_s$. (b) $J_{xy}^{\prime }$ dependence of spin gap $E_T$ for the $XY$ model on $3\times 3\times 4, 3\times 4\times 4$, and $3\times 4\times 8$ tori. Low-energy spectrum of the NN $XXZ$ model ($J_{xy}^{\prime }=0$) at $k=(0,0)$ and $(0,\pi )$ sectors for (c) $J_z=0.0$, (d) $J_z=0.5$, and (e) $J_z=1.0$ on the $3\times 3\times 4$ torus. The label $(\pm 1,\pm 1)$ denotes the quantum numbers related to spin inversion and lattice $\pi$-rotation symmetries.

Figure 6

Kagome cylinder on (a) $YC$ geometry and (b) $XC$ geometry. The cylinders are closed in the $y$ direction and opened in the $x$ direction.

Figure 7

Entanglement spectrum flow for $J_z=0.0,J_{xy}^{\prime }=0.1$ on the $L_y=6 XC$ cylinder, which is obtained by keeping 3000 states using iDMRG.

Figure 8

(a) Entanglement spectrum for $J_z=0.0,J_{xy}^{\prime }=0.1$ on the $L_y=6 YC$ cylinder, which is obtained by keeping 3000 states. The red arrow denoted as $v_c$ indicates the two lowest values in the $k=0$ and $2\pi /L_y$ sectors that are used to calculate the chiral velocity. (b) Near-degenerating pattern for the low-lying entanglement spectra with different relative momentum $\mathrm{\Delta }{k}_y$ and total spin $S^z$ for the same system in (a). $\mathrm{\Delta }{k}_y$ is in units of $2\pi /6$. All results are obtained using iDMRG.

Figure 9

ES for (a) $J_z=0.0,J_{xy}^{\prime }=0.1$, (b) $J_z=0.0,J_{xy}^{\prime }=0.0$, and (c) $J_z=1.0,J_{xy}^{\prime }=0.0$. The arrow denoted by $v_c$ in (b) indicates the two lowest eigenvalues in the $S^z=0$ sector which are used to calculate the chiral edge-mode velocity.

Figure 10

Entanglement spectrum for (left) $J_z=0.5,J_{xy}^{\prime }=0.3$ and (right) $J_z=1.0,J_{xy}^{\prime }=0.3$ on the $L_y=6 YC$ cylinder, which is obtained by keeping 1600 states in iDMRG calculations. The near-degenerating pattern for the low-lying entanglement spectra with different relative momentum $\mathrm{\Delta }{k}_y$ and total spin $S^z$ for the same system in the main text. $\mathrm{\Delta }{k}_y$ is in units of $2\pi /6$.

Figure 11

Entanglement spectrum for $J_{xy}^{\prime }=0.3$ on the $L_y=4 YC$ cylinder, which is obtained using iDMRG. $\mathrm{\Delta }{k}_y$ is in units of $2\pi /4$. The low-lying entanglement spectra have the degeneracy pattern $\{1,1,2,3\}$.

Figure 12

Size dependence of the spin gap obtained on the torus systems with $N=12–96$. The blue circles and red squares denote the gap of the pure Heisenberg kagome model and the pure $XY$ kagome model, respectively. Both data are fitted using the formula ${\mathrm{\Delta }}_{\mathrm{T}}=\alpha /N+\beta /N^{3/2}+\gamma /N^2$.