Distinct Spin Liquids and Their Transitions in Spin-$1/2$ $XXZ$ Kagome Antiferromagnets

By using the density matrix renormalization group approach, we study spin-liquid phases of spin-$1/2$ $XXZ$ kagome antiferromagnets. We find that the emergence of the spin-liquid phase is independent of the anisotropy of the $XXZ$ interaction. In particular, the two extreme limits—the Ising (a strong $S^z$ interaction) and the $XY$ (zero $S^z$ interaction)—host the same spin-liquid phases as the isotropic Heisenberg model. Both a time-reversal-invariant spin liquid and a chiral spin liquid with spontaneous time-reversal symmetry breaking are obtained. We show that they evolve continuously into each other by tuning the second- and the third-neighbor interactions. And last, we discuss possible implications of our results for the nature of spin liquid in nearest-neighbor $XXZ$ kagome antiferromagnets, including the nearest-neighbor spin-$1/2$ kagome antiferromagnetic Heisenberg model.

Figure 1

(a) Kagome model with first- ($XXZ$), second-, and third- ($XY$) neighbor interactions. (b) Phase diagram of the kagome $XXZ$ model.

Figure 2

(a) Two topological degenerate states of CSL. (b) Entanglement spectrum (sorted by quantum number $S_z$) of CSL on YC12 cylinder; here $\theta =0.016\pi$, $\tau =1$: (I) ${\psi }_1$, (II) ${\psi }_s$. 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). (c) Energy splitting between two topological degenerate ground states. (d) Transition from CSL ($\theta >0$) to superfluid ($\theta <0$). Here, we show results for the YC8 cylinder, $\tau =1$. I: Scalar chirality order parameter $\chi$; II: $\log$ plot of the correlation length $\xi$ (unit cells) extracted from the transfer matrix; III: fidelity $F$ [defined in Eq. (5)] between all pairs of neighboring points.

Figure 3

Results of NNKAXXZ ($\tau =0$) obtained on the YC8 cylinder. (a) Bond spin correlations along the $XY$ direction, $⟨S_i^x{S}_j^x+S_i^y{S}_j^y⟩-e_a$ ($e_a$ represents its average value). The transition from spin liquid ($\theta >0$) to superfluid ($\theta <0$). (b) Log plot of correlation length $\xi$ (measured in unit cells). (c) Fidelity $F$ [defined in Eq. (5)] between all pairs of neighboring points.

Figure 4

Transition from CSL ($\tau \sim 1$) to spin liquid in NNKAXXZ ($\tau =0$). We show results (YC8 cylinder) obtained in the Ising ($\theta =0.016\pi$), the $SU(2)$ ($\theta =\pi /4$), and the $XY$ ($\theta =\pi /2$) limits. (a) Chirality order $\chi$, (b) correlation length $\xi$ (measured in unit cells), (c) fidelity $F(\tau )$ [Eq. (5)] $F$ between pairs of neighboring points. The dashed line marks the critical point.